JHEP03(2014)043 N Z Springer March 7, 2014 : coefficients for February 7, 2014 : December 24, 2013 : orbifold of the near- N Published 10.1007/JHEP03(2014)043 Z Accepted Received doi: icrostate counting. We also c erboloids with conical singu- sity, ute the logarithmic contribution s to higher-spin fields over . , ICTP), , = 4 supergravity. We find that this Published for SISSA by N cone and logarithmic 2 and Somyadip Thakur b = 4 vector multiplet about a N [email protected] , Shailesh Lal . a 3 1311.6286 The Authors. c Black Holes in String Theory, AdS-CFT Correspondence

We develop new techniques to efficiently evaluate heat kernel , [email protected] [email protected] E-mail: Abdus Salam International Centre forStrada Costiera Theoretical Physics 11, ( 34151, Trieste,Department Italy of Physics andSeoul Astronomy, 151-747, Seoul Korea National Univer Centre for High EnergyC.V. Physics, Raman Indian Avenue, Institute Bangalore of 560012, Science India High Energy, Cosmology and Astroparticle Physics, b c a Heat kernels on the AdS Open Access Article funded by SCOAP Abstract: Rajesh Kumar Gupta, corrections to extremal black hole entropy larities. We then apply theseto techniques to black explicitly hole comp entropy from an the Laplacian in the short-time expansion on spheres and hyp horizon geometry of quarter-BPSvanishes, black matching holes perfectly in withdiscuss the possible prediction generalisations from of the m our heat kernel result orbifolds of higher-dimensional spheres and hyperboloids Keywords: ArXiv ePrint: JHEP03(2014)043 3 5 2 3 7 9 17 20 22 23 25 26 29 29 30 31 10 11 12 13 14 16 19 28 33 33 35 36 tion 2 N Z / 2 in ropy N N Z Z N 2 N / / Z 2 Z
/ / 2

S 2 2 S S N ⊗ N ⊗ Z 2 ⊗ Z / 2 2 / 2 S – 1 – 2 and AdS 2 AdS N = 4 string theory and their entropy AdS N Z Z / N / 2 and the analytic continuation to AdS 2 2 and AdS 2 by zeta function regularization 2 orbifold of S N Z 6.1.1 The6.1.2 bosonic determinants The fermionic determinants 5.2.1 Zero modes of the vector field in AdS 4.3.1 A check of the analytic continuation 4.2.1 The4.2.2 large radius approximation The Euler-Maclaurin formula and the heat kernel on S 6.1 The graviphoton background on 6.2 The analytic continuation to 5.1 The scalar5.2 field The vector field 5.3 The Dirac5.4 spinor A group-theoretic interpretation of these results 4.3 The analytic continuation to AdS 4.4 The Hodge Laplacian for vector fields 4.1 The 4.2 The scalar Laplacian on S 2.1 Exponentially suppressed2.2 corrections to black The hole quantum ent entropy function and their macroscopic orig function C Zero modes on AdS 7 Conclusions A The scalar Laplacian on S B Anti-periodic fermions on S 6 The one-loop determinants in the graviphoton background 5 The one-loop determinants on 4 The heat kernel Laplacian on AdS 3 The heat kernel method and logarithmic corrections to the parti Contents 1 Introduction 2 Quarter-BPS black holes in JHEP03(2014)043 i θ ]. A 25 nted (1.1) , 24 ] that the 16 ] of the black entropy of the 2 . In this picture 2 e horizon, and oposal is that the sal already passes two of posing the black hole um gravity has been to ], ] and more generally the ssed contributions to the = 8 string theory [ ometry. The finite part of 16 tring path integral over all 22 [ t of as saddle-points of the n extremal blackhole is the , nt of view of the quantum N – bout the saddle-point defined i cific context of exponentially 2 re the string answer explicitly f such black holes is always of q te coordinate transformations e Wald entropy [ ld formula and it is interesting h scale as log(charges), to the lack hole at the quantum level. 17 ted with the black hole horizon ]. This program has been very = 4 string theories as well, provided oles and a detailed matching of 1 ons is possible, at least in some , finite AdS ≡ [ ]. ). This proposal has been tested N rom the string computation can be ~q 15  15 i θ 1.1 = 4 and A ]. N ] for a review and a more exhaustive set 14 dθ i – q 11 12 I i  – 2 – exp  ≡ ) ~q ( is a compact manifold. Then, it was proposed in [ gauge field along the boundary of the AdS hor d K ) where the near-horizon geometry of the black hole is quotie th i 1.1 ]. We refer the reader to [ where 10 – 3 K However, the string answer is the full quantum answer for the ⊗ 2 1 ] for an overview. A particularly non-trivial test of this pr is the full quantum degeneracy associated with the black hol 23 orbifold which we shall shortly review and define. This propo hor d N Z At this point it is natural to ask if the exponentially suppre In the context of extremal black holes there is a parallel way This matching extends to the negative discriminant states in 1 string path integral of ( microscopic degeneracy haveentropy an function. interpretation from It is the proposed poi that these should be though where the macroscopic configurations are carefully identified [ by a This is obtained by expanding theby quantum entropy the function near- a horizon geometry of the black hole. in a variety oflectures ways, [ for which weleading refer quantum the corrections reader in tosemi-classical the Bekenstein-Hawking [ large formula as charge predicted limit, f reproduced whic from the quantum entropy function for is contained in thespacetimes quantum which entropy function, asymptote defined tothis as the path the integral black s is hole expected nearIn to contain particular, horizon the for ge entropy extremal of black the b holes carrying charges suppressed corrections to the microscopic degeneracy [ entropy problem. In particular, thethe form near AdS horizon geometry o full quantum answer for the microscopic degeneracy associa hole in question [ of references. black hole and therefore alsoto contains ask corrections to if the acases. Wa direct In physical this interpretation paper, of we these will correcti explore this question in the spe explicitly carried out for athe string class answer of has supersymmetric been black donematches in h with the the large Bekenstein-Hawking, charge or limit, more whe generally, th non-trivial tests. Firstly, it may be shown using appropria One of the mainprovide a successes statistical of interpretation string of theory black as hole entropy a theory of quant 1 Introduction is the component of the free energy corresponding to the partition function ( the entropy associated to the horizon degrees of freedom of a JHEP03(2014)043 ) 2 5 we ,S 6 2 -BPS Q, P and 1 4 ( 4 d ]. he sphere, = 4 string 25 N ero modes are , orbifold of the orbifolds of the 24 N N Z Z -BPS black holes in 1 4 orbifolds of AdS a quantum test of this = 4 vector multiplet to is a brief review of the e the heat kernel about N . Finally in section Z e. This paper is devoted n the N 3 hall review soon, the log zon. We then propose a 4 scalars and fermions have e = 4 vector multiplet. We we briefly review the string ty calculation is technically n ] for an indepth account and wer available from the string ropy of rom the partition function of ssentially mit us to solve this problem the quantum entropy function 2 udy the degeneracy of hole entropy, sections ectations from the microscopic stency check of this conjecture viour for it to be considered an N 11 ually vanishes in ingle opy obtained from string theory, = 4 string theories. In particular, N ] which is a prescription for computing 16 = 4 string theory and their entropy – 3 – N ] analogous to the tests performed in [ 16 ]. We refer the reader to [ . As a consistency check against this microscopic result, 10 N ), and secondly, it gives the correct leading term for , . We then conclude. 6 ]. 1.1 2 , 4 17 , 3 dependence, the final log contribution still vanishes when z N . The AdS results are reached by analytic continuation from t 2 S ⊗ 2 In this paper, we shall initiate a program aimed at providing A brief overview of this paper is as follows. In section = 4 string theories [ for the validity of which some evidence is provided in sectio and AdS near horizon geometry of the black hole in question. Section We begin with a brief review of black hole entropy for properly accounted for. This matchesresults perfectly reviewed with in our section exp 2.1 Exponentially suppressedIn corrections this to section black we hole briefly entropy review the stringya origin more of extensive the set ent of references. In particular, we will st we shall discuss the microscopic formulaand for the black hole Quantum entr Entropy Functionthe formalism full [ quantumquantum entropy test associated of the with proposal the of [ black hole hori very nontrivial 2 Quarter-BPS black holes in compute the contributionfind to that the though log individual term contributions from from a gauge single fields, proposal for their macroscopic dual. These geometriesheat are kernel e method as applieda to quantum extract theory. logarithmic As termsthese the f orbifolds overall and goal extract of out the log corrections paper to is the to black comput theory answer for the entropy of quarter-BPS black holes and N develop new techniques for computing the heat kernel in thes admissible saddle-point to ( that this configuration has the appropriate asymptotic beha theory for arbitrary values of we shall explicitly demonstrate that the contribution of a s the log term vanishesmore about involved, this and is saddle in point. progress. The full gravi in the large-charge limit [ proposal. The overall goal is to compute the log correction i computation. We doto not developing solve firstly this the problemefficiently, and necessary in secondly its techniques to carrying entirety thatas out her will a an natural per application firstcorrection consi of about these these techniques. exponentially suppressed terms In act particular, as we s attractor geometry and match it against the microscopic ans JHEP03(2014)043 . ) ≡ v of 10 or →  ) ) 2.1 2 ) P (2.3) (2.1) (2.6) (2.4) (2.5) (2.2) T P · Q, P × ( 3 m n j ~n, ~m, jQ  K stat + j 2 S , 2 ) 1 ˆ v . Q 1 m ] 4 1 . in degeneracy as one ˆ σ, − m 1 11 , ategy one follows in 1 = ρ, 1 , − (ˆ − 2 heory on 3 n 4 2 2 j , which is a three real- , 10  n C ] that the dominant con- P Φ 2 , 2 + 1 pared to the gravity side. ) 1 11 n which are located at n , and magnetic charge 2 given in terms of the Igusa P · ≤ ( saddle n Q = 0 2 due theorem one first reduces 2 − i n [ j 10 vQ atistical entropy iπ n 2 he poles of the integrand which
)) m , we can use symmetries of Φ cy at the saddle point is given by ≤  2 +2ˆ 2 m P 2 ]. The procedure is to deform the − + 0 n · q + 1 11 ( σQ exp 1 n , +ˆ ,Q O 1  2 1 2 0. In the remainder of this paper, we 2 ). For more details, we refer the reader ˆ ρm ˆ ρP v − ) of the integrand, while the contribution (ˆ ≥ ,P /n ]. We shall concentrate especially on the − 2 2 2 (1 + ˆ πi σ, 2 , m ) n 1 2.2 ˆ σ Q − 15 n − P 1 ρ, ) ≤ · n ˆ ve + 1 2 σ 1 ) – 4 – , the saddle point lies at [ ( Q Z + ˆ σd g ( P m 2 1 ˆ v · ˆ j ρd − − ∈ ≤ ) d Q 2 to ]. ) are symmetric under the transformation ( ρ ( 0 m n j ~n, ~m, C ( Further, for a given P ) + j Z 26 g 2 − 2 2.2 2 , j , ˆ i v 2 Q spanned by (ˆ +2 +1 1 ) from three to two by performing the integration over ˆ Z 1. ) of the order of 1/charge. In that case, the integral in ( κ P P − and 2 are performed using the method of steepest descent. It may 3 p · ) 2 − ∈ ) has been explicated in [ ˆ v ≥ ˆ 2.1 ρ 1 π . In this case the electric charge Q C , σ D 2 Q 2 6 σ  2 2 n 1) (ˆ 2.1 , n T ˆ n σ p 2 , n 1 , − 2 2 ≤ n ˆ 2 and ˆ Ω + exp m n ρ 1 2 ρ C , m Q n · 1 ) = ( P ) arises from the the poles ( 2 ≤ , n are complex variables, and the integral is over ) = det( n 1) 1 0 = 0 is also physically relevant. These poles capture the jumps ˆ v h v − Q, P 2 m − by ( Q, P × ( n ( d ]. The poles of the integrand are the zeros of Φ ˆ ). We use this symmetry to set σ, 10 d = j and ˆ 15 ρ, 2 to values of (ˆ (ˆ , σ ) and extract its behaviour in the large charge limit. The str − n ) C , ˆ ~n 11 ρ , − Q, P ( Q, P The case of 10 = 4 string theory obtained by compactifying Type IIB string t ~m, ( 2 d d − to [ ln dimensional subspace of the evaluating the integral in ( a dyon are 28-dimensional vectorscusp and form the Φ dyon degeneracy is heterotic string theory on We note here that the equations ( ( where ˆ shall focus on the case of were crossed during this deformation.tribution to It can further befrom show the deformed contour isthe subleading. integration Then variables using in the resi ( contour receives contributions from the deformed contour and from t N Using the above results, it is finally found that the degenera We shall focus on how these results are used to evaluate the st to restrict the range of moves across walls of marginal stability [ The integrations over ˆ This is essentially a review of the results of [ dyons in the large charge limit in which the results can be com where be shown that for a given choice of JHEP03(2014)043 in = 1. ∆ (2.8) (2.9) (2.7) (2.10) (2.11) 2 √ π we will n e 6 = 4 vector   ). Thus the 2 2 N . Thus we see 1 1 Λ Λ ( 2 BH  n O S O the contribution of ) grows as ! . In section ˆ ˆ v σ 2 pare with macroscopic 2 (1) + ) ˆ ˆ Q, P ρ v n (1) + = 4 string theory at any O ( ˜ P 2

O d · dle point for which , N  m are independent of Λ. Thus ) on for a single ˜ e of ea scales as harge vector to be very large. Q y where Λ is very large number. P ˆ ( Ω at the saddle point do not + ˆ ich goes like Ω = d contributions which grow as · ˆ ˜ ρ − + ln P 1 goes as are variables defined in terms of 2 saddle , Λ  ,Q | 2 m 1 ˜ 2 ) or 2 P ith the above result, and secondly it n v ˆ − v 2 ) ... → − ) ) /n ˜ − ,P Q = 0. The exponential term, apart from 2 ˆ σ + D 2 D ) 1 v ˆ σ, q and Q P Q, P n 2 π ( · ρ, (1) 2 σ ˆ ˆ d Λ Ω + Ω + + ˜ = ( Q, P Λ O Q , ˆ C v C ( 2 Λ ρ ) is at ( 1. In the context of extremal black holes there j )( + → – 5 – − et > → B  2 2.2 2 d ) 2 ) + , q 2 ) which are proportional to Λ will give logarithm 2 P Q P n 24 2 ˆ v /n · ) ˆ Ω + 2.6 2 Q ρ ) − ( A Q are chosen such that ˆ ( P p ρ η · π σ D − (ˆ  2 Q 2 ) = ( that while the statistical entropy n = 4 string theory there is no log corrections to the leading = ρ P ( and 2 − 2 = Ω = ( g = n N 2 Q 2.1 ) C v ! P , p 2 π B Q ρ v v σ Q, P , (

p A d π  = ln ∼ 2 ) is Dedekind eta function and . To prove this statement explicitly, we will need to include 2 n ρ ( n S η ) as in the same scaling limit, where ˆ v ∆ ˆ 2 σ, √ n π ρ, Here the matrices Thus in the un-hatted variables the pole ( phase factor, in the degeneracy gives the leading entropy wh Now in order toentropy from see gravity the side, we log takeWe correction all then the in uniformly components scale the of all the entropy charges c and as to com (ˆ In this case the leading contribution to entropy and hence ar Where entropy from the saddle point is given by we see that the term in the square bracket of the large charge limit,e it also has exponentially suppresse show from the quantum entropymultiplet function vanishes. that Firstly, the this log isgives correcti entirely evidence consistent w that there should be no log corrections for an correction to entropy. It is easy to see that (ˆ depend on Λ and hence the matrices A, B, C, D and Ω value of Thus the terms in the degeneracy ( where that dominant contribution to the entropy comes from the sad Thus in this example of exponential term. This statement is independent of the valu 2.2 The quantumWe entropy have function seen and in their section macroscopic origin the full gravity multiplet, which is work in progress. JHEP03(2014)043 ] 3 n 17 3. K ]. At (2.12) (2.13) 25 , . Due to the 24 [ is the radius K xponentially . a 5 , 2 | , 5  ) = 1 term extracted from vanishes even about τdx panding in quadratic , where a a a well-defined prescription a + κ, N llowing orbifolds of the + dual 4 ntropy from these saddle do not scale with the AdS e black hole, known as the dx dφ py function formalism, and ] for more details regarding | gcd ( ured by the string partition τ to log g saddle-point of the string 2 ∧ 2 f this proposal. In particular, 27 τ ]. By now, there is very non-trivial n geometry of the black hole. R ds to the Bekenstein-Hawking , , that the contribution of a single nal to log dψ 16 and + ifying Type IIB Supergravity on ons for them to be included in the sed corrections to the microscopic 17 πκ ∧ times a compact manifold N
, R 2 ed [ 2 4 2 . 15 ]. + v = 4 string theories with an arbitrary 25 5 ψdφ ψdx he degeneracy in the horizon degrees of freedom = . Additionally, it can also be shown [ x 2 N = 8 string theory yields a non-vanishing quantity u N sin 7→ N I 5 , in that both of them are zero ↔ P + sin 2 – 6 – 2 2.1 + n . , x dψ dφ π N 2 u ∧ are determined completely in terms of the electric and − + V dψ φ
2 ∧ ]. The proposal is that the degeneracy associated with the 7→ 5 and = constant ηdθ 16 r I 2 ). We refer the reader to [ ] for more details about this solution. Here we merely mentio ψdx , φ 27 1.1 ) of the black hole, and ] π sin ,V N 2 I 17 ] for a review. + sinh u, v, R, τ , Q + ], which precisely matches the leading contribution of the e 2 submanifolds. This matches precisely with the log 23 Q, P  factor, the string path integral diverges. There is however θ 2 15 17 2 2 dη , 1 π )[ 7→ 8 v 15 θ 4 [ and S = constant = = 2.12 2 i N 2 I A I 4 V . A particularly non-trivial test of this proposal is that ex G ds u The corresponding computation for black holes in We write the solution in the supergravity obtained by compact The near-horizon geometry of an extremal black hole is AdS 5 4 3 by which a finite part of this partition function may be extract presence of the AdS radius We refer the reader to [ magnetic charges ( these exponentially suppressed terms. It must therefore be evidence that this finite partof does the indeed correctly black capture hole, t see [ this orbifold. In thisas paper, we we have will seen performnumber above, a of further vector there test multiplets exist o for a which family the term of proportional It may be seen,points is for one, thatsuppressed the terms leading with contribution thethat identification to these geometries the obey e the appropriatestring fall-off path conditi integral in ( it turns out thatgeometry ( the answer is yes. These correspond to the fo this point, it is natural todegeneracy ask have if a the proposed exponentially suppres counterpart in the quantum entro that the parameters horizon degrees of freedomfunction of over the spacetimes extremal that black asymptote hole to is the near-horizo capt path integral quantum entropy function [ In particular, the dominant(more contribution, generally, which Wald) correspon entropy corresponds to the followin is a complementary formula for the full quantum entropy of th fluctuations about this saddle point yields a term proportio which is also found to match with the microscopic formula [ of the AdS the microscopic degeneracy as in section JHEP03(2014)043 . 2 a it is from (3.4) (3.5) (3.1) (3.2) (3.3) ] only a and S 28 , 2 at large artition 24 a correspond φ cl = 4 vector multiplet angles of AdS , which solve is sensitive only to the N φ cl e will set up heat kernel , guration Φ a ) x and ne-loop partition function is ( eld theory. In particular, we lied to extract the logarithmic θ carried out in the near horizon ves contributions from massless Dφ ce with the expectation from the on function may be extracted for large um of massless fields in four dimensions ) . quotient. This is true. However, we are is conjecture. In particular, we will x a ( 5 [Φ] . x S . Higher-order terms in ) . 1 ~ gφ φ n which scales non-trivially with ractor geometry and, as argued in [ D e to the log term. − ( √ φ, e ] and carry out the same computation on ] where more details and references may = 0 x 1 2 cl + 24 29 − quotients the Φ] +1 cl d D ]–[ N [ d + 1 dimensions, defined on a background with Φ=Φ | Z – 7 – 24 d Z Z = det S ℓ Φ = Φ Φ δ − ] + δ 1 cl part of the near-horizon geometry. Hence, for this purpose, Z [Φ] = 2 [Φ S Z S ]. We will therefore concentrate on the one-loop partition ⊗ 2 ≃ 28 -BPS black hole in [ [Φ] 4 1 S 6 dependence which would come from the 0, this is dominated by classical configurations Φ N → , and show how the term that scales as log ~ a geometry of a ). We will then demonstrate that the contribution of the is the kinetic operator for the fluctuation 2 S orbifold of the geometry where the D 2.13 ⊗ N 2 function Z The reader might worry that by computing only over the spectr = 4 vector multiplet to the log term vanishes. In this paper, w 6 the heat kernel. We refer the reader to [ This scale appears only in the AdS looking to compute the contribution to the partition functio where to higher-loop corrections andthen have been given by omitted here. The o two-derivative sector of thefields, theory, and and further at only one-loop recei only [ sufficient to reduce onto the four-dimensional part of the att the massless modes in this effective geometry will contribut be found. Firstly, it may be shown that the term scalingfunction as of log the theory. We consider, therefore, the partiti the one is missing the techniques to tackle this problem, andrevisit explicitly verify the th computation ofAdS the one-loop partition function In the limit length scale in this orbifolded background indeed vanishes,microscopic in results. accordan 3 The heat kernel method and logarithmic corrections to the p as in ( N This section is a brief reviewcorrections of to the heat the kernel method partitionshall as function app focus of on a an generic effective quantum field fi theory in We decompose the field Φ into small fluctuations about the confi in which case JHEP03(2014)043 f To is the (3.9) (3.7) (3.6) (3.8) 7 (3.10) (3.12) (3.13) (3.11) δ ]. 31 [ 0). In that are positive , p , 2 1 ) , where ith a residual n − x κ n δd ( e n + dδ )Ψ degenerate spectrum, x , . . . , λ ( 1 belonging to the eigen- ∗ n λ , ( i-definite. In particular, ) , Ψ D s non-trivial zero modes. x n n ( -forms on AdS of zero modes, consider the cept for the constant scalar tκ tκ p n diag − − D , e e ns for the path integral as well. j , s apparent. The final expression = )Ψ ) x t ) dt , given by ∆ = y x ij t , , ( ( ( M n M ∞ ∗ n i 0 ∗ n,b x zero K Ψ dx , Z n t n =1 dt e background metric by the Hodge star. ·Z x )Ψ Z κ tκ n i,j x 1 2 1 2 − ∞ ( +1 n ǫ P − d − Y Z
x e − D n,a gd ′ e = = i +1 M Ψ – 8 – ′ √ d D D dx n gd Z det X = det n ], and more generally . In that case, it is easy to see that =1 √ det i Y n ℓ D X = − 30 ) = Z [ 1 Z t log det = ; Z Z 2 n = n − X κ x, y Z is given by ( ) = log ab t Z . Hence, the path integral decomposes into the determinant o ( i K n κ X K 1 − − =1 n i = Q evaluated over the subspace of its non-zero eigenvectors, w D are the normalised eigenfunctions of the operator ) = are the eigenvalues of M n n M ( κ ′ has already been diagonalised. In particular, log det .Though this expression has been written for a discrete non- n − M ) is just the trace of the heat kernel for the operator is a UV cutoff. In the above analysis we have assumed that all th κ ǫ 3.7 For these fields, by ‘Laplacian’, we mean the Hodge Laplacian 7 mode, but thethere Laplacian exist on fields hyperboloidsThis on is includes hyperboloids only one-forms for positive on which sem AdS the Laplacian ha definite. This is indeed true for the Laplacian on spheres, ex case, it is easy to see that and zero-mode integral remaining. This isWe find precisely what that happe where det the operator illustrate how the abovefinite-dimensional analysis Gaussian is integral modified in the presence where the Ψ adjoint of the exterior derivative, defined with respect to th the generalisation to the continuousin and degenerate ( cases i where To define the determinant, we use the identity value where where the i.e. JHEP03(2014)043 ic 10 . 9 l of 2 and is the (3.17) (3.15) (3.16) (3.14) 8 zero Z ] on AdS 34 – = 4 vector mul- 32 -BPS black hole in N 1 4 he log term. For this, we mula [ and their product spaces. ions from the microscopic ..., 2 ns to the partition function , it is sufficient to determine + determinant of the Laplacian dle-points from the Quantum rnel about the quotient spaces a e the zero mode integral. . . , AdS ) . We then define a new variable is the order-1 term in the series log 2 s 2 1 (¯ 1 (1) a 1 ods are of course equivalent. a. ] for related work on spherical geometries. K K K O 38 n the zero mode contribution was subtracted ¯ zero modes and integration over a zero mode s ¯ s = N , + d 0 ] and we do not repeat it here. The final ) log Z ]. n 37 a 0 D and 2 ∞ ′ / ǫ 24 n which we shortly specify. In principle, this a 28 ], where the heat kernel was computed over the full 2 scale as , a Z 2 log has + 29 0 25 orbifolds of S 1 D = D ]–[ n – 9 – K log det N 24 D (1) [ = ′ Z 1 2 zero modes when acting over a vector field, then evaluated over its non-zero modes, and O = ( − 0 + zero n D log a = Z F log det log has log − β orbifold of AdS 0 ) becomes D = 0. We then find that the contribution to the free energy n non-zero N s = 3.9 orbifold of the near horizon geometry of a Z Z is a Laplace-type operator defined on a background with a metr ’ do not scale as log N zero , the eigenvalues of . This has been done in [ log Z Z a a D ... . As a warm-up, we will consider in this section the heat kerne we will see that this contribution from a single is given by 6 a 2.2 ) about ¯ s ] for related work in AdS geometries, [ (¯ then log 36 K is the determinant of , β a 35 D scales with ′ in terms of which ( zero 2 t Z a The more general result is that if an operator See also [ This is in slight contrast to the methods of [ = 4 supergravity, which precisely matches with our expectat 9 8 10 = ¯ Finally, in section In the rest offor this scalar, paper, spin-half we and will vector fields compute on the two contributio which scales as log expansion of gives a factor residual zero-mode integral. Weexplicitly will over therefore non-zero compute modes the only, and separately analys We therefore find that tiplet vanishes in the where the terms in ‘ In order to computeEntropy logarithmic Function, corrections we need about to these evaluatedefined the sad integrated in heat ke section s computations of the entropy of this black hole. 4 The heat kernel Laplacian on AdS answer can also be extracted by employing the Sommerfeld for set of eigenmodes ofout the to fields, give including the zero determinant modes, over and non-zero the modes. The two meth the scalar Laplacian on a In order to extract thehow zero mode contribution toresult the is log term that if the operator N where det Now, we need to examinefirstly how observe the that non-zero if modes contributewith to an t overall scale JHEP03(2014)043 2 ), χ the (4.4) (4.2) (4.3) (4.1) ≃ . The χ ia , that our 7→ orbifold of S a dependence can , nal spheres and and sinh . This has two N 2 ρ iχ Z N m and degeneracies ≃ 7→ ρ n ρ -dependence of the heat = 0 is the fixed point on h the at kernel of the Laplacian N χ heat kernel on these mani- , to AdS swer turns out to have an red for the heat kernel on el for higher-spin fields, in ll see that this strategy of ponding l efficients to arbitrary power cient method for computing on these conical spacetimes. nt one-loop determinants for the Laplacian on this space, -dependent terms in the heat ctrum and degeneracies of the ven by N , ]. Even in those cases the heat , suffers from a volume divergence

2 41 2 N , , Z 2 χdθ / 40 ρdθ 2 2 orbifold, and when taking into account . 2 dθ 2 π r N N 2 Z + + sin + + sinh 2 2 θ 2 and AdS ]. We propose here an alternate method, which dr – 10 – , its 2 dρ 2 2 dχ 7→ 39 = S 2 θ 2 2 a ⊗ a ds = 2 = ) by the analytic continuation 2 2 and AdS ds 4.1 ds 2 are fixed points on the sphere and , π = 0 as the case may be, i.e. in the neighborhood of the fixed point, ρ aχ ]. It is possible, though we do not carry out this program here orbifold of S or 30 Laplacian are very explicitly known on arbitrary-dimensio N s aρ Z with metric = 2 = 4 vector multiplet. Additionally, we note that the spectru r . In the neighborhood of each fixed point (small enough that si 2 N which needs to be regulated carefully to capture the finite kernel expansion. Thishyperboloids is quotiented analogous by to freely the acting problem orbifolds [ encounte folds. The integrated heat kernel on AdS geometry is that of a cone. We have to evaluate the integrated the corresponding geometries become where which can be obtained from ( Under this quotient, hyperboloids [ AdS and AdS of the spin- We will consider quotients of the two-sphere, with metric gi kernel suffers from a volume divergence, but the regulated an analysis may be extendedhigher-dimensional to spacetimes explicitly as well. compute the heat kern 4.1 The only be extracted ininvolved a series computing expansion the [ integrated heat kernel on the corres However, the answer is obtained in an integral form from whic quotient we consider is the effects of the graviphotonthe flux while computing the releva coefficients in the Seeley-de WitThis method expansion can of be the easily heatin employed, kernel the once set heat up, kernel to time. extract In co this manner, we will obtain the ful benefits. Firstly, at a technical level, we obtain a new and effi by explicitly enumerating theand spectrum performing and degeneracies an of analytic continuation, which we motivate on the higher-dimensional spaces S kernel coefficients to thecomputing the required heat accuracy. kernel byquadratic explicitly Secondly, operator enumerating we will the spe be wi of great use when we compute the he JHEP03(2014)043 , 3 ns (4.5) (4.8) (4.9) (4.6) (4.7) , where wer for , ] one may values are n 24 S Np m . The eigen- = =1 Laplacian on N 2 , which house X n . ℓ N m ≡ which we encounter Z tE  / ) − N 2 e N . Z ℓ + / d Z
]; we will carry out the Np 2 ∈ S n where the coefficients 2 1)( 41 X a 12 ⊗ − , aplacian on S t kernel of the scalar Lapla- , m 2 N = ints on S . ] + sult to the hyperboloid by a 40 ) n , ℓ Np AdS + +1) ( en by y smooth, and the heat kernel ℓ [0 t is an integer, the allowed ( onding to the quotient. To obtain Np ℓ − ,..., 2 e t 2 2 1)( a | ∈ a , − − + 1 n m e , | . However, using the methods of [ + q < N ... ≤ Np ( , = 0 t + + 1) − ℓ e imφ +2) ± (2 , ℓ e – 11 – Np ) ℓ 2 , where 0 + 1) a ρ X q N 2 p + 1) +1)( Z + a ) can be related to the integrated heat kernel over ℓ ) follows. / (2 Np ( ) = ( 2 (cos , where the heat kernel is computed over scalar modes t ℓ t Np 4.8 4.8 ; =0 ∞ N m − ℓ p X = e = 0 mode, which is a zero mode. As per the discussion in section = Z P ℓ / ℓ ℓ = x, x + + 1)-fold degenerate, with the corresponding eigenfunctio 2 ( E n ℓ ) = S , with some corrections which can be precisely computed. The +1) 2 gK 2 a Np ρ, φ = (2 ( √ ( ℓ x quotient. Clearly, these are just the modes for which Np 2 d t ℓm d N − Y . The expression ( ]. Intuitively, it should be possible to think of the final ans Z e Z The expression for the integrated heat kernel for the scalar  term extracted remains the same. For fields on 42 = a 11 I + 1) p N, . . . , Np we shall indeed treat the zero mode integral separately. 2 (2 5 in terms of eigenvalues and degeneracies is then given by dependence, we will adopt a strategy very similar to [ ,N, =0 ∞ p X N N It is easy to see that for a given This spectrum also includes the Z = 0 = / | is an integer. 12 11 S 2 m curvature singularities, theasymptotics remaining can space be isare computed completel well-known from [ the Seeley-de Wit expansio where values of the Laplacian are given by we should perform the integralshow over that the the zero log mode separately the unquotiented sphere S S computation on the sphere,prescription and we analytically develop below. continue the re 4.2 The scalar Laplacian on S final result should not be surprising; if we excise the fixed po It turns out that the expression ( in section | this interesting dependence on the chemical potentials corresp We begin by recalling the well-known spectrum of the scalar L The corresponding integrated coincident heat kernel is giv We will now write down acian corresponding over expression the for quotient the space hea S periodic under the being given by the spherical harmonics these eigenvalues are p JHEP03(2014)043 N n, Z roxi- (4.14) (4.11) (4.10) (4.15) (4.13) (4.16) (4.12) . . 2 ) is subleading el over the S , +1) ℓ . ( . For definiteness, + ) ℓ 2 b +1) 4.10 1 2 ℓ t ( a ( . S ℓ f − 2 t 1 2 a ≡ +2) ion ( ]. We do indeed recover − , i m e 43 dℓ ℓe . We will focus on the domi- 1) + 1 +1) +2) ul to consider first the heat mooth part of the manifold,  , while the other terms scale 0 n by ) ally, in the large radius limit, m m 2 +1)(2 − 13 t Z + 1) a a (2 m ties [ b ( h to us. ℓ 2 1 ( m (2 2 1) f (2 +1)(2 2 t − 2 is entirely analogous. In this case, − a t a m k + 0 − ( ∞ − (2 X e N f +1) e 2 t ℓ ... a 2 1 ( ℓ − − 2 t ) e ≃ a + 1) b is very large. + 1) ( − + , which is also useful for fixing the analytic a m e m 1) +1) + 1) + N ℓ (2 − ( (2 +1) Z – 12 – ℓ k a ( ( / 2 m t + 1) a 2 f f dm (2 ℓ dm  − m e ∞ k (2 ! ∞ 2 0 ) + 0 k 2 t B Z a a Z dℓ ( − + 1) times the integrated heat kernel over S + f =2 ∞ + ∞ e k X ℓ 1 2 0 t 2 h 1 2 1 2 N a may be approximated as Z (2 = ≃ − 2 2 1 ≃ e S dℓ E + 1) 2 1 1 . + dx expansion. In the next section, we make no such approximatio S ∞ ) N m  0 ≃ a x ) can be replaced by an integral.In particular, we use the app Z t and 2 Z ( 2 2 (2 a / 1 f 2 1 2 S − 4.8 expansion. Hence b S =0 e ∞ a ≃ X in a limit where the radius a m Z ), in ( S to 2; the computation for arbitrary N = expansion. Further, in this limit, the integrated heat kern 1 + p is just given by  Z 4.10 S a N / 2 1 2 2 ≃ S We would like to thank Justin David for suggesting this approac 13 orbifold of S By simple changes of variables, we can easily see that We will show that the error involved in making the approximat The error associated with making this approximation is give in the large- nant term in the large- mation that this structure in our final answer for S Of these terms, the integral from zero to infinity scales as we shall fix and write down thethe corresponding sum exact expression. over Essenti the integrated heat kernel is given by the sum as order 1 in the continuation to AdS with an additional contribution from the conical singulari the heat kernel on these singular spaces as arising from the s and Now, using ( kernel on S 4.2.1 The largeAs radius the approximation subsequent computations are rather formal, it is usef JHEP03(2014)043 2 2 x . Z / 2 (0) ) iself (4.20) (4.18) (4.17) (4.19) (4.21) x and its 1) . Using . ( , i.e. n − ) f 1 k n S ( expansion. n S (0) + f ), a , k 1) ! Nx k − B 2 N 4.8 1)( k a ( (0) n − Z 0. Hence a naive with respect to n f =2 / ∞ 1) + k 2 X k ! 2 7→ f − +1) k a ℓ B k Nx ( t − ( ( n ℓ t t f =2 ates that ∞ − − k ! 2 E, +1) k X e e a ℓ see that, to a given order k ( B ℓ − t nel over the quotient space times the heat kernel on S ) + , it is natural to expect that − . To complete the proof, we =2 ∞ b + 1) 2 k + 1) 2 X e +1) ( is the summand in a 1 ℓ N ( . ℓ x f ℓ − N dℓ t 2 1 (2 4.2.1 ) 1 − +1) n 2 e − x + th term in the sum ( n dl (2 N ) + n . To solve this further, we substitute 0 ) = (2 x Np x n Z 1 2  x ( , which enables us to explicitly evaluate 2 1)( 2 ( − t  a n a ) are subleading in the large- − n n E Taking derivatives of 2 n n 0 f − by the corresponding sum. Hence, we have 2 + Z e N dx f − ℓ N 14 b 4.11 − Np 1 − a ( 1 , making these terms further suppressed in the t Z 2 + 1) − – 13 – . 1 is justified. Now, it is easy to see that 2 a +1) e 2 a 1+ t ℓ 1 + x a ( ) + ℓ  E  t a ( − − + + 1) given in ( e f is finite (in fact, zero) in the limit p ) = (2 1 2 2 E +1) ) expansion. x a ℓ . Consider now, the E (2 ( + 1 a ( t ℓ N + 1) ℓ t f ) = dp ℓ 2 − for arbitrary values of n e (  (2 ∞ should finally be expressible as 0 2 f approximation, the heat kernel of the scalar Laplacian on S N dℓ ∞ ). From the analysis of section Z dl N a dl N a N b ∞ = 1 = + Z (or large- 0 X n − ∞ / ∞ x 4.11 t Z n 1) 2 0 Z 2 −  Z a n ( n + n + 2 t 1) N − − 1) 2 − 1 in the integral to find e a 2 1 − n ( a n 1 2 − ( n t n n 1+ t = − is given in (  − e + n approximation. This completes our proof. e 1 2 E + S 1 2 a , only a finite number of terms contribute to (1) in the small- = = Np 2 The dimensionless expansion parameter is t . To do so, we shall use the Euler-Maclaurin Formula, which st a n n O 14 = N S S is power-series expansion of the terms in This can be further simplified to obtain the Euler-Maclaurin formula, we find that derivatives vanish at with additional terms, which nowin have to be computed. We will where ℓ the heat kernel to that order in the heat kernel on S shown that in the large- Firstly, note that the expression Now, we return toZ our original goal of evaluating the heat ker where we have approximated the integral over where we have used the fact that the function large- will in general pull down more powers of 4.2.2 The Euler-Maclaurin formula and the heat kernel on S is given by half the heat kernel of the scalar Laplacian in S need to show that the error terms We shall explicitly demonstrate this for the function which JHEP03(2014)043 . of  1 (4.28) (4.24) (4.25) (4.27) (4.22) (4.23) (4.26) − 2 N without N 2 as follows. S  . . n 6 1 ) and discard

t 2 ditional terms, . , 2 ( t + t
2 O 2 over S 1 t O O n A + S + O 1 , (0) = 0 N t t  # + orbifold. The coefficient

= ing 2.7) and (2.8)) t with conical singularities 1 2 Σ) N +1) β ,
rms upto a integral in terms of a sum, ♯ ℓ  + 1 ( Z − ( 1 ℓ 2 1 l scalar field on the quotient 2 N t 2
M a term which we have obtained Nt 2 − − 1 − N N a N 2 N e 2 0 2 − t − a N N N
N dℓ 2 2 is given by = 180 a + 10  2 in this approximation, we find
N 180 ∞ 4 1 6 0 2 +1) a + 11 ℓ S Z 180 N ( π ℓ + 2  is the angle of the cone. For the sphere N 2 t + 11 3 1 N · N − π 2 = + · N e 2 Z 1 6 / N 1 2 + 2 N +1) = + a ℓ + 1 ( N 1 term in the heat kernel expansion on + 1) ℓ + – 14 – β 2 orbifold of S = 6 R t ℓ 0 − 1 t − N Σ N , we find that  N ). e 2 (2 6 2 − t − Z 1 a N β 2 + N = 0 and we can extract the short time asymptotics by N N dℓ 2 6 dl t A.5 M N − N + 2 ∞ Z +1) a ℓ 0 N ∞ ( 1 6 ℓ + Z 0 2  t +1) " Z 2 a ℓ  − ( s S π ℓ 6 π e 1 n t 4 =1 N K 4 2 − X n · e 1 N , and there are two singular points, the north and south poles N = − π 2 1 + 1) 2 4 1 a 1 = ℓ A + 1) + = (2 N ℓ 2 Z 1 + / a R 1 dℓ (2 2 is the coefficient of the  3 ) onwards. On evaluating the sum s S 2 2 ∞ · ∞ t =0 0 S 1 ∞ K ( ℓ 0 2 X in terms of the heat kernel on the unquotiented sphere, and ad Z A Z O 1 N N πa term is given for scalars on compact manifolds 1 N N 4 =1 N Z 0 X n · / = t = 2 π S 1 ]. Their answer for this term is (see their equations (1.2), ( S 4 37 = 1 Now, using the Euler-Maclaurinand formula again to working express to the linear above order in Firstly, and its quotients This is our finalspace result. S It expresses the heat kernel for a rea 4.3 The analytic continuation to AdS and This matches (term-by-term) with the coefficient of the any quotient, and is read off from ( doing a power series expansion of each term. We will retain te The rest of the terms are regular as Then the total integrated heat kernel can be obtained by summ The heat kernel for the scalar on the where Σ denotes the singular points and A the sphere. Hence the above expression becomes above. Here in [ which can be thought of as arising from the fixed points of the for the terms from JHEP03(2014)043 ]. is 46 2 and – ) on with S ] will 2 , each (4.29) (4.31) (4.30) (4.32) 44 β 2 37 4.25 M are slightly ]. As d account for e possible to 2 42 . , , )
) 2 t may be decomposed The other subtlety is . We will now outline (or its quotients), and and AdS β 2 . O x, x, t en removed), therefore, ed. By the above con- 2 x, x, t ( 2 # ( M S + on the smooth manifold
K 2 # K t est of the terms arise from i . The first term corresponds t ǫ ǫ . The discussion in [ anifold times a constant. On s into the smooth part and a . We expect that the answers Z
Z nel over a manifold N north and south poles of the O 1 \ i riant quantitites [ ion on 2 r points and an integral over the + − X ) + S ution to the heat kernel ( t 2 here the orbifold group acts freely on the be (arbitrarily) small disconnected
N i N ) + orbifold on S 2 1 ǫ a x, x, t
− N ( where the coefficients are expressible in 2 Z 180 t ent answers are obtained. See for example [ K x, x, t N N ( ǫ 2 + 11 a − 2
K N We first show how the differences in the global } \ N i 180 2 – 15 – ǫ 15 + 11 S . As there are two conical singularities on S − . Further, let AdS 2 i 2 −{ p Z 1 S β N − N M 2 ) = 6 Z + N 1 " ) for the quotiented sphere ) = − N 1 2 x, x, t , but there are two subtleties. Firstly, the answer on AdS 2 ( 6 ]. The analytic continuation acress the quotient spaces shoul . Then the integrated heat kernel over 4.25 ia = i N K 41 p x, x, t " N , ( 7→ N Z 1 2 \ / 40 2 a 2 K β AdS conical AdS M times the answer on Z Z K under this quotient has only one fixed point. It should again b 1 N admits an expansion in powers of 2 } i denotes the integrated heat kernel on the unquotiented S ǫ 2 s contains the origin, where the conical singularity is locat S S as orbifold is volume divergent, and hence has to be regulated. (once the singular points, the north and south poles, have be ǫ K N N N − { Such differences have already been encountered in the cases w Z Z Z β / / 15 2 2 Consider now our answer ( into integrals over the small neighbourhoodsrest of of the the singula manifold, which is smooth. regions enclosing points where siderations, we analytically continue the conical contrib these integrals on such manifoldsS are just the volume of the m carry out the above decompositionsmall of neighbourhood the containing integration the domain singularity. conical singularities located at points cone contributes homogeneous, these invariants are independent of the locat Now the AdS S integrating about thesphere. conical singularities We located argueM at this the as follows: the integrated heat kernel to the integral over the remaining smooth manifold, and the r properties of the quotientbe group useful action for may this be purpose. accounted for Consider the integrated heat ker sphere and hyperboloid [ terms of volume integrals of local general-coordinate inva how this answer is toshould be be analytically related continued to by AdS spaces the answer is where different, and this must be accounted for. due to the fact that the global properties of the its these differences, but once that is done, completely consist JHEP03(2014)043 may (4.39) (4.36) (4.33) (4.37) (4.34) (4.38) (4.35) N , Z
/ 2 2 t incident heat respectively. 2 O ), is presented . This is again 2 at a large value + . # volume integral is η t 4.18  2

1 may be expressed in 0 , η − 0 ) and AdS ] using the Sommerfeld − η 2 a dure to regulate all the . e . In this section, we will e are homogeneous spaces, 2 ( , ) the scalar on S N 39 N ) 2 s 2 N 2 2 t 2 a O t ( Z K
s briefly here. We state the ( / 2 a cutoff on O n be cancelled off by boundary , 2 O 180 existing literature. In particu-
1 + + +11 AdS η + 2 4 − cone in [ 4 0 and AdS is given by is given by N 2 t η πa t sinh 2 πa e 2 N 2 60 1 2 = Vol. − . 60 Z a 2 2  / ] for details. The AdS + 1 2 2 + πa 29 − to be s In particular, AdS N 2 πt 1 πa , as well. The analytic continuation of the first 2 πt 1 6 4 2 − dηdφ 4 16 24 N ia π – 16 – + = , 2 " + ,K 0 2 7→ ) 1 2 2 17 Z V 1) = 2 a , 1 a πa ( ]. Since both S + ], since the divergent term 1 ∞ πa − s 1 0 16 12  0 30 29 Z 12 2 η K − 2 a t = S 30 ) = V ) = a (cosh ( a − ( 2 s 1 1 6 s 2 ] for explicit expressions. We also emphasize that there the co = Vol. K πa 2 K 24 + s S t 2 ), again employing the Euler-Maclaurin formula ( K = 2 2 a are the coincident scalar heat kernels over S − reg s 2 4.34  V K 1 . The final step is to regulate the volume divergence of AdS N is computed independently of this analytic continuation. ) is well understood [ A 2 = and N 4.25 s 1 Z / K 2 We refer the reader to [ s AdS , which gives the regulated volume of AdS 16 K 0 η These are related by analytic continuation. where the volume divergence has been regulated4.3.1 as above. A checkIn of the the previous analytic section, continuation we have shown how the heat kernel for which clearly diverges. To regulate this divergence, we put we have verify this explicitly againstlar, expressions the scalar obtained heat in kernel the was computed on the AdS where kernel on AdS in appendix well understood by now.divergences we However, encounter since inmain our we steps, analysis, referring will the we use reader will this to review [ proce thi term in ( given by Now, following the discussion in [ and be analytically continued to obtain the heat kernel on AdS where we have analytically continued Hence, the integrated scalar heat kernel on AdS A derivation of ( terms of the radius of curvaturecounterterms. of the Hence, boundary the sphere, regularised it volume ca of AdS JHEP03(2014)043 α 2 H (4.44) (4.46) (4.47) (4.40) (4.45) (4.43) (4.42) (4.41) (4.48) is quotiented by . , ) 2 2  y ¯ s 2 α , − ) . µ e α −   ) .
− (1 g ... 2 , ¯ s DσA µ  y, α (1 + ρ ) becomes ) (
 2 2 = 1. D A O t ¯ s a O , yf ρσ 4.40 + N 2 (∆ 1 g ) 15 O s µ , µ ) + 1 expansion. This is precisely the 2¯ 15 ] for more details regarding these = cosh + − α A µν gA µ µ 39 ≈ − 6 1 F ¯ To evaluate the path integral, we will s − dy √ 20 A D 1 3  . ( x µν N  ∞ µ (1 d −  δd, g 0 1) d A gF  ) Z 1 3 √ ν + , − y α , and find that √ 2 1 x ∂ Z ν  d ) 1 x ¯ s N y ¯ s dδ 4 A − 2 1 N 4 d d 2 − − 2 – 17 – d − πs − ν µν e ≡ (1 e Z sinh 2 + ( (4 ) formally around , equipped with a metric A R 2 Z α α = 1 2 2 ∆ µ α − + ∂ ) was determined to be H M = − + + gf 1 + µ 4.39 K 2 2 S A  = 2 with our A H H y, α S Tr H + ( y  α K K gf f 2 1 K A N − S ], the path integral for a U(1) gauge field in the Euclidean 1 Tr Tr S 1 expansion, the expression ( Tr , then the traced heat kernel on the quotient space (called = α α ≃ 24 sinh α = ≃ + µ N 2 = ≃ ) R α S = H α A 2 ) = α ∈ 2 K H H (∆ α K Tr K y, α ( Tr Tr f ]. It was found that if the angular coordinate of AdS and the function 34 , where 2 – t is the U(1) field strength a πα 32 = µν F s + 2 ]) is given by θ 39 7→ fix gauge by adding the term where where is the Hodge Laplacian, expressed as where ¯ so that the total action becomes signature on a background manifold expressions. Now in the θ expression obtained on expanding ( in [ 4.4 The HodgeLet Laplacian us for consider, vector following fields [ We now identify the parameter formula [ where we have retained the leading term in the We refer the reader to equations (20) to (23) of [ JHEP03(2014)043 , c 2 is a e of φ (4.52) (4.51) (4.54) (4.50) (4.53) (4.49) ∇ ǫ , and our 1 cases. In φ and AdS 2 µ 2 ∇ and AdS = 1 when evaluating 2 . . ℓ
scalars, and hence )] 1 hese modes in the next h lead to zero modes in he second term represents − g scalar ghost fields whose is section, we will consider , ively conclude correspondence with scalar d longitudinal modes of the 2 s he vector Laplacian can be = 0 mode of the scalar is a s S ontribution of the gauge field rivial gauge field. We account x, x, s K ℓ ( oth the S K s ues from = 2 K , 2 2 . s = 2 1 φ orbifolds. We will use the fact that, , − K ν φ  1 may be written as ) φ N ∇ +  2 µ 2 +1) Z 2 a ℓ s 1 µν ( a ∇ ℓ ǫ K  t x, x, s ∓ + ( = − and S = Λ 1 e Λ v L φ 2 1 – 18 – v for the Hodge Laplacian over vectors is given by  K µ orbifold. These modes cannot be reached by the φ [ 2 µ K , or their ∇ = g + 1) N 2 are longitudinal and transverse modes of the gauge ∇ 1 + ℓ √ Z = φ L ∆ x T (2 v µ d v  is covariantly constant, we will concentrate on just d A K =1 ∞ ℓ µν X or AdS Z ǫ and = 2 s v T ds v K ) = 2 ∞ t is the Hodge dual defined with respect to the background metri ǫ ( Z ∗ v and 1 2 2 K 17 is the Hodge Laplacian over 1-forms, then we must have and its quotients is more subtle. It has to do with the presenc geometry and its is either S are scalars on the corresponding manifold. For the spaces S 2 2 , and δd S ∗ 2 M d + φ ⊗ s don’t commute over arbitrary tensors, they do commute over 2 and AdS dδ − ∗ ∇ 2 and = 1 case, there is a correction due to the fact that the φ δ 2 Though 17 . This gauge fixing requires us to introduce two anticommutin the S constant over the two-sphere and doesfor not this give rise by to explicitly a starting non-t the sum over scalar eigenval the heat kernel. So the heat kernel on S one can show that if transverse vector mode. Further, as field. However, there are corrections to this expression in b obtained from those of the scalar Laplacian, and one would na Hence, eigenmodes of theeigenfunctions. vector Laplacian Hence, are the in spectrum one-to-one and degeneracies for t analytic continuation we aresubsection, but working for with. the moment We we shall note that come for to transverse an t kinetic operator is theis usual given Laplacian. by Then the one-loop c g where ∆ = harmonic 1-forms in thethe spectrum full AdS of the Hodge Laplacian whic The case of AdS where where for both S the contribution due tothe the case ghost where fields. Now, in the rest of th The first term is the heat kernel of ∆ over vector fields, while t upto global issues, the vector field on AdS discussion goes through for transverse modes as well. JHEP03(2014)043 o ), and and 2 4.50 (4.56) (4.58) (4.59) (4.55) (4.57) (4.60) N ls just Z / T 2 in ( . For the v S 2 K φ 5.2.1 ,  by the methods and 1 1 n, there is a series − φ N Z / 2 s S continuation of the vector odes of the gauge field. In ode of the scalar field as it K . tail in section  N lar fields N Laplacian over the scalar, vec- ut also contribute to the gauge Z Z / ally continue from integrable. These configurations / 2 uation when we take into account = 2 . term in the heat kernel expansion.
2 . 2 N
. Firstly, vector fields on AdS  s AdS N 0 Z 1 S Z N t / s AdS K / geometry and analytically continuing 2 Z 2 v − S K ⊗ / = geometry. We will do this by computing O N 2 1 2 , N = s AdS Z N N / Z Φ 2  K Z / ,K d
2 / 1 2 L = v
AdS AdS S = L + 2 2 − – 19 – v AdS N K S ]. The discussion of the corresponding quotient ⊗ A and AdS 1 . Z N K / 2 6 Z = 2 ⊗ 24 2 − / [ S N 2 2 2 = 2 6 s S T,L 2 v AdS T N N K v AdS Z 2 K AdS / 2 K = to AdS ]. These arise from field configurations T N N v AdS Z , we can compute the Z / 30 L 2 K N / v S = 0 mode back to the heat kernel. This of course is equivalent t 2 Z ℓ / K 2 = there is no subtlety, and the heat kernel for these fields equa and S 2 N 2 Z / T 2 v S K have longitudinal and transverse modes as above, in additio term, which precisely matches with the expression obtained 0 after adding the N t Z N ]. We close this section by a brief discussion of the analytic / Z 2 / 37 L 2 v S evaluating the Laplacian over the full set of modes of the sca K of [ and the corresponding determinant over In this section we will computetor the and one-loop Dirac determinants spinor of field in the 5 The one-loop determinants on moment, we concentrate onthis the case, longitudinal the and transverse natural m analytic continuation is to analytic and we indeed find after this analytic continuation that to the heat kernel from S spaces is entirely analogous. We finally find that AdS the heat kernel of the scalar Laplacian. In particular, there isis no not need a to normalisable subtract out mode the on constant AdS m which are square-integrable, though Φare itself not is captured by not analytic square- continuationfield from path the sphere, integral. b We shall treat these configurations in de vector field on AdS For the vector field on S We find that the conical singularities contribute This is the procedure we shallthe follow for graviphoton the background analytic contin in section of harmonic modes as well [ JHEP03(2014)043 is re 2 tion S (5.4) (5.1) (5.2) (5.6) (5.5) (5.3) ⊗ , 2 . That is, over these 2 imθ e ) belongs to  5.4 2 η are obtained by , given — upto 2 2 2 S . We will consider, 2 , = 4 vector multiplet , sinh ⊗ S
plicit answers for the
2 , 2 2 − N with metric , Z , for the final computation rectrum of the Laplacian ) N ∈ ρdφ ρdφ + 1 Z 2 2 | / an for various spin-fields over . in AdS ρ, φ
( m arly suggestive form of the an- . 2 |  , m , S + sin y acting quotients of spheres and  + sin ℓ,n 1 2 may be written as 2 ∞ Y , and impose an orbifold projection 2 ). We shall then explicitly evaluate ⊗ π . The orbifold action on S 2 N + 1) 2 ) 2 . The eigenfunction ( 2 + dρ iη S dρ S 2 ℓ | S is given by 5.2 ( − ⊗ 2 η, θ 2 m ⊗ 2 2 ℓ 7→ | ( 2 a S a 2 < λ < , φ + χ + + ) are coordinates on ⊗ + π 0 , λ,m 1 4 N
2 2
a f 2 iλ 2 ρ, φ + + − 7→ 2 – 20 – , θ with the scalar eigenfunctions on AdS ηdθ ) = and AdS λ χdθ 2 1 2 2  2 2 a 2  and ( S , + 2 7→ 1 | 2 a ⊗ ia ) 2 m + sin | = + sinh η, θ, ρ, φ 7→ 2 ( AdS 2 θ, φ + 1 ( λ,ℓ dχ a dη , and indicate how the answer may be analytically continued iλ E N 2 1  2 a Z a 1 λ,m,ℓ,n that are left invariant under the orbifold action. We note he / orbifold of this geometry F Φ = 2
] = 2 ]. We begin with defining the geometry and the orbifold projec 2 S 2 η N 2 S 30 | 41 Z ⊗ ds . m ds | , ⊗ 2 N 2 ). We now turn to the evaluation of the integrated heat kernel are the eigenfunctions of the scalar Laplacian on AdS 40 Z S / 5.2
2 λ,m s are the usual spherical harmonics on S S , where the effects of the graviphoton coupling to the ) = sinh ) are coordinates on 6 ℓ,n ⊗ Y 2 η, θ η, θ ( AdS λ,m f where the Ψ normalisation — by [ the spectrum of scalar eigenfunctions on AdS tensoring the scalar eigenfunctions on S The eigenfunctions of the Laplacian acting over scalar field 5.1 The scalar field to that this geometry may be reached from the space the eigenvalue the heat kernelcompact over space this subset of modes. We mostly obtain ex In particular, we will computethe the modes heat in kernel of AdS the Laplaci where ( fields is accounted for.swer Additionally, closely we related will to find thosehyperboloids a obtained in particul previously [ for freel that will be imposed. In global coordinates, AdS and the via the analytic continuation of section the answer to the AdS case. This will serve as a useful warm-up still given by ( in what follows, the which maps to the subset of modes invariant under ( two geometries. Ourover strategy the will unquotiented geometries be S to write down the full sp JHEP03(2014)043 ) 2 n S 5.2 ⊗ (5.9) (5.7) (5.8) 2 (5.15) (5.14) (5.11) (5.13) (5.10) (5.12) and ] , 30 is related m .   2 S     ! ⊗ 2 +1) ℓ +1) ( . ℓ ℓ , ( 2 2 ℓ t ). Modes invariant is given by ) a +1) tion 2 2 may be expressed as t 2 ℓ a − ( 2 N ℓ 5.7 e − S 2 2 Z t i e . a / s ) by analytic continuation. ⊗ Z −
s  ρ, φ, a 2 genvalues and their degen- e ( 2 ( cian on AdS N +1) ∈ . We recognize the functions ) a. ℓ S N N ℓ πs . Z n χ  ant under the projection ( (2 ℓ,n +1 / ⊗ ˜ ion to AdS. ℓ 7→ π Y
( 2 ˜ =0 l on S h ℓ ∞ 2 ) 2 . sin ℓ X S and 2 1 1 + 1) t S + a

i ℓ are given by ) on modes ( sin Z m − ( ⊗ s iη, a e 2   ℓ 2  ∈ ℓ by the analytic continuation [ S 5.2 =0 2 2 ∞ , m, n, s +1) ℓ 1 χ, θ, a X − N N a ℓ πs 7→ which is nonvanishing when 2 ℓ s ( +1) N ⊗ = ˜  ℓ S X   ) (2 ( AdS n 2 + ˜ n π ℓ n ˜ ) on the modes on the compact space S ℓ ˜ ℓ,m ⊗ 2 1   h − t  sin − a Y 2 ˜ ℓ ia, χ m = 5.2 ( − X +1) sin e m ˜ and ℓ + 1 πi – 21 – ( 7→ Np , p ) 2 ) = ˜ ˜ ℓ ℓ s 2 e 1 2 1 =0 ∞ t (  m ˜ = ℓ a 1 X ) = ˜ ˜ ℓ ℓ s − , a − =0 n 2 1 χ ( s e X 1 =0 1 2 N ∞ a ℓ ℓ X i − χ =0 ∞ s 1 − ˜ , ρ, φ, a ℓ N X = = 1  m   2 +1) iλ = ˜ ℓ,ℓ N N s ˜ ℓ πs S representation of SU(2). The above answer may be expressed    E (2 ℓ 1 K 7→ π · − =1 χ, θ, a ˜ ℓ h s sin ( X 2 N s S m,n,N δ 1 sin N K = + =0 ∞ ˜ ℓ,m,ℓ,n ˜ ℓ X s 2 Ψ S   K ⊗   2 s S 1 − =0 K s X N 1 ). We choose the following representaion for this delta func N 1 N = 5.10 s = s K K in the following form, which suggests an analytic continuat as the characters in the spin- In particular, we will impose the projection ( From this, we can extract the heat kernel of the states invari by inserting a Kronecker delta over under the orbifold action obey the quantisation condition Firstly, note that the unconstrained integrated heat kerne which has been obtained by carrying out the sums over Then the scalar heat kernel over the orbifold geometry and compute the integratederacies. heat kernel Then, by we enumerating will the obtain ei the answer over to the spectrum of the scalar Laplacian on S We now begin with imposing the orbifold action ( The eigenfunctions of the scalar Laplacian on S which belong to the eigenvalue Now, we will use the fact that the spectrum of the scalar Lapla obey ( JHEP03(2014)043 ⊗ 2 , and (5.17) (5.19) (5.16) (5.20) (5.18) (5.21) in our ld heat ia . # 7→ . The rules  1 # 2 has only one a . Finally, the  . 2 2 . The second +1) 2 a ℓ
a ( 2 2 ℓ 4.3 1 nted space S +1) is given by a ℓ (  , ℓ t 2 − ) S −  ) 2 t e N 2 . a − ⊗ ) Z to the quotient space a ( / e ( . placian over the vector 2 ) 2 +1 2 2 v  ˜ S ℓ orbifold, this leads to the s S + 1) S ) v ( ˜ ℓ = 1, i.e. the unquotiented 2 K 2 ℓ K t + 1) 2 1 K that acts as a spectator, its t ) Z ( a (2 2 ℓ 1 N ) 2 e − el on S O a e, from which we’re doing this ver longitudinal and transverse 1 (2 ( e =1 ∞ a ual prescription of cified in section ℓ 2 +  X ( s (AdS s S =0 ∞ 2 2 4 1 ℓ X s K S K "  + 1 " K + ˜   , the contribution to the heat kernel ℓ 1 4   has to be replaced by 2  ) + N N   2 + Z 2 Z 2 1 / a +1) a 2 / ) + 2 2 1 a ˜ may be expressed as ) +1) ( ℓ s S a ˜ 2 =0 ( 2 ℓ ∞ 2 ˜ ˜ ( ℓ ℓ 2 X 2 ˜ S a ℓ s s S K S ( )  2  t 2 K ⊗ 2 t K − s 2 a S 2 − ) s ( e S – 22 – s e 1 AdS K 2  a K s S 
K ( v (AdS 1 2 1 2 K v . We will account for the presence of harmonic modes S + 1 K + 1 − ˜ ℓ N = K ˜ ℓ ) = 2 ) = Z to account for the global feature that AdS 1 2 2 which is being analytically continued to AdS 2  / Z a  N 1 2 2 / 2 ( ) = Z orbifold. This may be decomposed into 2 , a 2 / =0 2 ∞ S 1 ˜ =1 s ℓ ∞ ) S X ˜ ℓ X a ⊗ 2 N , a   ( 2 S K 2 1 Z 2 ⊗ a   + S 2 ( s . We will first consider the case of AdS ⊗ 2 2 N S K s S AdS ) = Z v ⊗ may be arrived at by analytic continuation from the vector fie ( ) = 2 2 2 K / ) may be regarded as arising due to the presence of fixed points 2 v and its S K N
, a 2 , a 2 K . Firstly, we note that Z 1 1 S S / 5.16 a N a (
( ⊗ Z ⊗ 2 2 is the integrated heat kernel of the scalar over the unquotie 2 / S 2 2 S S 2
S ⊗ ⊗ S 2 2 ⊗ 2 v ⊗ S S v S 2 2 s S K K ⊗ K 2 AdS . We will now perform an analytic continuation of this answer 2 AdS kernel on S spaces and specify the analytic continuation. The heat kern analytic continuation is trivial, just the radius on 5.2 The vectorWe field now turn to thefield evaluation on of AdS the heat kernel of the Hodge La fixed point under the actionanalytic of continuation has this two. orbifold, while For the the specific spher example of th subsequently. For the non-zero modes on AdS which may be written out as for this analytic continuation have been very precisely spe multiplying an overall factor of The first term is just the heat kernel for the scalar on the S This may be re-expressed as final expression quotient space. This is analytically continued using our us second term is ( To begin with, we consider the heatmodes kernel of evaluated the purely o gauge field on AdS term is the heat kernel on the S Here JHEP03(2014)043 2 is ) to and N 2 Z (5.24) (5.22) (5.23) (5.26) (5.25) S / may be 5.25
⊗ 2 2 N S Z / ⊗ . .
. } 2 . gets continued 2 0 ) S
S 1 2 +1) 1 ℓ a a ( ( ⊗ orbifold. Secondly, . ℓ − { − 2 ) 2 t N ) a Z N a Z a ( +1) − / Z ( ℓ ∈ 2 e 2 ( des invariant under the 2 ℓ S AdS s S  2 on of the modes ( t nic modes in this analysis. K a 2 We thus obtain AdS K , m rises out of gauge fixing. btracting out the constant of radius 2 answer on N − πs K e 2 ) or the transverse and longitu- der the orbifold, but no other  − 2 zero modes of the scalar on S imθ and its a ℓ ) e ( 2 1 − χ | which are not related by analytic 2 n,Np S a ) m ( 2 | − case. This involves the following =1 ∞ ⊗  ℓ X s m N AdS 2 2 η a, a δ Z ), in which case the modes invariant ( K /   2 m as above. Secondly, there is no need to 2 b η to produce zero modes in AdS 2 N , n S f πs N a Z 2 2 m / K ∗ ) ) + 2 a 2 im sinh . These are just modes invariant under the 2 f a e S quotients as well, where we have to be careful ( Z − | 1 + cosh ab ⊗ 2 ) = (0 ) 2 N  m s S ∈ | 2 – 23 – | Z 2 p K  , a of radius AdS m ) s m, n 1 | ( 0 1 2 a 2 a η π ( gdηdθg K ( 2  2 √ N s Z 0 = 4 should be apparent from the fact that the constant mode / η s AdS ) ), where 2 = N 2 tanh K Z S Z } ℓ / m 0 ⊗ ) . Now the analytic continuation to − 2 ℓ 2 , Np 2 S −{ = S s X ( ) = 2 n Z X ⊗ , φ a ∈ 2 K ( m =1 m ∞ 2 ℓ X φ ) = (0 S AdS   } = 4 a v ( ⊗ 0 1 2 ∇ N . These are − K =0 −{ Z 2 s X m, n N / Z = X v AdS ) ∈ ). We then obtain 2 1 1 N m K S m a a f ⊗ ( 2 = = 2 S ) are ( ′ v ( b K 5.2 orbifold. We will first calculate the heat kernel contributi K N orbifold on the other S Z N continuation to S Now we take into account the presence of modes on AdS We remind the reader that thisdinal expression modes is of the the heat vectorNeither kernel field. f have We we have subtracted not out included the the harmo ghost determinant5.2.1 which a Zero modes of the vector field in AdS The above discussion goes through on the obtained from continuing themode scalar on S heat kernels, and not su given by That the last twoorbifold terms projection are on the a single heat S kernel evaluated on the mo subtract out the constant mode of the scalar in the first term. to the heat kerne of the scalar on AdS to project onto the appropriatechange subset is of required. modes invariant In un this case, it is apparent that the final on one sphere corresponds to modes ( Z the non-zero modes replacements. Firstly, the heat kernel of the scalar on the S under ( Now, we will analytically continue this to the AdS they tensor with the zero mode of the scalar on S its These contribute in two ways.to Firstly, produce they tensor non-zero with modes non- of the vector field in AdS JHEP03(2014)043 and g off (5.30) (5.31) (5.29) (5.32) (5.34) (5.28) (5.27) (5.33) N Z / ated. We inate, and 2 with moding are obtained . expansion to | 1 2 p | 0 AdS N . η − N . is regularised in Z 1 2 ) gives the same / ke  that the only field N −
0 + 1 2 2 η ber of zero modes in ≤ 5.30 S N 5.2 N πis Z or field, it is sufficient 2 he homogeneous spaces ⊗ m / ber of zero modes. This ber of zero modes to be n 2 − 2 s S tanh e ≤ 
K 1 0 } , , 2 , se divergences, we retain the . − η 0 AdS ℓ n l do the large tor fields on AdS 1

ted term sums to 1 f 0 0 −{ = 2 − ∗ η η Z ℓ X m − # − ∈ f = e e p tanh N +1) mn to compute the above sum (which is = ℓ . Z O O ( / ℓ 1 − with the constant mode of the scalar 0 gg 2 2 η t Np 1 − n a 2 √ f 1 + 1 + − 0 AdS ∗ = e n − 1 + N  Np m 0 dθdη ≃ − Z = η f − / ) e N πs – 24 – 2 Z on AdS N mn m  } Z N 1 N πis ( ℓ 0 / 2 2 0 AdS ) gg χ e 2 n −{ S √
≃ 5.25 Z X ⊗ 0 =1 ∞ ∈ 2 ℓ ,T 2 η X p N 1
2 1 0
dθdη 2 − N = =0 η AdS 2 0 s 2 X 2 0 ( N 0 ) η is given by − at some large value 0 n 0 e n 2 η η 1 η N N AdS tanh Z Z " O − / tanh } 1 . Our strategy will be to compute the above integral by cuttin 0 2 1 − Z (tanh 1 + −{ − Z X ∈ = 1 − ∈ ) = ′ p p b s 0 , this is typically a divergent quantity which has to be regul ( η we outline how a zeta function regularization of ( limit. We drop the factor diverging with the AdS radial coord K N e = = 2 . Hence, we obtain T for 2 1 Z 0 Z C 0 / η n ∈
≃ where the integral (essentially the volume of the AdS space) Np 2 p S 2 , = (0) ⊗ ℓ T . It is straightforward to check from the orbifold projectio 2 Np 2 S = AdS this manner. is completely equivalent tolike the AdS way we compute zero modes on t pick out the order 1 term which will be our definition of the num now describe how this is done. We take our definition of the num where on completely convergent in that regularisation). Then we wil configurations that can contribute are harmonic modes of vec answer as obtained here. Finally, zero modes on the space the AdS radial coordinate by tensoring the harmonic modes Now, to compute theto contribution compute of the the number zero modes of to zero the modes. vect On non-compact spaces li in the large We will now explicitly evaluate this expression. The bracke Then, in accordance withorder-1 our term usual only rule to for find that regularising the m Further, we can show that the quotient space AdS In appendix keep the order 1 term as the number of zero modes. Hence the num JHEP03(2014)043 . , , N Z ! ! / ) ) ) (5.43) (5.39) (5.36) (5.42) (5.41) (5.40) (5.35) (5.37) (5.38) )
ψ ψ ψ 2 ψ S ⊗ (cos (cos (cos (cos 18 ) . 2 ) jorana fermion, . # +1) ,m . +1) ,m . 2 , which means we 2 , ± l,m / +1 i AdS 2 m D +1 η m m i m m,m n +1) m − m,m +1) ( − m ( − l ℓ ( − l ℓ + ( l ( l +1) ( P β ℓ P 2 2 P t 2 2 ( P ) t a 2 ψ a + 1) 2 ψ 2 2 ψ 2 t x ψ is given by a l − − ( m e e e of the Dirac spinor on 2 − m +1 +1 1 e S 1 2 m m − cos 2 1 (1 + − 2 1 cos ⊗ ℓ n − 2 + 1) ψ 2 1 + i a ℓ 2 − 2 cos ψ + cos X ℓ ℓ − + = α l. ± 2 ψ 2 X ℓ ψ ) +1 n = 4 ( +1 x m ≤ 2 = m m m m 2 − =0 ∞ , are still given by the quantization m ℓ sin X =0 ∞ sin sin ± l,m ℓ sin Z X " (1 η =0 i ∞ · ± ≤ i , which is the square of ℓ · X ± 2 h ∈ 2 5.2 2 S 0

/   n

D = n 2 6 D φ ) d φ − +1) 2 ) dx ′ ) , 1 2 ℓ are given by 1 2 0 ( β +1 Np, p ˜ ℓ + 2 2 1 +1) − t + ( ℓ a S ) ≥ m ( , 2 1 = m – 25 – t ( − x / 2 a ( t i D a i e n − − e ± l,m 1 2 − e e , l e χ − −  1 2 ′ (1 + Z ℓ α m + − ′ ∈ X − ℓ + 1)! + 1 = and analytically continue the answer to + 1) ) + 1) + 1)! quotients. We will again compute the answer explicitly on ˜ ℓ l x m ℓ m N (  m N 2 l, m Z − 4 ( 1 4 + Z / + − l ! =0 ∞ l ′
l (1 ! =0 X ∞ ℓ =0 l ∞ 2 ℓ ˜ X n ℓ ! i a )!( X S )!( , but we are also considering a Dirac fermion, rather than a Ma n   ± 1) m 2 1 n m ⊗ ) = − ) = 2 2 = − 2 ( a − ) = l S ( and their , a l ( 2 2 ( 1 ± l,m 2 f S a p , a ) = χ S ( p 1 2 x K ] we will compute the heat kernel of 2 2 a ( S × 2 ) ( S / 2 24 D ) are the Jacobi Polynomials: 2 πa ⊗ 1 πa S 2 1 4 x α,β ( f 4 ( S n ⊗ √ 2 AdS P √ f K S , α,β n = 2 K = P S Following [ ± l,m × ± l,m 18 χ 2 η Now the integrated heat kernel on the unquotiented space S where condition We first recollect that the eigenstates of The modes invariant under the orbifold action from which we obtain We finally examine howS the above discussion extends to the cas 5.3 The Dirac spinor which yields a factor of 2. These two factors cancel. the compact space should multiply a factor of and These spinors satisfy Each of the two products corresponds to Here JHEP03(2014)043 . N . Z 2 / , and (5.46) (5.45) (5.47) (5.44) to the
H +1) 2 ℓ ( S 2 2 t a ⊗ − G/H 2 e \ 2 Γ . On imposing 2 AdS . These are coset +1) may be expressed a ′ ℓ , ( which is quotiented 2 . Harmonic analysis 2 1 t a G +1) − G/H ℓ e ontribution of the fixed . ( \ 2 2 exploited to evaluate the t G/H   when restricted to a xpect that the heat kernel − trary-spin particles as well, S N N 19 πs πs rallel, we begin with a brief e ated heat kernel is given by inuation to tion of the smooth part of the 2   aces where the quotient group , 2 1 2 1 S R +1) + + y acting quotient groups and the ′ ℓ ℓ ℓ tE ( χ χ − symmetric as well leads to a few technical 2 1 t e a   ) is a subgroup of − γ e N N πs πs (
ts. H   ]. In this section, we draw an interesting R which contain , ℓ 2 1 1 2 χ ′ 41 ℓ + + Γ G ′ ′ from the left. Further, the action of Γ is such , ℓ ℓ ∈ f X γ χ ] would hold for quotients of homogenous spaces as well. χ 40 G – 26 – 1 D 41 R − =0 ∞ =0 , X ℓ X s X N =0 ∞ · 40 ℓ X using the method of images. For a field transforming v =0 4 ∞ ′ N X ℓ 2 =0 ∞ ′ = S 1 X ℓ = ⊗ − S =1 Γ 2
s X = . In that case, the heat kernel on Γ N ]. Consider a symmetric space G/H f S K , ℓ G are Lie groups, and N ′ K 4 41 N ℓ Z , the integrated heat kernel for the Laplacian on the quotient , = 0 contribution as for the scalar, and write / 1 N , which acts on H + ) f s 2 H 40 G S D ⊗ of ⊂ 2 and ) = S 2 f ( S G are the representations of , a K 1 R a ( , N are the simplest examples of the so-called symmetric spaces where Z 2 / ) G/H 2 is the azimuthal quantum number on the sphere with radius is a volume factor corresponding to the ratio of the volume of S ⊗ G/H v n 2 and S S f ( 2 Strictly speaking, most of the results of [ K 19 that it has no fixed points in by a discrete group Γ in terms of the heat kernel on The additional requirement thatsimplifications, the and space minor begin simplifications quotiented in be the final resul results obtained here.may be This explicitly parallelin evaluated also particular, on encourages in these us higher-dimensional conicalreview to spaces. of spaces e the for To results arbi draw of the [ pa parallel between the results obtained previously for freel We can separate out the points. This form of the answer is suitable for analytic cont The first term on thequotient right-hand space, side represents while the the contribu second term may be regarded as the c in a representation spaces this projection as for the scalar field, we find that the integr space was found to be where where However, we do not carry out the analytic5.4 continuation here. A group-theoreticAdS interpretation of these results where volume of on such spaces hastraced a integrated group heat theoretic kernel, on structureacts quotients freely which of has on symmetric been the sp sphere or hyperboloid [ JHEP03(2014)043 , t 2 N 20 to N Z Z Z / , and cit as onical (5.50) (5.49) (5.48) 2 2 ) could may also . We will 2 2 5.47 Z / 2 and AdS s determined by were assumed to N .... ) was carried out ), and multiplying Z N + / H 2 2 ). In particular, the a 5.48 t SO( ( o not fully explore this = 2 there, which verifies fact produce the correct and / 16 5.47 N placing the Weyl character 1) G − s in the context of the rnel can be explicitly solved ]. +1) 1 8 ] seem to carry over to these ℓ bout these quotient spaces for N, ] that the answer ( ( = 2. For this, we require the 47 estion further. ℓ 41 ). This is given for the scalar, ≃ as 2 t 41 , res and hyperboloids in these a ) N , ,R 4 1 − dλ, 40 5.1 le in [ e + 40 λ 2 which we have considered on S  λ ) times the heat kernel on AdS ) on setting ( SL(2 ) θ ) for 2 N 2 πs t 2 1 = 0 term corresponds to the contribution a N Z ≃ πλ  − s − 4.39 ℓ 4.39 e 1) π χ ) , =0 πλ cosh ( ∞ – 27 – ℓ X 1 1. The cosh ( 1 , =0 2 1 s X cosh ( 1 2 = = 0 , which is just dλ s = N dµ 2 ∞ Z 0 Z ). This is possibly related to the fact that for the scalar hea / / Z 2 2 s S ), have remarkable similarities to ( has fixed points, the final forms of the answer thus obtained 1 2 · K 5.47 2 = 2 and S 1 2 ]. It would be very interesting to make this connection expli 5.44 ) by the Global (or Harish-Chandra) character. It is natural N ⊗ 34 = [ = 1 term, which is responsible for the contribution from the c 2 s 5.47 N . Firstly, we observe that the heat kernel for the scalar on S to account for the relative number of fixed points in S conical ), and ( 2 1 2 K . For us, being summed over in these expressions are precisely the one 5.13 N πs ℓ = ), ( θ ] is the eigenvalue of the Laplacian. To obtain this result, 4.25 47 The corresponding expression for the fermion is also availab 20 S R concentrate on the be compact, and the extension to non-compact spaces SO( singularities. We therefore find These are precisely the conical terms found in ( by analytic continuation. Though the action of the branching rules as in ( in ( leads to the asymptotic expansion found in ( our conjecture. Thus, many of the central results of [ orbifolds as well. It would be interesting to explore this qu quotient spaces as well,for leading arbitrary us rank to tensors, expect on that arbitrary-dimensional the sphe heat ke from the smooth part of AdS by a factor of which appears in ( ask if the samequestion extension here, is we possible provide in some this preliminary case. evidence for Though we thi d kernel over the coneanswer in for flat integer space, theit method would of provide a images very doesarbitrary-spin explicit in particles. solution to Finally, the itbe heat was extended kernel a to observed the in non-compact [ case of the hyperboloids by re values of the product space S where by [ E orbifold of AdS We will now show that replacing the Weyl character of SU(2) in be expressed, using the projection methods of section Harish-Chandra character of the group SO(2 JHEP03(2014)043 ] - 2 2 2 , , 1 4 φ 1 on 24 # = 4 . In 2 β 6 do with (6.3) (6.1) (6.2) (6.4) . The 2 , A N and ≤ 5 2.2 S β # a β T D ⊗ α 2 A A ≤ β along the S D α D along AdS A α D A ld − A α where 3 ! ] a β 2 − A φ n the fermionic kinetic φ A 24 f this mixing is that the ! Firstly, we note that the ion function of an , two Dirac fermions Ψ β , for modes of 1 the end of section ar field minimally coupled A φ A ! levant kinetic operator is [ he analysis of section β l is therefore known from the . or is [ . The following is a summary to 2 nt with the microscopic results. ! D
α e standard Laplacian over scalar fields β 1 4 + 2) D D submanifolds. We refer the reader ℓ + γ α do not mix with the scalar, and their . + and S 2 2 2  D D λ 2 4 1 αβ ) of the near-horizon geometry of a + 1) to γβ γ + ε 2 + 1) ( R . ℓ 1 + D a 2 2 a ( ℓ and S 5.2 αβ 2 ( + ℓ ) γβ − 2 2 R i on AdS = 4 vector multiplet in the attractor geometry 2 ε 1 a 1  i a a 2 + ± A – 28 – N αβ λ g  , ( −  αβ 6. Of these, the four scalars 1) orbifold ( 2 g γ 1 2 ]. This four-dimensional background is AdS a 2 mixes with the transverse modes of a ≤ − − D N 24 ℓ 1 a 2 Z γ ( + 2 αγ φ a ε ℓ D ≤  ] for details. Hence, to determine the spectrum, we can 2 a 2 − 1 a αγ − − 24 ε  i a 2

− 

α  A , where 1 α . The field 2 a ] which we will need for our analysis. 5 A φ φ = 4 vector multiplet has a single vector field 1 24  φ " 1 the eigenvalues shift from N  g of the kinetic operator shift from " ≥ g ℓ 21 det det p on the other hand mixes with transverse modes of the vector fie 2 ] and section 4.2 of [ p Z φ We will first consider the kinetic operator for bosons and the Z 1 2 11 We remind the reader that these are the eigenvalues of both th 2 1 21 − − labelled by contribution to the partition function is again known from t account of the graviphoton flux. The relevant kinetic operat to [ where the last term is again the gauge fixing term. In this case spectrum of the quadratic operator for the vector multiplet. We find thatWe it now vanishes, which give is an consiste overview of our strategy for this computation. graviphoton fluxes running through the AdS field of the results of [ separately analyse the quadratic operators on AdS particular, we will consider the direction again on account of the graviphoton flux, and the re has been completely solved for in [ BPS black hole and compute the log term in the one-loop partit where the last termeigenvalues is the gauge fixing term. The consequence o The longitudinal and zero modes modes of In this section we will carry out the computation proposed at 6 The one-loop determinants in the graviphoton background operator. The and 6 real scalars to background gravity. Theiranalysis contribution to of the section heat kerne not mix with any other field. Their action is that of a real scal as well as the Hodge Laplacian over vector fields in AdS JHEP03(2014)043 , . . 2 e is N 2 to N 2 a 2 Z Z 2 new / φ (6.7) (6.5) (6.6) ) E ) 2 the S s. The S 2 s,s ⊗ to ) 2 + 2 , and the S v 2 ( cases, even ( , a a old respectively, 1 2 K + 1) on S E a 2 ( ℓ a ( N 1 Z − / radius and S ) and ) ia 2 2 2 with the graviphoton S 1 ± s,s a 2 ⊗ ), while the scalar 2 + S ]. To compute the heat S v ( , which are again related ( 6.3 ⊗
48 . We will compute the heat i K 2 2 those of the Laplacian, the tion of the ghost fields. We ue shifts from 5 s [ e from ± = 4 vector multiplet in a via ( ) + λ 1 2 N a N , a 1 . Further, on comparing the kinetic , Z s have radii 1 − / 1 2 2 a ). We will then analytically continue old a
( ia 2 − invariant states and compute the heat E . We shall now compute N ± 6.4 S 7→ Z iλ N / 2 5.1 ⊗ ) of radius to orbifold of AdS Z ) 2 ]. The analysis for the fermions is similar. 2 s 2 S 7→ . The effect of this coupling on the spectrum 2 S N 2 + 30 ⊗ . [ a 2 Z ) and ( S s,v N – 29 – 2 ( ia, a ( Laplacians are to each other. Finally, the shifted Z 6.3 . 2 K / ia, ℓ ) 7→
on AdS 2 on the S in section 2 1 , where the two S ) + and S a 7→ S λ , a A N 2 2 1 1 and S of radius 2 a Z S ⊗ a − / , a 2 2 ( ) 2 1 ⊗ 2 ia a N S is as per ( 2 ( ± Z ) ⊗ / 2 2 AdS N ) S s,s 2 Z ( ), it is apparent that they are related to each other by the sam ( S / ) are related to each other by the same analytic continuation ) ) on the S ⊗ K 2 2 6.3 does not mix, but its eigenvalue does get shifted from zero to is equal to the degeneracy of S do not mix with the scalar. In both the AdS 6.4 S s,s background with a graviphoton flux running through both S A ) ( ( ⊗ 2 2 L 2 φ S s,s K new S denotes the heat kernel of the vector-scalar mixed system on ( ( A 2 , and from E orbifold of S K ⊗
) and ( N − = 4 vector multiplet in the ) and ( 2 1 2 Z N / ) 6.1 Z N multiplied with the heat kernel for the scalar on the S ± ) 2 6.2 ) =4 S 2 1 s,s ⊗ a 2 couples to the gauge field + , a + 1 S v 1 ( 1 ( ℓ a = 0 mode of φ ( K 1 b ℓ − K ia kernel over them only.have The already last computed term represents the contribu subscript means that we have to pick out the of radius has already been reviewed above. 6.1.1 The bosonicWe determinants consider the first the boson contribution which is given by operators in ( Here the degeneracy of and the mixing along each S though the eigenvalues ofdegeneracies the do quadratic not. operator That shift is to from say, that though the eigenval Again, the modes identically coupled to kernel in the the answer thus arrived at via While the degeneracies do not change, the eigenvalues chang kernel of the analytic continuation as the AdS 6.1 The graviphoton background on scalar eigenvalues ( to each other through the analytic continuationflux, for we fermion will therefore employ the same strategy as in section In this section we will compute the heat kernel for an as those of the Laplacian on AdS The to arrive at the answer on orbifold of an S ± JHEP03(2014)043 . is
N . . 1 2 Z 2  / 2 (6.8) (6.9) +1) ) (6.13) ± ℓ
(6.15) (6.10) (6.11) (6.12) (6.14) 2 1 ( , 2 ℓ + S 2 ℓ t . . a ( + 1 ′ ′ +1) 2 2 ′ ℓ ℓ 2 2 . ⊗ a t ℓ − ℓ a ′ uce this to ( . With the E E e ℓ ′ 2 2 , 2 2 2 2 ℓ , − t t E S 1 a a t ℓ a  e  2 2 t − E − − − a + e 2 2 e e N t πs 2 ia ↔ a − ℓ ℓ N ) # πm e  E E 2 3 − ± ℓ 1 ℓ 2 1 2 1  e t t a E + E a a ′ ℓ ) ℓ 2 1 2 1 ′ t cos ( t a − − a E 2 2 t e e  ℓ,ℓ a 2 1 − t − ( a e kernel on e − χ ) that N e − πs N N ′ + 1) to 1 πm πm e  N ℓ − =0 round for fermions on S 2 2 πm   ℓ ( ℓ, ℓ X 2 m N ( 2 χ 1 ) s sin sin 2 1 − N 1 N πm D +   =0 ∞ cos ′ ia ℓ X  ℓ  ( ≡ 2 ± 2 =1 ∞ N N 2 1 t t ℓ a X πm πm a i 4 N   − cos πm + − m ) ) e e ′ ′    + ) +1 +1) ℓ,ℓ ℓ,ℓ ′ ℓ 2 N N ′ ( ( πm ) ℓ N E = 2 πm ℓ,ℓ 2 3 χ χ  2 1 ( (2 t a +   π χ ′ =0 =0 ) 2 ℓ − h ′ ′ sin ′ ( ) ∞ ∞ e – 30 – X X 1 2 2 1 =0 t ) ℓ,ℓ ℓ,ℓ ℓ,ℓ ′ a ′ is given by ∞ ( + sin 1 1 X ℓ − ) ℓ,ℓ χ − − =1 =1 ( i e 1 ℓ, ℓ 2 X X  s,s t ( m m N N m a − =1 =0 s ∞   ′ + is just related by the replacement X X ℓ m N − v 8 4 N N ( D e +1) N N N ℓ πm N 8 πm =0 N ∞ K Z  − − ℓ + X (2 =0 ∞  / ℓ X π ) 1 2 ) + N N m ) 2 1 ) 2 h − sin Z Z s 2 3 2 + S = + / / X s S N + ⊗ ) ) 1 + ,ℓ 2 2 2 ⊗ sin ℓ 2 2 1 2 − 2 ( S t S S s,v ℓ a 1 ( s ( + S 4 2 E ⊗ ⊗ t ′ a − =0 ℓ 2 2 ), we obtain as our final result 1 2 − t K K ( S S X a m N − e s s ( ( χ e 8 2 1 − N t a 1 e K K N  5.15 4 =0 ∞ ) ℓ X ′ =  − e = 8 = 3 =0 +1). By using standard trigonometric identities one can red N ∞ ′ ℓ, ℓ ) = N πs X ), the heat kernel ℓ 4 ′ Z ℓ ( N N N / ( 1 s  Z Z ) ℓ =0 − − ) 2 / / 6.3 ℓ, ℓ D X ) ) 1 2 S ( m N ) 2 2 s = ⊗ = + S S =1 1 2 ∞ ℓ N s,s ℓ D X ⊗ ⊗ S ( ℓ K b 2 2 + ( " − χ S S v E Again, using standard trigonometric identities we can show b ( ( ( K = =0 =0 ∞ ∞ ′ K K ℓ X X ℓ K changes to Finally, on using ( Putting everything together, we find that The degeneracies do not change. As a result, the fermion heat to shift the eigenvalues by of the Dirac operator from spectrum ( where the form The computation for 6.1.2 The fermionicAs determinants mentioned previously, the effect of the graviphoton backg We therefore find and JHEP03(2014)043 a is . 2 1 = 0 ) 7→ ], we ℓ πa . The 2 (6.17) (6.16) (6.18) (6.19) (6.20) e non- 2 24 a . ia, a ( # ator with ℓ N , -BPS black E and Z ernel by an N 2 2 / t Z a ) ) / ia 2 on AdS s − ) , S , which is 4 2 e 1 2 + ) ollowing [ ⊗ ′ S 7→ ℓ φ 2 a ⊗ S ( s,v E 1 2 ( ( 2 1 a t ) N a K Z continued to the heat − AdS s,s / 2 1 ( ( ) e r field 2  S K , ide above reminds us that 2 ⊗ ) ) = ′ 2 ) 2 a N − πm ( ,s the terms is straight forward. 0 , a following prescription. We will  N N AdS v 1 which there are no subtleties of 2 ( ( Z Z a / S quotient is half the number of fixed / ( ) K ) N 2 cos 2 N S S Z  Z ) ) / ⊗ . Hence the overall factor of half, with the ⊗ e contribution from the conical singularities / ) + s 2 2 2 )
n from the smooth part of the manifold, the a + l times the volume. The two coincident heat s,s N 2 2 ( πm πa S above and continue + S 2 AdS s,v AdS v  ⊗ N ( ( ( ) ( 2 − N Z ) ⊗ ,ℓ S / ′ b K Z ( ) 2 ℓ s,s / ( 2 K S + +
χ ,K 2 2 L . This may be decomposed as ⊗ 2 t ) a – 31 – v N 2 = 4 multiplet the overall fermion contribution is . Further, we have to be careful to add the S 4 N =0 AdS Z ∞ + ′ Z / X N − ℓ T / ⊗ ) , which is N e AdS v Z ) 2 2 ia, a ( ( 2 2 1 2 S / ( =0 ∞ t a S S ℓ ) X 4 ⊗
K N ) s ⊗ 2 1 2 Z − 2 + / S − =1 e s,s ) gets continued again as before because the field for which when we analytically continue the vector heat kernel from X 2 m N ) = + AdS s,v − ⊗ ) ( AdS S v ( a s ( ( ) 2 ( + ⊗ ]. In the analytic continuation, the volume of the S = K + 2 4.4 K 2 N S s,s f 24 S ( ( Z [ s,v + ( / ⊗ ). = 0 term, we find AdS K ) 2 K N ia 2 s S K s 1 2 This will yield the regulated heat kernel of the kinetic oper Z S / K ) ⊗ ) gets continued to the corresponding scalar heat kernel in th 6.12 7→ 2 2 22 " ) S s,s a 2 2 ) = t a ⊗ + a 4 s,s 2 AdS v ( . ( ( ( = 4 vector multiplet in the near horizon geometry of a quarter ) − N ia e . We will multiply the resulting functional form of the heat k K K Z AdS s,s 5 2 1 / N ( ( t a 7→ ) 4 2 a K . We begin with the bosonic contribution to the heat kernel. F S − 2 e ⊗ 2 = 4 4 ) N b AdS s,s − ( ( K The origin of the factor of half is as follows. Firstly, for th to AdS = K 22 2 K given by where we have used ( prime on the superscript of the second term on the right hand s kernel of the We will now focus on how the above results may be analytically continuation replaced by the (regularised) volume of AdS the graviphoton flux on S have to compute overall factor of half. 6.2 The analytic continuation to As there are two Dirac fermions in an modes mentioned in section zero modes. Hence the analytic continuation being done is just the scalar, for as in section it accounts for thepoints fact on that the the sphereintegrated number quotient. heat of kernel Secondly, fixed is forkernels points given the are on by contributio related the the by Ad coincident heat kerne and separating out the It should be clear thatIn the particular, analytic continuation for most of compact geometry and hole. To carry out thetake analytic the continuation, we heat will kernels use computed the on It now remains to compute since the zero modes of the gauge field do not mix with the scala JHEP03(2014)043 ) gitu- 6.17 (6.22) (6.23) (6.26) (6.25) (6.21) (6.24) (6.29) (6.28) (6.30) , + 1 ,   ) ) . This has already a . We therefore find 2 ( we have to add back ia, a N . 4.4 ( Z try, which is -1, so 2 ogarithmic contribution / N 2 s S Z nsoring the zero modes of nel is finally . / ) + 1 es is ) K ) 2 S ed from the expression ( . a. s,s . ⊗ to AdS ) + 1 ia, a 2 + ) ( a ) + 2 S v 2 ( ( ( N ) + 1 N Z K ia, a / Z ia, a = +1 (6.27) ( ) / ( a. ) 2 2 f ) s ia, a N S S ) + N ( s,s Z ⊗ K Z s,s K / . (+1) = log 2 + / N ) S + v ¯ s ) 2 − Z + ( ¯ ( s ia, a 1) log L 2 d S / ( v b S = 0 ) K − ⊗ ) 2 2 + ⊗ ǫ 2 s as discussed in section K a 2 1 S ) = 2 T S + log b S v – 32 – ( a ⊗ Z 2 = ( ( ( 2 ( = S ) together, we find that the bosonic heat kernels φ 1 2 S s,v K ( b K ) = ( N 1 2  Z a zero = log K K / ( − 6.23 2 1 ) S 2 1 and 2 N S = Z 1 non-zero ) + = / ⊗ ′ φ f ) ) is given by ) = 2 ) K b 2 a a ,s S K non-zero ( ( log 0 K ia, a ) ) and ( ⊗ AdS ) that v S ( N 2 N ( ( s,s Z Z ) ) N / / K with the non-zero modes of the scalar on S + Z ) ) 6.19 AdS v 6.22 s,s s,s / 2 2 ( ( 2 ) S ) and found to be, S + + 2 L L ⊗ K ⊗ ), ( S v v 2 2 being added is because when we analytically continue the lon ⊗ + + 2 5.29 T T S N AdS v s AdS v ) and ( 6.18 ( Z ( ( ( ( / K 2 K K s 2 S 6.21  K 1 2 ), ( = b 6.20 = 0 modes of the scalars K ℓ the dinal and transverse modes of the vector fields from S where the 2 Therefore the contribution to the log term from non-zero mod Finally, putting ( Then the contribution from the non-zero modes to the heat ker Now the fermionic contribution may be analytically continu to obtain contribute from ( The first term the vector field on AdS this is the heat kernel over the non-zero modes obtained by te which simplifies to Thus the two terms,vanishes. when added, cancel each other and the net l We’ve already counted the number of zero modes on the 4d geome been evaluated in ( JHEP03(2014)043 . , 2 2 N Z , in ]. In 2 N 51 – 49 e analysis of this = 4 supergravity. tch the logarithmic N rnel expansion by explic- constructed in [ am “Higher Spins, Strings Krishnan, Sameer Murthy, k, SL was supported by a totic behaviour of the heat ey, and Ashoke Sen for sev- uld especially like to thank -0093843, 2010-220-C00003 = 8 string theory along the 4 rried out. SL’s work is sup- HRI, Allahabad, NORDITA, placian and using the Euler- ces results for the integrated k hole in = 4 vector multiplet in a N ould firstly be very interesting heat kernel over arbitrary-spin To carry out this computation, so be used for computations of ess. In addition, these methods or freely acting quotient groups to the logarithmic correction to ]. We start with the expression of the quantum entropy function N 24 of radius different than the other S vanishes for arbitrary values of 2 a = 4 and N orbifolds. Though we worked mostly with S N – 33 – Z orbifolds, we also obtained group-theoretic forms ]. Additionally, we will recover the result for the s to an AdS and the analytic continuation to AdS N 2 24 Z 2 have different radii which in our expressions can be obtained 2 ]. S 52 and 2 ] for the leading saddle-point. This requires us to extend th AdS 25 , ]. Finally, the motivation for this work was to concretely ma 24 41 , , their product spaces and their 2 40 The techniques developed in thisentanglement paper entropy [ could potentially al by analytically continuing one of the S itly enumerating the eigenvaluesMaclaurin and formula, degeneracies we of will theheat show La kernel that arrived this at precisely previously reprodu in [ As a zeroth order check of our method of evaluating the heat ke contribution of a single masslessthe minimally entropy coupled of scalar an extremal black hole also obtained in [ postdoctoral fellowship at the ICTS-TIFR. A The scalar Laplacian on S Shamik Banerjee, Justin David, Rajesheral Gopakumar Soo-Jong discussions, R comments, andStockholm, correspondence. NIKHEF Amsterdam, SL and thanks theand GGI, Florence’s Dualities” progr for hospitalityported while by part National of Research thisand Foundation of work 2012K2A1A9055280. Korea was grants During ca 2005 the initial stages of this wor Acknowledgments We would like to thank Pallab Basu,Suvrat Archisman Ghosh, Raju, Chethan and Spenta Wadia for helpful discussions. We wo would also be useful inthis the case case the of extremal black holes in AdS corrections about exponentially suppressed saddlewith points the corresponding microscopic answers for paper to the gravity multiplet as well, which is work in progr In this paper we computed the heat kernel for a single 7 Conclusions lines of [ AdS in [ for the heat kernel suggestivefields of in a higher-dimensional possible spheres solutionto and for make hyperboloids. this the connection It to w the results previously obtained f orbifold of the near-horizon geometry of a quarter-BPS blac kernel over spheres and hyperboloids with accordance with the expectation fromwe microstate developed counting. new techniques to compute the short-time asymp We found that the contribution proportional to log JHEP03(2014)043 is uct 2 S (A.5) (A.1) (A.2) (A.4) (A.7) (A.6) (A.3) (A.8) × , 2 y analytic  ) , 2  t AdS ( O (0) . Also, to extract . 1) ng so, we find + . − ∞ 2 k 2 +1) ( a ℓ  a t ( and therefore the sum ) f ℓ expansion of the scalar 2 t 15 t t . We know that the heat x − y analytically continued − ( 2 e e ) + from the above expression. O on note that 2 2 precisely with the series ex- e this, as there is a volume ∞ g e Sitter case. Note that this t + a ( + 1 , S πa 4 1) . ℓ 4 + ) 2 − (0) . 2 . t k πa 1 3 t ( 1) ( 2 =0 f ∞  +1) − 60 2 +1) ℓ a X ℓ O k  · a ( x ( ( ℓ ) + x t k ! 2 f +  t ) onwards in this expansion. Expanded S k − ) k ! B 2 2 − 2 e πt k 1 t t e B a ( t 4 ( =2 ∞ k X 15 O O =2 ∞ + + 1) k X will be polynomial in + + 1) + + 2 = (Vol. ℓ − – 34 – x 2 4 f t (2 1 ) this can be expressed as πa 2 a 2 +1) 2 (2 t a +1) ℓ and all its derivatives vanish at t a a πa ( ℓ 12 ( ℓ =0 ∞ + ℓ ℓ t ≡ f X t can be done to obtain 60 4.18 + − − 3 1 ) ]. Then the integrated heat kernel on − ℓ e =  1 2 x e + ( ) 24 = g times the scalar heat kernel on S 2 f = S g 2 πt 1 g + 1) S 4 + 1) AdS S ℓ ℓ + , the function (2 t (2 2 . The scalar heat kernel on this product space is just the prod dℓ 2 =0 ∞ 1 πa ℓ S X ∞ = (Vol. 0 12 ⊗ g = of (2.15) of [ Z 2 − A g 2 t  S a 2 so we will neglect terms from (0) + t 1 = πa f 2 s 2 1 . The integrated heat kernel on AdS can hence be written down b − ia = g above is a finite sum and can be evaluated completely. Upon doi ) = 7→ S t k ( a K Firstly, note that the integral over is the integrated heatdivergence kernel, in AdS and which we will cannot be naively missed. continu To do this continuati Now we would like to analytically continue this to the Anti-d Using the Euler-Maclaurin formula ( given by where where we have also used the fact that The expression for thepansion coincident in heat kernel ¯ above matches out to any finite power of for the integrated heat kernel on the two-sphere over The last sumvia is the coincident heat kernel, which can be safel We will now extract the short-time asymptotic behaviour of the log term from the heat kernel, we need the order 1 term in th coninuation as of the scalar heat kernel on AdS heat kernel on AdS kernels go as JHEP03(2014)043 as oop and N (B.1) (B.2) (A.9) Z N (A.11) (A.10) / Z 2 / , we shall 2 6 onsider the 1. There are . − ] . Therefore the  and AdS
2 37 N ¯ s N t of the anti-periodic Z ≤ ] for the unorbifolded ). Then, the one-loop / , . . . , r O s 2 1 the Dirac operator has e integrated heat kernel 48 , , , + ubset of the anti-periodic + 4.38 ≤ n the integrand, given by + n S ¯ Z s rmions as well. The main be extracted e integrated heat kernel on Z 15 = 0 N ∈ ] to verify this statement. We riodic fermions on S ∈ nsidered in section p + Z e more degeneracies than this. 48 / ¯ s [ 1 2 + . , 2 , where 0 ) , p, q 1 3 ǫ , p, q s t a ( 2   + · 2 1  K log 1 2 as obtained in (  t + dt
2 Nr ]. and AdS 1 2 + q ¯ 180 s ∞ q 24 ǫ = + N − Z + – 35 – ℓ Z O  2 1 = , corresponding to / 1 2  + q 2 1 2 = + ¯ + s BH 60 + p δS δS p  + Np  s N 1 4¯ N   − ± = 1 = 12 λ λ . While these are not relevant to the computation of the one-l  2 ¯ s . The spectrum of the Dirac operator is given by [ ¯ s d N ], the same analytic continuation as found in [ Z 2 ∞ ǫ a / 48 2 Z is the regularized volume of AdS and AdS = . The last expression is, to our knowledge, new. We will first c 2 2 N Z . Unlike scalars and spin-1 fields, these modes are not a subse πa BH / 2 N 2 and further prescribe an analytic continuation to obtain th δS − Z . We therefore obtain / 2 N 2 Z πa 1 +1) ways of realising this as / 2 r 180 ( determinants in orbifolds of the graviphoton background co and the degeneracy ofeigenvalues each eigenvalue is 2. Then the square of In this appendix, we will compute the heat kernel formodes anti-pe on S This matches with the expression found in [ AdS where each eigenvalue is four-foldTo degenerate. see the But there point, ar consider an eigenvalue calculation on S well. It is straighforward to follow through the analysis of can therefore use theS Euler-Maclaurin formula to compute th briefly outline theresult main which elements we of shall use themodes is found analysis that in for though [ these thesecase modes can fe be are used not to a relate s the spectrum of the Dirac operator o B Anti-periodic fermions on S effective action is given by on AdS The log contribution, as before, comes from the order 1 term i where from which the one-loop correction to black hole entropy may JHEP03(2014)043 2 by N (B.5) (B.6) (B.4) (B.3) (C.2) (C.1) (C.3) Z /
2 S . jorana fermion, ⊗
. 2 2 , which means we , t
at any point on s / 2 D . We also analyt- k t ℓ i n ,..., O N f AdS 2 ∗ =0 ∞ Z ℓ O m ± s + X / , f 2 # + 1 t and t mn ±
≡ − g 2 the conical singularity, as
1 modes contribute. Then 1 this is typically a divergent ℓ 1 S =  . ± . We will first evaluate this 2 2 − 2 2 ) P − ) , l 2 = dS , 2 , +1 2 ℓ +1 n 2 ℓ iℓθ N 1 N N f N N N ]. In this case, we note that AdS 2 e ∗ 2 − | + a ℓ
+ m ℓ a s
| 24 s f +  + 720 ) = η 720 , which is the square of mn 2 Nr + 17 23 Nr is given by 2 ( + 17 ( η / 2 2 gg D t 2 2 ]. Then, the analytic continuation of this t a N a − N √ Z − N − 37 , we use the Euler-Maclaurin formula to find 7 e / sinh 7 e 2  1 + cosh 4.2 − dθdη – 36 – (Vol.AdS  + 1 | + 1) 2 1 Z ℓ + 1) | r 2 − Z r πa N π − 1 N ∈ 2 4 6 4 ( 2 ℓ 2 X 6 4 ( N 1 p = N =0 = ∞ " in section − r ], and then show how a zeta function regularisation gives X =0 0 0 by zeta function regularization s X = + 2 1 N n N 24 n = ℓ 2 in these terms, as per the rules outlined in the main text. 2 Z f S + s =0 ∞ / r X k 2 ia K 2 term computed in [ − , φ 1 0 N ℓ , but we are also considering a Dirac fermion, rather than a Ma f AdS t 2 1 φ = 7→ − K = m N a 1 N Z N ∇ / Z 2 / f = = S 2 f S ] we will compute the heat kernel of ℓ K N m K Z f 24 / 2 has only one such singularity, not two, as in the case of S f AdS N Z K / 2 Following [ 23 quantity which we define as a different regularization. On non-compact manifolds like A ically continued via In this section we will evaluate the number of zero modes on Ad C Zero modes on AdS This matches with the Here, we introduced aAdS factor of half in the contribution from Now, as for the scalar on S where rise to the same result.is We a begin homogeneous with space the and method of we [ can evaluate the integrand answer to the AdS case is that the heat kernel for Dirac fermions on S are the discrete zero modes of the vector Laplacian on AdS where answer using the methods of [ the space, in particular, the origin. In that case only the heat kernel for the Dirac operator can be given by which yields a factor of 2. These two factors cancel. should multiply a factor of the number of zero modes is given by JHEP03(2014)043 . ∞ , , (C.4) (C.5) , integral ,..., ommons system 2 ] η , 5 ± , . The modes D 1 2 , − ± 1 = D string theory p (1993) 3427 on AdS ting of Microstates = 4 . π (2006) 018 N 1 2 N hep-th/0412287 D 48 [ − string theory + ]. 04 , where . Now we will evaluate the ]. θ supersymmetric Type II string redited. 2 r as obtained by regularizing ]. n find, on doing the = 4 Np Asymptotic degeneracy of dyonic 7→ ]. (0) = ]. = 4 N SPIRE θ = ζ ]. JHEP SPIRE IN ℓ , N SPIRE IN Phys. Rev. (2004) 075 ][ IN , ][ SPIRE SPIRE 1 = 2 ][ 12 ) obtained in the main text. SPIRE IN IN ]. ]. IN ][ . ][ =1 ∞ ℓ X Counting dyons in 1 ][ 5.33 − JHEP SPIRE SPIRE = 2 = , IN IN – 37 – 0 π 1 ][ ][ n 2 hep-th/0603066 Product representation of Dyon partition function in Dyon spectrum in [ Recounting Dyons in dθ hep-th/0602254 [ π 2 arXiv:0708.1270 [ 0 hep-th/9607026 Spectrum of dyons and black holes in CHL orbifolds using hep-th/0607155 Z [ [ ), which permits any use, distribution and reproduction in hep-th/9601029 0 [ Microscopic origin of the Bekenstein-Hawking entropy − ]. Z ∞ ]. X ∈ as the regularized volume of AdS (2007) 077 ℓ Dyon spectrum in CHL models CHL dyons and statistical entropy function from (2006) 064 2 hep-th/0605210 hep-th/0505094 = 11 [ [ SPIRE πa 06 0 SPIRE (2008) 2249 (1997) 543 2 (2006) 073 IN (1996) 99 n IN CC-BY 4.0 − ][ 40 ][ 11 JHEP This article is distributed under the terms of the Creative C JHEP , Black hole entropy is the Noether charge B 484 , B 379 (2006) 072 (2006) 087 JHEP ]. Black Hole Entropy Function, Attractors and Precision Coun , string states and black hole entropy 11 10 = 4 SPIRE IN hep-th/0510147 gr-qc/9307038 JHEP Gen. Rel. Grav. Borcherds lift CHL models [ JHEP [ [ Nucl. Phys. N Phys. Lett. theories which survive the projection are the ones for which ℓ [1] A. Strominger and C. Vafa, [5] D. Shih, A. Strominger and X. Yin, [7] A. Dabholkar and S. Nampuri, [8] J.R. David, D.P. Jatkar and A. Sen, [9] J.R. David, D.P. Jatkar and A. Sen, [6] D.P. Jatkar and A. Sen, [3] R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, [4] G. 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