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1. log LA O w a 9 yy se tlw tap lostLA gage tO where we have introduced a lower cutoff of e e 7A on the heat kernel integral in eq 2 21 and ignored a p independent but quadratically divergent in A terms Under reg ularization of the AdS2 volume we indeed find that our expression 2 18 matches this full eigenfunction expanded heat kernel result in 2 27 Although we have not shown the calcu lation here these two expressions will continue to be equal term by term in powers of 1 u 3 Two interesting properties of AdS zero modes Before extending our results to other dimensions or to thermally identified spacetimes we pause to study two important properties of the zero modes given by 2 6 and 2 7 First these modes have a natural physical interpretation as the Wick rotation of quasinormal modes of an AdS black hole Second these modes are manifestations of the discrete series representations of SO 2 1 on the space of the St parameterized by 0 For convenience we set the AdS length to one in this section 3 1 Zero modes Wick rotate to quasinormal modes So far we have been working entirely in Euclidean space we have really calculated the one loop determinant for the Euclidean hyperbolic space H If we wish to study the partition function in the context of a Lorentzian spacetime we can either first study its Euclidean section or we can locate the poles in the partition fu
2. 1 Our most important reason to discount these odd dimensional modes is group theoretic We expect these modes should not be considered because they are matrix elements for a discrete series representation but discrete series representations of the motion group for Aq only exist in even dimensions Regardless the delicacy of deciding what boundary conditions should be applied in order to determine acceptable zero modes when we consider unphysical general complex A is the most important disadvantage of the zero quasinormal mode method for computing partition functions Moving into the complex A plane means we are away from the physical A gt 0 case and so some of our intuition about which modes to include may break down Conversely the major advantage is that we only require information about the location and multiplicity of the zero quasinormal modes in the complex A plane we do not need to normalize the modes or provide a complete set of states as in the heat kernel eigenfunc tion approach of 10 or compute characters of representations as in 2 We believe this advantage is sufficient to motivate further study and application of the method in other spacetimes Acknowledgments We would like to thank F Denef T Hartman S Hartnoll F Larsen P Lisbao and J Maldacena for valuable discussion C K is supported in part by the US Department of Energy under grant DE FG02 95ER40899 G N was supported by DOE grant
3. 2 11 and 2 12 Following 1 we introduce a lower cutoff of e 7A on the heat kernel integral in 2 11 where y is the Euler number and A labels the IR cut off For the scalar operator we study 1 as dt 2 gt R f dt 2 VAN ETR d r Rn f a a imn O SATT Ja a o p mea ee ee 1 Ta nl u 1 R la d Ne An I val 2 Selar 412 6 PAo 1 2 pe i Mpg R ARG log et 2 1 F In x oval rz 08 72 g O8 A2 O u 2 13 where we have dropped an m independent quadratically divergent term and introduced an arbitrary renormalization scale Arg As in 1 we recognize the last line here as renor malization of the classical couplings We can thus account for their effect by using the logarithmically running couplings in evaluating the classical contribution to the action and we do not need to consider these terms further here In order to compare the remaining expression with our zero mode method result in eq 2 10 we need to evaluate the volume of the hyperbolic space Following discussions in the context of holographic renormalization in AdS CFT the regularized Euclidean action involves the regularized volume of H2 which is given by 1 at 5 Volreg Han T n 2 2 L Veen Voan 2 14 bo With the regularized volume Volreg H2 2nL the first line in the second equality of eq 2 13 evaluates to be l 2 2 l 2 1 2 l 2 a Z log LA 21 a s 5 ogu 5 s
4. 75 los LAra 2 15 where we have also used Rags 2 L Comparing eq 2 10 and eq 2 15 we find Pol A to be 1 Pol u 1 log LAra u 19 log LAra or Pol A 1 log LApgg A A 1 log LARG gt 2 16 where in the second line we have reexpressed the polynomial in terms of A Combining this result with 2 8 the final expression for Z A becomes Z A A A a A J log LArg 2 1 A 2A 1 0 A 2 17 See e g 23 25 For latter comparison with the heat kernel method we expand this final Z in terms of u 1 1 1 1 T e Zu 50 5 1 town 1 n 5 bela 5 24 O u 4 2 18 2 2 940 Before comparing with the full eigenfunction heat kernel computation let us make few comments on the nature of our computation Here we have actually computed the inverse of the one loop determinant 1 det V m on Euclidean AdS2 with a Dirichlet boundary condition This determinant is equivalent to the partition function for a complex scalar on the same space via Gaussian integration as long as we are evaluating the partition function away from a zero mode If we need the value of Z at one of the A we would need to more carefully evaluate the path integral We also note that our calculation here is not just a direct extension of the prescription given in 1 since the proper discrete set of modes to sum over in Euclidean AdSg is not a priori cle
5. coordinates of the above type for T 1 27 We can thus examine the Wick rotation of the zero modes in 2 6 under 0 it We also use the similar coordinate r cosh n related 10 to p by r 1 p In these coordinates the horizon of the Wick rotated spacetime will be at r 1 2 2 2 dr ds r 1 dt 3 4 rT We recognize this spacetime as an AdS black hole Here the modes 2 6 become 1 paa e 1 r3 PLA h Agil 3 5 and these modes satisfy the boundary condition p7 p gt when A L lt o l TEA 3 6 In the Wick rotation above modes with gt 0 correspond to quasinormal modes while those with l lt 0 correspond to antiquasinormal modes Hence when identifying with the mode frequency w in the Lorentzian spacetime for l gt 0 we set l iw while for l lt 0 we instead have l iw Using this identification and the identity 15 3 30 in 26 we find A iw iw 1 A Fiwt iw p PA X e pP 9 9 1 iw p7 3 7 where indicates that we expect a anti quasinormal mode Near p 0 the candi date quasinormal modes are purely ingoing while the antiquasinormal modes are purely outgoing While the Euclidean zero modes satisfying the p boundary condition only exist for values of A A where Z A has a pole anti quasinormal modes are present for any physical A Instead we have to find the particular values of w where 3 7 satisfies
6. od p and only at special A will iw be an integer To examine this behavior further we now fix A be a generic but physical conformal dimension Using 15 3 7 in 26 to expand 3 7 as p gt oo we find 9 iar a TCL iw P GG A ate Aqiw l A Pes rie 9 3 9 9 2 ot AE iw T A fl A iw 1 A iw 3 1 Fiwt A 1 2 saaa ANN For A physical that is real and positive da satisfies p gt when A iw L lt 3 9 For a given A any w satisfying this equation will be a quasinormal frequency Note that when we are considering physical A our boundary condition at infinity is merely normalizability when we are working in the complex A plane we instead insist on removing entirely one of the two behaviors at infinity We can now see explicitly that the quasinormal modes in Lorentzian space Wick rotate into zero modes in Euclidean space so we can find the poles in Z A from either perspective 1 lt To find the poles in a Lorentzian spacetime with a horizon we first fix a physical A We then find the anti quasinormal modes i e modes which are purely out ingoing at the horizon and satisfy Dirichlet boundary condition at infinity The frequencies of these modes can be written as function w A which can then be analytically continued to the complex A plane Poles in Z A are at A A such that iw A Z where the integer is positive for quasinormal modes and negative for antiquasinormal mod
7. 3A 1 A a 5 2 A 5 3 A 5 8 Or 25 1 1 log Z u Pol p pt Sy 240u 1204 17 log p le Co og Z u Pol u aH gg ggg 240p 1204 17 log w O U 5 9 Similarly to what is done for AdSg in section 2 3 from the heat kernel local curvature expansion on H4 we obtain l 1 log Z u T a de A L p TS 240 1204 17 log A L 1 son 240p 1204 17 log u O u 5 10 Here we have again eliminated a divergent constant term although we are left with a polynomially divergent term dependent on u Matching this expression with the zero mode expression allows us to find Pol 1 2 1 1 Pol w 5 6 log LAro u F 1 log LA rG 2e7L Aka la 17 log LARG 2880 ry 2 i 3 Pol A 45 log LA Ka Pol A 5 5 1g LAna 4 5 1 p 3N ELE E 5 11 9880 u Thus the full 1 loop partition function on Hy is given by log Z A 1f 2 1 3 i Wee a _ _ 2i ee 9 9 z 108 Ara a 5 ZE 24 pl log LARG 2e L ig 4 sy 7 1 E log LAra 2A 9A 13A 6 c 0 a 3A aver A 3 1 A n gt Ou a 5 12 16 5 2 Comparison between thermal partition function in quasinormal mode and heat kernel method Proceeding as in section 4 we now compute the 1 loop partition function on thermal AdSog According to the quasi normal mode method using normal modes
8. all possible zero modes from Lorentzian signature spacetimes it may be necessary to consider quasinormal modes visible in each such Wick rotation as in the thermal AdS case here e We have highlighted the relationship between zero modes and the group structure of the spacetime It turns out in odd dimensions we will need to consider this information Thus far we have not discussed odd dimensional AdS First in odd dimensions the calculation of the thermal partition function follows the correct AdS3 case done explicitly in 1 As odd dimensional AdS has no logarithmic terms in the large mass expansion of log Z the thermal sum as in 4 6 is sufficient in this case If we wish to compute the 8 scalar partition function for H n 1 without a thermal compactification the entire result is in fact polynomial in A and so can be captured just by setting Pol A according to the heat kernel curvature expansion However this means in the odd dimensional case we should not consider the modes in section 5 to be zero modes This is at first a bit surprising because they appear to satisfy h i n As we argue in the our condition of having only sinh n behavior and no sin appendix the distinction between the behaviors is not so clear because they differ by an in teger power of sinh 7 Additionally the Laplacian equation 5 1 under the restriction 5 2 has an abelian monodromy group for odd d 1 but it is nonabelian for even d
9. computed in 2 is poly nomial see e g ref 28 In the even dimensional case however it is not a polynomial and we conjecture that it could be obtained by the zero mode method above which in the AdS case results in eq 5 12 If so by combining the normal modes on thermal AdS with the zero modes we have found here the quasinormal or zero mode method can be used to compute the full AdS2 partition function EE an 6 Discussions and conclusions We have calculated the scalar one loop determinant for even dimensional AdS spacetimes via a zero mode method In the case of AdS2 these zero modes are the Wick rotation of quasinormal modes for the AdS2 black hole In all even dimensions our results match with previous results using purely heat kernel methods Our method is a nontrivial extension of the zero mode method developed in 1 to the case of even dimensional non compact spaces In a compact space such as S the condi tion on the zero modes is normalizability via square integrability Since the condition on the complete set of eigenmodes used in the heat kernel eigenfunction expansion is the same the modes are very similar Conversely in the AdS n case the eigenmodes used in the heat ker nel computation are constrained to be delta function normalizable whereas the zero modes in our zero mode method are instead constrained by having only A behavior at infinity This application of the boundary condition at infinity
10. coordinates 2 1 can be gained by relating each to the embedding coordinates of the hyperboloid Hz in Rt The embedding measure is dx2 dx du and the two sets of coordinates are given as Lcoshn u Lcosho cosh Tg 4 2 Lsinhysin zo L cosh o sinb Tg Esimhycoss 7 Lsinhe In both cases the Lorentzian spacetime is obtained by Wick rotation of x9 iv However the two rotations are not really equivalent In the disc coordinates we rotate 0 which is initially a compact direction In the global coordinates we rotate Te 17 global Lorentzian AdS usually refers to allowing 7 to range over all real values As the coordinates 4 1 make clear translations in Tg are a Killing symmetry Ther malizing along this direction is thus quotienting by the subgroup generated by discrete translations of size T in the Tg direction 4 1 T dependent portion of Z A in thermal AdS2 Working in global coordinates 4 1 1 finds the temperature dependent piece of the par tition function on general thermal AdS spacetimes by first finding the global normal mode frequencies in Lorentzian signature The frequencies are normal because there is no dissipation so they are purely real k These frequencies then correspond to zero modes which manifest as poles in the thermal partition function Summing over these poles 1 propose log Z A r Pol A 2 log 1 ei 4 6 k 0 Since the thermal identificat
11. on Lorentzian global AdSy the thermal partition function of a complex scalar on thermal AdS is given by log Z A r Pol A 2 Y 21 1 log 1 eT EE n l gt 0 k 1 k 2 log i er 5 13 k gt 0 On the other hand from 2 the 1 loop partition function of a complex massive scalar on thermal AdS4q using the heat kernel method is log Z A r X I eila 5 14 p 1 e Str evr 1 p In particular for d 3 we obtain 2 p lo Z A e LT 8 T 2 e T Y k 1 k 2 log 1 ez 5 15 k gt 0 So we see that the temperature dependent piece of the heat kernel answer in eq 5 15 agrees with the thermal normal mode method answer in eq 5 13 for AdS4 Again as for AdSg if we wish to show Pol A is truly a polynomial we need to compare the full partition function for thermal AdSyz i e including the temperature independent part To do this computation we add to eq 5 15 the contribution from the zero mode method on H4 which is just eq 5 12 multiplied by Vol H4 Z Volreg H1 As a side note in fact we can show that for general AdSg the quasinormal mode computation including only thermal normal modes on thermal AdSg agrees with the heat kernel answer in eq 5 14 d 2 log Z A r Pol A 2 Y A oe 2 log 1 e rr d 2 d 3 p20 y 2 te 4 8 5 16 d p 1 e rr ex 1 p For odd dimensional AdS the temperature independent term not
12. the zero mode method to AdS In this section we use the zero mode method to calculate the 1 loop partition function for a complex scalar o with mass m in Euclidean AdS with AdS length L We begin by setting notation and briefly reviewing the zero mode method from 1 We will extend their method for compact spaces to the non compact AdSo We work in coordinates do L dn sinh n dO 0 0 27n gt 00 2 1 where 7 is the boundary Writing the partition function in terms of the scalar determinant we have 1 ZIA Doe d V eee A J pe gt det V2 m2 2 2 where A solves mL A A 1 Here we choose the root A according to the stan dard Dirichlet boundary condition for AdSg i e we impose the Dirichlet type boundary condition sinh 7 for n gt o 1 1 1 4 202 2 5 atm z TH 2 3 A at the AdS boundary In order to calculate the determinant we will use the method developed in 1 If Z is a meromorphic function of A then we can use the Weierstrass factorization theorem from complex analysis to find the function Under this theorem if we know the poles and zeros and their multiplicities of a meromorphic function then we know the function itself up to a factor which is an entire function The determinant we are interested in will have no zeros only poles so we can write Z A PA A A 2 4 where the A are poles of Z A each
13. with degeneracy d Importantly Pol A is a Pol A is entire polynomial in A since e We find the poles in Z A by searching for values of A where det V m becomes zero These A occur whenever there exists a solving v a ob 0 2 5 where is smooth in the interior of the Euclidean AdS and satisfies the boundary condition 2 3 That is at these generally complex A is a zero mode Thus at A A Z A has a pole the degeneracy of this pole is the number of smooth solutions to 2 5 which also satisfy 2 3 For a spacetime with a compact direction these A can be labelled by a discrete index i we call this set the A As we will show for AdS2 even a noncompact space may have a discrete set of poles when we choose the appropriate boundary condition As shown in 1 for spacetimes which are Euclidean black holes these zero modes Wick rotate to become quasinormal modes of the corresponding Lorentzian black holes We will discuss the details of this Wick rotation in the Euclidean AdS spacetime in section 3 1 below As a preview the zero mode method for calculating the 1 loop determinant has three steps which we will explain in detail as we proceed e Find A where there exists a zero mode and the multiplicities d of these zero modes e Use eq 2 4 to express Z A in terms of these A and d Zeta function regularize log Z A to perform the infinite sum e Find the polynomial
14. DE FG02 91ER40654 and the Fundamental Laws Initiative at Harvard A Finding zero modes in AdSgi for d 1 even We work in spherical coordinates dsj dn sinh nd0 A 1 where dO is the line element of S We are looking for eigenfunctions of the Laplacian satisfying v ea ao d 0 n s00 sinh 7 A 2 Following 10 the eigenfunctions of the Laplacian regular at the origin are given by o Vima A 3 19 d d d 1 di isinh n F t 5 7 7 5 l ae sinh 2 l where Yim is a spherical harmonic on S whose total angular momentum is set by the nonnegative integer l m is a set of d 1 integers with gt ma 1 gt gt mi mi E Z For the specific case of d 1 2 dimensions there is no m instead for every nonnegative integer l we have two modes exp i and exp 110 We want to rewrite the eigenvalue as A A d L Reparameterizing the hypergeo metric function in terms of A we find A 4 d 1 1 cosh hia x Yim isinin F 1 AU d As ae 2 2 Using the quadratic transformation 15 3 30 in 26 we rewrite q now dropping the Y as I A l d A d 1 qa x isinhn F A SH ae sinh n i A 5 As long as A d 2 is not an integer we can use 15 3 7 in 26 to expand q near 7 co wh n 4 i 4 r A _Ji aA 1 A d 1 sal i S U A A 1 I A ew eae 2 2 1 A Z n A 6 sinh 7 1 A S sinh n These modes already satisfy the conditi
15. PUBLISHED FOR SISSA BY SPRINGER RECEIVED March 6 2014 REVISED May 2 2014 ACCEPTED May 30 2014 PUBLISHED June 17 2014 Partition functions in even dimensional AdS via quasinormal mode methods Cynthia Keeler and Gim Seng Ng Michigan Center for Theoretical Physics Randall Laboratory of Physics The University of Michigan Ann Arbor MI 48109 1040 U S A e Center for the Fundamental Laws of Nature Harvard University 17 Oxford Street Cambridge MA 02188 U S A E mail keelerc umich edu gng fas harvard edu ABSTRACT In this note we calculate the one loop determinant for a massive scalar with conformal dimension A in even dimensional AdSg 1 space using the quasinormal mode method developed in 1 by Denef Hartnoll and Sachdev Working first in two dimensions on the related Euclidean hyperbolic plane H2 we find a series of zero modes for negative real values of A whose presence indicates a series of poles in the one loop partition function Z A in the A complex plane these poles contribute temperature independent terms to the thermal AdS partition function computed in 1 Our results match those in a series of papers by Camporesi and Higuchi as well as Gopakumar et al 2 and Banerjee et al 3 We additionally examine the meaning of these zero modes finding that they Wick rotate to quasinormal modes of the AdS black hole They are also interpretable as matrix elements of the discrete seri
16. Pol A by matching the zeta function regularization at large A to the local heat kernel curvature expansion expression for log Z A The initial version of the Weierstrass factorization theorem only applies to analytic functions but it can be extended to meromorphic functions Additionally convergence of infinite products will not concern us so we do not worry about using the proper Weierstrass factors If we include information from the growth behavior at large A which we will have from the heat kernel curvature expansion we could use instead the Hadamard factorization theorem here we will not need these complications 2 1 Finding the zero modes The first step is to find the zero modes in the complex A plane Following 10 the solutions to 2 5 which are regular at 7 0 in Euclidean AdS2 can be written as ba e isinh n F A li li 1 As i 1 sink 2 2 6 where F is the hypergeometric function also written 9f We have already imposed part of the boundary condition at the origin by choosing the solution of the hypergeometric equation which is regular at 7 0 To impose that a be single valued due to periodicity of 0 0 27 we additionally impose Z Now rather than imposing delta function normalizability as in 10 we impose the boundary condition by insisting the modes have only the Dirichlet like behavior sinh j as 7 oo That is we wish to turn the behavior sinh7 off entir
17. R Gopakumar R K Gupta and S Lal The heat kernel on AdS JHEP 11 2011 010 larXiv 1103 3627 1nSPIRE 3 S Banerjee R K Gupta and A Sen Logarithmic corrections to extremal black hole entropy from quantum entropy function JHEP 03 2011 147 arXiv 1005 3044 INSPIRE 4 A Sen Quantum entropy function from AdS2 CFT correspondence Int J Mod Phys A 24 2009 4225 arXiv 0809 3304 INSPIRE 5 I Mandal and A Sen Black hole microstate counting and its macroscopic counterpart Nucl Phys Proc Suppl 216 2011 147 Class Quant Grav 27 2010 214003 larXiv 1008 3801 minSPIRE 6 D V Vassilevich Heat kernel expansion user s manual Phys Rept 388 2003 279 lhep th 0306138 INSPIRE 7 R Camporesi Zeta function regularization of one loop effective potentials in anti de Sitter space time Phys Rev D 43 1991 3958 mnSPIRE 8 R Camporesi and A Higuchi Stress energy tensors in anti de Sitter space time Phys Rev D 45 1992 3591 INSPIRE 9 R Camporesi and A Higuchi The Plancherel measure for p forms in real hyperbolic spaces J Geom Phys 15 1994 57 10 R Camporesi and A Higuchi Spectral functions and zeta functions in hyperbolic spaces J Math Phys 35 1994 4217 iNSPIRE 11 J R David M R Gaberdiel and R Gopakumar The heat kernel on AdS3 and tts applications JHEP 04 2010 125 arXiv 0911 5085 mivSPIRE 12 F Denef S A Hartnoll and S Sachdev Quan
18. angular momentum labeled by k Following 1 we denote this degeneracy Q k where 2k df kt d 1 k i es aa 2E HE sa We can then express Z A as in 2 4 log Z A Pol A J gt Q k log k A 5 4 k gt 0 where the conformal dimension is related to the mass via d A 3 u u ym L d 2 5 5 Defining 6 as an operator on a function f s via dsf s f s 1 5 6 we regularize 5 4 as log ZAqs A Pol A Q A 6 s A s 0 5 7 To finish the computation we need to compare the large expansion of 5 7 to the heat kernel curvature expansion as in 2 11 and 2 12 In the next section we will explicitly evaluate Pol A for the specific cases of d 1 4 showing agreement with previous computations of Z A in both cases We only sum over the solution for A d 2 u because we impose Dirichlet like boundary conditions at the AdS boundary Conversely for the sphere case studied in 1 they must sum over both A d 2 iv v mLas d 2 This difference accounts for the discrepancy between the analytic continuation of sphere results and the actual AdS results as studied via a Green function method for d 1 4 in the appendix of 7 bp 5 1 Finding Pol A ford 1 4 In this subsection we repeat the analysis in section 2 3 for AdS to obtain Pol A From the zero modes obtained in the previous section we find log Z A Pol A 2A 9A 13A 6 0 A T a
19. ar as the space is non compact Here we have provided a prescription for finding this discrete set of modes 2 4 Comparison with full eigenfunction heat kernel method We now compare our result with previous results for the scalar partition function on AdS spacetimes In contrast to the direct computation of the partition function developed in 1 which we have adapted for Euclidean AdSgi1 the other methods described here make use of the heat kernel on Hg More details about the heat kernel can be found in 6 while its application to Hq can be found in 10 The heat kernel satisfies O O K x 2 t 0 K x 2 0 5 x 2 2 19 where Ll is the scalar Laplacian acting only on x where x is a point on AdS We can relate this object to the corresponding complex scalar partition function via log Z A 2 108 kn 5 a 12 E 2 20 f faia akea 2 21 Here kKn are the eigenvalues of the scalar Laplacian and the sum is done over these eigenvalues for a complete set of delta normalizable modes Since the heat kernel satisfying equation 2 19 can be written in terms of an eigenvalue weighted sum over a complete set of normalizable eigenstates we can use this information to construct the heat kernel and thus the partition function This is the method most closely followed by Camporesi and Higuchi in their series of papers 7 10 it is also the method used in 3 For later usage we will briefly generalize
20. com munity particularly since the advent of the AdS CFT correspondence nearly two decades ago Even leading one loop effects can contain important physical results For example quantum corrections to the entropy of black holes can be computed using the quantum entropy function 3 5 which requires the one loop effective action due to the matter fields on an AdS x M spacetime As such there is interest in developing efficient methods for various one loop calcula tions Several methods for calculating the one loop determinant for a particular field fluc tuation are already known mostly based on the heat kernel method see e g 6 There are several known ways to calculate the heat kernel First if one is only interested in large mass behavior for a massive field of mass m the curvature expansion in terms of local invariants is known for many field types up to 10th order 6 If we wish to know the heat kernel outside of a 1 m expansion then we can expand the heat kernel as a sum over normalized eigenfunc tions of the Laplacian weighted by their eigenvalues this is the approach used in the series of papers 7 10 and again by 3 for AdS2 x S spacetimes These eigenfunctions are gener ally special functions and the details particularly for higher spin fields can be challenging Of course these eigenfunctions of the Laplacian themselves arise from the group theory structure of the spacetime the Laplacian is the quadratic Casimir f
21. ely Using the result A 8 from appendix A this is accomplished whenever A A where as lzi Z lt A 2 7 In other words at each nonpositive integer A we have a zero mode of multiplicity 2A 1 There are no other zero modes for any A in the complex plane Since these zero modes occur at integer values A they are discrete In order for the zero mode method to work we require a discrete set of poles if the set of poles had an accumulation point other than infinity the Weierstrass theorem would not apply As we will discuss further in section 3 below these modes have several interesting physical and mathematical properties They are the Wick rotation of quasinormal modes for AdS black holes as well as the manifestation of the discrete series representation of the SO 2 1 symmetry of AdS2 on the space of smooth functions on the circle St 2 2 Summing the contributions of the zero modes For now as we have found a discrete set of zero modes we continue our calculation of the AdS complex scalar partition function by evaluating log of 2 4 log Z A Pol A XC 2A 1 log A A A EZ lt 0 Pol A S 2k 1 log A k k 0 Pol A 2 1 A 2A 1 0 A 2 8 We use this boundary condition in accordance with AdS CFT intuition We will show in section 2 4 below that this choice agrees with the results from the heat kernel eigenfunction expansion done over delta func
22. eparate the two series from each other The expansion of the top line can contaminate that of the bottom line If we do continue by studying the degenerate hypergeometric equation see 29 30 we will in fact find that a as in A 4 with A 1 Z lt o become polynomials in sinh n with only positive powers As such a will contain none of the behavior sinh fy which would have to be a negative power for A lt 0 As we argue in the conclusion we nonetheless do not need to include these putative zero modes in the sum for odd dimensions As further evidence for the difference between even and odd modes we again mention that the modes in A 4 are matrix elements of the discrete series representation for SO d 1 1 These representations should only be unitary in the case of d 1 even again for more details see 27 B Euclidean disc vs global coordinates We begin with the embedding coordinates for Hq 1 d Sap 25 w Sel B 1 i 1 with line element ds dx drg du B 2 i 1 To recover the Euclidean disc we choose u Lcoshn B 3 xo Lsinh nsin 69 x1 L sinh n cos bo cos 6 x2 Lsinh7 cos bo sin 01 cos 2 d 2 Lq 1 L sinh y cos 69 TI sin 3 cos 04 1 i 1 d 1 tq L sinh n cos 0o Jli sin 6 i 1 To recover the usual measure on dQ4 we should additionally take 69 7 2 6 Wick rotating 09 it produces the black hole coordinates for the case of AdSg This rota
23. es If we instead wish to find the poles in Z A directly from Euclidean space we allow A to range throughout C from the beginning We instead fix a boundary condition of smoothness and singlevaluedness in the interior of the spacetime and then find the A which satisfy p gt as we near the boundary p oo 3 2 Zero modes as discrete series representations Since quasinormal modes are related to the group theory structure of a spacetime we now consider the group theoretic interpretation of the zero modes 2 6 These modes are mani festations of the discrete series representations of SO 2 1 on the circle St parameterized by 0 Rotating the modes in 3 7 back into zero modes in Euclidean AdS2 or H we obtain ba e i sinh n lF A l U 1 A E aA 1 lls sink 7 3 10 2 which can also be interpreted as a matrix element of a representation of SO 2 1 with Laplacian eigenvalue set by A Before we assign a particular value to A we can see this function is quite similar to a spherical harmonic for SO 3 the 0 dependent piece has weight under transformations by Og the generator corresponding to the SO 2 subgroup A sets the eigenvalue under the Laplacian or quadratic Casimir for the entire SO 2 1 Next for the zero modes A is restricted to be a positive integer For the group SO 2 1 representations with A E Zso are unitary but with respect to an unusual norm These repre
24. es representations of SO 2 1 in the space of smooth functions on St We generalize our results to general even dimensional AdSo2 again finding a series of zero modes which are related to discrete series representations of SO 2n 1 the motion group of Haj KEYWORDS Gauge gravity correspondence AdS CFT Correspondence 1 N Expansion ARXIV EPRINT 1401 7016 OPEN ACCESS The Authors Article funded by SCOAP doi 10 1007 JHEP06 2014 099 Contents 1 Introduction 1 2 Applying the zero mode method to AdS2g 3 2 1 Finding the zero modes 5 2 2 Summing the contributions of the zero modes 5 2 3 Finding Pol A via local curvature invariants 6 2 4 Comparison with full eigenfunction heat kernel method 8 3 Two interesting properties of AdS2 zero modes 10 3 1 Zero modes Wick rotate to quasinormal modes 10 3 2 Zero modes as discrete series representations 12 4 The thermal partition function on AdS2 13 4 1 T dependent portion of Z A in thermal AdS2 13 4 2 Partition function on thermal AdS via heat kernel on quotient of hyperbolic space 14 5 Extension of story to all even dimensions 14 5 1 Finding Pol A ford 1 4 16 5 2 Comparison between thermal partition function in quasinormal mode and heat kernel method il 6 Discussions and conclusions 18 A Finding zero modes in AdSq 1 for d 1 even 19 B Euclidean disc vs global coordinates 21 1 Introduction Quantum effects in AdS spacetimes are of great interest to the theoretical physics
25. for a non compact space is novel For thermal AdS in even dimensions our results augment those of 1 which provided only the temperature dependent piece of the partition function Since the temperature independent piece of log Z A has logarithmic components the polynomial Pol A cannot account for these terms Instead in order to compute the partition function for thermal AdSq 1 in even dimensions we should consider both the zero modes of Euclidean Hq 1 as well as the thermal zero modes arising from compactification in Tg This paper concen trates on the Hg modes while 1 considered the temperature dependent thermal modes considering both effects together results in a Z A which is meromorphic as expected Our interest here is not just in reproducing the AdS even dimensional calculation but instead in further developing the zero quasinormal mode method with hopes that it may be useful for more general spacetimes This particular case does give a few caveats which may be of importance for more general applicability e The boundary condition of normalizability on the zero modes A 2 may be delicate In the AdS2 case the zero modes appear for A lt 0 which means the modes satisfying sinh n are not normalizable in the usual sense e A Euclidean spacetime may be Wick rotatable into Lorentzian signature in more than one way as we explicitly explored for AdS2 in section 4 and discuss for general d in the appendix To capture
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27. ion is global not local the curvature expansion of the heat kernel will not change In particular this means that the large A behavior of Z A should be temperature independent we again expect 2 13 which contains terms of the form logy Z A p in 4 6 however does not have any logarithmic terms Pol A cannot provide these terms because it must be a polynomial a We can resolve this apparent conflict by interpreting log Z A r as only giving the temperature dependent behavior of the partition function Since the temperature depen dence here is a global identification 7 will not show up in the local curvature expansion of the heat kernel If we wish to compute the full partition function on thermal AdS2 and verify that Pol A is in fact a polynomial then we need to include both contributions 4 2 Partition function on thermal AdS via heat kernel on quotient of hyper bolic space From 2 using the heat kernel on a quotient of Hz constructed from the method of images the 1 loop partition function for a scalar field of mass m on thermal AdS is given by log Z A r a oe 1 e er 4 7 which agrees with eq 4 6 without Pol A Therefore we see in this concrete example that the normal mode method as used in 1 provides only the temperature dependent piece log Z A r To obtain the full log Z A via a quasinormal or zero mode method including the local behavior we need to include the temperature inde
28. lobal Lorentzian AdS As such we will use the term zero mode method when working in Euclidean signature but quasinormal mode method when working in Lorentzian signature In fact 1 does provide a formula for thermal AdS using a normal mode method referencing the global normal modes For odd dimensional AdS spacetimes these modes are sufficient and as shown for AdS in 1 they correctly reproduce the partition function in 21 However as we will show in this paper on even dimensional AdS spacetimes the normal modes are insufficient Although the normal modes capture all temperature dependent information which is often the behavior of primary interest if one wants to obtain the temperature independent piece the normal modes are insufficient Since the temperature Although we will primarily discuss the case of AdS n in this note this method actually applies to the case of de Sitter space as well as was discussed in the original paper of 1 It would be interesting to understand the relations between the interpretation of de Sitter quasinormal modes as highest weight representations 13 14 and their roles in this 1 loop partition function method This method has been generalized to fields of arbitrary spin on AdS in 18 20 3This assumption is reasonable we will apply boundary conditions in terms of A and there is no reason to expect a branch cut or essential singularity in Z A independent piece contains
29. logarithmic behavior its presence is needed to show Pol A is truly polynomial To properly compute this piece it is necessary to include a set of modes whose boundary falloff is set by A However these modes are nonetheless not normalizable because they are located in the complex A plane on the negative real axis In the case of AdS we will see that they are still interpretable as a set of quasinormal modes In section 2 we apply the zero mode method to compute the scalar one loop determi nant on Euclidean AdS2 or H In the process we review the details of the method We then compare our results to previous results finding agreement In section 3 we explore the physical and mathematical meaning of the novel zero modes we have found with the hope of better understanding when such modes might be present in other spacetimes In section 4 we study the thermal AdS as a quotient space of H We show our results from section 2 can be combined with those of 1 to match the thermal partition function as computed via the methods of 2 including the temperature independent piece In sec tion 5 we extend our analysis to general even dimensional AdS spaces again matching with previous results In the conclusion we discuss application of this method to odd dimensional AdS spaces and also highlight important caveats to consider when applying the zero quasinormal mode method The appendix contains further calculational details 2 Applying
30. nction in the complex A plane directly in the Lorentzian spacetime As shown in 1 the zero mode method Wick rotates to a quasinormal mode method because the zero modes themselves Wick rotate into quasinormal modes We choose coordinates on Euclidean space so 0 0 27 is the Euclidean time circle and p is a radial coordinate such that the time circle shrinks smoothly at the origin p 0 The zero mode boundary conditions become pi pre asp 0 din p as p gt o 3 1 where n is a nonnegative integer Under Wick rotation 6 gt 27T 7t the condition at infinity does not change Additionally replacing n w 27T the condition at p 0 becomes p fg pet exp Fanat p log exp W ii m x 108p iz Ft 3 2 so e 82 a2 where in the last line we set wn iz We can now recognize z as the anti quasinormal mode frequencies since these modes are purely out ingoing here For a black hole space time these are true anti quasinormal modes as the origin of Euclidean space Wick rotates to the horizon of the corresponding black hole spacetime Whenever the complex confor mal dimension is tuned to A A so that one of the anti quasinormal mode frequencies satisfies 2 Jr 3 3 for integer n then the Euclidean section will support a zero mode at A and hence Z A will have a pole at Ay Specifying to the case of AdS2 the coordinates p sinh n and 0 in 2 1 are in fact
31. on of regularity in the interior of Hq 1 this is how we chose to consider the solution A 3 to the second order differential equation in A 2 In order to satisfy the 7 co boundary condition in A 2 we must choose A such that the second line in A 6 vanishes This occurs whenever one of the T functions in the denominator has a pole that is whenever l A Ze lt g Since l is already restricted to be a nonnegative integer as it labels the total angular momentum of the spherical harmonic Y m in A 3 this forces A to be a negative integer For d 1 even A d 2 is not an integer and our analysis is complete there is a zero mode for A Z lt o with degeneracy equal to the sphere degeneracy for a mode on oo with total angular momentum A Specifying to the case d 1 2 we find ba e isinh n F A U 1 A 1 sinh 2 A 7 has only sinhy and no sinhn 4 behavior at 1 oo whenever A A for A l Z and A l lt 0 Since here we are allowing Z to account for both exp 2zl0 and exp zl0 we can rewrite this requirement as Acla eZ We A 8 The case of odd dimensions is more complicated d is an even integer and the analysis using A 6 is insufficient because the formula 15 3 7 in 26 is invalid when A d 2 Z In OY this case the top line of A 6 differs from the bottom by an integer power of the argument of F sinh 7 so we cannot truly s
32. or the motion group of the space As such Gopakumar et al developed a method for computing the heat kernel from group theory characters 11 which they extended to AdS4a for d gt 2 in 2 This powerful method works well for spacetimes where we understand the group theory structure In two papers 1 12 Denef Hartnoll and Sachdev developed a method not directly based on the heat kernel approach In this method developed from an argument in 15 also see 16 17 the partition function Z7 A of a massive scalar of mass m is calculated via a complex analysis trick Rather than specifying a physical mass m corresponding to a particular conformal dimensional A and then calculating the partition function at that mass this method studies Z7 A as a function in the complex A plane If we assume Z7 A is a meromorphic function in complex A plane then to find the function completely we need only the location and multiplicities of its poles and zeros as well as one entire function which we term exp Pol A where Pol A is some polynomial of A In the case of Z A this extra function is completely constrained by the behavior of the function at large mass that is we can use the heat kernel curvature expansion to completely determine Z A As shown in 1 for scalars Z A has only poles these poles are due to zero modes in Euclidean space which Wick rotate to quasinormal modes in Lorentzian signature or normal modes in g
33. pendent piece This term is d eok lat log Z2 A i H3 Z t a Volreg H2 k Vol H2 Z vo 1 log LArg A A 1 log LARG 4 2 1 A 2A 1 0 ay 4 8 So the full 1 loop partition function on thermal AdS is log ZAA X log 1 ear Vol H gt Z Volreg Ha 1 log LArg ACA 1 log LArG 42 1 A 2A 1 0 a 4 9 This expression now has the expected logarithmic behavior at large A as seen in the heat kernel curvature expansion and as expected from the anomaly in even dimensions Note that the new term in the second line of 4 9 is temperature independent except for the volume contribution thus it did not show up in the explicitly temperature dependent contribution calculated in 2 nor in the equivalent calculation done for general AdS in 1 5 Extension of story to all even dimensions We now extend the procedures of section 2 to all even dimensional AdS gi for d 1 2n e As shown in appendix A there are modes regular at the origin with sinh n behavior v a bs 0 5 1 at 7 oo that solve These modes are given by A 4 under the restriction D 2S mg s m3 gt mil Apl m E Z 2 2 This is the same pattern of quantum numbers for spherical harmonics on S Thus the degeneracy for the AdSq labeled by the integer k Ay is the same as for a spherical harmonic in St with total
34. re we have replaced Pol A with Pol u as a polynomial in A is also a polynomial in pu 2 3 Finding Pol A via local curvature invariants Now we still must compute Pol A which we do by studying the behavior of Z7 A at large A via a heat kernel curvature expansion Specifically we set Pol A by requiring at kod 1 log Z A const Paf gir O m 2 11 where a are the kernel curvature coefficients in our case for the operator V When there is no boundary contribution the curvature invariants ag 0 for odd k while for even k ay f d x ga x can be expressed following 1 as 1 12V R 5R 2R R 2Rvpo R 2 12 ag z 1 ag ixz as 360 6 where we have included a4 for use in future sections Setting 2 11 equal to the large u expansion 2 10 we solve for Pol A which must be a polynomial If it is not possible to set these two expressions equal by fixing Pol A to be a polynomial than either some zero modes have not been included in the sum or the meromorphicity of Z A is suspect On the other hand if we can equate these expressions with a polynomial Pol A then that provides a nontrivial check on our calculation More complicated spacetimes might require use of a generalized zeta function such as the multiple Barnes zeta function but we will find the Hurwitz function sufficient here Now we expand log Z A at large mass using the heat kernel curvature expansion in terms of
35. sentations are equivalent to the discrete series representations on SO 2 1 For physical A that is A real and nonnegative we would not encounter these representations at least not while considering scalars fields with spin can have these eigen modes even for physical A as in 10 In our method where we have continued A into the complex plane we must con sider these zero modes although they are nonphysical Even though they are not square integrable they do obey our boundary condition sinh gt n at 7 oo and more impor tantly they do not have any of the behavior sinh n Additionally they do correspond to matrix elements of the discrete series unitary representations on SO 2 1 In even dimensional hyperbolic spaces such as Hz SO 2 1 SO 2 these discrete representations continue to be present so we should not be surprised by their appearance at generic A C For further details on special functions on homogenous spaces as representations of associated Lie groups see 27 especially 12 of 9 4 2 and 4 of 7 4 4 2 4 The thermal partition function on AdS We now consider the thermal partition function for Euclidean AdS at temperature T where the Euclidean spacetime is constructed from global coordinates ds cosha drz do 4 1 by identifying the Euclidean time coordinate Tg Tg 1 T Some intuition for the relationship between these coordinates and the Euclidean disc
36. tion also sets zo iv in the embedding coordinates For d gt 1 there is a bispherical set of coordinates on St which are useful for Wick rotation u Lcoshn B 4 i xo Lsinh sin j sin o x Lsinh7 sin 1 cos o x2 L sinh n cos j cos 02 x3 Lsinhn cos sin 62 cos 63 d 2 Lq 1 Lsinhncos TI sin 3 cos 64 1 i 2 d 1 tq Lsinh n cos 4 Ll sin 0 B 5 i 2 The metric for these coordinates is de wi D dn sinh nde sinh 7 sin 1 dds sinh 7 cos 1 dO o B 6 From these coordinates we can again Wick rotate now setting o it and defining ras cos 1 2 d 2 ds _ L dn L sinh 7 en 1 r g dt r2gdQ2_3 B 7 dS which is just a foliation of AdSgi with the d dimensional static de Sitter space It is very curious that the modes in A 4 Wick rotate to quasinormal modes of this foliation of AdSq 1 It would be interesting to explore further the interpretation of these quasinormal modes In fact for an even dimensional AdS space we can pair the embedding coordinates xo through x2 _ It is possible to pick coordinates so that we can Wick rotate easily by the angle between each of the n pairs of coordinates In fact written in these bi spherical coordinates we explicitly have written a full commuting set of generators for AdS n For odd AdS n 1 such a pairing isn t possible To get a maximal set of commuting gen era
37. tion normalizable modes The agreement is a consistency check that our boundary condition produces the same result It would also be interesting to understand why these two prescriptions are equivalent One might recognize these expressions as similar to that of the computations of dS in 1 There are two important differences first due to the AdS Dirichlet boundary condition we are only summing over the A boundary condition here and not 1 A Second the relationship between A and the mass for AdS differs by a sign from that for de Sitter where we have used zeta function regularization as in 1 Here s x is the Hurwitz zeta function defined by analytic continuation of C s x X g a k and s x O s x whose large A behavior is given in 1 which is taken from appendix A of 22 as 1 lt 2 kta l DEH A 2 9 0 T s Akt s 1 where B s are the Bernoulli numbers Since we will find Pol A by expanding the expression 2 8 at large A and comparing to Z A as found from the heat kernel curvature expansion we now present the expansion of the zeta function expression In order to compare more easily with the heat kernel expansion we also expand in terms of u A 1 2 as in 2 3 Expanding eq 2 8 at large A in terms of u using 2 9 gives 1 log Z A Pol A A J 4 A 5 log A DR F e POA 12042 36048 Pol u TA i logu a OG 2 10 Z 2 12 940 whe
38. to Euclidean AdSgij or Hap In 10 the eigenstates for Hg which satisfy d2 Og Q Z d 2 22 are written as d d d 1 D i sinh n F t 9 T l t 9 I l l a sinh 5 Vie 2 23 where F a b c z is the hypergeometric function the Yp are the orthonormal spherical harmonics on S satisfying wW UL d 1 ia 2 24 and m is a d 1 dimensional vector of integers distinguishing among distinct harmonics with the same eigenvalue For real A these are the only eigenfunctions which are delta function normalizable and thus they can be used to compute the heat kernel via a sum over eigenstates However for real these eigenfunctions do not obey the Dirichlet boundary condition at the boundary Ag41 and thus here they are not candidates for the zero modes used in the zero mode method of section 2 1 Since A is continuous we must extend the sum over states in 2 20 to an integral this is done via the spectral function f A The coincident heat kernel then becomes K a 2 t J GO ox z 3 m 2 25 where ji A is given by 1 d 2 2 For odd d 1 AA to aT h d 2 2 F TERS E N 7 2 2 orevend 1 f A aa say X tan TA x IE j 2 26 2dr eT SS j 1 2 When d 1 2 the product should be omitted In order to compare with our previous results we specialize again to the case d 1 2 and use expression 2 21 to find log Z We obtain log Z A 2 27 p
39. tors we must include one noncompact generator such as Tg below It is this difference between even and odd AdS that causes even AdS to have discrete series representations while odd AdS does not For the global coordinates in any dimension we choose u Lcosho cosh Tg B 8 zo L cosh o sinh Tg x Lsinho cos 0 z L sinh sin 6 cos b d 2 4 4 Lsinho Jli sin 6 cos g 1 i 1 d 1 Lg Lamho Ll sin 6 i l In this case we Wick rotate by taking Tg it which takes x9 iv as before This rotation produces the usual global coordinates on Lorentzian AdSgi for general d We can see that is a noncompact generator so this Wick rotation is fundamentally different from those considered above Thermal AdS usually refers to global AdS with time identification that is setting TE Tg 1 T for some temperature T In Lorentzian signature this can more properly be thought of as taking 7 to the covering space that is allowing oo lt T lt oo and then identifying up to 1 T Open Access This article is distributed under the terms of the Creative Commons Attribution License CC BY 4 0 which permits any use distribution and reproduction in any medium provided the original author s and source are credited References 1 F Denef S A Hartnoll and S Sachdev Black hole determinants and quasinormal modes Class Quant Grav 27 2010 125001 arXiv 0908 2657 INSPIRE 2
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