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Heat kernel expansion: user's manual

1. where one can find some further references It is very well known that the conditions 6 11 6 14 which we call transmittal are not the most general matching conditions which can be defined on a surface In general boundary values of a function and of its normal derivatives are related by a 2 x 2 transfer matrix Vis st ot 0 St WVi s 5 6 20 Note that the transfer conditions 6 20 do not assume identification of and on In other words there is no ad hoc relation between the restrictions of the vector bundles Vt s and V sy We can even consider the situation when we have dimVt dimV i e the fields on M7 and M can have different structures with respect to space time and internal symmetries S are some matrix valued functions on X one can even consider the case when they are differential operators It is clear from the notations on which spaces they act For example St V y gt Vt y The matching conditions 6 20 arise in heat transfer problems some problems of quantum mechanics and in conformal field theory In a formal limiting case Stt S t S ST oo while v 2 Stt St is kept finite one arrives at transmittal boundary conditions 6 11 6 14 The heat kernel coefficients are divergent in this limit In a particular case of spherical X the heat kernel expansion with transfer boundary conditions was evaluated in General expressions for a
2. In some cases the term by term integration of the heat kernel expansion gives good estimates of the vacuum energy even if no large parameter is ex plicitly present in the model see e g calculations of quantum corrections to the mass of two dimensional solitons 8 2 Covariant perturbation theory Suppose that the matrix potential the Riemann curvature and the field strength 2 are small but rapidly varying Can we get any information on the heat kernel coefficients containing a fixed power of the quantities listed above and arbitrary number of derivatives For the linear and quadratic orders on a manifold without boundaries the answer is positive and the results can be obtained either by the functorial methods of sec 4 1 or by solving the DeWitt equation cf sec 4 3 Retaining only the leading terms we have for k gt 1 az z D 40 ai k AR c h APE 8 3 higher order terms where klk 2k Ges S T a One can sum up the expansion 8 3 and corresponding higher order terms to obtain an information on the behaviour of the full heat kernel in the limit described above There is however a more straightforward method to control non localities which is called the covariant perturbation theory To make the idea of this method most transparent we consider a simple example of flat M and zero connection w The exponent exp tD exp t A E with A OF can be expanded in a power s
3. Consider a differential operator Q not necessarily of the second order Let is separate the part containing highest derivatives and replace ikp like in doing the Fourier transformation In this way we obtain an object Ag a x kuky kp which is called the leading symbol of Q If the operator Q acts on fields with indices i e in a vector bundle the index structure of Q is inherited by Ag In other words for fixed k the leading symbol A is a matrix valued function an endomorphism of the vector bundle If Ag x is non degenerate for all k 0 the operator Q is called elliptic For the Dirac operator Q D the leading symbol is Ag y k Therefore P is obviously elliptic The Laplace type operators are also elliptic since Ap k Ellipticity means that at large momenta the operator Q is dominated by its highest derivative part This is the property which guarantees that on a compact manifold without boundaries Laplace operators have at most finite number of negative and zero eigenvalues and which ensures existence of the heat kernel On manifolds with boundaries just ellipticity is not enough to ensure nice 52 properties of the spectrum There is an additional requirement called strong ellipticity which should be satisfied by the boundary operator see for details Dirichlet and Robin boundary conditions are always strongly elliptic Oblique boundary conditions are strongly elliptic if and only if T lt 1 If
4. to quantum field theory and quantum gravity which became dominant for many years Heat kernel is a classical subject in mathematics 1 Asymptotics of the heat kernel are closely related to the eigenvalue asymptotics found by H Weyl and studied further in The problem as it was formulated by Kac reads Can one hear the shape of a drum In other words this is the problem of recovering geometry of a manifold from the spectrum of a natural differential operator Heat kernel coefficients proved very useful in this context On the other hand the heat kernel is also an adequate tool to study the index theorem of Atiyah and Singer By about 1990 the heat kernel expansion on manifolds without bound aries or with boundaries and simplest local boundary conditions on them was well understood Also the heat kernel became a standard tool in calcula tions of the vacuum polarisation the Casimir effect and in study of quantum anomalies Later on progress in theoretical physics especially in string the ory and related areas and parallel developments in mathematics made this field highly specialised New results on non standard boundary conditions as e g containing tangential derivatives on the boundary or non localities or on non standard geometries domain walls were scattered in large amounts of physical and mathematical literature The aim of this report is to present a unifying approach to the heat kernel exp
5. 4 32 is the Killing form of the gauge algebra Let us remind that the indices 7 j k in 4 28 refer to a local orthonormal frame In flat space they may be identified with the vector indices u v p In the case of vector fields the trace in 4 30 and 4 31 is taken over pairs consisting of a gauge index a 8 or y and of a vector index u v or p For the ghost operator we have to put R U 0 in 3 6 and identify the connection with the background gauge field according to 3 49 Consequently Ele 0 and ij ij pu pv try QQE Fo FY Ks 4 33 ae Next we substitute 4 30 4 31 and 4 33 in 4 28 to obtain o vec gh afo qlved 2a f d z JG F F Ks 4 34 The Yang Mills gauge group G has usually a direct product structure G G X Go XxX so that on each of the irreducible components G the Killing form K is proportional to the unit matrix Therefore the one loop divergence 4 34 reproduces the structure of the classical action with different charges for each G We also recover the coefficient 11 3 which is familiar from computations of the Yang Mills beta functions 4 2 2 Free fields in curved space Consider free quantum fields on a Riemannian manifold without bound aries Free means that we neglect all interactions except for the one with the background geometry The heat kernel coefficients can be expressed then in terms of local invariants of the metric In particular 1 me
6. 5 FO FO m 1 B EXD s 1 F D k F Y 0 Remm n 8 32 where the reminder Rem satisfies Rem n lt 2 21 is f ROM r dr 8 33 B are the Bernoulli numbers If F i is taken to be N exp t one can take limit k oo in 8 32 and restrict the summation to some finite m to obtain asymptotic series for the heat kernel For example one can easily recover the expansion 8 31 33 The spectrum of differential operators on coset spaces can be calculated exactly even if one adds a homogeneous gauge field see and references therein 82 In many cases calculation can be done by means of the Mellin transform This method will be clear from the following example Consider K t 55 k exp tk 8 34 The Mellin transform of K t is oe K ef dx x exp xk XO kT s Cr 2s 1 8 k 0 k 1 8 35 where Cpr is the Riemann zeta function Inverse Mellin transformation gives 1 Oni K t f t Cr 2s Dr e ds 8 36 c The contour C covers all poles of the integrand at s 1 0 1 2 By calculating the residues we obtain the desired expansion K t O t 8 37 Actually the techniques introduced above should only be used for com plicated multi parameter sums see e g For a simple one parameter sum it is easier to use combinations of known asymptotic series cf Appendix of Ref In this method com
7. This equation can be integrated with the initial condition W p 0 0 1 By sei K 7 A AS f dx pn 0 0 p 9 4 By returning to the covariant notations we obtain the famous Polyakov action see also 1 1 Note that on a compact manifold one has to take into account contribu tions from the zero modes which lead to an additional term in the Polyakov action The problem of finding an action whose conformal variation reproduces the conformal anomaly can be posed in higher dimension as well and some physical consequences of such actions can be studied Complete expres sions for the conformal anomaly induced effective action in four dimensions were obtained in Conformal action in the presence of boundaries was constructed in One should keep in mind that unless the background is conformally flat the conformal action represents only a part of the full effec tive action Applicability and limitations of the conformal action approach are discussed in 9 2 Duality symmetry of the effective action In this section we deal with variations of the effective action which can be expressed through zeta functions of several operators in contrast to genuine anomalous transformations of sec 7 which involve a single zeta function each A rather simple example can be found in the paper which considered two operators of Dirac type D ixte de pt ive 0 e 9 6 on flat two dimens
8. Vat Sill vlau 0 5 Deal Il 1 IL 3 37 We remind that La is extrinsic curvature of the boundary The boundary conditions 3 30 3 37 with 3 35 are mixed cf 2 17 Spectral geometry of the Dirac operator with these boundary conditions has been thoroughly studied by Branson and Gilkey We can generalise the boundary conditions presented above by consider ing non hermitian projectors I 4 II Then the boundary condition for the conjugated spinors reads PI lau 0 3 38 Instead of 3 34 we have the condition 1 T y 1 TI_ 0 3 39 which yields 1 nS r a y gt Has 3 1 i e 3 40 with an arbitrary real function or even with an arbitrary hermitian matrix valued function r x In the Minkowski signature space the spinor conjugation includes Therefore the boundary conditions on the conjugate spinor 3 31 are changed Roughly speaking to continue II to the Minkowski signature space time one 25 has to replace y y by 7 in 3 35 and 3 40 and to take into account powers of i which appear in the Dirac gamma matrices In particle physics interest to the boundary conditions defined by 3 35 is due to the MIT bag model of hadrons proposed in This model was manner similar i 3 40 for a review see Renormalization of quantum field theory with the boundary conditions defined by 3 35 was considered by Symanzik As we have already mentioned above bag boundary co
9. x D B Obviously the canonical mass dimension of ax j is k 7 1 In addition to the usual bulk invariants F R etc ay can also contain specific boundary quantities as the extrinsic curvature La or S for Robin boundary conditions Note that La and S are defined on the boundary only and therefore can be differentiated only tangentially 1 Canonical mass dimension of Las and S is 1 In this section we explicitly calculate the first three heat kernel coefficients ao a and az Basing on the considerations of the preceding paragraph we may write aol f D B 4n es dx Gtty f 5 17 a f D B 4n D2 f Teva betty f 5 18 aol f D B alan d a yatrv 6fE fR elev try bE f Laa bE fn BE fs 5 19 where B denotes either Robin B or Dirichlet B boundary operator b are some constants To keep uniform notations we formally included S also in the expression for Dirichlet boundary conditions It should be assumed that S 0 for that case Since the product formula 4 2 is still valid with obvious modifications for the boundary contributions we can again consider the case M M x St and repeat step by step the calculations 4 3 4 6 thus arriving to the same conclusion that the constants b do not depend on dimension of the manifold all explicit n dependence of the heat kernel is contained in the power of 47 This property is crucial for our calc
10. 2 4 where o op 1 Fu 9 3 ugo OvGup pIu 2 5 is the Christoffel symbol Iy is the unit operator on V 5 This simply means that we assume that there is a positive definite metric tensor Ju on M Many physicists strongly dislike vector bundles Nevertheless there are two good reasons for using the fibre bundles in this paper in parallel with more familiar nota tions of matrix valued functions gauge fields etc First our simplifying comments and examples may help the reader to understand mathematical literature on the subject Second one of the main ideas of this report is to reveal some universal structures behind the heat kernel expansion In particular we shall see that there is not much difference between different spins and symmetry groups The vector bundle language seems to be the most adequate language to achieve this goal The reader may consult the excellent review paper by Eguchi Gilkey and Hanson 11 It is important to understand how general the notion of the Laplace type operator is The most obvious restriction on D is that it is of second order i e it contains second derivatives but does not contain higher derivative parts In this paper we shall also consider first order operators Dirac operators for ex ample Second the operator D is a partial differential operator This excludes negative or fractional powers of the derivatives The operators containing such structures are called pseudo differen
11. 58 SE uy OO O t 4 25 From the equation 4 25 we conclude that aio 30 We have obtained also an independent confirmation for the values of ao a1 a3 and as The method we used above in eqs 4 21 4 25 can be applied to more general manifolds and operators as well The key ingredient is a convenient basis which should be used instead of the plane waves For the case of a box with anti periodic boundary condition such a basis is rather obvious see On curved manifold M one has to use the so called geodesic waves although calculations with ordinary plane waves are also possible For calculation of the coefficient ag we refer to The results for the leading heat kernel coefficients are summarised in the following equations 39 aof D 4ny deyar f 4 26 az f D 4r 267 l d zy gtry f 6E R 4 27 a f D 4n 236071 i dx gtry f 60E GORE 180E 12R gk 5R 2R Riz 2Rije Rije 300013 4 28 ag f D 4x angry ESR sirm Rea OR Rs 4R knRjnk 9RijkinRijnin 23RRinn 8RjkRjk nn 24 Rip Rjnskn 12RijklRijkinn 35 9R 14 3RR Ri 4 14 3R Rijn Rijxi 208 9RjrRjnRen 64 3Riy Rei Rirsi 16 3R i RiniRenui 44 9RijknPijtpRenip 80 9RijknRikpRjinp 360 f 8Q4 4Qj amp Qij Nikk 412 5 pj 20 O GRO a AR in ia H5ROkn Nn 6E zj COLE 30E E 608 H30ENQ Qij 10RE kk ARE jx 12R E 30EE
12. Of course the heat kernel expansion can be used also on Lorentzian man ifolds at least for the renormalization theory where non local terms are of less importance Counterterms are still defined by the same heat kernel coefficients with the same functional dependence on the Lorentzian metric Explicit defi nitions with the imaginary proper time 7 can be found in One should note that some background fields receive an imaginary phase when being con tinued to the Euclidean domain cf secs 3 2 and 3 3 18 3 Relevant operators and boundary conditions The operator of Laplace type is not necessarily the scalar Laplacian In fact in almost all models of quantum field theory the one loop effective action is defined by an operator of this type This can be demonstrated by bringing relevant operators to the canonical form 2 2 In this section we give ex plicit construction of the connection w and the endomorphism matrix valued potential for scalar spinor vector and graviton fields We also describe appropriate boundary conditions 3 1 Scalar fields Consider first the example of the multi component real scalar field 4 in n dimensions The action reads pe f de Salg VV 04 U EROAG4 3 1 where is the conformal coupling parameter U is a potential The covariant derivative V contains the background gauge field Gi V 0 04 GPBP G is antisymmetric in internal indices A B To evaluate the one l
13. The relations 7 28 and 7 28 are called the Wess Zumino consistency con ditions One can check by a direct calculation that the anomaly defined in this section indeed satisfies these conditions For this reason A is called the consistent anomaly as opposed to the covariant anomaly which we do not consider here Let us turn now to calculation of the chiral anomaly A Since the smearing function y y in the heat kernel coefficient 7 27 is matrix valued we need more information than we possess at the moment To recover the missing terms one can adopt the strategy of the paper Consider Tr Q exp tD 69 with arbitrary matrix valued function Q and arbitrary Laplace type operator D There is an asymptotic expansion Trz2 Q exp tD Y t a Q D 7 30 k gt 0 where the coefficients a are locally computable This means that they can be represented as integrals of local invariants constructed from Q and local invariants of the operator D These local invariants enter with some coefficients universal constants which are to be defined Since Q does not commute with E and Q there are more invariants than before In the particular case Q Iy f where Jy is the unit matrix and f is a function we should recover the old result 4 26 4 29 For k 0 2 4 this last requirement is strong enough to recover a Q D completely Therefore calculation of the chiral anomaly in the dimensions n 2 and n 4 is quite simple ta
14. Vv v Gigs 2 8 The condition V e 0 yields kl VTP pk _ pv k Oy e hes E Ope 2 9 Let Q be the field strength of the connection w Quin OpWy OW WuWy WW yu 2 10 The covariant derivative V acts on both space time and internal indices For example Vouw Ona ae Ogs Pi Que lwp Qu 2 11 py ov 12 If the boundary M is non empty we have more invariants Let e be inward pointing unit vector field Let Roman indices a b c and d range from 1 to n 1 and index a local orthonormal frame for the tangent bundle of 0M Let La r be the second fundamental form extrinsic curvature of the boundary We use the Levi Civita spin connections and the connection w to covariantly differentiate tensors of all types Let denote multiple covari ant differentiation with respect to the Levi Civita connection of M and let denote multiple tangential covariant differentiation on the boundary with respect to the Levi Civita connection of the boundary the difference between and is measured by the second fundamental form Thus for example E a E a since there are no tangential indices in EF to be differentiated On the other hand E a Z F a since the index a is also being differentiated More precisely Egy FE ab LabE n Since L is only defined on the boundary this tensor can only be differentiated tangentially Consider an example of the circle St in the plane R The
15. be a complete basis of orthonormal eigenfunctions of the operator D corresponding to the eigenvalues A Then K t 2 y D gt gt d 2 dr ye 2 20 We shall almost exclusively work either on manifolds without boundaries or on manifolds with boundaries with the fields subject to local boundary conditions 2 15 2 16 or 2 17 In all these cases there is an asymptotic 14 expansion as t 0 7 Trr2 f exp tD X t 2q f D 2 21 k gt 0 This expansion is valid for almost all boundary conditions appearing in appli cations to physics There are however some exceptions which will be discussed in sec 5 4 The coefficients a and the coefficients bg introduced in the previous sec tion cf 1 13 are related by the equation m 0 for the simplicity a f D 4r J d z gbz z f 2 2 22 Note that the definition 1 13 is valid on flat manifolds without boundary only although generalisations to other cases are possible The key property of the heat kernel coefficients a is that they are locally computable in most of the cases This means that they can be expressed in term of the volume and boundary integrals of local invariants For a positive operator D one can define the zeta function by the equation C s f D Tre f DB 2 23 The zeta function is related to the heat kernel by the integral transformation Cls f D T 8 f dtt KG f D 2 24 0 This relation
16. this inequality is violated infinite number of negative eigenmodes appears a simple example can be found in Appendix B of Note that the boundary conditions 5 34 correspond to a symmetric D if the matrices Ia are anti hermitian Therefore rT I is typically negative Another complication stems from the fact that I is dimensionless Con sequently arbitrary powers of may enter az so that instead of the unde termined constants cf 5 17 5 19 one has to deal with undetermined functions of I The problem becomes more tractable if we suppose that s commute among themselves a To 0 For this case the coefficients a and az have been calculated by McAvity and Osborn ao is still given by 5 17 alf D 4r 09 2 T Pev htrv fA 5 35 aa f D 4 daygtav 6fE FR F f dlaeV htry bo T f Laa bi D fin ba T fS fo L Lalals Hsi where 1 2 _ 1 l as 1 1 6 artanh vV R io 14 7 Tl 6 by sala v T 3 5 37 pa D 2 1 Es T2 1 o me bo We see that if approaches 1 the heat kernel blows up indicating violation of the strong ellipticity condition As an example let us consider the open string sigma model of sec 3 2 To simplify the subsequent calculations we put BAs 0 In the zeta function regularization divergent part of the effective action 2 31 reads 1 s a2 D 1 s a 1 D On general grounds we expect
17. with k 0 1 2 3 4 were obtained in Somewhat surprisingly the calculations 62 for 6 20 are easier than for the singular particular case 6 11 6 14 We should note that not all singular limiting cases of 6 20 are described by the transmittal conditions 6 11 6 14 The heat kernel expansion for a generalisation of transmittal condition is known in the spherically symmetric case only Very little is known about the heat kernel if the transfer matrix contains differential operators on conformal walls of ref belong to this class of problems The case when the singular potential is located on a surface of co dimension larger than one i e when dim lt n 1 is rather complicated Even a care ful translation of this problem to the operator language was done only in 1960 s Direct calculations show that the heat kernel asymptotics may contain very unusual Int terms More references can be found in 6 4 Non smooth boundaries Rectangular region in a plane is probably the simplest manifold with boundaries as far as eigenvalues of the Laplacian are concerned However the formulae 5 29 5 33 are not valid for this case because of the presence of corners The heat kernel expansion on manifolds with piecewise smooth boundaries was considered by Kac in his famous paper He demonstrated that for a region in R each corner with the inside facing angle a contributes
18. D 7 7 where we have used the relation 2 27 between the zeta function at zero argument and the heat kernel coefficient ap We compare 7 3 with 7 7 to see that Th x an x D 7 8 Note that 0 D as well as the pole term in 2 31 is conformally invariant Consequently the conformal anomaly is not divergent and 7 8 does not contain the normalisation scale pu The anomaly 7 8 is defined by the same coefficient a as the divergent part of the effective action in the zeta function regularization Important dif ference is that the divergence is given by the integrated coefficient a D while the conformal anomaly is defined by the localised coefficient an x D The use 25 Strictly speaking this property holds up to a similarity transformation D eP De which does not change the functional determinant 26 To derive this formula rigorously we have to use the method of We first assume that s is sufficiently large to keep us away from the singularities then use the Mellin transformation 2 24 perform the variation then perform the transfor mation back and then continue the result to s 0 This is a perfectly standard procedure which allows us to work with variations of positive integer powers of D see for further detail 66 of the integrated coefficient in 7 8 is insufficient to recover total derivatives in the anomaly As an example consider quantum scalar field in two dimensions c
19. T a 2474 a2 corner 6 21 to the coefficient a gt while ap and a are still defined by their smooth expres sions The formula 6 21 looks similar to the contribution of a conical singu larity 6 7 The reason for this similarity is that the cone can be obtained from the wedge by gluing the sides together and imposing the periodicity conditions The study of boundary discontinuities was continued by Apps and Dowker who calculated the coefficients a3 and a4 for piecewise smooth bound aries We also refer to where functional determinants on simplicial com plexes were analysed and to where divergences in the Casimir energy found in were attributed to non smoothness of the boundaries A recent study should also be mentioned When the angle a goes to 0 we obtain a cusp In this limit 6 21 is di vergent Presence of the cusp is an essential singularity which modifies powers of the proper time t which appear in the asymptotic expansion of the heat kernel 6 5 Dielectric bodies Calculations of the Casimir energy of a dielectric body have attracted much attention and created many controversial results A rather large liter 63 ature on this subject is reviewed in Quantum field theory formulation of this problem is known for a long time already cf However the heat kernel analysis of divergences in the Casimir energy in dielectric is a relatively new subject Wave pr
20. an appropriate surface action 3 2 Bosonic string Our next example is the non linear sigma model in two dimensions de scribed by the action cel if Px GGan X gMO X40 X e Bap X O X40 X M B f ABdX 3 10 From the point of view of two dimensional world sheet the fields X4 x are scalars In string theory they are interpreted as coordinates on a d dimensional target manifold with the metric Gapg X e is the Levi Civita tensor density e e 1 Byp X is an antisymmetric tensor field on the target space Ap X is the electromagnetic vector potential The action 3 10 describes charged open stings For simplicity we absorb the inverse string tension a into a field redefinition We do not include tachyon and dilaton couplings in the bulk or on the boundary Usually the term with the B field gets an imaginary coefficient in the Euclidean space Since the physical space time has Minkowski signature it is not especially significant which way of continuation to the Euclidean space has been chosen provided the results are properly continued back to Minkowski space after the calculations As we will see below real coefficient in front of the B field leads to a well defined spectral problem This situation is in close 20 analogy with the continuation rules for the axial vector field in the spinor determinant The other way to deal with the field B is to keep the coefficient of the B term in 3 10 imaginary
21. at the expense of introducing a more sophisticated conjugation operation containing the sign reversion of the B field The same refers to the electromagnetic potential Apg The field X enters the action 3 10 at many places making the back ground field expansion a quite cumbersome procedure The most economic way to arrange such an expansion and to calculate higher derivatives of the action 3 10 is to introduce the geodesic coordinates in the target space A detailed explanation of the method as well as further references can be found in Consider the target space geodesics defined independently at each point of the two dimensional world surface and parametrised by the arc length s in the target space They satisfy the usual geodesic equation 2 d A A d B d C Aa x s YBo X 7X z s g x 5 ga 0 3 11 where y o X is the Christoffel connection corresponding to the target space metric Gag Let us supplement the equation 3 11 by the initial conditions X 0 X47 2 L XA 0 E4 z 3 12 where X is the background field 4 parametrises deviations from X and therefore can be identified with quantum fluctuations The k th order term of the expansion of the action 3 10 around the background X is given by 1 d Er as L X s s 0 3 13 Higher order derivatives of X x s with respect to s can be traded for the first derivatives by means of the geodesic equation 3 11 and then replaced by at
22. by the condition fu 0 3 69 If the background admits conformal Killing vectors these are the vectors which are annihilated by the operator L 3 66 the condition 3 69 is not enough and one should impose one more gauge condition on the trace part see e g We suppose that conformal Killing vectors are absent 30 The Jacobian factor induced by the change of variables 3 65 hj gt h En hi is J det L L 3 70 where the determinant is calculated on the space of the vector fields exclud ing the conformal Killing vectors which we do not take into account It is convenient to shift the scalar part of the metric fluctuations by V so that the decomposition 3 65 becomes 1 hy G 0 2V Ep Guu LE Pity 3 71 Since the change of the variables h o does not introduce any Jacobian factor we conclude that the path integral measure is Dhu det Lt L DoDEDh 3 72 To simplify the discussion we suppose that the background metric gy satisfies the Einstein equations Ryw g Agu 3 73 The quadratic part of the action reads 1 l iw j l 167G faeva je AgyupGue 2Rypvo h ar a z o 3 74 Due to the gauge invariance 3 74 does not contain Functional integration over produces an infinite constant equivalent to the volume of the diffeomor phism group which will be neglected The kinetic term for has a wrong negative sign This represents the well known confor
23. can be inverted REEDS fast T s C s D 2 25 where the integration contour encircles all poles of the integrand Residues at the poles can be related to the heat kernel coefficients akl f D Res maja T s s f D 2 26 In particular an f D 0 f D 2 27 Zeta functions can be used to regularize the effective action The T These papers contain also a method allowing to calculate the coefficients of the expansion The method is however too complicated to use it on practice 15 regularization is achieved by shifting the power of t in 1 18 1 dt W jw K t D 2 28 2 o tis where is a constant of the dimension of mass introduced to keep proper dimension of the effective action The regularization is removed in the limit s 0 Eq 2 28 can be considered as a definition of the regularized effective action without any reference to 1 18 One can also rewrite 2 28 in terms of the zeta function W SH T s C s D 2 29 where s D s 1 D The gamma function has a simple pole at s 0 ra hp Ole 2 30 where yp is the Euler constant The regularized effective action 2 29 has also a pole at s 0 17 1 1 W 5 Z 7s m CO D 300 D 231 According to 2 27 the divergent term in the zeta function regularization is proportional to a D cf 1 21 for another regularization scheme The pole term in 2 31 has to be removed by the r
24. condition 3 58 for the normal component is also gauge invariant The key observation is that near the boundary the scalar Laplacian can be represented as D Vn Laa Vn E where E does not contain normal derivatives and therefore E oy 0 under the boundary conditions 3 58 for the Consequently Vn Laa b Anlam Vn Laa Vn lom DE E 8 am 0 3 59 where D 9 a 0 on the eigenfunctions of the operator D9 which is enough for our purposes The boundary term in the action 3 43 can be rewritten as 1 fd aVJhlAg Ama LesAs Aan 3 60 2 Jam This boundary action vanishes for both types of the boundary conditions 3 57 and 3 58 Hence the operator D is symmetric for these boundary conditions Another remarkable fact is that the fields A satisfying the gauge condition 3 44 are orthogonal to the gauge transformations Indeed lt AL VE gt f de RALE 0 3 61 OM for both 3 57 and 3 58 We should also check whether the gauge condition 3 44 is indeed com patible with the boundary conditions 3 58 and 3 57 For an arbitrary A there should exist unique such that VH A Vue 0 3 62 We rewrite 3 62 as VHA AE 3 63 Let us start with relative boundary conditions 3 58 In this case the left hand side of 3 63 satisfies Dirichlet boundary conditions The scalar Laplacian A 11 The equation to follow contains two types of covariant derivatives Defin
25. global heat kernel may be changed As an example of non integrable potentials consider the harmonic oscil lator in one dimension The Schrodinger operator reads D 8 72 6 3 22 On a curved manifold the subtraction procedure is more subtle On has to define a reference metric which differs from the physical one on a compact submanifold 57 If we consider the problem on the whole real axis M R the potential term is not integrable Already the expression 4 27 for a2 1 D diverges There fore analytic expressions of sec 4 cannot be used in this case However the integrated heat kernel can be easily calculated Eigenvalues of the operator 6 3 are contained in almost any textbook on quantum mechanics A v 2j 1 7 0 1 2 6 4 The integrated heat kernel reads etr CI s sinh vt A 6 5 Me K t D j 0 As t 0 it behaves like 1 t while for smooth rapidly decaying potentials in one dimension the leading singularity in the heat kernel is 1 vt This state ment may be generalised to higher dimensions If D 0 P x x with a non degenerate matrix P on M R the leading term in K t D is 2t der P 6 2 Conical singularities Conical space is defined as M 0 1 x N where N is an n 1 dimensional manifold called the base The metric of the cone has the form ds dr rad 6 6 where r 0 1 and dQ is the line element on the base N Thi
26. line element has the form ds dr r d0 2 12 where 0 lt r lt 0 lt 0 lt 2r Then gw diag 1 r St is defined by the condition r ro We may choose e r e 1 the minus sign appears because e is an inward pointing unit vector for the disk with the boundary S1 Then the second fundamental form of S is Ly eeb hT 2 13 T ro In general on S considered as a boundary of the ball in R the extrinsic curvature is Lab Sab If the boundary OM is non empty one has to define boundary conditions for the field A convenient way to write them down is Bo 0 2 14 where B is called the boundary operator In general the operator B calculates a linear combination of the boundary data for any given function If D is of Laplace type the boundary data include value of the function at the boundary and value of it s first normal derivative The most frequently used choices are the Dirichlet and Neumann boundary operators which we denote B and Br respectively Bo dlom 2 15 where S is a matrix valued function defined on OM The boundary conditions 2 16 are also called Robin or generalised Neumann In some literature they 13 are called mixed boundary conditions We shall not use this latter terminology The name mixed is reserved for another type of boundary conditions Let II and II be two complementary projectors defined on Vay II I It II J There
27. locally symmetric manifolds i e vanishing Q co variantly constant Riemann curvature and constant In this case formulae similar to the ones presented above are available Covariantly constant curvature means that locally the manifold M is a symmetric space Various approaches to the heat kernel on such manifolds are described in detail in monographs and survey articles In particular very detailed infor mation may be obtained for group manifolds see for example and for hyperbolic spaces 8 4 Heat kernel on homogeneous spaces In this section we briefly explain how one can find the spectrum of some natural differential operators on homogeneous spaces by purely algebraic methods We start with some basic facts from differential geometry and har monic analysis Consider a homogeneous space G H of two compact finite dimensional Lie groups G and H The Lie algebra G of G can be decom posed as G HOM 8 23 80 where H is the Lie algebra of H and M is the complement of H in G with respect to some bi invariant metric We have H M cM 8 24 where is the Lie bracket on G If moreover M M C H then G H isa symmetric space We do not impose this restriction M can be identified with the tangent space to M G H at the origin i e at the point which represents the unit element of G Eq 8 24 tells us that H acts on M by some orthogonal representation This action defines the embed
28. potentials 2 22 0004 6 2 Conical singularities 2 0 0 2 2 oani a addaa ee 6 3 Domain walls and brane world 0 20 0004 6 4 Non smooth boundaries 2 0 0 ee 6 5 Dielectric bodies s al ao d yatai p eot na a A a o a a aaa Anomalies 7 1 Conformal anomaly aoaaa a T2 Chiral anomaly ss ennan o kk ies ee ee ER i 7 3 Remarks on the Index Theorem oaoa a a Resummation of the heat kernel expansion 8 1 Modified large mass expansion ooa a 8 2 Covariant perturbation theory ooo a e 8 3 Low energy expansion sooo a a 10 10 13 16 18 18 19 22 25 29 32 32 39 39 40 41 42 44 44 46 49 5l 51 53 56 56 57 58 62 62 64 64 67 71 8 4 Heat kernel on homogeneous spaces 9 Exact results for the effective action 9 1 The Polyakov action 9 2 Duality symmetry of the effective action 10 Conclusions 1 Introduction It was noted by Fock in 1937 that it is convenient to represent Green functions as integrals over an auxiliary coordinate the so called proper time of a kernel satisfying the heat equation Later on Schwinger recognised that this representations makes many issues related to renormalization and gauge invariance in external fields more transparent These two works introduced the heat kernel to quantum theory DeWitt made the heat kernel one of the main tools of his manifestly covariant approach
29. that az repeats the structure of 53 the classical action 3 10 az S dr pila XA bulk terms 5 38 where Bal is a beta function We put La 0 otherwise we have had to introduce a dilaton coupling on the boundary to achieve renormalizability We also suppose that the target space metric G 4p is trivial With these simplifying assumptions the beta function can be easily calculated from 5 38 5 36 5 37 3 21 1 E 5 OaFac 1 F ph 5 39 By a lengthy but straightforward calculation one can demonstrate that the condition BE l 0 is equivalent to the equations of motion following from the Born Infeld action on the target space Lpi J aR aea iF pax exp jena F 5 40 where i appeared due to our rule of the Euclidean rotation for the gauge fields The Born Infeld action has been derived from the beta functions of the open string sigma model in confirming an earlier work which used different methods The heat kernel analysis was performed by Osborn and then repeated in for more general couplings 5 4 2 Spectral or Atiyah Patodi Singer APS boundary conditions Spectral boundary condition were introduced by Atiyah Patodi and Singer in their study of the Index Theorem These boundary conditions are global i e they cannot be defined by using local data only Consider a Dirac type operator D in VitE 5 41 where V is a covariant derivative with a compatible connec
30. that it delivers necessary information in terms of just few geometric invariants This method does not make distinctions between different spins gauge groups etc Even dependence of the space time dimensionality is in most cases trivial Therefore on one hand just a single calculation serves then in many applica tions On the other hand calculations in simple particular cases give valuable information on the general structure of the heat kernel This property is espe cially useful when one deals with complicated geometries like in the presence of boundaries or singularities During the last decade many models which lead to such complicated geometries were very actively studied in theoreti cal physics The Dirichlet branes and the brane world scenario are the most popular but not the only examples We have to mention also the limitations of the heat kernel formalism It works less effectively in the presence of spinorial background fields i e when there is mixing between bosonic and fermionic quantum fields This problem is probably of the technical nature so that the corresponding formalism may be developed some time in the future A more serious drawback is that the heat kernel expansion is not applicable beyond the one loop approximation It is not clear whether necessary generalisations to higher loop could be achieved at all It is not possible to write a review paper on heat kernel which would be complete in all respects especiall
31. the heat kernel expansion many interesting problems still remain open There are many opportunities to extend and generalise the results presented in this report This refers especially to the material of sec 5 9 where one could add new types of the operators boundary conditions geometries and singularities There is a completely new field of research related to the heat kernel expansion where very little has been done so far This is an extension to non commutative geometry This problem is an especially intriguing one since one can expect very unusual properties of the spectral functions because of very unusual properties of corresponding field theories in the ultra violet asymptotics Acknowledgements I am grateful to my collaborators S Alexandrov M Bordag T Branson E Elizalde H Falomir P Gilkey D Grumiller K Kirsten W Kummer H Liebl V Lyakhovsky V Marachevsky Yu Novozhilov M Santangelo N Shtykov P van Nieuwenhuizen and A Zelnikov who contributed in many ways to the material presented in this report I have benefited from enlight 92 ening discussions with I Avramidi A Barvinsky G Esposito D Fursaev G Grubb A Kamenshchik V Nesterenko and with my friends and colleagues at St Petersburg Leipzig and Vienna I am grateful to all readers who suggested their comments on the previous version of the manuscript I also thank the referee for useful critical remarks This work has been supported by P
32. vary the action 3 16 with respect to the fluctuation field 1 5082 Pa Va se DE dr 6e BE 3 20 2 M aM Now we require that the boundary integral in 3 20 vanishes for arbitrary d Hence we arrive at the boundary conditions 3 18 with the operator S given by 1 S 30V VT S Ta Ba Fy 3 21 io c 7X9 HP o Bps Fpp H gc Bp Fp 1 5 0 X D4 Bac Fc Dp B4c F4o Note that the operator in 3 21 is not of the ordinary Neumann or Robin type since it contains tangential derivatives on the boundary cf sec 5 4 1 22 The variation 3 20 vanishes also if we choose Dirichlet boundary con ditions for some of the coordinates of the string endpoints Namely we can take a projector IT and impose 3 18 on I1 and 1 IT 6 Ja 0 Physically this means that the endpoints of the bosonic string are confined to a submanifold in the target space Such configurations are called the Dirichlet branes 38 8 Spinor fields The action for the spinor fields Y es I dx GoD w 3 22 contains a first order operator D of Dirac type In Euclidean space the conju gate spinor w is just the hermitian conjugate of y y yt By definition an operator D is of Dirac type if its square D p is of Laplace type Spectral theory of general operators of Dirac type both on manifolds without bound aries and with local boundary conditions on manifolds with boundaries can be fo
33. we concentrate on main ideas of the method For the details an interested reader can consult the original paper and the monographs We start with very general properties of the heat kernel coefficients Let us consider a smooth compact Riemannian manifold M without boundary To be able to define functions on M which carry some discrete spin or gauge indices we need a vector bundle V over M Let D be an operator of Laplace type on V and let f be a smooth function on M There is an asymptotic expansion 2 21 and 1 Coefficients with odd index k vanish a2 41 f D 0 2 Coefficients a2 f D are locally computable in terms of geometric invari ants Already the existence of the power law asymptotic expansion 2 21 is a non trivial statement We postpone the discussion of this property to sec 5 4 The second statement above is very important It means that the heat kernel coefficients can be expressed as integrals of local invariants alf D try fy CEVI Oale D E tev f Pevi ALD 4 1 where A are all possible independent invariants of the dimension k con structed from E Q Ryvps and their derivatives We use usual assignments of the dimensions when F has dimension two any derivative has dimension one etc u are some constants For example if k 2 only two independent invariants exist These are and R Note that we can always integrate by parts to remove all derivatives from f The first statement a2 0
34. 5 31 as f D B a 3 4r VP f a aVhitry f 96xE 16yR 8 fX Ranan 380 70 bag Lob Olle 10M Lab Lab 969 Laa 1928 12X aX a fin 64 30I Laa 969 24X finn 5 32 aa f D B 4r Gh dx gtry f 60E 60RE 180E 300 2 12Rii 5R 2Ri Rij 2RijzxRijn n d ta vhtry f 2401L 12011 En 491 181T_ Rey 24Laa bb OLeb ab 120E Laa 20RLaa 4Ranan Lon ania 4RabcbLac 1 57 2801L 4011 LeaLopLee 16814 SAN Lap Lab Lec 04 4290 Lap LocLac 720S E 1205R 0S Ranan 1445 Laa Lon 485 Lab Lab 4809 Laa 4809 1209 00 60XX aQan 12X aX a Ltb 24X aX bLab 120X aX a9 fan 180XE 30XR ORanan 1 644 18011 Lag Lop 8414 60 Lab Lab 729 Laa 2408 18X aX a finn 24Laa 1208 30X fiin 5 33 Dirichlet and modified Neumann Robin boundary conditions are recovered when II 0 or rT 0 ope iye A different algorithm was Fiesected in For pure faci iced boundary conditions the coefficient a was calculated by Branson Gilkey and Vassilevich for the special case of a domain in flat space or of a curved domain with totally geodesic boundaries Kirsten generalised these results for arbitrary manifolds and boundaries Branson Gilkey Kirsten and Vassile vich calculated the coefficient a for mixed boundary conditions l 5 4 Other boundary conditions From the tec
35. 5 E diana AY VV AS PAET AS AAR a a B v 2F 2 B c3 AP AY 1 5 fd aVvh Agv as Aer Ad 3 43 2 Jam 26 The covariant derivative V contains both metric and gauge parts V Aji A7 Bfc A T 4 One should impose a gauge condition on the fluc tuation Aji We choose Vi At 0 3 44 In the gauge 3 44 the bulk term in 3 43 defines an elliptic operator of the Laplace type with wy 28 Bric 45 12 63 3 45 E RP5S 2F BY ch 3 46 The field strength corresponding of the connection 3 45 reads Qu 83 R puvdg F B 2 G48 3 47 Note that all the quantities above 3 45 3 47 are matrix functions with both gauge and vector indices On a manifold without boundary the operator D with 3 45 and 3 46 is symmetric with respect to the standard inner product in the space of the vector fields lt AY A gt gde ADH AQe 3 48 M The ghost operator corresponding to the gauge 3 44 is just the ordinary scalar Laplacian D9 V lohe igh with the connection E A 3 49 The one loop path integral reads Z B det D 1 det D1 3 50 where the first determinant is restricted to the fields satisfying the gauge condition 3 44 Note that pure gauge fields A VE are zero modes of the total operator D VV of the bulk action 3 43 only on shell i e when the background field B satisfies the classical equation of motion V F B 0 Therefore the path integr
36. Heat kernel expansion user s manual D V Vassilevich 2 Institut fiir Theoretische Physik Universitat Leipzig Augustusplatz 10 D 04109 Leipzig Germany bY A Fock Insitute of Physics St Petersburg University 198904 St Petersburg Russia Abstract The heat kernel expansion is a very convenient tool for studying one loop diver gences anomalies and various asymptotics of the effective action The aim of this report is to collect useful information on the heat kernel coefficients scattered in mathematical and physical literature We present explicit expressions for these co efficients on manifolds with and without boundaries subject to local and non local boundary conditions in the presence of various types of singularities e g domain walls In each case the heat kernel coefficients are given in terms of several geometric invariants These invariants are derived for scalar and spinor theories with various interactions Yang Mills fields gravity and open bosonic strings We discuss the re lations between the heat kernel coefficients and quantum anomalies corresponding anomalous actions and covariant perturbation expansions of the effective action both low and high energy ones Key words heat kernel functional determinants effective action boundary conditions anomalies PACS 04 62 v 11 10 z 02 40 k Email address vassil itp uni leipzig de D V Vassilevich Preprint submitted to Elsevier Sci
37. P eyatrv a FE aF a 4 17 The equation 4 17 shows that as 180 ay 60a2 To proceed further we need local scale transformations defined in 4 7 and 4 9 These scale transformations look similar to the local Weyl trans formations but are not exactly the same The scale transformations in 4 7 are designed in such a way that the operator D always transforms covariantly This is not the case of the Weyl transformations of an arbitrary operator of Laplace type For example the scalar Laplacian 3 4 is conformally covariant for a special value 3 5 of the conformal coupling only Therefore some of the basic quantities are transformed in a somewhat unusual way The metrics transforms as guy e fg thus defining standard conformal properties of the Riemann tensor the Ricci tensor and the scalar curvature The functions a and b in eq 2 1 transform homogeneously Transformation properties of w and E are then defined through 2 3 and 2 4 One can obtain the following relations a c 0 9 nfV 9 e0 Rijki 2f Rage Oj fax Our bgt Ou fjr jk fii lt E 2fE s n 2 fii R 2f R 2 n 1 fui 1 e 0F kk 4f Ekk 2f E 5n 2 fiji n 6 fE de 2 1 d 7 10E gt 4f E oF n F 2 fa 36 e 0 Rk 4f Rk 27h 2 n That n 6 fake a R 4f R A n fuk u de X de Let us remind that the indices 7 j k l are flat so we ca
38. R 12E Ri 5ER 2ER Ri 2E Rijn Rijx 4 29 Everyone who ever attempted calculations of the heat kernel coefficients on curved background for arbitrary spin should appreciate that the method presented here is a quite efficient one One should also take into account that some of the universal constant were indeed calculated twice The coefficients apg a4 are contained in ag was first computed by Gilkey The next coefficient ag has been calculated by Amsterdamski et al for the scalar Laplacian and by Avramidi for the general operator of Laplace type The coefficient a1 has been calculated by van de Ven Higher heat kernel coefficients in flat space were studied in 4 2 Examples Here we consider several simple physical systems in four dimensions and calculate the heat kernel coefficient a4 which defines the one loop divergences in the zeta function regularization 4 2 1 Yang Mills theory in flat space Our first example is pure Yang Mills theory in flat space We are inter ested in the total heat kernel coefficient att defined by 3 53 Let us start with the first term describing contribution from the vector fields We choose the gauge 3 44 The only non vanishing invariants are EF and Q The coef ficient a4 is quadratic in these quantities By using eqs 3 46 and 3 47 we obtain 40 try E Boe Ee SAF Fo Ks 4 30 try UN 4F F Koy 4 31 where Kip O ge
39. al is gauge independent on shell but it depends on the gauge choice off shell For example the Feynman gauge path integral Zr B det D det D 9 3 51 where the first determinant is calculated on the space of all vector field is equal to the Z B defined in 3 50 only on shell However physical predictions of the two path integrals are of course equivalent 27 The path integral 3 51 can be obtained by adding the gauge fixing term 1 Lot 4 ax GV yA 3 52 to 3 43 with x 1 The case k 4 1 yields a non minimal operator on the gauge field fluctuations cf sec 4 4 One can define total heat kernel coefficients for the path integral in a certain gauge y A aj ax D 2a D9 3 53 where DX is defined by the action 3 43 with the gauge fixing term y A and the ghost operator is De y V Even on shell only the coefficient a is gauge independent Only this coefficient contains information on the one loop divergences in a gauge invariant regularization like the function one On a manifold with boundary one should impose boundary conditions on gauge fields and ghosts The boundary conditions should be gauge invari ant Consider a more general set up when we have some quantum fields and a linearised gauge transformation P with local parameter Boundary operator 6 defines gauge invariant boundary conditions Bd 0 3 54 if there exist boundary conditions for the ga
40. ansion and to supply the reader with a user friendly guide to the field The main idea which we shall pursue is the universality of the heat kernel A single calculation though sometimes quite involved may help in a large variety of applications regardless of such details as spin gauge group etc As well just a single universal object in fact describes counterterms anomalies some asymptotics of the effective action and much more To illustrate the use of the heat kernel in quantum field theory let us consider the generating functional for the Green functions of the field in the path integral representation Z J f Deexp L J 1 1 The heat kernel methods are almost exclusively used for the one loop calcula A historical survey of the mathematical literature on the heat kernel expansion can be found in tions In this approximation it is enough to expand the action to up to the quadratic order in quantum fluctuations L La J DO 1 2 where La is the action on a classical background denotes an inner product on the space of quantum fields Usually this inner product is just an integral over the underlying space For real one component scalar fields it reads 61 02 d e G01 2 o2 c 1 3 The linear term in 1 2 contains in general contributions from the external sources of the field and from the first variation of the classical action If the background satisfies classica
41. ased on the Gauss Bonnet theorem see We use here a more lengthy way which however works perfectly on flat manifolds without boundary and of trivial topology of R Note that R is non compact To make the heat kernel well defined we should suppose certain fall off conditions on the background fields and on the smearing function f The basis in the space of the square integrable functions is given by the plane waves exp ikx Therefore for M R with flat metric the heat kernel reads K f t Trz2 f exp tD 7 a s ee ve x exp tD e fae ja L try f x exp t V ik B 4 21 The following integrals will be useful a k g tk __ 1 On T na d k p hea a g Qn inate J GPK RP pap poge l L gge 4 ghp geo 4 guage 4 22 27 4tr 2 4t Now we isolate exp tk on the right hand side of 4 21 and expand the rest of the exponent in a power series of t K fit fare 2 Pete E V E EV E RvB E kV kV E KY 2 2 agi V kV kV V7 kV cee xe 4 23 ety ry ro h t V E Eav We use the integrals 4 22 to obtain 38 KDZ capa Merry F a 1 tB 2 5 V V E EV E 2 t z VR EV VEV t FVV V V V 2 VEV VV EVV EEN Va Y ot 4 24 All derivatives combine into commutators Finally we get K f t J aatry 1 r 14 tE 4r t o 2 1 2 1 1 pv 3
42. ated geometries as e g the spherical cap Note that here we consider local boundary conditions without tangential derivatives only For other types of boundary operators see sec 5 4 5 2 Dirichlet and Neumann boundary conditions Let us now find analytic expressions for the heat kernel coefficients in terms of geometric invariants Here we follow the method of Branson and Gilkey For both Dirichlet and modified Neumann Robin boundary conditions the heat kernel coefficients are locally computable This means that a may be represented as a sum of volume and boundary integrals of 15 A general discussion of the representation of the zeta function by contour integrals can be found in 16 Some of these works consider contributions of the so called physical modes only As explained in complete answer for the effective action and for the scaling behaviour must include contributions from ghosts and non physical modes This applies also to the spherical cap case considered below 47 some local invariants k 1 ax f D B f Peyatlejale D E fa evhf Oale DB j 0 5 16 where f0 denotes j th normal derivative of the smearing function f Note that we cannot now integrate by parts to remove normal derivatives from f This reflects the distributional nature of the heat kernel asymptotics The volume terms a x D are the same as in the previous section see 4 26 4 29 Here we evaluate the boundary terms ax
43. ay be interpreted as mixtures of D branes and open strings General form of the heat kernel expansion for spectral boundary condi tions was established by Grubb and Seeley n 1 oo K t f D F agt anteparo 5 45 k 0 j n In the contrast to all previous cases the expansion 5 45 contains loga rithms of the proper time t Although such terms appear typically for pseudo differential operators they may lead to rather unpleasant physical con sequences As follows from 2 25 non zero a means that the zeta function has a pole at s 0 and therefore the expression 2 32 for the renormalised effective action does not make sense Fortunately for the APS boundary con ditions a 0 if f 1 near the boundary Consequently the integrated zeta function is regular at s 0 and eq 2 32 still can be used However all calcu lations involving localised heat kernel coefficients remain problematic Many logarithmic terms vanish if the manifold M has a product structure near OM In non product cases strong criteria of partial vanishing of logarithms have been found recently Another problem with the heat kernel expansion 5 45 is that the coef ficients may have a more complicated dependence on n than just a power 21 Let us remind that to prove the simple dependence on n we used the product formulae 4 3 4 4 We assumed that the spectral problem can be trivialised in one direction In t
44. brane in a five di mensional space According to the Israel junction condition the metric in such models cannot be smooth on X Typical form of the metric near is ds de e daa 6 15 where a is a constant and where ds _ is a line element on the n 1 dimensional hypersurface X Due to the presence of the absolute value of the n th coordinate in 6 15 the normal derivative of the metric jumps on X One can think of two smooth manifolds M and M glued together along their common boundary X Neither Riemann tensor nor matrix potential F must be continuous on Also the extrinsic curvatures L and L of considered as a submanifold in M and in M respectively are in general different All geometric quantities referring to M respectively M7 and their limiting values on X will be supplied by a superscript respectively For the case at hand there is still an asymptotic expansion 2 21 for the heat kernel The heat kernel coefficients can be decomposed as ax f Dlv ag f D a f D aK f D v 6 16 24 A similar scenario was proposed earlier in see for a review 60 where a f D are known volume contributions corresponding to M cf 4 26 4 29 The coefficients aF are given by integrals over of some local invariants Note that a V V3 6 17 being a difference of two connection is a pseudo vector with respect to all spac
45. cause of great technical complexity only the first two heat kernel coefficients were analysed Some calculations in various physical systems with non minimal operators can be found in 13 This case covers most of the physical applications 44 5 Heat kernel expansion on manifolds with boundaries 5 1 Two particular cases We start our analysis of manifolds with boundaries with two simple ex amples First let us consider a one dimensional manifold M 0 7 Let D 0 We consider both Dirichlet 2 15 and Neumann 2 16 boundary conditions taking S 0 in 2 16 to simplify the calculations The eigen functions of D are Dirichlet sin l z L 1 2 Neumann cos lx L 0 1 2 5 1 The eigenvalues are l in both cases where l is a positive integer for Dirichlet boundary conditions and l is a nonnegative integer for Neumann boundary conditions The heat kernel asymptotics can be calculated with the help of the Poisson summation formula 4 4 K t 82 B7 Dee i 1 0 e 5 2 K t 82 B aE i 1 O e 5 3 1 gt 0 where for the later use we explicitly mention the boundary operators B and B which define Dirichlet 2 15 and Neumann 2 16 boundary conditions respectively Let us modify a little bit the example above by allowing for a non zero S at one of the components of the boundary Oz0 2 0 0 Ox S o z r 0 5 4 Note that at m the derivative with re
46. crepancies was clearly stated by Dowker who confirmed the result 7 16 More extensive literature can be found in 68 The gauge invariance can and should be retained in quantum theory while the chiral invariance is typically broken by the quantisation Let us study these effects in the zeta function regularization Define determinant of the Dirac operator as a square root of the determinant of the associated Laplace operator 1 1 log det P gt log det H 5 log det D 7 23 The effective action W corresponding to the path integral 3 41 reads 1 1 ne eke W log Z 3 les det D z600 D 5 nu 0 D 7 24 where u is a normalisation scale By virtue of 7 4 5 s D sTr D AD Tr D A 0 7 25 This proves gauge invariance of the effective action For the chiral transformation we have Alp pW 2iTr 72yD 0 7 26 A y measures non invariance of the effective action with respect to the chiral transformations It is called the chiral anomaly One can express A through the heat kernel coefficients Alp 2ia y y D 7 27 where a y v D is defined as in 2 21 but with a matrix valued smearing function instead of the scalar one The chiral anomaly A y should satisfy certain consistency conditions following from the Lie algebra identities 7 20 7 22 and gauge invariance of the effective action Alp AA 7 28 do A y2 p Al 0 7 29
47. ct relations between the effective actions in dual theories In this section in contrast to section 8 we don t have to neglect derivatives or powers of the background fields Section 10 contains concluding remarks 10 2 Spectral functions heat kernel zeta function resolvent 2 1 Differential geometry and the operators of Laplace type Let M be asmooth compact Riemannian manifold of dimension n with smooth boundary 0M We shall also consider the case when the boundary OM is empty Let V be a vector bundle over M This means that there is a vector space attached to each point of the manifold For example this could be a representation space of a gauge group or of the space time symmetry group Sections of V are smooth functions bearing a discrete index which could correspond to internal or spin degrees of freedom We study differential operators on V We restrict ourselves to second order operators of the Laplace type Locally such operators can be represented as D g 3 3 a7 b 2 1 where g is the inverse metric tensor on M a and b are matrix valued func tions on M There is a unique connection on V and a unique endomorphism E of V another matrix valued function so that D gH V V_ B 2 2 where the covariant derivative V VF w contains both Riemann V and gauge bundle w parts We may express 1 Ws Iva 9 T u ly and 2 3 E b g bywy WyWy Wel yy
48. ding H C so n 8 25 All physical fields are classified according to certain representations of the Lie algebra so n Restrictions of these representations to H define transformation properties of the field with respect to H From now on we work with each irreducible representation of H separately The field 64 belonging to an irreducible representation 7 H can be expanded as see e g a z Vo gt T Tea DE 8 26 Jd where Vol is the volume of G H dr dim7 H We sum over the repre sentations TY of G which give 7 H after reduction to H labels multiple components 7 H in the branching T H d dim TO q runs from 1 through d The matrix elements of 7 0 have the following orthogonality property f PevaTIRUG VTE Mac Voljdz dr cc pa y 8 27 Therefore to construct the harmonic expansion on G H it is necessary to have powerful methods for reduction of the representations from G to H There are several standard and less standard techniques which may be used depending on the particular homogeneous space It is important that not only the harmonic expansion but also the spec trum of the invariant operators can be analysed by the group theoretical meth ods The covariant derivative on G H reads V VA rA 8 28 Here V is the canonical covariant derivative on G H At the origin Vii can be identified with the tangent space generators from M taken in the representation J Th
49. e aC uvpo C b Fw FY R eR dR 4 35 1 28807 where Chpo is the Weyl tensor 1 Cpa OOP Ruy R 2Ryy R SR 4 36 pv po a b cand d are some constants depending on the spin The first two structures in 4 35 which appear together with a and b are conformally invariant in four dimensions This explains our choice of the basis in the space of invariants 41 Table 1 a4 for various spins The constants a b c and d can be evaluated by substituting particular expressions for and Q obtained in sec 3 in 4 28 Alternatively one can use the analysis of Christensen and Duff who calculated a4 for arbitrary spin fields see also The results are collected in Table 1 Some comments are in order Spin 1 2 means 4 component Dirac spinors For spin 1 and spin 2 fields we took into account contributions from corresponding ghost fields Note that vector Yang Mills fields and vector ghosts for gravity interact differently with the background geometry The cosmological constant is taken to be zero As a physical application of the heat kernel expansion in curved space we may mention for example the asymptotic conformal invariance phenomenon which was studied by using this technique in A similar analysis can be performed also in the presence of boundaries 4 8 DeWitt iterative procedure The iterative method by DeWitt uses separation of the heat kernel with non coinciding arguments
50. e part ri depends on the invariant metric on G H and on the structure constants of G restricted to M On symmetric spaces such 81 structure constants are zero and therefore the Laplace Beltrami operator has a particularly simple form D x Villy C3 G C2 H 8 29 where Ch are quadratic Casimir operators of G and H which depend on the representations J and T H respectively On general homogeneous spaces the expressions are a bit more complicated see e g for explicit exam ples In any case eigenvalues of D are given by a second order polynomial Q m Mp of several natural numbers m These eigenvalues are in general degenerate with multiplicities defined essentially by dimensions of the repre sentations of G and H They are also polynomials in m The heat kernel is then represented as an infinite sum K t D gt gt N mi mx exp tQ mi Mx 8 30 There exist several tricks which can be used to evaluate the t 0 asymptotics of such sums For example one may use the Poisson summation formula 4 4 By taking derivatives with respect to t one obtains YP exp tP V2 2r 1 O e 8 31 lEZ for r N More general polynomials of l may be treated by the Euler Maclaurin formula which reduces sums to the integrals Let F T be a function defined on 0 lt T lt If the 2m th derivative FC r is absolutely integrable on 0 00 M ll fos F0 7 Far
51. e the de Rham complex for which V consists of p forms on M and P is the exterior derivative the Dolbeaux complex which deals with the forms on a complex manifold and the signature complex which treats selfdual and anti selfdual forms More details can be found in Another very important construction of this type is the Witten index which constraints the supersymmetry breaking 74 8 Resummation of the heat kernel expansion The heat kernel coefficients define the one loop counterterms in the back ground field formalism In many cases the heat kernel can also give a useful information on the finite part of the effective action Just one of the examples is the large mass expansion 1 22 which is valid when all background fields and their derivatives are small compared to the mass of the quantum field In order to get the effective action in other limiting cases one has to re arrange the heat kernel expansion 8 1 Modified large mass expansion In many physical applications there is a quantity M which is large com pared to the rest of the background fields and their derivatives Therefore it is a well motivated problem to construct an expansion of the effective action in a power series of M t To do so one has to re express the heat kernel as KED See eee ae 8 1 k To obtain the effective action one has to integrate the heat kernel over t cf eq 1 18 To simplify the argumentation we assume the cut off regulariza ti
52. e time and gauge symmetries Therefore a can be used for constructing the surface invariants To make the formulae more symmetric we introduce two inward pointing unit normals y and vy to in M and M respectively We do not suppose that the smearing function f is smooth on X therefore there is no relation between f and f but we assume continuity of f ft fT f on X The surface invariants can be constructed from Lay Rijko E7 V a and their derivatives This gives much more invariants than we usually have for a boundary value problem There are however some properties of a which simplify the calculations considerably First of all a must be invariant with respect to interchanging the roles of M and M Also a must vanish when the singularity disappears The first property excludes for example the term f E E which changes sign under M M The second requirement excludes the invariant f E E because it survives even if there is no singularity on X These simple arguments show that af does not contain E even though such terms are allowed on dimensional grounds It is also very helpful that in some particular cases the problem in question can be reduces to a sum of Dirichlet and Robin boundary value problems The coefficients az k 0 1 2 3 read Gi Da 0 a Die 0 Spey ar f av Tirv 2f Et Ln 6fo 6 18 1 3 az f D v e A d aV htry ARL Laali 207 Low 3
53. e with respect to d d d d Nips A TD ead e ee eed 4 12 0 sao leaotn 1 Dle 8 Tlecogrleaotn 1 D 65 412 Finally eq 4 8 yields 4 9 Eq 4 8 restricts dependence of the heat kernel coefficients on the poten tial E while 4 7 and 4 9 describe properties of the heat kernel coefficients under local scale transformations To calculate the heat kernel coefficients we adopt the following strategy First we write down a general expression for a containing all invariants A of dimension k with arbitrary coefficients u The constants u are then calculated by using the properties derived above The first three coefficients read alf D 4ry draygtrv aof 4 13 ag f D ar J d xy gtrv f ai E a2k 4 14 1 aa f D e L d x gtrv f azE kk RE aE a6 R kk a7 R ag Rij Ri Ag Rijn Rizr 449245 N45 4 15 Instead of the u we use rescaled constants ar By 4 6 the coefficients a are true constants i e they do not depend on n One can check that indeed 35 no more invariants exist For example R is proportional to R due to the Bianchi identity The coefficient ag follows immediately from the heat kernel expansion for the free scalar Laplacian on St see eq 4 4 We obtain ap 1 Let us now use 4 8 First take k 2 Then f d x gtry aF J de gtry F 4 16 This gives a 6 Take k 4 to see 1 1 zgo Jy PPV IVF R 2asFE 3 d
54. easy to calculate the part of Wa which diverges in the limit A oo m I div _ _ n 2 n A 2 m l Wy 4r fa L G Le bo x x na 3 21 X In A m l by 2 2 O A 1 21 2 jt l n We see that the ultra violet divergences in the one loop effective action are defined by the heat kernel coefficients b x x with k lt n On non compact manifolds the integral of b x x is divergent This di vergence is removed by subtracting a reference heat kernel see sec 6 1 Contributions from higher heat kernel coefficients bg k gt n to the effective action are not divergent and can be easily calculated yielding in the limit A oo 4r mt f deya 5 a D9 n 1 22 2j gt n This is nothing else than the large mass expansion of the effective action This expansion is valid for relatively weak and slowly varying background fields We have seen that the heat kernel expansion describes e short distance behaviour of the propagator e one loop divergences and counterterms e 1 m expansion of the effective action We shall see below that heat kernel provides a natural framework for studying e quantum anomalies sec 7 e various perturbative expansions of the effective action sec 8 e selected non perturbative relations for the effective action sec 9 Of course in all these applications the heat kernel methods have to com pete with other techniques The main advantage of the heat kernel is
55. effective action It describes the quantum effects due to the background fields in the one loop approximation of quantum fields theory To relate W to the heat kernel we shall use the arguments of For each positive eigenvalue A of the operator D we may write an identity hn PM ai 1 17 o t This identity is correct up to an infinite constant which does not depend on and therefore may be ignored in what follows 4 Now we use In det D Trln D and extend 1 17 to the whole operator D to obtain 1 ec Wee Kt D 1 18 0 where K t D Tr e dx GK ts x x D 1 19 3 We shall mostly use the coefficients a which differ from b by a normalisation factor 4 To prove this statement one has to differentiate both sides of eq 1 17 with respect to A Here we have only presented some heuristic arguments in favour of eq 1 18 A more rigorous treatment of functional determinants can be found in sec 2 2 The integral in 1 18 may be divergent at both limits Divergences at t oo are caused by zero or negative eigenvalues of D These are the infra red divergences They will not be discussed in this section We simply suppose that the mass m is sufficiently large to make the integral 1 18 convergent at the upper limit Divergences at the lower limit cannot be removed in such a way Let us introduce a cut off at t A 1 re dt Wa K t D 1 20 a 0 KD 1 20 It is now
56. elds in the path integral The action given in eq 9 23 is invariant under the gauge transformation which sends A to A dyAp 1 This means that the p forms which are dg exact have to be excluded from the path integral but that a Jacobian factor corresponding to the ghost fields A _ has to be included in the path integral measure Next we note that dg exact p 1 forms do not generate a non trivial transformation of A Hence such fields must be excluded from the ghost sector Then we have to include ghosts for ghosts This goes on until the zero forms have been reached By giving these arguments an exact meaning one arrives at the Faddeev Popov approach to quantisation of the p form actions We note that the procedure of is valid also in the presence of a dilaton interaction if one simply replaces the ordinary derivatives by the twisted ones As a result we have the following expression for the effective action W Nl rR P gt gt 1 In det A Wp 9 27 k 0 wep depends on certain topological characteristics of the manifold the Betti numbers We shall neglect W gt and some other topological contributions in what follows By combinatorial arguments similar to the presented above one can show that 5 Wo Wr p 2 8 1 an 58 AS 0 28 k 0 One should distinguish between 6 which is a variation of the dilaton and de which is a twisted co derivative The
57. ence 9 September 2003 Contents 1 2 Introduction Spectral functions heat kernel zeta function resolvent 2 1 Differential geometry and the operators of Laplace type 2 2 Spectral functions a atores aa ane a u Go ee ee 2 3 Lorentzian signature 3 6 gus c04 a p aae eed eee ee he AY Relevant operators and boundary conditions cL Scalar fields p r 94 piv ae Be ee a e aE oA 3 2 BOSONIC StH e a 2 4 an Gastar ot ae gee dot Ste ae ae Be Hae 3 3 Spor fields sa os a ho ae Ree ee ee ae ee ee aA 3 4 Vector field8 2 24 44 40 wae de eat aS OS ke ee eS D MGEAVAIGON S 208 een 3 ES ee coe Bagh BOR Cet Unt BAe Sak el SY Heat kernel expansion on manifolds without boundary 4 1 General formulae 2 2 ee 4 2 Examples saruni steak y Ate PSR ACP Go Ae Eee ee BG 4 2 1 Yang Mills theory in flat space 4 2 2 Free fields in curved space 2 2 0004 4 3 DeWitt iterative procedure 2 a 4 4 Non minimal operators 0 0 0000 eee eee Heat kernel expansion on manifolds with boundaries 5 1 Two particular cases 2 a a 5 2 Dirichlet and Neumann boundary conditions 5 3 Mixed boundary conditions sooo a 5 4 Other boundary conditions ooo a 5 4 1 Boundary conditions with tangential derivatives and Born Infeld action from open strings ooo aaa 5 4 2 Spectral or Atiyah Patodi Singer APS boundary conditions Manifolds with singularities 6 1 Non integrable
58. enormalization The remaining part of W at s 0 is the renormalised effective action w 50 0 D 5 In u C 0 D 2 32 where we have introduced a rescaled parameter u e 7 ji In this approach u describes the renormalization ambiguity which must be fixed by a suitable normalisation condition Let us remind that here we are working on a com pact manifold On non compact manifolds s D may have divergent contri butions proportional to the volume Such divergences are usually removed by the subtraction of a reference heat kernel see sec 6 1 Together with 1 16 equation 2 32 yields a definition of the functional determinant for a positive elliptic second order operator which is frequently used in mathematics In det D 0 D In 0 D 2 33 Note that the definitions 2 23 2 33 are valid for positive operators only Elliptic 2nd order differential operators have at most finite number of 16 zero and negative modes which must be treated separately However one can extend the definition of the zeta function to operators with negative modes Ct DI aS 2 34 where the sum extends over all non zero eigenvalues One can also define another spectral function in a similar way n s D So sign A A 2 35 This function is especially useful in spectral theory of Dirac type operators where 0 2 measures asymmetry of the spectrum Another function which is frequently used e
59. eries in FE 2 t t e P e dsc Bet J dsa daet A Fetma pena 0 0 0 8 5 The heat trace can be also expanded Re Dte sy KO 8 6 where K contains the jth power of E 76 Covariant perturbation theory approach prescribes to take the Oth order heat kernel in the free space form cf eq 1 12 for m 0 Ko a y t 4nt exp e 8 7 which is an exact kernel on M R only This formula neglects all global contributions and therefore is valid only for sufficiently close x and y and for small t We have Kolt a dx gtry Kola x t 4nt A dx gtty Ivy 8 8 where tryly simply counts discrete spin and internal indices of D This formula reproduces the ao contribution to the heat kernel In the next order we have t t Ky t Tr dset 95 BeA Tr dse E 0 0 ttry f d xy gKolz x t E x t CE f dz gtry E x 8 9 where we used the cyclic property of the trace and the expression 8 7 for the Oth order heat kernel This expression is consistent with the k 1 term of 8 3 The terms with k gt 1 are total derivatives and therefore they do not contribute to the integrated heat kernel The quadratic order of the heat kernel reads t S2 Ko t Tr ty f doet ea gelea ena 0 0 t S2 try A dy f dz ds f ds1Kolz y t s s1 E y 0 0 x Koly z 2 1 F z 8 10 Next we introduce the rescaled variables s t and get rid
60. es if any are such that the power law expansion 2 21 exists Then expanding both sides of 7 41 in a power series of t one obtains ax D1 ak D2 0 for k n an D1 an D2 index P 7 42 12 The last equation provides a simple way to calculate the index from known heat kernel expansion It also allows to understand a very important property of the index namely that it is a homotopy invariant under quite general as sumptions Indeed suppose that P depends on a parameter a in such a way that all geometric quantities including the metric connection the matrix po tential the boundary conditions etc corresponding to D and D are smooth functions of a Clearly no essential deformations like changing the order of the operator or adding higher derivative terms to the boundary conditions or including new types of the singularities are allowed Under these smoothness assumptions a D and a D2 are smooth functions of a as well Therefore the index is also a smooth function of a Since the index is an integer it can only be a constant Hence index P is invariant under the deformations described above The relations 7 42 are also useful for calculations of the heat kernel coefficients as they give restrictions on the universal constants appearing in front of independent invariants see e g Let us consider an example of the spin complex Let M be an even di mensional manifold admitting a spin str
61. everal topics are very close to the subject of this review but are not included 1 The heat kernel expansion can be successfully applied to quantum field theory at finite temperature A new interesting development in this field is related to the so called non linear spectral problem see for an overview 2 The heat kernel expansion has interesting applications to integrable mod els and in particular to the KdV hierarchies see for an elementary introduction 3 Recently some attention has been attracted to the so called N D or Zaremba problem which appears when one defines Neumann and Dirichlet boundary conditions on two intersecting components of the boundary It is unclear whether this problem may have applications to quantum theory 4 Instead of considering asymptotics of the heat trace Tr fe one can also consider an asymptotic expansion for individual matrix elements of the heat kernel f e P f2 which are called the heat content asymptotics since they remind short time asymptotics of the total heat content in a manifold with the specific heat f and the initial temperature distribution f2 Such asymptotics do not contain negative powers of t More details can be found in 5 Many results on the heat kernel asymptotics can be extended to higher order differential operators see e g and to differential operators in superspace Although quite a lot is already known about
62. f Lele Lala 2L La a Lia Lin fo fo 48 fv Af Da a 24f Li Lade 24 f3 fz The coefficients af and a are too long to be presented here in full generality Therefore we restrict ourselves to the case of smooth geometry Ry Rib Lt L smooth connection 0 and smooth smearing function fat fo fn In other words the only singularity comes from the surface potential v 61 1 1 f D v any a tavhitry fo Zf Ro f Ev 1 1 1 Sg vea Eg gft Laa a Fn 1 1 az f D 4a f d aVhtry afe qi he 6 19 oy op 24 pe Ly Ly LPL L nn Q DEG aa sao IU Dabba fog hr ee age ee Tg Ea 1 5 beh 2 1 A aa aa TIRA ava Doido Laa Rhynn tggi aat ggz ava seq Fin Daa Ey Finn The heat kernel coefficients for 9 7t C R and v const were cal culated in Generic X with arbitrary v was considered in Moss added a non smooth connection see also Calculations on a particular brane world background can be found in The heat kernel coefficients in the general setting described here were calculated in This latter paper also considered renormalization of the brane world scenario and predicted a non standard Higgs potential on the brane Related calculations in wormhole models were done in The function for brane world geometries with matching conditions 6 11 6 14 was considered recently in
63. f the action 3 22 is that it contains first order derivatives only Consequently boundary conditions should be imposed on a half of the spinor components Let these be Dirichlet boundary conditions T_vlom 0 3 30 where H_ is a hermitian projector H2 II I II_ Due to the hermiticity WIl_lau 0 3 31 Following Luckock let us consider a family of the projectors 1 2 I_ 5 1 7 exp iq7 3 32 where q is a scalar which can depend on the coordinate on the boundary To make the operator D formally self adjoint including the boundary we must require that gee em ea 3 33 OM 10 The present author is grateful to Valery Marachevsky for his help in deriving and checking eqs 3 27 3 28 and 3 37 see also 24 for all 12 satisfying the boundary conditions 3 30 Since and Wz are arbitrary the projector II_ should satisfy 1 1_ y 1 I_ 0 3 34 This condition yields q 7 2 The projector 3 32 takes the form T Z TE 3 35 To formulate the spectral problem for the second order operator D p we need boundary conditions for the second half of the spinor components The relevant functional space should be spanned by the eigenfunctions of the Dirac operator D It is clear that on this space the functions Py should satisfy the same boundary conditions 3 30 as the w s themselves OP vlom 0 3 36 Let us adopt the choice 3 35 for I By commuting I with A in 3 36 we obtain
64. h the example of spectral boundary conditions we illustrate appearance of non standard ln t asymptotics in the heat kernel The results of sections 4 and 5 are valid on smooth manifolds with smooth potentials and gauge field In section 6 we consider the case when either the background fields or the manifold itself have singularities In particular conical and domain wall singularities are consid ered Sections 7 9 are devoted to applications In section 7 we relate confor mal and chiral anomalies to certain heat kernel coefficients and re derive the anomalies in several particular models This is a very spectacular but also rather well known application of the heat kernel expansion In section 8 we go beyond the power series in the proper time t We consider mainly two par ticular cases The first one is the case when derivatives are more important than the potentials so that we sum up leading non localities The second one is the so called low energy expansion which neglects derivatives of the back ground fields starting with certain order but treats all powers of background potentials and curvatures exactly In that section we also review some results on the heat kernel on homogeneous spaces where spectrum of relevant oper ators may be found exactly In section 9 we consider two examples when the heat kernel can be used to obtain exact results for the effective action The first one is the famous Polyakov action The second example is exa
65. he present case all tangential coordinates enter the operator 5 43 on equal footings Therefore the proof does not go through 59 of 47 After this long list of troubles it is not a surprise that only the coef ficients a with k lt n are locally computable The coefficient ap is given by 4 26 a and az have been calculated in ee ee n 1 a 47 B n 1 datrvhtry f 4 aM az 4r f d x gtry F ZR E T dt aV htry Fe 1 573 n Laaf E t i fa E The coefficient ag can be found in where 5 46 5 47 Some string theory applications suggest that spectral boundary con ditions can be defined directly for a second order differential operator Exis tence of the asymptotic expansion 5 45 and vanishing of leading logarithms for such problems have been stated in 56 6 Manifolds with singularities All results on the heat kernel expansion formulated in the previous sec tions are valid on smooth manifolds only If there are boundaries they also have to be smooth As well any singularities in the potential term or in the field strength are strictly speaking forbidden However many physical models deal with singular backgrounds Even if such backgrounds may be represented through certain limiting procedures from smooth configurations the heat ker nel coefficients are not given by limits of their smooth values The most visible manifestation of failure
66. he right hand side of 6 10 Let us integrate 6 10 over 23 The present author is not aware of any simple example where the Int terms actually appear 59 a small cylinder C C x e e I d ea V2b Vea E A oa l deV huga 0 6 12 We now take the limit as 0 Since the expression in the square brackets in 6 12 is bounded the contribution that this term makes vanishes in the limit We obtain 0 f evh Vnbaleno VnPrler 0 09 6 13 Since C and X are arbitrary we conclude that a proper matching condition for the normal derivatives is Vndlar o F VnGlan 0 F vo 0 6 14 Physically this problem corresponds to two domains separated by a pen etrable membrane X a domain wall In many cases penetrable membranes are better models of physical boundaries then just boundary conditions which are imposed on each side of X independently and thus exclude any interaction between the domains potentials are being used in quantum mechan ics where one studies the Schr dinger equation which is nothing else than the imaginary time heat equation The Casimir energy calculations have been performed e g in In the formal limit v oo one obtains Dirichlet bound ary conditions on X although the heat kernel coefficients are divergent in this limit see below Further generalisations are suggested by the brane world scenario which assumes that our world is a four dimensional mem
67. hiral transformations y const this formula was obtained in 70 As soon as we know the anomaly 7 34 for A 0 many more terms containing Ay can be restored by using the Wess Zumino condition 7 29 Some comments on the Fujikawa approach to the chiral anomaly are in order Consider the path integral 3 41 Since the action is invariant under the chiral rotations the only source of the chiral anomaly could be non invariance of the path integral measure The Jacobian factor appearing due to the change of the spinorial variables 7 18 can be formally represented as J det 1 2ipy 7 35 Then to the first order of y Alp 6 W log J 2iTr y7 7 36 where J has appeared due to the negative homogeneity of the fermionic measure The operator yy is not trace class on the space of square integrable spinors though formally its trace is zero at every point The right hand side of 7 36 is therefore ill defined Fujikawa suggested to replace 7 36 by a regularized expression Aly 2i Jim Tr ype OR 7 37 The chiral anomaly 7 37 has now the heat kernel form with the identification t M If we suppose that all positive powers of the regularization parameter M in the small t large M asymptotic expansion are somehow absorbed in the renormalization we arrive at Alp 2ia y y D 7 38 that is just the expression 7 27 obtained above in the zeta function regular izatio
68. hnical point of view boundary conditions for a Laplace type operator are needed to exclude infinite number of negative and zero modes and to ensure self adjontness In principle any linear relation between the boundary data m and aaz is admissible as long as it serves this purpose Here we consider two physically motivated examples of boundary conditions which contain tangential derivatives on the boundary These examples should give the reader an idea of what can be expected for a more general boundary value problem 5 4 1 Boundary conditions with tangential derivatives and Born Infeld ac tion from open strings The boundary condition 1 Vn ENEAS s 6 5 34 2 OM is the simplest condition containing both normal and tangential derivatives I and S are some matrix valued functions defined on the boundary Such or similar structures appear in open strings cf sec 3 2 and in chiral bags cf sec 3 3 They also describe photons with the Chern Simons interaction term concentrated on the boundary and may be relevant for solid state physics applications The conditions 5 34 appeared in the mathematical literature FER Several heat kernel coefficients for the boundary conditions 5 34 which are called oblique have been calculated by McAvity and Osborn and by Dowker and Kirsten Avramidi and Esposito lifted some com mutativity assumptions and proved a simple criterion of strong ellipticity see below
69. ino action We do not consider this action here since such a consideration would require a lot of additional technical devices The reader can consult excellent original papers 4 The next subsection is devoted to another well know example which is the Polyakov action obtained through integration of the conformal anomaly in two dimensions In sec 9 2 below we consider a more complicated situation when relevant variation of the effective action leads to a linear combination of the heat kernel coefficients of several different operators Typically in this case one obtains exact relations between the effective actions rather than the effective actions themselves 9 1 The Polyakov action Let us consider a two dimensional Riemannian manifold M without bound ary and a scalar field minimally coupled to the geometry This means that we choose 0 in the action 7 9 and Y 0 in the inner product 7 10 In two dimensions any metric is conformally flat i e by a suitable choice of the coordinates one may transform it to the conformal gauge Juv ernn Nw diag 1 1 9 1 In this gauge JGR 2q v p 9 2 The effective action W depends on p only since there is no other external field in this problem We substitute 7 16 with Y 0 in 7 3 to obtain 2 pv W Br E x pn bup 9 3 34 Calculation of the effective action in two dimensional QED by integration of the anomaly may be found in 84
70. into a leading part which is non analytic in t and the power law corrections For a flat manifold this separation has been de scribed in sec 1 cf eqs 1 12 and 1 13 Let us consider a massless scalar field on a curved compact Riemannian manifold M The DeWitt ansatz reads in this case o x y Kt D at AYE ye ep E2 Sta 437 where a x y is one half the square of the length of the geodesic connecting x and y In Cartesian coordinates on a flat manifold ogat z y y Ayvva is the so called Van Vleck Morette determinant det 3 amp gt 02 y 4 38 g x g y on Avyva 2 y As a consequence of the heat equation 1 10 the kernel should satisfy EE AV OV Ayn DA u E 0 4 39 42 with the initial condition 0 x y D 1 4 40 The essence of the DeWitt method is to look for a solution of the equation 4 39 in the following form atat Y rbaa D 4 41 j 0 The initial condition 4 40 yields The recursion relation j V 0 V p bog Ayyu DAY mbz5 1 0 4 43 which follows from 4 39 and 4 41 allows in principle to find the coinci dence limits x y of higher heat kernel coefficients b2 An important ingredi ent of such calculations is the coincidence limits of symmetrized derivatives of the geodesic interval o x y This method becomes very cumbersome be yond a4 A refined nonrecursive procedure to solve the DeWitt equation 4 43 was used by Avramidi t
71. ion 1 6 is true only if the operator D is self adjoint This means that D is symmetric or formally self adjoint with respect to the scalar product 1 Do2 Do1 2 1 7 for any 1 2 and that the domains of definition of D and its adjoint coincide We will not care about the second requirement since it involves mathematical machinery going beyond the scope of the present report The first require ment 1 7 poses important restrictions on admissible boundary conditions To become convinced that 1 7 is really necessary one can calculate a finite dimensional Gaussian integral with ab D 1 8 cd and a b c and d being real constants first by completing the squares in the exponent and then compare the result with det D of course one should remember a factor of 7 The two results coincide if b c Let us return to the generating functional 1 6 To analyse the two mul tiplies on the right hand side of 1 6 which depend on the operator D it is convenient to introduce the heat kernel K t x y D x exp tD y 1 9 This somewhat formal expression means that K t x y D should satisfy the heat conduction equation Dz K t x y D 0 1 10 with the initial condition K 0 x y D 06 9 1 11 For D Do 1 4 on a flat manifold M R the heat kernel reads K t x y Do 4rt exp a in 1 12 The equations 1 10 and 1 11 can be checked straightforwa
72. ional manifold without boundary depending on two scalar functions and Y Let us now consider small variations of and Y According to 7 4 the zeta function of corresponding Laplacian changes as 5 s Pp 2sTr Ppt sv Pp 9 7 Variation of 0 reads 5 0 Pp 2 0 6 Pp 0 6 p p 2 a2 5V Pp a2 50 p p 9 8 Here we used that pp and pip are operators of Laplace type and therefore 85 the corresponding zeta functions are regular at s 0 In the last line we replaced 0 by the heat kernel coefficients by means of 2 27 Next we have to rewrite PD and PtP in the canonical form 2 2 with the help of 2 3 2 4 For Pip we have gi e2 Wu Y _ yy vy E 0 R 2 6 W 9 9 where comma denotes ordinary partial derivatives all contractions are per formed with the effective metric g Note that in two dimensions If g in the conformal coordinates To obtain relevant geometric quantities for pip one has to replace by W and vice versa Equation 4 27 gives 6 0 Pp J dx OU AV 248 50 AS 2AV 9 10 where A is the flat space Laplacian Let us recall that 0 D defines the functional determinant of D see eq 2 33 The variation 9 10 can be in tegrated to give 3 In det Pp 0 Pp dPe VAYV 4VAS GAS 9 11 We see from 9 11 that Indet Pp Indet P P which is obvi
73. ions 5 24 and 5 25 are enough to define these constants b7 b 2 b b 3 5 26 This completes the calculation of a2 The coefficients az and aq can be obtained as particular cases of more general formulae of the next subsection 18 For M 0 7 the boundary integral reduces to a sum of two contributions from c 0 2nd 2 x 49 5 38 Mixed boundary conditions Let us now turn to mixed boundary conditions 2 17 which as we have seen in sections 3 3 and 3 4 are natural boundary conditions for spinor and vector fields These boundary conditions depend on two complementary local projectors II_ and Il 1 II which define subsets of components of the field satisfying Dirichlet and Robin boundary conditions respectively More precisely Helam 0 Vn SIl elom 0 5 27 Consequently there is one more independent entity as compared to Robin case for example on which the heat kernel coefficients for mixed boundary conditions can depend This makes the calculations somewhat more compli cated It is convenient to define y IL I_ 5 28 Calculation of the coefficients a up to k 4 can be found in see also for some corrections The result reads 19 Examples of such projectors are given by eq 3 35 and below eq 3 58 50 alf D B 4r kK d xy gtry f 5 29 OD By aan tev ite Oxf 5 30 ath D By 4m n 2 reviatev 6FE fR f d av htry 2f Laa 3X fn 123
74. is a decomposition V m Vn Vp where Vy p I V m Decompose also 6 and set Bb 6 amp 6 SO lon 2 17 The matrix valued function endomorphism S acts on Vy only S IIS SII In other words we define Dirichlet boundary conditions on Vp and gen eralised Neumann boundary conditions on Vy For obvious reason the bound ary conditions 2 17 will be called mixed In sec 3 we shall see that natural boundary conditions for spinor and vector fields are of this type 2 2 Spectral functions For the boundary conditions considered in this section as well as on man ifolds without a boundary the operator exp tD with positive t is trace class on the space of square integrable functions L V This means that for an auxiliary smooth function f on M K t f D Trz2 f exp tD 2 18 is well defined We also write K t f D f dx gtty K t a 2 D f x 2 19 where K t x x D is an y gt limit of the fundamental solution K t x y D of the heat equation 1 10 with the initial condition 1 11 If there is a boundary the kernel A t x y D should also satisfy some boundary condi tions B K t x y D 0 in one of the arguments We stress that K t x y D is a matrix in the internal indices try denotes the trace over these indices Let D be self adjoint This implies that in a suitable basis in the internal space the matrix w is anti hermitian and EF is hermitian Let
75. is clear now One cannot construct an odd dimension invariant on a manifold without boundary Let us study further relations between the heat kernel coefficients which will turn out to generate relations between the constants u Consider now the case when the manifold M is a direct product of two manifolds M and Mp with coordinates x and x2 respectively and the operator D is a sum of 33 two operators acting independently on M and My D D 8 1 18 Dao This means that the bundle indices are also independent As an example one can consider the vector Laplacian on M x M One can write symbolically exp tD exp tD exp tD2 Next we multiply both sides of this equation by f 1 2 fi x1 fo x2 take the functional trace and perform the asymptotic expansion in t to get alz D 5 ap z1 D agl 2 Do 4 2 p q k The consequences of eq 4 2 are very far reaching In particular eq 4 2 allows to fix the dependence of u on the dimension of the manifold M Consider an even more specialised case when one of the manifolds is a one dimensional circle M S 0 lt x lt 2r Let us make the simplest choice for D D 0 gt Al saie invariants associated with D are defined solely by the D2 part Moreover all invariants are independent of x Therefore by the equation 4 1 alfe D f p POVIE tt f e ul ALD f d e yg Y trv f w2 ut AL Ds 4 3 Here dependence of the constants
76. itions can be found in sec 2 1 29 for these boundary conditions is invertible Therefore a solution for 3 63 always exists and it is unique The case of absolute boundary conditions is a bit more involved One should take care of a zero mode in the ghost sector We leave this case as an exercise A more extensive discussion of compatibility of gauge and boundary conditions can be found in 3 5 Graviton We start this section with the Einstein Hilbert action on a four dimen sional Euclidean manifold without boundary 1 4 64 L i6 a 4 g R 2A 3 64 where R is the scalar curvature G y is the Newton constant A is the cosmolog ical constant As usual let us shift the metric gv guv A From now on gy will denote the background metric hu will be the quantum fluctuations We can decompose the h further in trace longitudinal and transverse traceless part 1 hyv gow am LE wy ag ia 3 65 where g hi V h 0 and 1 LE ay V Ev T Vuby zI So 3 66 The decomposition 3 65 is orthogonal with respect to the inner product lt h h gt J dir SJG huh 3 67 po 1 GP s ghegh gigi Calg Here C is a constant For positivity of 3 67 C should be greater than i Under the action of infinitesimal diffeomorphism generated by a vector the components of 3 65 transform as Eu gt Enten R gt oh 2V e hi oh 3 68 One can fix the gauge freedom 3 68
77. ke the expression 4 27 or 4 28 and substitute 2iy y for f and 3 27 3 28 for w E and Qu Computation of some y matrix traces is still required but nevertheless calculations are considerably easier than presented in many papers In flat space and with A 0 the result is particularly simple n 2 A y gt Patro Fu 7 31 J atxte pe Fy Foo 7 32 n A4 Aly where tr denotes trace over internal flavour or colour indices We can easily generalise this result for arbitrary even dimension n First we observe that the only way to construct a pseudoscalar density of appropriate dimension for A 0 is to contract n 2 tensors F with the Levi Civita tensor 12 Such tensor structure can be produced only by a trace of 7 and the maximal number n of the gamma matrices The only invariant having the required form is J d xtr QE 7 33 This term does not contain derivatives Under the trace Q and E commute Therefore we can calculate the coefficient in front of 7 33 by considering the particular case E Iy x const For this simple case the dependence of the heat kernel on E is given simply by e By picking up an appropriate term in the expansion of the exponential we find that the coefficient in front of 7 33 in Gy is 4r n 2 7t Next we substitute E 4 y F and take trace over the spinorial indices to obtain a ase A y aa j d xtr ee oe Frite 7 34 For global c
78. l equation of motion for the field the latter part of the linear term vanishes though the former one external sources should be kept arbitrary if one wishes to study correlation functions of We stress that the background and quantum fields may be of completely different nature For example it is a meaningful problem to consider pure quantum scalar fields on the background of pure classical gravity D is a differential operator After a suitable number of integrations by part it is always possible to convert the quadratic part of the action to the form given in 1 2 We postpone discussion of possible boundary terms to the next sections In the simplest case of quantum scalar field on the background of a classical geometry D is a Laplacian with a mass term D Do V V m 1 4 Just this simple example is enough to illustrate the material of this section Note that in this case J has no contribution from the first variation of the classical action since gravity is not quantised The path integral measure is defined by 1 f Doexp 9 1 5 Strictly speaking the right hand side of 1 5 is divergent The essence of the condition 1 5 is that this divergence does not depend on external sources and on the background geometry and therefore may be absorbed in an irrelevant normalisation constant The Gaussian integral 1 1 can be evaluated giving 1 1 Z J e det D exp 372 1 6 We stress that the equat
79. lem to find exact dependence of det D on WV by using the methods of sec 7 86 Let us introduce two first order differential operators L gt Ones L e 0 e 9 12 and associated second order operators Dells Doe 9 13 We restrict ourselves to flat two dimensional manifolds Therefore position of the indices up or down plays no role We calculate again variation of the function with respect to variation of 5 s D 2sTr 88 Lt Lu D75 L DFL 9 14 The operators in the first term here recombine in D_ The situation with the second term is more subtle Strictly speaking to treat this term one has to perform the Mellin transform and use analytic continuation in s see footnote 26 in sec 7 1 However the result of such manipulations is almost obvious Tr 88 L DIL Tr 68 H7 9 15 where Hw Ly Li 9 16 is a matrix operator acting on the space V of the vector fields which can be represented as v L f with some scalar function f The operator 9 16 is not of Laplace type as it contains a complicated differential projector on V Therefore even though the variation 9 14 looks very similar to 9 7 for example it cannot be used directly to evaluate 0 D since we cannot even guarantee that 9 15 is regular at s 0 in fact it is not regular see We also have 5 s D_ 2sTr 5 D H 9 17 where the operator Aix Epp Erv Lly Lys 9 18 act
80. m 2 2 with the help of 2 1 2 3 and 2 4 However one can first consult sec 3 Probably relevant expressions can be found there If the problem in question contains neither boundaries nor singularities one can look in sec 4 for an expression for the heat kernel coefficient or for a reference or for a method In the case of the boundaries one has to proceed with sec 5 in the case of singularities with sec 6 Relations between the heat kernel coefficients and quantum anomalies can be found in sec 7 The results going beyond the standard heat kernel expansion are collected in sec 8 In particular in this section we explain how one can extract leading non localities from the effective action and what the heat kernel looks like if the background is approximately covariantly constant in this context we also consider invariant operators on homogeneous spaces Exact results for the effective action which can be obtained with the help of the heat kernel expansion are reviewed in sec 9 The Casimir energy is one of the most classical applications of the heat kernel and zeta function technique cf It follows from the locality of the heat kernel expansion that the divergences in the Casimir energy are given by volume and surface integrals of some local invariants Therefore if the boundaries are being moved in such a way that the boundary values of the background fields remain unchanged the boundary contributions to the dive
81. mal factor problem of quantum general relativity Different explanations of this phenomenon suggest roughly the same remedy the conformal mode must be rotated to the imaginary axis o ia The path integral can be written in terms of functional determinants corresponding to vector fields indicated by the subscript V scalars indicated by S and transverse traceless tensors indicated by T L 1 R me Z det L L det A det Aguydue 2Rupwe 3 78 This expression is in principle suitable for calculations on some homogeneous spaces since the harmonic expansion on such spaces usually respects separa tion of tensors to transverse and longitudinal parts However we must warn 31 the reader again that we have neglected the presence of Killing vectors and of conformal Killing vectors The ways to treat these vectors created a long discussion in the literature The representation 3 75 is not convenient on generic manifold since the tensor operator is restricted to the transverse modes To remove this restric tion one can either multiply 3 75 by a compensating vector determinant or add a suitable gauge fixing term to the classical action 3 64 and repeat the quantisation procedure right from the beginning The result can be found elsewhere in the literature see e g Boundary conditions for one loop Euclidean quantum gravity must be diffeomorphism invariant and must lead
82. n This method based on calculations of the regularized Jacobians can be applied to conformal anomaly as well The most essential ingredient of the anomaly calculation presented in this section is the homogeneous transformation low 7 19 for the Dirac operator This homogeneity allowed us to restore the power s in the transformation law 7 4 for the zeta function and to obtain a simple expression for the anomaly 7 26 It is clear therefore that as long as the operators transform homoge neously we shall obtain relatively simple local expressions for corresponding anomalies understood as variations of the effective action with respect to infinitesimal transformation of the background fields This suggests to con sider extensions of the chiral group For example one of such extensions identifies the group parameters with the diquark fields 29 Applications of this method to topological anomalies can be found in 71 7 8 Remarks on the Index Theorem This report is mainly devoted to local aspects of the heat kernel expansion There is however one global application of the heat kernel which cannot be ignored This is the Index Theorem In this section we briefly sketch formal mathematical aspects of the index construction and its relation to the heat kernel For more details we refer to Physical applications of the index theorem to gravity gauge theories and strings are so numerous that we cannot eve
83. n mention them all The index theorem was first formulated by Atiyah and Singer and the heat kernel approach appeared later in Roughly speaking their con struction is as follows Consider two vector bundles V and V over a manifold M Let the operator P map Vi to V2 and let Vi2 have non degenerate in ner products 2 so that one can define an adjoint Pt by the equation 2 P 1 2 Pte 1 1 As an example one can keep in mind V respec tively V2 describing positive respectively negative chirality spinors In this example P is a part of the Dirac operator see eq 7 44 below Let us define two operators D P P and Dy PP acting on smooth sections of Vj and V2 and let us suppose that D and D are elliptic In this case we deal with an elliptic complex Since elliptic operators may have only a finite number of zero modes say N and N for D and D respectively we may define the index by the following equations index P M No dim ker D dim ker D3 dim ker P dim ker Pt 7 39 We also have the intertwining relations DiP P D PD DP 7 40 which tell us that non zero eigenvalues of D and D coincide Consider now the heat kernels K t D K t Dz X e Soe N N index P 7 41 1 az where A 9 are eigenvalues of D42 To be more specific let us suppose that D and Dz are of Laplace type and that boundary conditions and singulariti
84. n put them all down and sum up over the repeated indices by contracting them with the Kronecker e 0 Rijk Rage 4f Rage Raga 8 faj Riz 0 4f 4 18 Let us apply 4 9 to n 4 eaotto e2 F eID 0 By collecting the terms with try fy dx g F fjj we obtain a 6a and consequently s 1 and a4 60a2 60 This completes calculation of av Let M M x M with a product metric and let D A1 Ag where Aj are scalar Laplacians on M and Mp respectively Eq 4 2 yields a4 1 A lt A gt aa 1 Aj ao 1 Ag a l A a2 1 A2 ao 1 Aj a4 1 Ag 4 19 It is clear from 3 6 that E 0 and Q 0 By collecting the terms with R R where R and R are scalar curvatures on M and M respectively we obtain 2 Q2 2 5077 FZ Consequently a7 5 Let us apply 4 9 to n 6 We obtain with the help of the variational equations 4 18 0 try f d e G F 2a3 1004 405 fak E M 2a3 1006 f iijj 2a4 206 20a7 2as fiR 8ag 809 faj Riz 4 20 The coefficients in front of independent invariants in 4 20 must be zero We determine a3 60 ag 12 ag 2 and ag 2 12 More precisely we assume that in 3 4 the potential U 0 the conformal cou pling is minimal 0 and there is no gauge coupling G 0 on both M and M 37 The most elegant way to calculate the remaining constant o is b
85. nditions defined by the projector 3 35 are a particular case of mixed boundary conditions 2 17 They will be considered in detail in sec 5 3 see also sec 5 1 for further references to calculations in a ball Chiral bag boundary conditions defined by 3 40 with r 4 0 are considerably more complicated because the equation 3 36 contains a mixture of normal and tangential derivatives We refer to for more calculations of spectral functions in this latter case The boundary conditions considered in this section are local i e they treat the fields at each point of the boundary independently One can also define global boundary conditions for the Dirac operator cf sec 5 4 2 Due to the fermionic nature of the spinor field the path integral over w and w gives determinant of the operator D to a positive power Gis i Dib Dipexp L det 3 41 where the action is given by 3 22 and 7 y are complex Dirac spinors 3 4 Vector fields Consider the Yang Mills action for the gauge field A Greek letters from the beginning of the alphabet label generators of the gauge group 1 n Q Va fa ry JFS Fe 3 42 where as usual Fo 0 Ay 0 Ali c3 Al AY c3 are totally antisymmetric structure constants of the gauge group Let us introduce the background field Bri by the shift A Bi Afi From now on Af plays the role of quantum fluctuations The quadratic part of the action reads 1 m QV Q Q V Q Q Q V
86. ntributions of the singularities from the smooth part For example if N 1 dQ dy with 0 a only as receives a contribution from the tip of the cone 4r Q 24ra e az tip In many particular cases of conical singularities a very detailed analysis of the heat kernel expansion has been performed One loop computations on general orbifolds were considered recently in 6 3 Domain walls and brane world Delta function is an example of an extremely sharp background potential Let us consider a manifold M and a submanifold of the dimension n 1 Let Div D v s 6 8 D is an operator of Laplace type 2 2 Let h be the determinant of the induced metric on X Then s is a delta function defined such that s dx G6nf x a daVhf a 6 9 The spectral problem for D v on M as it stands is ill defined owing to the discontinuities or singularities on X It should be replaced by a pair of spectral problems on the two sides MF of together with suitable matching conditions on X In order to find such matching conditions we consider an eigenfunction of the operator 6 8 Dlvld Ada 6 10 Let us choose the coordinate system such that e is a unit normal to and x 0 on It is clear that 6 must be continuous on X Plan 0 P an 0 6 11 Otherwise the second normal derivative of 6 would create a 0 singularity on X which is absent on t
87. o calculate ag see also for a short overview The method of DeWitt can be naturally extended to treat coincidence limits of the derivatives of bo x y D The recursion relations can be generalised to the case of manifolds with boundaries However for a practical use the functorial methods of sec 5 seem to be more convenient 4 4 Non minimal operators Quantisation of gauge theories oftenly leads to second order differential operators which are not of the Laplace type For example by taking 1 in 3 52 one obtains the following operator acting on the gauge field fluctuations Dm Agy 1 VV a 4 44 where for simplicity we suppose that the manifold M is flat R vpo 0 and the gauge group is abelian cg 0 The leading symbol of the operator 4 44 the part with the highest derivatives has a non trivial matrix structure Such operators are called non minimal In some simple cases see e g the spectral problem for non minimal operators can be reduced to the Laplacians If the leading part of a non minimal operator has a form similar to 4 44 but the lower order part 43 is more or less arbitrary necessary generalisations of the DeWitt technique were suggested by Barvinsky and Vilkovisky The technique was further developed in Complete calculation of a4 required extensive use of the computer algebra Most general non minimal operators were considered in where be
88. oefficients Through this report we work on Euclidean manifolds A short remark on the analytical continua tion to the Lorentzian signature is given at the end of section 2 In section 3 we consider the most widely used models of quantum field theory and open bosonic strings The one loop dynamics in each of these models is defined by a second order differential operator which depends on an effective connection and on a matrix valued potential The connection and the potential serve as a basis of an invariant description of all that models in the language of spectral geometry These quantities are written down explicitly for each model We also define suitable boundary conditions In section 4 we consider the heat kernel expansion on manifolds without boundary We introduce a simple and very powerful method and illustrate it by calculating several leading terms in the heat kernel expansion We also briefly discuss some other methods and non minimal operators Section 5 is devoted to manifolds with boundary The heat kernel expansion for standard Dirichlet Neumann and mixed boundary conditions is considered in some detail We also describe less known oblique and spectral boundary conditions these are the ones which contain tangential derivatives on the boundary or non local projectors We discuss loss of the so called strong ellipticity for oblique boundary conditions which corresponds to the critical value of the electric field in string physics Wit
89. of redundant integrations to obtain 2 i Kalt strv dy f dz f deKo z wit amp E M M J x Koly 23 t6 E z 8 11 3 The original paper contained a curved space generalisation of 8 7 This however does not improve the global issues discussed below TT Now we use the identity Ko z y t 1 Koly z t Ant Ko z y t amp 1 8 12 and relate the heat kernel on the right hand side of 8 12 to a matrix element of exp t 1 A to obtain the final result K t cae f evaER AA E 8 13 where the non local form factor f reads fla 5 f ageso Sel afqbra ya 814 As we have already discussed above applicability of this formula is limited by our choice of Ko x y t Namely the potential E x should have a compact support and t should be reasonably small Note that the small t approximation in the expansion 8 5 is self consistent if t is small the integration variables s are even smaller The main difficulty in constructing an expansion in powers the Riemann curvature and of the field strength is to organise the procedure in a covariant way The details of the construction and higher order form factors can be found in From further developments of this method we mention the work of Gusev and Zelnikov who demonstrated that in two dimension one can achieve considerable simplifications in the perturbation expansion by using the dilaton parametrisation of
90. of the smooth field approximation is that a with sufficiently large k are divergent Usually the presence of singularities changes even the structure of the heat kernel kernel expansion as compared to the smooth case 6 1 Non integrable potentials According to 1 21 divergences in the effective action are defined by integrated heat kernel coefficients Although the formulae 4 26 4 29 for the localised heat kernel coefficients are valid on non compact manifolds pro vided the smearing function f falls off sufficiently fast transition to the integrated heat kernel is not that straightforward Already the coefficient ao D ao 1 D which is proportional to the volume is divergent This divergence is usually removed by replacing det D in 1 6 by det D det Do 6 1 where the operator Dg 0 m describes a free particle propagation in an empty space It is argued that since Do does not depend on essential variables division by det Do does not change physical predictions of the the ory In all subsequent formulae the heat kernel K t x y D is then replaced by the subtracted heat kernel In flat space the coefficient ag corresponding to K yp is identically zero 7 If the field strength Q and the subtracted potential E m have a compact support or decay sufficiently fast at the infinity the small t asymptotic expan sion of Kult 2 x is integrable on the whole M If not the very structure of the
91. on 1 20 although in other regularization schemes the result will be essen tially the same The divergent and finite parts of the effective action are given by 1 21 and 1 22 respectively up to the obvious replacements m M 47 b Gy After a suitable renormalization this procedure indeed gives a large M expansion of the effective action Therefore the problem is reduced to calculation of a If M commutes with D the coefficient a are the heat kernel coefficients for the operator D M This case returns us to the standard large mass expansion If M does not commute with D then e tD Ze t D M 6 tM 8 2 To achieve an equality the right hand side of 8 2 must be corrected by commutator terms In this case calculation of a requires some amounts of extra work Heavy particles of non equal masses described by the mass matrix M is probably the most immediate example of a physical system to which the modified large mass expansion should be applied Corresponding technical tools were developed recently The next example is a scalar field in curved space cf sec 3 1 Parker and Toms suggested to use the modified large mass expansion with 30 A diagrammatic technique which can be used in resummations of the heat kernel expansion is described in 75 M m ER which partially sums up contributions of the scalar curvature R to the effective action This formalism was developed further in
92. oop effective action one should expand 3 1 around a background field 4 and keep the terms quadratic in fluctuations Ly f d x Go vev i 5 U F ERs p z i do 2 Jhb Vio 3 2 M h is the determinant of the induced metric on the boundary The inner product for quantum fields reads lt pn h gt J da goed 3 3 The operator D is defined by the bulk part of the action 3 2 1 5 p48 V V P ey ER6AP 3 4 For a special choice of the parameter n 2 ees 3 5 a 3 5 the operator D 3 4 is conformally covariant if also U 0 To bring the operator 3 4 to the canonical form 2 2 we introduce 1 P a SGre te O nae 3 6 19 For this case Q is just the ordinary Yang Mills field strength The operator 3 4 is symmetric with respect to the inner product 3 3 if the surface integral ott VRE 68968 3 7 vanishes for arbitrary and 2 belonging to its domain of definition This may be achieved if one imposes either Dirichlet oom 0 3 8 or modified Neumann Vind 479 lon 0 3 9 boundary conditions S4 is an arbitrary symmetric matrix Note that the integral 3 7 vanishes also if S4 is an arbitrary symmetric differential op erator on the boundary For the Dirichlet conditions 3 8 the boundary term in 3 2 vanishes automatically To ensure absence of the surface term for the modified Neumann conditions 3 9 one should add to 3 1
93. opagation with variable speed of light c a is described by the operator D c x V where we neglected the lower order terms In a dielectric medium c is expressed in terms of the dielectric permittivity and of the magnetic permeability u c x 1 e x u x For a smooth distribution of and u the operator D is a particular case of curved space Laplacian with an effective metric defined by c x The heat kernel coefficients can be calculated in the standard way Consider now a dielectric body bounded by X Typically c x and the effective metric jumps on X This singularity is much stronger than the one considered in sec 6 3 Thus the geometric interpretation of this problem is very difficult Very little is known about the heat kernel expansion in a dielectric body of an arbitrary shape There are calculations for a dielectric ball and for a dielectric cylinder These calculations exhibit a puzzling property of the heat kernel expansion in dielectrics in dilute approximation e 1 for a dielectric body the Casimir energy in the ultra violet limit behaves better than for the smooth case The heat kernel expansion for a frequency dependent e was considered in refs 64 7 Anomalies The most immediate application of the technique developed in the pre vious sections is calculation of quantum anomalies which are defined as non zero variations of quantum effective action with respect to symmetry tran
94. oupled to the background dilaton 7 f Page 3 6 0 d 9 7 9 On dimensional and symmetry grounds the inner product may also contain an arbitrary function WV of the dilaton nay J dr sge gi x 2 7 10 Such couplings and inner products appear for example after the spherical reduction of higher dimensional theories to two dimension see the review paper for more details and further references The rescaled field e possesses the standard dilaton independent inner product 3 3 In terms of the action 7 9 reads ce J x GoD6 7 11 D e g V V 2 U V U V 2V 8 7 12 where comma denotes covariant differentiation with V UV V Y We can bring D to the standard form 2 2 D 0 V Vi E 7 13 by introducing the effective metric g e tive g and the covariant deriva A A Vaz Oty ton Op Vy Oy 7 14 where is the Christoffel connection for the metric g Here the potential reads E gh 8 F 7 15 Now we combine 7 8 with 4 27 and the definitions given above to obtain Pe f Y el i 5a E 6 V 4V 2V V 7 16 27 Much work on conformal anomalies on two dimensional manifolds was done in the context of string theory Here we like to mention the papers 67 In the case Y the expression 7 16 was first obtained in and in the general case in 8 In fou
95. ously true up to possible topological contributions from zero modes which have been neglected in this calculation The result 9 11 was confirmed in by the methods of covariant perturbation theory This example is clearly of at least an academic interest since it adds up to very few cases discussed above in this section when a closed analytic expression for the determinant may be obtained without any assumptions on smoothness of the background fields The operator ppt shares some simi larities with the kinetic operator for non minimally coupled scalars in two dimensions 7 12 There are good grounds to believe the functional de terminant 9 11 may describe the spherically symmetric part of the Hawking flux in four dimensions Let us consider the dilatonic operator 7 12 We take V for sim plicity We shall be interested in properties of det D under reflection of the dilaton The quantity In det D In det D defines the dilaton shift under the T duality transformations in string theory In the string theory context this problem was solved in basing on earlier results of Here we follow presentation of the paper 35 The heat kernel coefficients az x Ppp and ag x PP are total derivatives Therefore the second term on the right hand side of 2 33 does not contribute 36 Variation of det D with respect to V may be expressed through the scale anomaly Therefore it is not a prob
96. p g According to 7 1 variation of the effective action reads 1 ie uv n H w f d z JGlw5g J d e y gT p 7 3 It is clear from the equation above that the trace of the energy momentum ten sor measures conformal non invariance of the theory If the classical action is conformally invariant classical energy momentum tensor is traceless How ever even in this case conformal invariance is typically broken by quantum effects For this reason quantum T is called trace or conformal anomaly 65 If the classical action is conformally invariant the fluctuation operator D is conformally covariant This means that D transforms homogeneously D e D under 7 2 We restrict ourselves to the one loop level and employ the zeta function regularization in which the effective action is expressed through the zeta func tion of the operator D see 2 32 Hence we have to study the conformal properties of s D One can prove that the variation of the zeta function with respect to variation of the operator D reads 7 6 s D sTr 6D D 7 4 Next we use that under infinitesimal conformal transformations the op erator D transforms as D 2 dp D 7 5 This equation yields 5 8 D 2s s 6p D 7 6 For the operators which we consider in this section the zeta function is regular at s 0 Consequently the variation of the effective action 2 32 reads W 0 dp D an p
97. plexity of calculations of the heat kernel coefficients ax is almost independent of k Therefore the algebraic techniques were applied to higher dimensional theories where one needs higher heat kernel coefficients to perform renormalization or to calculate the anomalies Rather naturally the most simple toroidal spaces were considered first and then computations on spheres were performed Other homo geneous spaces were considered for example in In the same approach non minimal operators on homogeneous spaces were treated in We refer to for a more extensive literature survey 83 9 Exact results for the effective action In the previous section we have seen that the heat kernel can be calculated exactly for all values of t if the background satisfies certain symmetry or smoothness conditions In this section it will be demonstrated that for some classes of the operators the effective action can be calculated exactly without imposing any restrictions on behaviour of the background fields These are the cases when the variation of 0 with respect to the background can be reduced to the heat kernel coefficients which are locally computable Then the variation is integrated to the effective action The most immediate example of the variations admissible for this scheme is quantum anomalies cf sec 7 Historically the first action obtained by integration of an anomaly was the chiral Wess Zum
98. r dimensions can be found in A generalisation to arbitrary dimension has been achieved recently Let A be a p form field with the field strength F dA Consider the classical action fe I e FAF 9 23 M where x is the Hodge duality operator Such actions appear for example in extended supergravities and bosonic M theory Instead of the dilaton also a tachyon coupling may appear We are interested in the symmetry properties of the effective action under the transformation p gt n p 2 In higher dimensional supergravity theories this is a part of the S duality transformation It is convenient to define the twisted exterior derivative da e de 9 24 and the associated twisted coderivative and twisted Laplacian dp e 5e As da e 9 25 Since d 0 we have an elliptic complex The restriction of Ag on the space of p forms will be denoted by A This twisted de Rham complex was intro duced by Witten in the context of Morse theory and supersymmetric quantum mechanics 88 Any p form can be decomposed as the sum of a twisted exact twisted co exact and twisted harmonic form Ay dgAp 1 de Ap 41 Yp NG s 0 9 26 The projections on the spaces of twisted exact and twisted co exact forms will be denoted by the subscripts and L respectively We assume that the fields A e A have a standard Gaussian measure and are to be considered as fundamental fi
99. r dimensions the conformal anomaly for different spins can be read off from 4 35 with the numerical coefficients given in Table 1 7 2 Chiral anomaly Chiral anomaly was discovered in 1969 by Adler Bell and Jackiw and since that time plays a crucial role in understanding of the low energy hadron physics A detailed introduction to the field and extensive literature can be found in The spinor action 3 22 with the Dirac operator given in 3 26 is invari ant under the gauge transformations byw AY Oy wr 6A A A y Ap OLA An Al 7 17 and local Euclidean chiral transformations ob ipp Sp ibp Dp Ap Op An P Op Ay An y 7 18 with anti hermitian local matrix parameters and y The Dirac operator transforms as dp z p A S i yy D f 7 19 The Lie algebra structure of the transformations 7 17 and 7 18 is encoded in the following relations a 9X2 Ep On204 IA A2 7 20 a 90a l 7 21 00199 p O iia 7 22 where all transformation parameters are taken at the same space time point If the matrices A z and y x belong to a finite dimensional compact Lie algebra of some Lie group G the transformations 7 17 and 7 18 generate locally the group G x G with gauge transformations belonging the diagonal sub group 28 Different values for the numerical coefficients in the conformal anomaly 7 16 were reported in The reason for these dis
100. ra Let us briefly formulate the results of Consider a flat manifold Ryvpo 0 with the background fields satisfying the low energy conditions K t D e tisinh Bt 8 18 ViQw 0 V V V E 0 8 19 with usual definitions of F and Q see 2 1 2 10 Moreover let us suppose that the background is approximately abelian i e that Qu E and all their covariant derivatives are mutually commuting matrices 32 We like to mention also the paper which treated the effective action in external electromagnetic field from a different point view 79 With all these assumptions a closed expression for the heat kernel may be obtained 3 K t x 2 D 4nt exp ve t TEWE 8 20 where and Y are complicated functions of t E and Q If V VLE 0 t 5 Indet W t tA Q coth tQ 1 8 21 where Q has to be understood as a space time matrix Q so that multipli cation in 8 21 is the matrix multiplication and det is the determinant of an n x n matrix These formulae generalise the equation 8 18 and justify the choice of the degeneracy factor made to derive it li 0 t 5 Indet pa W t vP tanh tv P tV P 8 22 where P 1 2 V VLE In the particular case of one dimensional har monic oscillator these formulae reproduce 6 5 In curved space the best one can do in the framework of the low energy expansion is to consider
101. rdly Let us con sider a more general operator D which contains also a potential term or a gauge field Then 1 12 still describes the leading singularity in the heat kernel as t 0 The subleading terms have a form of the power law corrections K t x y D K t x y Do 1 tba x y Pba x y 1 18 The coefficients bg x y are regular in the limit y x They are called the heat 2 On a curved space even the leading term must be modified cf sec 4 3 On manifolds with boundaries also half integer powers of t appear in the expansion and consequently b2j 1 0 kernel coefficients At coinciding arguments b z x are local polynomials of background fields and their derivatives The propagator D x y can be defined through the heat kernel by the integral representation D e y a K t 2 y D 1 14 which follows from 1 9 if we suppose that the heat kernel vanishes sufficiently fast as t gt oo We can formally integrate the expansion 1 13 to obtain F T lz y 4n j 1 Desy afa E T K pulle whiten j 0 1 15 where b 1 By examining the behaviour of the Bessel function K z for small argument z we conclude that the singularities in the propagator at coinciding points are described by the first several heat kernel coefficients bp Let us consider the part of the generating functional 1 6 which contains det D The functional 1 W z Indet D 1 16 is called the one loop
102. refore we have related the variation of the effective actions with respect to to a combination of the heat kernel coefficients which is called the supertrace of the twisted de Rham complex A somewhat surprising feature of the supertrace is that it can be calculated for any n with or without boundaries For example the volume term in 9 28 does not depend on and therefore is the standard Euler density which is given by pan E E jij E 4r gral e 1etm EJ cae gee eee ET In 1injnjn 1 9 29 with n n 2 for n even En 0 for n odd 89 It is known that for 0 the dual theories are quantum equivalent W 0 Wr p 2 0 0 9 30 By using this equation as an initial condition we can integrate the variation 9 28 to obtain W Wn p 2 8 1 i d x GPE 9 31 Further terms gt 7 1 a a A are of some interest in mathematics They can be calculated for k lt n 2 and any n 90 10 Conclusions Here we present a short guide in this report In other words we are going to answer the following question What should one do if one likes to calculate one loop counterterms anomalies an expansion term in the effective action or something else which is defined by the heat kernel expansion The first step is to find the bulk part of the variation of the classical action 1 2 and corresponding operator D Next one has to bring this operator to the canonical for
103. rgences also remain constant This leads to the well known conclusion that there are no boundary divergences in the Casimir force which is defined roughly speaking as a variation of the Casimir energy under infinitesimal trans lations of the boundary Consequently one can assign a unique value to the Casimir force see e g This observation however does not mean that the quantum field theory on a manifold with boundary is finite In general some surface counterterms are required at one loop order they may be read off from sec 5 Moreover if the background field are non trivial the boundary divergences will not be constant Similar arguments created certain scepticism towards reliability of the Casimir energy calculation This point has not been settled so far for a generic theory We may add that in supersymmetric theories cancellations between divergences in the bosonic and fermionic sec tors appear if the boundary terms are considered together with the volume terms therefore separation of boundary and volume contributions is not always natural for that theories Renormalization of self interacting theories on manifolds with boundaries was considered in where one can find further references Some aspects of the relationship between the Casimir 91 energy calculations and the heat kernel coefficients have been clarified recently by Fulling Of course not everything can be found in this report S
104. rical harmonics Their degeneracies are 21 n 2 1 n 3 N _ il n 2 5 13 To proceed further it is convenient to consider the zeta function which 46 may be presented through a contour integral 1 dk o D X N k pm 14 eD OM fF e 5 14 where is a function which has zeros at the spectrum k WV For Dirichlet boundary conditions this function reads b k J k 5 15 For Robin boundary condition is given by a somewhat more complicated combination of the Bessel functions The contour y runs counterclockwise and encloses all the solutions of 0 on the positive real axis Note the presence of k in 5 15 which is included to avoid unwanted contributions coming from the origin k 0 The next step is to rotate the contour to the imaginary axis and to calcu late residues of I s s D as prescribed by 2 26 It is interesting to note that the heat kernel coefficients are defined by several leading terms in the uniform asymptotic expansion of ivz for large v and fixed z For further details we refer to Euclidean ball was frequently used in calculations of the heat kernel co efficients and functional determinants Apart from the papers already quoted above also the computations for a scalar field spinors abelian gauge fields should be men tioned A similar technique works also for more complic
105. roject BO 1112 12 1 of the Deutsche Forschungsgemeinschaft I am grateful to E Zeidler for his kind hospitality at the Max Planck Institute for Mathematics in the Sciences where a part of this work has been done 93
106. s 0 with the help of the initial conditions 3 12 It is convenient to introduce the Riemann curvature of the target space metric GAB Ragcp and a 3 index field strength Hago OaBpc OpBoa c Bega 3 14 The covariant derivative V is Vat Ob afl XOXE Seu Awc 8 18 The quadratic part of the action 3 10 reads 21 Lo f bah Gan X V EAE Rason RARP 1 8 XO 0 XP Da Hac 7 X O X H ancl pE g 3 16 ie dr D 4 Bas Fap 0 X Da Bao Fpo S where 7 is the arc length along the boundary Fgc OpAc cApsg The covariant derivatives D and D contain the Christoffel connection on the target space but not H4gc as the full covariant derivative V eq 3 15 The natural inner product in the space of fluctuations reads lt a ga gt PoyaGan X e 6h EH 2 817 The volume part of the action 3 16 has now the canonical form D with the operator D 2 2 which is obviously of Laplace type The connection w is defined in 3 15 and the endomorphism F can be easily extracted from 3 16 The operator D is formally self adjoint with respect to 3 17 if we impose the boundary conditions of Neumann type BE Vn SRE lom 0 3 18 with arbitrary operator S which should be symmetric with respect to the restriction of 3 17 to the boundary ue dr EG S4 Ep Ea Sa EB 0 3 19 There is a preferable choice of the boundary operator Let us
107. s formations of the classical theory In this section we consider two most impor tant examples of quantum anomalies These are conformal scale and chiral anomalies We also discuss briefly the Index Theorem There exists also a broader view on the anomalies which includes any qual itatively new phenomena of a quantum theory which are absent in its classical counterpart An example is the so called dimensional reduction anomaly For two operators D and D the quantity det D det D2 det D D2 is called the multiplicative anomaly since det D D2 4 det D det Dz2 is an anomalous property of infinite dimensional operators It is interesting to note that the heat kernel is also useful for treatment of these non standard anomalies To simplify the discussion in this section we work on manifolds without boundaries 7 1 Conformal anomaly Conformal invariance is one of the symmetries which are usually broken by quantisation This phenomenon called conformal or trace anomaly is known since mid 1970 s see for more extensive literature The vacuum polarisation induced by quantum effects is described by the energy momentum tensor 2 W pv Ig 7 1 where W is the quantum effective action calculated on the background with the metric tensor g Consider the conformal transformation Juw eP gy 7 2 for an infinitesimal value of the parameter g gt 1 2dp gu g gt 1 26
108. s metric is in general singular at r 0 However if we take the unit sphere S t N with standard round metric the singularity disappears and we obtain the n dimensional unit ball 5 10 If a manifold has singular points where the metric can be approximated by 6 6 we say that this manifold has conical singularities Conical singularities appear in many physical applications First of all with N S the metric 6 6 is the Euclidean version of the Rindler metric Conical singularities appear in classical solutions of the Einstein equations and in the supermembrane theory Gravitational field of a point mass in three dimensional gravity is a conical space There are evidences that conifolds dominate the path integral for quantum gravity in topological sectors Sommerfeld was probably the first to consider the heat kernel in the presence of conical singularities The mathematical theory of the heat kernel asymptotics with conical singularities was developed almost 100 years later There two peculiar features of these asymptotics First the heat kernel expansion contains in general both integer and half integer powers of t even without boundaries Second a non standard Int term may be contained in the 58 asymptotic series 7 On a manifold with conical singularities no closed analytical expression for the heat kernel coefficients is available However usually it is possible to disentangle co
109. s on the space V_ of the vector fields of the form 0 eel All epsilon tensors cancel after taking the trace in 9 17 and we arrive at the same formula as for D but with the replacement 6 The spaces V and V_ are orthogonal and V V_ V is the space of all vector fields on M we neglect possible zero modes Therefore 5 s Ds s D_ 2s s 60 H F s 6 D s 6 D_ 9 19 87 The operator H A yw w Dt P 9 20 is of Laplace type We can now act in the standard manner to obtain 5 0 D s D_ 2 ao 6 H F a2 5 D a2 5 D_ 9 21 The right hand side can be evaluated by using 4 27 We leave it is an exercise to show that the variation 9 21 is zero alternatively one may look up in This leads us to the conclusion that 0 Di 0 D_ Indet D lIn det D_ 0 9 22 In the one dimensional case a similar relation may be obtained by meth ods of supersymmetric quantum mechanics In this simplest case and in higher dimensions if depends on one of the coordinates only D and D_ are isospectral up to zero modes and 9 22 follows immediately In two dimen sions D and D_ are not isospectral in general Although D and D_ have coinciding determinants other spectral functions can be different Extensions of 9 22 to the case of matrix valued manifolds with bound aries and the dilaton Maxwell theory in fou
110. specially in the mathematical literature is the resolvent or more precisely its powers Rig D 237 2 36 If D is on operator of Laplace type subject to good boundary conditions and if sufficiently large l gt n 2 there is a full asymptotic expansion T 1 Tr R z gt 23 U E uoa 2 37 as z oo The coefficients a are the same as in the heat kernel expansion 2 21 2 8 Lorentzian signature Locality of the heat kernel coefficients in the Euclidean domain can be easily understood by examining the free heat kernel 1 12 For small t the first term in the exponential strongly suppresses non local contributions For Lorentzian metrics the squared distance function x y is no longer positive definite Therefore the simple arguments given above do not work A partial solution to this problem is to consider the Schrodinger equation i D K T x y D 0 2 38 for the kernel K instead of the heat conduction equation 1 10 Then oscillates at large distances However even though non local contributions to K oscillate furiously as r gt 0 they are not small Consequently local asymptotic series do not exist in many cases A discussion on this point can be found in chapter 9 of 8 On manifolds with boundaries one has to require strong ellipticity of the boundary value problem cf sec 5 4 A more precise meaning of this restriction will be explained in sec 5 4 17
111. spect to an inward pointing unit normal is The eigenfunctions Qp cos kx 5 5 satisfy the boundary conditions 5 4 at zx 0 The spectrum is defined by the condition at x m which reads 14 ksin kr Scos kr 5 6 In this example we restrict ourselves to the linear order in S We suppose that S is small and that the spectrum 5 1 is only slightly perturbed kp l z 5 7 14 For S gt 0 also a negative mode cosh kx appears Here we take S lt 0 45 The equation 5 6 gives in this approximation i se for l gt 0 lr y for l 0 5 8 T We expand also the heat kernel in a power series in S and use 5 8 and 5 2 to obtain K t gt exp tk7 I gt 0 2St St exp tl 14 4144 0 5 1 gt 0 iG a 5 Z 1 s t O S 5 9 where we also dropped exponentially small terms in t Our next example is the heat kernel expansion for scalar fields in a ball with Dirichlet and Neumann boundary conditions The metric of the unit ball in R reads ds dr r dQ 0 lt r lt 1 5 10 where d9 is the metric on unit sphere The scalar Laplace operator has the form EE a pE 5 11 where YA is the Laplace operator on S The eigenfunctions of the op erator 5 11 are well known PL x pe Foy ot r Vay ZE 5 12 Jp are the Bessel functions The eigenvalues A are defined by boundary condi tions Y x are n dimensional scalar sphe
112. t M R Consider a scalar particle in constant electromagnetic field with the field strength Fiz F 2 B As a potential we choose A 0 Ay Bx Then the operator acting on quantum fluctuations is D 8 3z iB1 m 8 15 This operator commutes with 02 Therefore we can look for the eigenfunctions of D in the form x exp ikx d z D x x 8f B a k B m dy z 8 16 In the x direction we have the one dimensional harmonic oscillator potential cf 6 3 Therefore the eigenvalues are Akp 2p 1 B m These eigenvalues do not depend on k For this reason the heat kernel E t kp 8 17 is ill defined To overcome this difficulty it was suggested to put the system in a box without specifying any boundary conditions however and replace the sum over k by the degeneracy factor B Vol 27 where Vol is volume of the box The degeneracy factor is chosen in such a way that the resulting heat kernel BVol Ar behaves as Vol 47t in the limit B 0 Calculations of the effective action in covariantly constant background gauge fields by the spectral theory methods have been performed by many authors Such calculations were motivated at least partially by various models of quark confinement see and references therein There exists an algebraic method which allows to evaluate the low energy heat kernel by using exclusively the commutator algeb
113. the potential cf eq 7 15 Recently Barvinsky and Mukhanov suggested a new method for calculation of the non local part of the effective action based on the resummation of the perturbation series for the heat kernel This method was extended in to include late time asymptotics of the heat kernel in curved space Let us stress that the expansion 8 5 can be used also for singular po tentials For example it is very effective for the calculation of the heat kernel for d potentials concentrated on a co dimension one subsurface cf sec 6 3 If the 6 potential has its support on a submanifold of co dimension greater than one the expansion diverges With a suitable choice of the zeroth order heat kernel Ko zx y t and of an operator to replace E in 8 5 one can treat manifolds with boundaries where the perturbative expansion takes the form of the multiple reflection expansion 8 8 Low energy expansion Let us now turn to the opposite case when the derivatives are less impor tant when the potential and the curvatures In this case one has to collect the terms which are of a fixed order in the derivatives but contain all powers of E R and Q Since the derivatives are sometimes identified with the energy 78 this approximation is being called the low energy expansion This scheme goes back to Schwinger s calculations in constant electromagnetic fields 3 Let us consider a simple example Le
114. tial operators More information on spec tral theory for these operators can be found in Third the operator 2 1 has a scalar principal part This means that the second derivatives in 2 1 are contracted with the metric and the internal index structure of the second derivative term is trivial Such operators are also called minimal Non minimal operators will be briefly considered in sec 4 4 We can define local invariants associated with w and g Let Pepe a Th z Onl og BARA _ BARE 2 6 be the Riemann curvature tensor let Rv R uvo be the Ricci tensor and let R R be the scalar curvature With our sign conventions R 2 on the unit sphere S in the Euclidean space Let Roman indices i j k and l range from 1 through the dimension n of the manifold and index a local orthonormal frame vielbein e1 e for the tangent space of the manifold In components we have e guy jk ef ey g The inverse vielbein is defined by the relation elet i These two objects Wy and e will be used to transform curved indices u v p to flat ones i j k and back In Euclidean space there is no distinction between upper and lower flat indices As usual the Riemann part of the covariant derivatives contains the Christoffel connection so that Villy Oyu ev 2 7 for an arbitrary vector v To extend this derivative to the objects with flat indices one has to use the spin connection o
115. tion Vu 9 5 42 i e the gamma matrices are covariantly constant is a zeroth order opera tor a matrix valued function We also suppose that the connection in V is unitary This means that the connection one form is represented by an anti hermitian matrix in a suitable basis We restrict ourselves to the case when Et E so that the operator P is formally self adjoint in the bulk Note that 20 Tn this case the subscript A is a target space vector index while the superscript A indicates the coupling Here we do not consider other beta functions Biel gl etc 54 compatible unitary connection is not unique Consider a first order differential operator on the boundary P Vat 5 Ern mE O 2 5 43 where O x is a hermitian matrix valued function on 0M The operator P is a self adjoint operator of Dirac type on the boundary All functions on OM can be decomposed in positive negative and zero modes of P Let us define I as a projector on the space spanned by non negative eigenspaces of P Then the equation Il_dlou 0 5 44 defines the APS boundary conditions Of course relative complexity and non locality limits physical applica tions of spectral boundary conditions However they appeared in a number of axial anomaly calculations quantum cosmology and in works on the Aharonov Bohm effect In brane models spectral bound ary conditions describe T selfdual configurations which m
116. to good i e symmetric and elliptic operators describing metric and ghost fluctuations The problem of finding a suitable set of boundary conditions in gravity appeared to be much harder that for the lower spin fields There exist several proposals on the market However neither of these proposals is fully satisfactory It is clear now that proper boundary conditions must depend on tangential derivatives Moreover they probably must be non local The material of this section can be generalised to the case when an in dependent torsion field is present Explicit expressions for relevant operators acting on fields with different spin in the Riemann Cartan space can be found in 32 4 Heat kernel expansion on manifolds without boundary 4 1 General formulae For physicists the most familiar way to calculate the heat kernel coeffi cients is the DeWitt iterative procedure which will be briefly described in sec 4 3 Here we take a different route following the method of Gilkey Advantages of this method are most clearly seen on manifolds with boundaries However even if boundaries are absent we use the same method for the following rea sons i this will ensure a smooth transition to more complicated material of the next section ii we believe that calculations of the coefficients a f D are a little bit easier even without a boundary We are not going to give com plete proofs of all statements Instead
117. u on the dimension n of the manifold is shown explicitly On the other hand we can use 4 2 Spectrum of the operator D is known The eigenvalues are l l Z The heat kernel asymptotics for D can be easily obtained by using the Poisson summation formula 1 exp tl Dew nl t leZ lEZ i O ter 4 4 Since exponentially small terms have no effect on the heat kernel coefficients the only non zero coefficient is a 1 D1 yr Therefore we obtain by eq 4 2 afa D VE fd eT E tvde AD 48 There are no restrictions on the operator D2 or on the manifold My By com paring the equations 4 3 and 4 5 we obtain This proves that the constants u depend on the dimension n only through the overall normalisation factor 47 34 To calculate the heat kernel coefficients we need also the following varia tional equations see Lemma 4 1 15 d qel eD n k ax f D 4 7 d e 0a 1 D eF ap 2 F D 4 8 de Jea0ttn 24 F eo D035 4 9 where f and F are some smooth functions To prove the first property 4 7 we note that c 0Tr exp e 4D Tr 2ftDexp tD Of exp tD and expand both sides of this equation in the power series in t Eq 4 8 can be checked in a similar way To prove 4 9 consider the operator D e 6 e D F 4 10 We use first 4 7 with k n to show d 0 qel De 8 4 11 Then we vary the equation abov
118. ucture so that one can define spinor fields on M In a suitable basis the chirality matrix may be presented in the diagonal form at this point it may be useful to consult sec 3 3 Ys 7 43 0 I Let V correspond the 1 eigenvalue of ys positive chirality and V2 to 1 The Dirac operator 3 26 anti commutes with ys and therefore may be represented in the form 0 Pt D 7 44 P 0 We also have 5 PUR 0 p 7 45 0 PP We conclude that index P which now measures difference between dimen sions of the null subspaces of D with positive and negative chirality is equal to global chiral anomaly index P Tr err Tr ere Tr rseP Gn ys f0 7 46 Therefore the index can be calculated by the methods of the previous section In particular the index of the Dirac operator in two and four dimensions on 73 flat manifolds without background axial vector fields Ae 0 can be read off from 7 31 and 7 32 with y i 2 We also see that chiral anomaly provides very important information on topology of the background One can associate an index with any elliptic complex which must not necessarily contain only two vector bundles In general one has a sequence of bundles V and a family of operators P which map smooth sections of V to smooth sections of V 1 One requires that P P 0 and that the corresponding Laplacians D PIP Pakai are elliptic Important examples includ
119. uge parameter amp B 0 3 55 such that Boe 0 3 56 The equation 3 56 means that the functional space defined by the boundary conditions 3 54 is invariant under the gauge transformations provided the gauge parameter satisfies 3 55 Upon quantisation 3 55 become boundary conditions for the ghost fields The condition 3 56 ensures validity of the Faddeev Popov trick on a manifold with boundary and guarantees gauge independence of the on shell path integral Since the boundary term in 3 43 is diagonal in the gauge index a we can consider the case of an abelian gauge group and drop a from the notations The general non abelian case does not offer considerable complications There are two sets of local boundary conditions which satisfy the gauge invariance requirements above The first set is called absolute boundary conditions It reads Anlam 0 On Aalam Vindab Law Avlam 0 Vn lom 0 3 57 28 The second set Vn Daa Anlam 0 Aalom a 0 Elau 0 3 58 is called relative boundary conditions The projectors on the Dirichlet and Neumann subspaces and the endomorphism S are II dindjn II On in jn Sij Lab ia jb and II On a Oin0 i IIF Crore Sij Laa in jn for the absolute and relative boundary conditions respectively It is straightforward to check gauge invariance of the absolute boundary conditions 3 57 A bit more job is required to show that the
120. ulations 17 See sec 2 1 for more information on differential geometry of manifolds with boundary 48 The one dimensional examples 8 5 2 and 5 3 immediately give ane bb bp F 5 20 Our next example 5 9 controls linear terms in S and gives bt 12 5 21 For Dirichlet boundary conditions we have set S 0 Therefore b plays no role To define the constants by and b3 we use the conformal variation equation 4 7 Variations of E and R are given by 4 18 Conformal properties of the second fundamental form La are dictated by that of the metric E f Laa lt n as Ufa 7 5 22 We wish to keep the boundary conditions invariant under the conformal trans formations This implies that the boundary operators B must transform ho mogeneously This property holds automatically for Dirichlet boundary condi tions 2 15 In the case of modified Neumann boundary conditions conformal transformation of S should cancel inhomogeneous term in the connection wp This yields d 1 qels soli Roa z 2 fn 5 23 Next we substitute the variational formulae 4 18 5 22 and 5 23 in 4 7 with k 2 We collect the terms with f n on the boundary to obtain n 4 n 1 b3 n 2 b3 5 24 for Dirichlet conditions and n 4 n 1 b 6 n 2 n 2 bf 5 25 for generalised Neumann Robin ones Since the constants b do not depend on n the two equat
121. und in Here we consider some physically motivated particular cases only Let us introduce the Euclidean Dirac y matrices which satisfy the Clif ford commutation relations Yu v VW Yv Loy 3 23 The y matrices defined in this way are hermitian qi We also need the chirality matrix which will be denoted y independently of the dimension It satisfies A P Wr 3 24 From now on we suppose that the dimension n is even We fix the sign in y by choosing 1 V y a E E 3 25 Let D be the standard Dirac operator in curved space with gauge and axial gauge connection 1 P iq a gle AG Au iaf 3 26 Here is the spin connection 2 9 A and A are vector and axial vector fields respectively taken in some representation of the gauge group Both A and A are antihermitian in the gauge indices The operator P 3 26 is for mally self adjoint in the bulk The operator D p is of Laplace type 2 2 23 with 1 1 V a V wu glen Yog Ap sh nA Tes a BSH RE oy tat A SHA AY 4 4 1 H AY 5 5 with obvious notations Fy 0 A 0 A A A D A gt 0 A gt T8 AR A A3 The expression for Q is a little bit lengthy 1 ie Py me A A gt 7S q V Ropu a ie PDA g WDA 5 45 5 45 5 45 HY Apy F EM APW 9 A gt AYP Yu P Ay A PA Any 3 28 where A A 8 A An Ap A AP 3 29 Consider now manifolds with boundary The specific feature o
122. y in the bibliography A more comprehensive treatment of many mathematical problems related to the heat kernel expan sion can be found in The book by Kirsten considers also specific phys ical applications as the Casimir energy and the Bose Einstein condensation The recent review paper is devoted to the Casimir effect see also The monographs by Birrell and Davies and Fulling remain standard sources on quantum field theory in curved space Quantization of gauge theories is explained in In the heat kernel expansion is treated from the point of view of quantum gravity and quantum cosmology Useful information about properties of the zeta function can be found in One may also con sult The DeWitt approach to the heat kernel and its generalisations are described in The path integral point of view on the heat kernel can be found in Our primary goal is local formulae for the heat kernel coefficients There fore in some cases global aspects will be somewhat neglected This report is organised as follows The next section contains necessary preliminary information on spectral geometry and differential geometry There we define main geometric characteristics of the manifold and of the bound ary We discuss the zeta function which defines the effective action and the resolvent which is a generalisation of the propagator We relate the asymp totics of these functions to the heat kernel c

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