Logarithmic corrections to twisted indices from the quantum entropy function

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Abstract

We compute logarithmic corrections to the twisted index B ^sub 6^^sup g^ in four-dimensional ...... and ...... string theories using the framework of the Quantum Entropy Function. We find that these vanish, matching perfectly with the large-charge expansion of the corresponding microscopic expressions.

Published for SISSA by Springer

Received: July 8, 2014 Revised: September 18, 2014

Accepted: October 21, 2014 Published: November 4, 2014

Logarithmic corrections to twisted indices from the quantum entropy function

Abhishek Chowdhury,a Rajesh Kumar Gupta,b Shailesh Lal,c Milind Shyania,d and Somyadip Thakure

aHarish-Chandra Research Institute,

Chhatnag Road, Jhusi, Allahabad 211019, India

bICTP, High Energy, Cosmology and Astroparticle Physics, Strada Costiera 11, 34151, Trieste, Italy

cDepartment of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea

dBirla Institute of Technology and Science Pilani,

K.K. Birla Goa Campus, India

eCentre for High Energy Physics, Indian Institute of Science, C.V. Raman Avenue, Bangalore 560012, India

E-mail: [email protected], mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We compute logarithmic corrections to the twisted index Bg6 in four-dimensional N = 4 and N = 8 string theories using the framework of the Quantum

Entropy Function. We nd that these vanish, matching perfectly with the large-charge expansion of the corresponding microscopic expressions.

Keywords: Black Holes in String Theory, AdS-CFT Correspondence

ArXiv ePrint: 1404.6363

JHEP11(2014)002

Open Access, c

The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2014)002

Web End =10.1007/JHEP11(2014)002

Contents1 Introduction and review 1

2 The heat kernel for the laplacian on AdS2 S2

[parenrightbig]

/ZN with twisted boundary conditions 52.1 The heat kernel for scalars on AdS2 S2

[parenrightbig]

/ZN 9

2.3 The heat kernel over discrete modes 9

3 Logarithmic corrections to the twisted index 103.1 The heat kernel for the N = 4 vector multiplet 11

3.2 The heat kernel for the N = 4 gravitino multiplets 13

3.3 The zero mode analysis 153.4 Logarithmic corrections to the twisted index 16

4 Conclusions 16

A The g-charges for the N = 8 supergravity elds 17

1 Introduction and review

Indices carry important information about the spectrum of dyons in string theory. In particular, in four dimensional string theories the helicity trace index, dened by [1, 2]

B2n = 1

(2n)!Tr

h(1)2h (2h)2n[bracketrightBig]

receives contributions only from those BPS states in the string theory which break less than 4n supersymmetries. Here the trace is over all states in the string theory that carry some specied electric and magnetic charges. This has now been computed exactly for a wide class of N = 4 and N = 8 string theories [315]. In an expansion in large charges it

may be shown that this reproduces the correct semiclassical entropy of an extremal black hole carrying the same charges as the dyons. In many cases, higher-derivative and quantum corrections have also been computed on the macroscopic side and the results have been successfully matched with the corresponding corrections computed from the microscopic formula. We refer the reader to the reviews [1619] covering various aspects of this program for details and a more complete set of references. The computation of the quantum corrections is performed using the formalism of the Quantum Entropy Function [20, 21]. This proposal exploits the fact that the near-horizon geometry of extremal black holes always contains an AdS2 factor [22, 23]. In particular, for spherically symmetric black holes

1

/ZN 7

2.2 The heat kernel for fermions on AdS2 S2

[parenrightbig]

JHEP11(2014)002

(1.1)

in four dimensions, the near-horizon geometry, embedded in 10-dimensional supergravity, contains an AdS2 S2 factor coupled to background U(1) uxes and scalar elds. The

entire conguration is completely determined by the SO(2, 1)SO(3) isometry of the solu

tion, along with the electric and magnetic charges carried by the black hole. In Euclidean signature, this conguration is given by

ds2 = a2 d2 + sinh2 d2

[parenrightbig]

+ a2 d2 + sin2 d2

[parenrightbig]

, 0 < , 0 < 2,

F (i) = ei, F (i) =

(1.2)

where the background has r U(1) uxes and s scalar elds, and a is a function of the electric and magnetic charges of the black hole, determined in terms of the ei, pi

.

Using this fact it has been argued that the quantum degeneracy dhor (~q) associated

with the horizon of an extremal black hole carrying charges ~q qi is given by the unnor

malized string path integral, with a Wilson line insertion, over all eld congurations that asymptote to the attractor geometry of the black hole. In particular, [20, 21]

dhor (~q)

exp

pi4 sin , w = uw, 1 i r, 1 w s.

JHEP11(2014)002

i

[contintegraldisplay]

qidAi[bracketrightbigg]

nite

AdS2

. (1.3)

The subscript nite reminds us that the path integral naively contains a volume divergence due to the presence of the AdS2 factor. Regulating this divergence is carried out in accordance with the AdS/CFT correspondence. Though (1.3) computes a degeneracy rather than an index, it may be shown that one may use this expression to compute the helicity trace index as well, which can then be compared with the microscopic results [24].

Since its proposal, the conjecture of [20, 21] has been put to a variety of tests. Firstly, the leading saddle-point of the path integral is the attractor conguration (1.2) itself, and it may be shown that the value of the path integral (1.3) at this saddle-point is the exponential of the Wald entropy associated with the black hole. Further, by expanding the massless elds of four-dimensional supergravity in quadratic uctuations about this saddle-point, the logarithmic correction to the Wald entropy may be extracted from (1.3) and matched with the microscopic answer [25]. This has been successfully carried out for the 14 -BPS black holes in N = 4 supergravity and

18 -BPS black holes in N = 8 supergravity [26, 27]

and for rotating extremal black holes in [28]. The corresponding expressions for 12 -BPS

black holes in N = 2 supergravity have also now been obtained [29], however in this case

the microscopic results are so far not available. Recently, [30] presented a new approach to the computation of logarithmic terms from (1.3) which greatly simplies the intermediate steps encountered in the calculations of [2729]. We also note here that (1.3) has been exactly evaluated for N = 4 and N = 8 string theories using localisation in [3135] and

the answer obtained precisely reproduces the microscopic expressions computed from the indices Bn.

Further, if we restrict ourselves to special subspaces of the moduli space which admit discrete symmetry transformations generated by an element g and also require that the charges of the dyons be g-invariant, then we may dene twisted indices as

Bg2n

1 (2n)!Tr

hg (1)2h (2h)2n[bracketrightBig]. (1.4)

2

The group generated by g is taken to be isomorphic to ZN. These indices were computed in [36, 37], and a proposal for their macroscopic interpretation was also presented in [36]. In particular, [36] considered Type II string theory compactied on M T 2, where M

could be either T 4 or K3, and g was the generator of a geometric ZN symmetry that acts on M and preserves 16 supercharges. The twisted index Bg6, which receives contributions

from dyonic states which preserve 4 supersymmetries all of which are g-invariant, was then computed. It was found that the answer in the large-charge limit takes the form [17]

Bg6 (Q, P ) = e

SBH

N + O (1) , (1.6)i.e. the logarithmic correction to the entropy vanishes. Here

SBH =

qQ2P 2 (Q P )2, (1.7)

the Wald entropy of an extremal black hole carrying electric and magnetic charges (Q, P ). This is also the asymptotic expansion arrived at from Type IIB string theory on the CHL orbifold [37]. In this paper we shall show how this result arises from a macroscopic computation of the kind performed in [26, 27, 38, 39] for the entropy of the black hole.

Before we do so, we briey review the proposal made in [36] regarding the macroscopic interpretation of the index Bg6. The key ingredient of the proposal is that Bg6 is indeed captured by a string path integral of the type (1.3) in AdS2. However, the path integral must now be carried out over elds which obey twisted boundary conditions along the -circle of the AdS2. In particular, as shifts by 2 the elds must transform by g. This partition function was denoted by Zg in [36]. When we impose these boundary conditions then the attractor geometry itself is no longer an admissible saddle-point of the path integral as the -circle is contractible in the interior of AdS2, which leads to a singularity. Let us instead consider the following ZN orbifold of the attractor geometry (1.2), generated by the identication

: (, ) 7

+ 2

N , 2N

3

Q2P 2(QP)2

N (O (1) + . . .) . (1.5)

Therefore, if we assign an entropy to the index by taking its logarithm then we nd that

ln |Bg6 (Q, P ) | =

JHEP11(2014)002

. (1.8)

Then it may be shown by an appropriate change of coordinates that the resulting eld conguration still asymptotes to the full attractor geometry (1.2). Additionally, this orbifold preserves enough supersymmetry that its contribution to the path integral (1.3) does not automatically vanish by integration over the fermionic zero modes associated to broken supersymmetries. For these reasons, these eld congurations are also admissible saddle-points of the quantum entropy function (1.3).1 Using these inputs, [36] proposed that Zg would receive contributions from the saddle-point obtained by imposing a ZN orbifold generated by the action of on the attractor geometry, with g-twisted boundary conditions

1These orbifolds have xed points at the origin of the AdS2 times the north or south poles of S2 and a priori it is not clear whether or not this is a consistent orbifold of string theory in the presence of background uxes. If however the 10-dimensional attractor geometry also contains a circle C which is non-contractible

imposed on the elds. It was further shown that the value of Zniteg at the saddle-point was

given by e

SBH

N , in agreement with the asymptotic growth of Bg6 from the microscopic side. In this paper we will show that the correspondence between Zg and Bg6 exists even at the quantum level. In particular, we will compute the log correction to the entropy

given by log Zg by expanding about the ZN orbifold of the black hole attractor geometry generated by the action of, where we impose g-twisted boundary conditions on the elds. We will nd that the answer vanishes, in accordance with the microscopic results. In order to compute log corrections, we shall use the fact that the contributions of the form log a to the partition function of a theory dened with a length scale a are completely determined from the one-loop uctuations about the saddle-point, where we may focus exclusively on massless elds and further neglect higher-derivative terms [25]. Therefore the only elds that can contribute to the log term in log Zg are the massless elds about its admissible saddle-points. We shall compute the log correction, focussing on modes which obey appropriate twisted boundary conditions, and nd that the answer vanishes. While we do this computation explicitly for N = 8 string theory obtained by compactifying

Type II string theory on T 6, this is only for deniteness and we shall see that the results obtained would carry over to the N = 4 case as well. We now give a brief overview of

the computation, emphasizing the overall strategy and the important di erences from the analyses previously carried out in [38] and [39]. We will decompose the N = 8 supergravity

multiplet into irreducible representations of the N = 4 subalgebra which commutes with

g. These are one N = 4 gravity multiplet, four N = 4 gravitini multiplets and six N = 4 vector multiplets, each of which are charged under g as enumerated in appendix A.

Importantly for us the N = 4 gravity multiplet is uncharged under g, and therefore obeys

untwisted boundary conditions. Its contribution to the logarithmic term in the large charge expansion of Zg is therefore identical to that computed in [39]. The contributions of the gravitini and vector multiplets are however di erent from [39], and are computed in this paper.

A brief overview of the paper is as follows. In section 2 we compute the heat kernel for scalars, Dirac fermions and discrete modes of the spin-1 and spin32 elds on

AdS2 S2

[parenrightbig]

/ZN with twisted boundary conditions. This is an extension of the analysis of [38] where the heat kernel over orbifold-invariant modes on these spaces was computed. We nd that the answer again assembles into a global part, which obeys untwisted boundary conditions, plus conical contributions which are nite in the limit where the heat kernel time t approaches zero. We put these results together to evaluate the contributions of

N = 4 vector and gravitino multiplets that obey twisted boundary conditions in section 3.

We nd that the contribution to the log term vanishes for any non-zero value of the twist. These results demonstrate explicitly that the log term in Bg6 vanishes for N = 8 string

at the origin of AdS2, then one way to avoid this potential pitfall is to accompany the orbifold (1.8) by a translation by 1

N units along C. The orbifold group then acts freely over the 10-dimensional attractor geometry. If the radius of the circle C does not scale with the AdS2 and S2 radii a, the precise details of the shift will not be relevant for us [25]. We do assume tacitly in our analysis that the generator includes such a shift along the internal directions as well. Such orbifolds have been explicitly dened in the 10-dimensional theory in [24, 40].

4

JHEP11(2014)002

theory and N = 4 string theory. We then discuss how our results also prove that the log

term vanishes even about exponentially suppressed corrections to the leading asymptotic formula for Bg6 and conclude.

2 The heat kernel for the laplacian on AdS2 S2

[parenrightbig]

/ZN with twisted

boundary conditions

The goal of this paper is to compute logarithmic corrections to the partition function Zg dened as the path integral (1.3) with g-twisted boundary conditions. These corrections only receive contributions from the one-loop uctuations of massless elds over the ZN orbifold of the attractor geometry generated by. The one-loop partition function about this background is determined in terms of the determinant of the kinetic operator D evaluated over the spectrum of the theory. We shall dene this determinant by the means of the heat kernel method [41]. The discussion below has has also been reviewed in the present context in [38, 39] so we shall mainly recapitulate the key elements of the method.

We shall focus on operators of Laplace-type dened over elds on a manifold M with

a length scale a. The eigenvalues of such operators scale as 1

a2 and are denoted by na2 and the corresponding degeneracies are dn. With these inputs we may dene the integrated heat kernel (referred from now on as simply the heat kernel) as

K (t) =

Xndneta2 n. (2.1)

Then the determinant of D may be dened via

ln det D =

[integraldisplay]

JHEP11(2014)002

ds

1 s K (s) , (2.2)

where is a UV cuto and s = t

a2 . Therefore, ln det D contains a term proportional to

ln a, given by

ln det D = 2K1 ln a + . . . , (2.3)

where K1 is the O s0

[parenrightbig]

term in the small s expansion of the heat kernel K(t) and the . . . denote terms that are not of the form ln a. From this expression, the term proportional to ln a in ln Z may be extracted. Logarithmic corrections to black hole entropy have been

computed from the quantum entropy function in this manner in [2629, 38, 39]. We remind the reader that the small s expansion of the heat kernel is in general non-trivial and contains

1sn terms which have to be carefully computed. We will however nd useful simplications which enable us to analyze the problem e ciently.

Before proceeding further, we remind the reader that the analysis presented above has subtleties when the operator D is only positive semi-denite, i.e. has zero modes. In that case the one-loop partition function contains the determinant of D evaluated only over non-zero modes. The zero mode contribution needs to be analyzed separately [25 27, 42]. The kinetic operator for which we compute the heat kernel is the one studied in [26, 27, 38, 39]. This has zero modes over spin-2, spin32 and spin-1 elds. However, the

5

zero modes of the graviton and gravitino arise only within the N = 4 gravity multiplet [27]

which obeys untwisted boundary conditions in the path integral Zg and have therefore already been accounted in the analysis of [39]. Additionally, it may be shown that the log term for vectors may as well be extracted out by dening the heat kernel over all eigenvalues n, including the zero eigenvalue, and extracting the O s0

[parenrightbig]

term as before [26]. We will therefore ignore the presence of zero modes in our present analysis.

We now turn to the main computation of this section, which will provide us with the essential tools we need to compute logarithmic corrections to the partition function Zg.

These are the heat kernels of the Laplacian over scalar elds and of the Dirac operator over spin-12 elds on AdS2 S2

[parenrightbig]

/ZN, where the ZN orbifold is generated by. The heat kernel over the uctuations invariant under the-generated ZN orbifold was computed and the log term extracted in [38, 39]. The analysis of this section is entirely analogous, with the only di erence being that we now focus on modes which obey twisted boundary conditions under the orbifold. We nd that the essential steps carry over directly from [38, 39] with only minor modications. For this reason, we shall focus on the scalar on AdS2 S2

[parenrightbig]

to illustrate the steps and main modications and then mostly enumerate nal expressions for the spin-12 eld. Further, as has been shown in [26, 27], the higher-spin elds in the supergravity multiplets may be expanded in a basis obtained by acting on the scalar with the background metric and covariant derivatives and acting on the spin12 eld with gamma matrices and covariant derivatives. It turns out that the heat kernel over all quadratic uctuations may be organised into the heat kernel over scalars and spin12 fermions with appropriate multiplicities and shifts in eigenvalues. This will also be of great utility in our present analysis. Finally, we note that the heat kernel expression (2.1) contains both eigenvalues and degeneracies of the kinetic operator D. On manifolds like AdS2 the notion of degeneracy is subtle and requires a careful denition. It takes the form of the Plancherel measure [4345]. On quotients of AdS spaces, it turns out to be useful to exploit the fact that harmonic analysis on AdS is related to the sphere by an analytic continuation [4345]. By exploiting this analytic continuation, one may obtain the heat kernel and degeneracies of the Laplacian on these orbifolded spaces as well [38, 39, 46, 47]. We shall adopt this approach in this paper as well. In particular, we will consider the geometry given by

ds2 = a21 d2 + sin2 d2

[parenrightbig]

which is related via the analytic continuation

(a1, a2) 7(ia, a) , 7i, (2.5)

to the AdS2 S2 [parenrightbig]

The ZN orbifold generated by acts on both these spaces via

: (, ) 7

+ 2

N , 2N

6

JHEP11(2014)002

/ZN

+ a22 d2 + sin2 d2

[parenrightbig]

, (2.4)

/ZN geometry

ds2 = a2 d2 + sinh2 d2

[parenrightbig]

[parenrightbig]

+ a2 d2 + sin2 d2

. (2.6)

. (2.7)

Following the strategy of [38, 39, 46, 47], we will do the computation on S2 S2

/ZN. We will however need to be mindful of an important subtlety while performing this analytic continutation which arises due to a class of discrete modes of the vector and spin32 elds in AdS2 [43, 44]. These are normalisable eigenfunctions of the Laplacian over AdS2 which are not related to normalisable eigenfunctions of the Laplacian over S2. Their contribution is computed separately in section 2.3.

2.1 The heat kernel for scalars on AdS2 S2

[parenrightbig]

In order to compute the heat kernel for the scalar Laplacian on AdS2 S2

2 + 14 + ( + 1)

and the corresponding eigenfunctions are given by [43]

,,m,n (, , , ) = f,m (, ) Y,n (, ) , (2.9)

where, omitting normalisation factors,

f,m (, ) =

Z, (2.10)

and the Y,ns are the usual spherical harmonics on S2. We will impose the projection (2.7) generated by on the modes (2.9) as in [38]. The modes invariant under this orbifold are those for which m n = Np, where p is an integer. The heat kernel was computed over

such modes in [38]. We will look at the more general case for which

m n = Np + q, p

Z, 0 q N 1, q

1 a21

which is related to E by the analytic continuation

~

= i

1

2, (a1, a2) 7(ia, a) . (2.14)

7

[parenrightbig]

/ZN and

analytically continue the result to AdS2 S2

/ZN, we will

rst enumerate its spectrum [43]. The eigenvalues of the scalar Laplacian are

E, = 1 a2

[parenrightbig]

/ZN

[parenrightbig]

JHEP11(2014)002

, (2.8)

sinh|m|

[parenrightBig] 2F1

i + |m| +12, i + |m| +12, |m| + 1, sinh2 2

eim,

0 < < , m

Z. (2.11)

We will refer to these as q-twisted boundary conditions. However, as mentioned above, we will carry out the computation by imposing the projection (2.7) on eigenfunctions of the scalar Laplacian on S2 S2, which are given by

~,m,,n (, , a1, , , a2) = Y~,m (, , a1) Y,n (, , a2) . (2.12)

The corresponding eigenvalue is given by

E~, =

~

~ + 1

[parenrightBig]

+ 1

a22

( + 1) , (2.13)

Using the methods of [38], we nd that the heat kernel on q-twisted modes on S2 S2

[parenrightbig]

/ZN

is given by

Kqs = 1

N Ks +

1 N

N1

Xs=1

s N

1

X,~=0 ,~

e

2iqs

N etE~

[parenrightBig][parenleftBigg] , (2.15)

where Ks is the scalar heat kernel on the full unquotiented S2 S2 space and the sum from

s = 1 to N 1 represents the contribution from the conical singularities and is expressed

in terms of ,~, the SU(2) SU(2) Weyl character

,~

s N

sin (

2~

+1)sN

sin s

N

[parenrightBig]

s N

[parenrightBig]

~

s N

[parenrightBig]

sin (2+1)sN sin s

N

[parenrightbig]

, (2.16)

where and ~ are SU(2) Weyl characters. The analytic continuation proceeds in the same way as for the untwisted case [38, 39]. Firstly, the heat kernel over the unquotiented S2S2 gets continued to the heat kernel over AdS2 S2. Then the eigenvalue E~ gets continued

to E via (2.14), and the Weyl character ~ gets continued to the Harish-Chandra (global) character for sl(2, R) [48]

b

s N

[parenrightbig]

[parenrightBig]

= cosh 2sN

[parenrightbig]

, (2.17)

and the conical terms get multiplied by an overall half [38]. The factor of half accounts for the fact that under the ZN orbifold (2.7), AdS2 S2 has half the number of xed points as

does S2 S2. Finally, the sum over

~

cosh () sin s

N

JHEP11(2014)002

gets continued to an integral over . We then obtain the heat kernel for the scalar on AdS2 S2

[parenrightbig]

/ZN with the q-twisted boundary condition

to be

Kqs = 1

N Ks +

1

2N

N1

Xs=1

[integraldisplay]

1 0 db, [parenleftBig]

s

N

[parenrightBig]

1

e

2iqs

N etE, (2.18)

where

b,

s N

[parenrightBig]

= b

s N

[parenrightBig]

s N

[parenrightBig]

. (2.19)

By doing the integral over and the sum over as in [39] we nd that (2.18) reduces to

Kqs = 1

N Ks +

1

2N

Xs=114 sin4 sNe2iqsN + O (t) . (2.20)

This is the expression we shall use to compute logarithmic corrections. It contains two terms. The rst is the heat kernel of the untwisted scalar evaluated on the unquotiented space AdS2 S2. The second term is the contribution of the conical singularities. As

observed in [39] for the untwisted modes, this term is nite in the limit where t approaches zero. Hence the contribution of this term to the O t0

[parenrightbig]

N1

term in the heat kernel expansion is independent of the eigenvalue E. This will be of great utility in our further computations.

Finally we note that the expressions (2.18) and (2.20) are divergent due to the innite volume of AdS2. However, using the prescription of [20, 21] this divergence may be regulated and a well-dened nite term extracted even on these quotient spaces [38, 39]. Once this

8

is done, we obtain a well-dened expression for the degeneracy ds of the eigenvalue E in the q-twisted set of modes on AdS2 S2

[parenrightbig]

/ZN. This is given by

ds =

1N ( tanh ) (2 + 1) +

1

2N

N1

Xs=1b,

s N

[parenrightBig]

e

2iqs

N . (2.21)

2.2 The heat kernel for fermions on AdS2 S2

[parenrightbig]

/ZN

We will turn to the heat kernel of the Dirac operator evaluated over Dirac fermions on AdS2 S2

[parenrightbig]

/ZN with q-twisted boundary conditions. The computations are entirely similar to those carried out in [38, 39] once the q-twist has been accounted for as we have for the scalar in section 2.1, we shall just mention the nal result for the degeneracy of eigenvalues labelled by the quantum numbers , in the q-twisted set of modes on AdS2 S2

[parenrightbig]

/ZN.

df =

8N ( coth ) ( + 1) +

2 N

N1

Xs=1 f,+12

s N

[parenrightBig]

e

2iqs

N , (2.22)

where we have dened

f,+12

s N

[parenrightBig]

= f

s N

[parenrightBig]

+12

s N

[parenrightBig]

, (2.23)

and f is the Harish-Chandra character for sl(2, R) given by [48]

f

s N

JHEP11(2014)002

[parenrightBig]

= sinh 2sN

[parenrightbig]

. (2.24)

We may use this degeneracy to obtain the heat kernel for the Dirac operator over the q-twisted Dirac fermions. We nd that2

Kqf =

1N Kf

sinh () sin s

N

2 N

N1

Xs=1

[integraldisplay]

1

1 0 d f,+

[parenrightBig][parenleftBigg]N etE. (2.25)

As for the scalar, we may expand the conical term in a power series in t omitting the O (t)

and higher terms, carry out the integral and the sum over to obtain

Kqf =

1N Kf

e

2iqs

1

2

s N

1

2N

N1

Xs=1cos2 s

N

sin4 s

N

N + O (t) . (2.26)

We will use (2.26) in our computations for the log term in section 3.

2.3 The heat kernel over discrete modes

Vectors, gravitini and gravitons on the product space AdS2 S2

[parenrightbig]

e

2iqs

/ZN may be expanded in a basis contructed from the background metric, Gamma matrices and covariant derivatives, allowing us to express the heat kernel of the kinetic operator over supergravity elds in

2We use the conventions of [26, 27] in which the fermion heat kernel is dened with an overall minus sign and is added to the bosonic heat kernels.

9

terms of the heat kernel over scalars and spin12 elds [26, 27]. However, this analytic continuation fails to capture a set of discrete modes, labelled by a quantum number , on the AdS space for the spin-1 and higher spin elds [4345]. The heat kernel over such modes needs to be computed directly on AdS2 S2

[parenrightbig]

/ZN. Using the methods of [38, 39], we nd that the degeneracy of an eigenvalue E of the Laplacian over vector discrete modes obeying q-twisted boundary conditions is given by3

dvd =

N1

Using the degeneracies (2.27) and (2.28), we can write down corresponding expressions for the heat kernels over these modes, though we do not do so explicitly here.

3 Logarithmic corrections to the twisted indexWe now turn to the computation of logarithmic corrections to Zg. We will carry out

this computation for Type II string theory on T 6. This compactication preserves 32 supercharges of which 16 commute with g. Also, as we have previously discussed, the only elds which can contribute to the log a term are the massless elds in AdS2 S2.

These are just the elds of four-dimensional N = 8 supergravity. We will therefore nd it

useful to organise the spectrum of N = 8 supergravity in terms of representations of the N = 4 subalgebra which commutes with g. All the elds in a single N = 4 multiplet are

characterised by a common g-eigenvalue which in turn dictates which twisted modes on AdS2 S2

[parenrightbig]

/ZN should the heat kernel be computed over. This information is summarised in table 1. In this section we shall compute the contribution of each multiplet in table 1 to the log term in Zg, which requires us to compute the contribution to Zg from quadratic

uctuations of massless elds about the ZN orbifold generated by the action (2.7) of on the attractor geometry of the black hole. To do so, we shall compute the heat kernel of the kinetic operator derived in [26, 27] about this orbifolded background, imposing g-twisted boundary conditions on the elds as we act on the background with. Therefore, the results of section 2 will be useful for us.

Finally, as in [26, 27, 38, 39], we need to compute the heat kernel over the supergravity elds taking into account their couplings to the background graviphoton uxes and scalar elds. As shown in [26, 27], the heat kernel over the various quadratic uctuations can be expressed in terms of the heat kernel over scalars, spin-12 fermions and discrete modes of

3We point out here that the modes with = 0 correspond to vector zero modes of the kinetic operator [26] and hence dvd=0 corresponds to the regularised number of vector zero modes of the kinetic operator.

Explicitly evaluating (2.27) with = 0, so that [notdef][notdef]

s

N

10

1

N

N1

Xs=1

2 + 1

N

s

N

[parenrightBig][parenleftBigg]N . (2.27)

The degeneracy over the q-twisted gravitino discrete modes is given by

dfd = 8

+ 1 N

e

2iqs

JHEP11(2014)002

[parenrightbigg]

4

N

Xs=1sin 2s(+1)Nsin sNcos sN e2iqsN . (2.28)

[parenrightbig] = 1 s, we nd that dvd=0 vanishes when q-twisted boundary conditions are imposed. This is in contrast to the untwisted case, where dvd=0 = 1 [38].

higher-spin elds. The coupling to the background elds however changes the eigenvalues of the kinetic operator from those when elds are minimally coupled to background gravity. The new eigenvalues can in principle be computed by rediagonalising the kinetic operator. However, the ux does not change the degeneracy of the eigenvalue. Hence, to compute the heat kernel over the supergravity elds with our choice of background and boundary conditions, we can use the shifted eigenvalues computed in [26, 27] and the degeneracies computed in section 2. On doing so, we nd two more simplications that are of great benet. Firstly, as observed in [39], the contribution of the conical terms to the heat kernel is nite in the t 70 limit. Hence the contribution to the O t0

[parenrightbig]

term from the conical terms is insensitive to the eigenvalues and can be computed from the degeneracies. Secondly, the other contribution to the O t0

[parenrightbig]

term in the heat kernel originates from the O t0

[parenrightbig]

term in

the heat kernel computed for the full attractor geometry without imposing any twist on the boundary conditions. This has already been computed in [26, 27]. Using these results, and the g-charges computed in table 1, we can now compute the heat kernel over the various supergravity elds and extract the O t0

[parenrightbig]

term in the heat kernel, which will yield the log term. With these results, we now turn to the main computation of this paper.

We rstly note that the N = 4 gravity multiplet is g-invariant, and hence its heat

kernel should be computed over untwisted modes. It has already been shown in [38] that the contribution of these modes to the log term vanishes. Additionally, the contribution of any g-invariant N = 4 vector multiplet to the log term also vanishes [38]. Therefore we

shall concentrate on the gravitino multiplets and the N = 4 vector multiplets which carry

a non-trivial g charge, which corresponds to a non-zero twist in the boundary conditions. We nd below that the contribution of these multiplets also vanishes for any arbitrary choice of twisting. This is in contrast to the untwisted case where while the contribution of the vector multiplet did vanish, the gravitino multiplet contribution was non-vanishing and was responsible for the non-zero log correction the entropy of 18 -BPS black holes in

N = 8 supergravity [39].3.1 The heat kernel for the N = 4 vector multiplet

We will now put the results of section 2 together, using the arguments presented above, to prove the rst of our main results: the log correction in Zg receives vanishing contribution

from any N = 4 vector multiplet with q-twisted boundary conditions. As in [26, 38], the

heat kernel for any N = 4 vector multiplet receives contributions from two Dirac fermions,

6 real scalars and one gauge eld, along with two scalar ghosts. We will focus on the contribution of the conical terms to the O t0

[parenrightbig]

JHEP11(2014)002

term in the heat kernel. We denote this contribution by Kc (t; 0). Firstly the contribution from the two Dirac fermions is given by

KFc (t; 0) =

1 N

N1

Xs=1cos2 s

N

sin4 s

N

N . (3.1)

We now turn to the contribution from the integer-spin elds. These are the 6 real scalars, the gauge eld and two scalar ghosts. Two of the scalars mix with the gauge eld due to the graviphoton ux [26] and we have

KB = 4Ks + K(v+2s) 2Ks, (3.2)

11

e

2iqs

where Ks is the scalar heat kernel along AdS2 S2

[parenrightbig]

/ZN with q-twisted boundary conditions, and K(v+2s) is the heat kernel of the mixed vector-scalar elds due to the background graviphoton ux. As we have previously argued, to extract the t0 term from the xed-point contribution to the heat kernel, we dont have to take into account the coupling of the gauge eld to the scalars via the graviphoton ux and can just add the various contributions piecewise. We therefore nd that (3.2) reduces to

KBc = 6Ksc + Kvc 2Ksc = 4Ksc + Kvc. (3.3)

Ksc can be read o from (2.20), but we need to compute Kvc. As shown in [26], the heat kernel Kv of a vector eld over AdS2 S2 may be decomposed into K(v,s), which is the heat

kernel of a vector eld along AdS2 times the heat kernel of a scalar along S2 and K(s,v), the heat kernel of a vector eld along S2 times the heat kernel of a scalar along AdS2. Further, the modes of the vector eld along AdS2 and S2 may be further decomposed into longitudinal and transverse modes. There is an additional discrete mode contribution from the vector eld on AdS2. These statements carry over to the case of the ZN orbifolds with twisted boundary conditions as well. Kv therefore receives the following contributions.

Kv = K(vT +vL+vd,s) + K(s,vT +vL). (3.4)

Now the modes of longitudinal and transverse vector elds along AdS2 and S2 are in oneto-one correspondence with the modes of the scalar with the only subtlety being that along S2 the = 0 mode of the scalar does not give rise to a non-trivial gauge eld [26]. We therefore have

K(vT ,s) = K(vL,s) = Ks, K(s,vT ) = K(s,vL) = Ks K(s,=0), (3.5)

where, as we have mentioned previously, Ks is the scalar heat kernel along AdS2 S2 [parenrightbig]

with q-twisted boundary conditions, and K(s,=0) is again the scalar heat kernel along AdS2 S2

[parenrightbig]

/ZN, however we only sum over the modes with = 0 along the S2 direction. We therefore nd that the contribution of the conical terms (3.4) reduces to

Kvc (t; 0) = 4Ksc (t; 0) + K(vd,s)c (t; 0) 2K(s,=0)c (t; 0) . (3.6) Further, using (2.21), we may show that

K(s,=0)c (t; 0) = 1

4N

and that Ksc (t; 0) is given by

Xs=114 sin4 sNe2iqsN . (3.8)

Finally, using (2.27), K(vd,s)c (t; 0) is given by

K(vd,s)c (t; 0) =

12

JHEP11(2014)002

/ZN

N1

Xs=11 sin2 s

N

e

2iqs

N , (3.7)

Ksc (t; 0) = 1

2N

N1

1

2N

N1

Xs=11 sin2 s

N

e

2iqs

N . (3.9)

Using (3.3) and (3.6), and then putting (3.7), (3.8) and (3.9) together, we nd that the total integer-spin contribution is given by

KBc (t; 0) = 1

N

N1

Xs=1e2iqsN 1 sin2

s

N

!. (3.10)

Then the total contribution of the conical terms from bosons and fermions is obtained by adding (3.1) and (3.10) to obtain

Kc (t; 0) = KBc (t; 0)+KFc (t; 0) = 1

N

N1

Xs=1e2iqsN 1 sin2

sin4 s

N

s

N

cos2

s

N

[parenrightBigg]

= 0. (3.11)

This vanishes for arbitrary values of q. Now, using the arguments at the beginning of the section, the heat kernel for the N = 4 vector multiplet about the-generated

JHEP11(2014)002

sin4 s

N

ZN orbifold

of the attractor geometry is given, on imposing q-twisted boundary conditions, by

Kq = 1

N K + Kc (t; 0) + O (t) , (3.12)

where K is the heat kernel on the unquotiented near-horizon geometry. We therefore have, for the t0 term in the heat kernel expansion,

Kq (t; 0) = 1

N K (t; 0) + Kc (t; 0) (3.13)

We have shown in 3.11 that Kc (t; 0) equals zero. In addition, it was shown in [26] that K (t; 0) also vanishes. This implies that Kq (t; 0) also vanishes, which proves that the contribution to the log term from the vector multiplet vanishes even for q-twisted boundary conditions.4

3.2 The heat kernel for the N = 4 gravitino multiplets

We now compute the contribution of the N = 4 gravitino multiplets to the log term in Zg for N = 8 string theory. From table 1, we see that the N = 4 gravitino multiplets

obey q-twisted boundary conditions. There are four such multiplets, where the highest-weight eld is a Majorana spin32 fermion, which we organise into two multiplets where the highest-weight eld is a Dirac spin32 fermion. One multiplet obeys twisted boundary conditions with q = +1, and the other with q = 1. Further, since we are considering

quadratic uctuations, the background ux in the attractor geometry does not cause gravitino multiplets with di erent g-charge, and hence di erent q-twist, to mix with each other. We will therefore focus on the contribution of the log term from one q-twisted multiplet where the highest-weight eld is a Dirac spin32 fermion.

4We emphasize here that though the contribution of an N = 4 vector multiplet vanishes for both twisted and untwisted boundary conditions, the origin of the result is di erent in both cases. For the untwisted case, the zero and non-zero modes of the kinetic operator give non-vanishing contributions to the log term which cancel against each other [38], while for the twisted case these contributions are individually zero as shown in this section and in footnote 3 of this paper.

13

Now we shall compute the contribution of the conical terms to the t0 term in the heat kernel expansion for this multiplet. Firstly, we focus on the integer-spin elds. There are 8 gauge elds and 16 real scalars. Further, gauge xing introduces two ghost scalars for every gauge eld. Hence the contribution of the integer-spin elds to the O t0

[parenrightbig]

term from

the conical terms in the heat kernel is

KBc (t; 0) = 8Kvc (t; 0) + 16Ksc (t; 0) 16Ksc (t; 0) = 8Kvc (t; 0) , (3.14) which therefore implies that

KBc (t; 0) = 4

N

N1

Xs=11sin4 sNe2iqsN8 N

N1

Xs=11sin2 sNe2iqsN . (3.15)

We have used (3.6) with (3.7), (3.8) and (3.9) to arrive at this expression. We now turn to the contribution of the half-integer spin elds. We will focus on the contribution of one Dirac gravitino multiplet, which contains one Dirac gravitino and 7 Dirac spin-12 elds.

The degrees of freedom reorganise themselves into in 4 Dirac fermions with 0, 6 Dirac

fermions with only = 0 modes along the S2, 7 Dirac fermions with only 1 modes

along the S2, one discrete Dirac fermion, and 3 ghost Dirac fermions [27, 39]. We can then show that

KFc (t; 0) = 8Kfc (t; 0) K(f,=0)c (t; 0) + Kfdc (t; 0) , (3.16) where Kf is the heat kernel for the Dirac fermion, K(f,=0) is the heat kernel for the Dirac fermion with only = 0 modes along the S2 and Kfd is the heat kernel over one discrete Dirac fermion. Now

Kfc (t; 0) =

1

2N

JHEP11(2014)002

N1

Xs=1cos2 sNsin4 sNe2iqsN , (3.17)

and

N1

Xs=1cos2 sNsin2 sNe2iqsN . (3.18)

Further, using (2.28), we nd that the discrete mode contribution from the conical terms is given by

Kfdc (t; 0) = + 2

N

N1

K(f,=0)c (t; 0) =

2 N

Xs=1cos2 sNsin2 sNe2iqsN . (3.19)

We nally obtain that the full half-integer spin contribution is given by

KFc (t; 0) =

4 N

N1

Xs=11sin4 sNe2iqsN + 8 N

N1

Xs=11sin4 sNe2iqsN4 N

N1

Xs=1e2iqsN . (3.20)

Adding (3.15) and (3.20), we nd that the conical contribution to the t0 term in the heat kernel for a given value of q is

Kc (t; 0) =

4 N

N1

Xs=1e2iqsN = + 4N , (3.21)

14

which is independent of q. Then the contribution of the g-twisted N = 4 gravitino multi-

plets to the log term in Zg is given by

Kg (t; 0) = 1N K (t; 0) + 2Kc (t; 0) , (3.22)

where K (t; 0) is the coe cient of the t0 term in the heat kernel expansion of the gravitino multiplets about the unquotiented near-horizon geometry. This was computed to be 8

in [27]. We therefore nd that Kg (t; 0) is given by

Kg (t; 0) =

8N +

8N = 0. (3.23)

Hence, the contribution of the N = 4 gravitini multiplets to the logarithmic term in Zg

also vanishes.

3.3 The zero mode analysis

We will now take into account the presence of zero modes of the kinetic operator for N = 8

supergravity elds expanded about the black hole near horizon geometry. The nal result, as mentioned above, is that the zero mode analysis of [39] goes through unchanged, but since the zero mode analysis is an important part of the computation, we shall present the result explicitly. The following general result [26, 27], see also [42], will be useful for us. Consider a theory with a length scale a and elds i such that the kinetic operator for quadratic uctuations about a given background has n0i 0 number of zero modes.

Further, let the zero mode contribution to the path integral scale with a as

Z an

0i i

JHEP11(2014)002

Z0, (3.24)

where Z0 does not scale with a, and the numbers i have been explicitly determined in [27]

for the vector eld (see also [26]), the gravitino and the graviton. In particular

v = 1, 32 = 3, g = 2. (3.25)

In that case, the log term for the partition function is given by

ln Zlog = K (0; t) +

Xin0i (i 1) [parenrightBigg]ln a, (3.26)

where K (0; t) is the coe cient of the t0 term in the heat kernel expansion of the kinetic operator over of all elds i, evaluated on both zero and non-zero modes. Therefore, as far as the vector eld is concerned, we may simply evaluate the heat kernel over all modes, extract the t0 coe cient from there, and ignore zero modes. Further, for the N = 8

kinetic operator, all the zero modes of the spin32 and spin-2 elds are contained in the

N = 4 gravity multiplet [27]. This is quantised with untwisted boundary conditions and

its contribution has already been evaluated on the orbifold space in [39], where it was determined that

n3

2 = 2, ng = 2. (3.27)

15

3.4 Logarithmic corrections to the twisted index

Now we are in a position to put together the above results to show that the logarithmic corrections to the partition function Zg vanish for the N = 8 theory. To do so, we will

need the coe cients K (0; t) from the N = 4 vector, gravitini and gravity multiplets, as

well as the corresponding zero mode contributions. It has already been proven in [38] that an untwisted N = 4 vector multiplet has a vanishing contribution to the log term

about our background. Further, we have seen in section 3.1 that K (0; t) for the N = 4

vector multiplet with twisted boundary conditions vanishes, and in (3.23) that K (0; t) for the N = 4 gravitini multiplets with twisted boundary conditions also vanishes. Hence,

the only non-vanishing contributions to ln (Zg)log come from the N = 4 gravity multiplet,

which obeys untwisted boundary conditions. For this multiplet (see eq. 5.46 of [39])

K (0; t) = 2. (3.28)

Putting these results in (3.26) with (3.27), we nd that

ln (Zg)log = 0, (3.29)

which completes the proof that the logarithmic term in Zg vanishes, in accordance with

the microscopic results for Bg6 for N = 8 string theory.

4 ConclusionsIn this paper we exploited the heat kernel techniques developed in [38] to compute the logarithmic terms in the large charge expansion of the twisted index Bg6 in N = 8 string

theory. These vanish, matching perfectly with the microscopic computation. Further, the result may be extended to the N = 4 case as follows. Firstly, since g commutes

with all 16 supercharges in this case, we continue to classify elds into multiplets of the four-dimensional N = 4 supersymmetry algebra. Secondly, we need to focus only on the

massless supergravity elds over the near-horizon geometry as only these can contribute to the log term. Finally, the g action on the various N = 4 multiplets can be found out using

techniques similar to the ones employed in the N = 8 case. Since g acts geometrically

on the compact directions, the N = 4 gravity multiplet still does not transform, and

its contribution to the log term vanishes as per the analysis of [39]. The N = 4 vector

multiplets would carry non-trivial g-charges, corresponding to non-trivial q-twists for these elds in the path integral Zg. We have already seen that the contribution to the log term

from N = 4 vector multiplets vanishes for arbitrary twists q. Therefore, the log term

vanishes even for N = 4 string theory.

As a nal observation, we note that the microscopic expression for Bg6 contains exponentially suppressed corrections of the form

Bg6,p (Q, P ) e

JHEP11(2014)002

Q2P 2(QP)2

Np (O (1) + . . .) , p

Z+, p 2. (4.1)

Using the arguments of [38] for the untwisted index we nd that the logarithmic correction vanishes about these saddle-points as well. Following through the arguments of [36], a

16

natural candidate for the macroscopic origin of these corrections corresponds to a saddle-point of Zg obtained by taking a

ZNp orbifold of the attractor geometry, where again g-twisted boundary conditions should be imposed on the elds in the path integral. From the analysis presented in this paper, it follows that the log corrections to Zg vanish about

these saddle-points as well, which matches with the expectation from the microscopic side.

Acknowledgments

We would like to thank Justin David, Rajesh Gopakumar and especially Ashoke Sen for several very helpful discussions and correspondence. AC would like to thank ICTP, Trieste for hospitality while part of this work was carried out. SL would like to thank IACS, Kolkata and ICTS-TIFR, Bangalore for hospitality while part of this work was carried out. ACs work is supported by the DAE project 12-R&D-HRI-5.02-0303. SLs work is supported by National Research Foundation of Korea grants 2005-0093843, 2010-220-C00003 and 2012K2A1A9055280. MS is supported by a J.C. Bose fellowship awarded to Ashoke Sen by the Department of Science and Technology, India. Finally, SL would like to dedicate this paper to the memory of Avijit Lal (14.12.1979 10.04.2014) gratefully acknowledging his constant encouragement to pursue research in theoretical physics.

A The g-charges for the N = 8 supergravity eldsIn this appendix we shall review how the g-twist acts on the elds of four-dimensional

N = 8 supergravity. As g commutes with the N = 4 subalgebra of the full N = 8

algebra, we expect that the N = 8 gravity multiplet will decompose into N = 4 multiplets,

each of which carry some charge under g. We shall obtain these charges by working with Type IIB supergravity compactied on T 4 T 2 and studying the action of g on the

supergravity elds, which are the graviton hMN, the two-form BMN, the three-form ux CMNP and two 16-component Majorana-Weyl spinors.5 This action of the g-twist on the Type IIB supergravity elds compactied on T 4T 2 can be realised in an appropriate

complex coordinate system(z1, z2) on T 4 and (z3) on T 2 as [16].

dz1 e

17

JHEP11(2014)002

2i

N dz1 dz2 e

2i

N dz2 dz1 e

N dz1 dz2 e

2i

2i

N dz2 (A.1) dz3 dz3 dz3 dz3 (A.2)

These transformations can be thought of as individual rotations along the two cycles of T 4.

The g-action on the ten dimensional elds is realised as a eld transformation under the di erent representations of the Lorentz group. In the four dimensional theory obtained on compactication, the g-action may be thought of as an internal symmetry.

The compactication of the N = 2 supergravity elds on T 4T 2 gives one N = 8

gravity multiplet in 4 dimensions. This contains one graviton h, 8 spin32 Majorana elds, 28 spin-1 elds, 56 spin12 Majorana elds and 70 real scalars.

5The indices M, N take values 0, . . . , 9, while , will take values 0, . . . , 3 which label the non-compact directions. The indices m, n will take values 4, . . . , 9 and label the compact directions.

Multiplet Number of Multiplets g-Eigenvalue

Gravity 1 1

Gravitino 2 e

2i

N

Vector 4 1

Vector 1 e

4i

N

Table 1. g-Charges of the N =4 multiplets. It is natural to expect the gravity multiplet to remain

invariant since the 4D spacetime metric h is a spacetime eld and is una ected by coordinate transformations on the internal directions.

The spin-2 eld h is just the spacetime metric. The spin-1 elds come from Gm, Bm, Cmn and A. The scalars come from Gmn, Bmn, Am, Cmnp, dualizing the components Cm of the three-form eld, and the axion and the dilation. The origin of the 8 spin32 elds and 48 spin12 elds lie in the spin32 and spin-1/2 m multiplets obtained on compactication of the two 16 component Majorana-Weyl spinors over T 6. 8

of the remaining spin12 elds come from the compactication of the two ten-dimensional [10] spinors.

The g-twist commutes with 16 of the 32 supersymmetries. Hence we split the N = 8

gravity multiplet into one N = 4 gravity multiplet, four gravitino, and six vector multiplets.

All the members of a given N = 4 multiplet carry the same g-charge since g commutes

with the N = 4 subalgebra. The g-charge of every eld has been found to conform with

the g-charge of the multiplet it belongs to. The nal results of this computation have been summarised in table 1.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References[1] C. Bachas and E. Kiritsis, F (4) terms in N = 4 string vacua,http://dx.doi.org/10.1016/S0920-5632(97)00079-0

Web End =Nucl. Phys. Proc. Suppl. 55B (1997) 194 [http://arxiv.org/abs/hep-th/9611205

Web End =hep-th/9611205 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9611205

Web End =INSPIRE ].

[2] A. Gregori et al., R2 corrections and nonperturbative dualities of N = 4 string ground states, http://dx.doi.org/10.1016/S0550-3213(97)00635-4

Web End =Nucl. Phys. B 510 (1998) 423 [http://arxiv.org/abs/hep-th/9708062

Web End =hep-th/9708062 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9708062

Web End =INSPIRE ].

[3] R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Counting dyons in N = 4 string theory, http://dx.doi.org/10.1016/S0550-3213(96)00640-2

Web End =Nucl. Phys. B 484 (1997) 543 [http://arxiv.org/abs/hep-th/9607026

Web End =hep-th/9607026 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9607026

Web End =INSPIRE ].

[4] J.M. Maldacena, G.W. Moore and A. Strominger, Counting BPS black holes in toroidal type II string theory, http://arxiv.org/abs/hep-th/9903163

Web End =hep-th/9903163 [http://inspirehep.net/search?p=find+EPRINT+hep-th/9903163

Web End =INSPIRE ].

18

2i

N

Gravitino 2 e

4i

N

JHEP11(2014)002

Vector 1 e

[5] G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Asymptotic degeneracy of dyonic N = 4 string states and black hole entropy, http://dx.doi.org/10.1088/1126-6708/2004/12/075

Web End =JHEP 12 (2004) 075 [http://arxiv.org/abs/hep-th/0412287

Web End =hep-th/0412287 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0412287

Web End =INSPIRE ].

[6] D. Shih, A. Strominger and X. Yin, Recounting dyons in N = 4 string theory, http://dx.doi.org/10.1088/1126-6708/2006/10/087

Web End =JHEP 10 (2006) 087 [http://arxiv.org/abs/hep-th/0505094

Web End =hep-th/0505094 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0505094

Web End =INSPIRE ].

[7] D. Shih, A. Strominger and X. Yin, Counting dyons in N = 8 string theory, http://dx.doi.org/10.1088/1126-6708/2006/06/037

Web End =JHEP 06 (2006) 037 [http://arxiv.org/abs/hep-th/0506151

Web End =hep-th/0506151 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0506151

Web End =INSPIRE ].

[8] D.P. Jatkar and A. Sen, Dyon spectrum in CHL models, http://dx.doi.org/10.1088/1126-6708/2006/04/018

Web End =JHEP 04 (2006) 018 [http://arxiv.org/abs/hep-th/0510147

Web End =hep-th/0510147 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0510147

Web End =INSPIRE ].

[9] A. Dabholkar and S. Nampuri, Spectrum of dyons and black holes in CHL orbifolds using Borcherds lift, http://dx.doi.org/10.1088/1126-6708/2007/11/077

Web End =JHEP 11 (2007) 077 [http://arxiv.org/abs/hep-th/0603066

Web End =hep-th/0603066 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0603066

Web End =INSPIRE ].

[10] J.R. David, D.P. Jatkar and A. Sen, Product representation of Dyon partition function in CHL models, http://dx.doi.org/10.1088/1126-6708/2006/06/064

Web End =JHEP 06 (2006) 064 [http://arxiv.org/abs/hep-th/0602254

Web End =hep-th/0602254 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0602254

Web End =INSPIRE ].

[11] J.R. David, D.P. Jatkar and A. Sen, Dyon spectrum in N = 4 supersymmetric type II string theories, http://dx.doi.org/10.1088/1126-6708/2006/11/073

Web End =JHEP 11 (2006) 073 [http://arxiv.org/abs/hep-th/0607155

Web End =hep-th/0607155 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0607155

Web End =INSPIRE ].

[12] J.R. David and A. Sen, CHL dyons and statistical entropy function from D1-D5 system, http://dx.doi.org/10.1088/1126-6708/2006/11/072

Web End =JHEP 11 (2006) 072 [http://arxiv.org/abs/hep-th/0605210

Web End =hep-th/0605210 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0605210

Web End =INSPIRE ].

[13] J.R. David, D.P. Jatkar and A. Sen, Dyon spectrum in generic N = 4 supersymmetric ZN orbifolds, http://dx.doi.org/10.1088/1126-6708/2007/01/016

Web End =JHEP 01 (2007) 016 [http://arxiv.org/abs/hep-th/0609109

Web End =hep-th/0609109 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0609109

Web End =INSPIRE ].

[14] A. Sen, N = 8 dyon partition function and walls of marginal stability, http://dx.doi.org/10.1088/1126-6708/2008/07/118

Web End =JHEP 07 (2008) 118 [arXiv:0803.1014] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0803.1014

Web End =INSPIRE ].

[15] A. Sen, Arithmetic of N = 8 black holes, http://dx.doi.org/10.1007/JHEP02(2010)090

Web End =JHEP 02 (2010) 090 [arXiv:0908.0039] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0908.0039

Web End =INSPIRE ].

[16] A. Sen, Black hole entropy function, attractors and precision counting of microstates, http://dx.doi.org/10.1007/s10714-008-0626-4

Web End =Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0708.1270

Web End =INSPIRE ].

[17] I. Mandal and A. Sen, Black hole microstate counting and its macroscopic counterpart, http://dx.doi.org/10.1088/0264-9381/27/21/214003

Web End =Nucl. Phys. Proc. Suppl. 216 (2011) 147 [arXiv:1008.3801] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1008.3801

Web End =INSPIRE ].

[18] A. Dabholkar and S. Nampuri, Quantum black holes, http://dx.doi.org/10.1007/978-3-642-25947-0_5

Web End =Lect. Notes Phys. 851 (2012) 165 [arXiv:1208.4814] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1208.4814

Web End =INSPIRE ].

[19] A. Sen, Microscopic and macroscopic entropy of extremal black holes in string theory, http://dx.doi.org/10.1007/s10714-014-1711-5

Web End =Gen. Rel. Grav. 46 (2014) 1711 [arXiv:1402.0109] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1402.0109

Web End =INSPIRE ].

[20] A. Sen, Quantum entropy function from AdS2/CF T1 correspondence, http://dx.doi.org/10.1142/S0217751X09045893

Web End =Int. J. Mod. Phys. A 24 (2009) 4225 [arXiv:0809.3304] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0809.3304

Web End =INSPIRE ].

[21] A. Sen, Entropy function and AdS2/CF T1 correspondence, http://dx.doi.org/10.1088/1126-6708/2008/11/075

Web End =JHEP 11 (2008) 075 [arXiv:0805.0095] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0805.0095

Web End =INSPIRE ].

[22] H.K. Kunduri, J. Lucietti and H.S. Reall, Near-horizon symmetries of extremal black holes, http://dx.doi.org/10.1088/0264-9381/24/16/012

Web End =Class. Quant. Grav. 24 (2007) 4169 [arXiv:0705.4214] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0705.4214

Web End =INSPIRE ].

[23] P. Figueras, H.K. Kunduri, J. Lucietti and M. Rangamani, Extremal vacuum black holes in higher dimensions, http://dx.doi.org/10.1103/PhysRevD.78.044042

Web End =Phys. Rev. D 78 (2008) 044042 [arXiv:0803.2998] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0803.2998

Web End =INSPIRE ].

[24] A. Sen, Arithmetic of quantum entropy function, http://dx.doi.org/10.1088/1126-6708/2009/08/068

Web End =JHEP 08 (2009) 068 [arXiv:0903.1477] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0903.1477

Web End =INSPIRE ].

19

JHEP11(2014)002

[25] A. Sen, Logarithmic corrections to Schwarzschild and other non-extremal black hole entropy in di erent dimensions, http://dx.doi.org/10.1007/JHEP04(2013)156

Web End =JHEP 04 (2013) 156 [arXiv:1205.0971] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1205.0971

Web End =INSPIRE ].

[26] S. Banerjee, R.K. Gupta and A. Sen, Logarithmic corrections to extremal black hole entropy from quantum entropy function, http://dx.doi.org/10.1007/JHEP03(2011)147

Web End =JHEP 03 (2011) 147 [arXiv:1005.3044] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1005.3044

Web End =INSPIRE ].

[27] S. Banerjee, R.K. Gupta, I. Mandal and A. Sen, Logarithmic corrections to N = 4 and N = 8 black hole entropy: a one loop test of quantum gravity, http://dx.doi.org/10.1007/JHEP11(2011)143

Web End =JHEP 11 (2011) 143 [arXiv:1106.0080] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1106.0080

Web End =INSPIRE ].

[28] A. Sen, Logarithmic corrections to rotating extremal black hole entropy in four and ve dimensions, http://dx.doi.org/10.1007/s10714-012-1373-0

Web End =Gen. Rel. Grav. 44 (2012) 1947 [arXiv:1109.3706] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1109.3706

Web End =INSPIRE ].

[29] A. Sen, Logarithmic corrections to N = 2 black hole entropy: an infrared window into the microstates, arXiv:1108.3842 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.3842

Web End =INSPIRE ].

[30] C. Keeler, F. Larsen and P. Lisbao, Logarithmic corrections to N 2 black hole entropy,

http://dx.doi.org/10.1103/PhysRevD.90.043011

Web End =Phys. Rev. D 90 (2014) 043011 [arXiv:1404.1379] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1404.1379

Web End =INSPIRE ].

[31] A. Dabholkar, J. Gomes and S. Murthy, Quantum black holes, localization and the topological string, http://dx.doi.org/10.1007/JHEP06(2011)019

Web End =JHEP 06 (2011) 019 [arXiv:1012.0265] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.0265

Web End =INSPIRE ].

[32] A. Dabholkar, J. Gomes and S. Murthy, Localization & exact holography, http://dx.doi.org/10.1007/JHEP04(2013)062

Web End =JHEP 04 (2013) 062 [arXiv:1111.1161] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.1161

Web End =INSPIRE ].

[33] R.K. Gupta and S. Murthy, All solutions of the localization equations for N = 2 quantum black hole entropy, http://dx.doi.org/10.1007/JHEP02(2013)141

Web End =JHEP 02 (2013) 141 [arXiv:1208.6221] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1208.6221

Web End =INSPIRE ].

[34] S. Murthy and V. Reys, Quantum black hole entropy and the holomorphic prepotential of N = 2 supergravity, arXiv:1306.3796 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.3796

Web End =INSPIRE ].

[35] A. Dabholkar, J. Gomes and S. Murthy, Nonperturbative black hole entropy and Kloosterman sums, arXiv:1404.0033 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1404.0033

Web End =INSPIRE ].

[36] A. Sen, A twist in the dyon partition function, http://dx.doi.org/10.1007/JHEP05(2010)028

Web End =JHEP 05 (2010) 028 [arXiv:0911.1563] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.1563

Web End =INSPIRE ].

[37] A. Sen, Discrete information from CHL black holes, http://dx.doi.org/10.1007/JHEP11(2010)138

Web End =JHEP 11 (2010) 138 [arXiv:1002.3857] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1002.3857

Web End =INSPIRE ].

[38] R.K. Gupta, S. Lal and S. Thakur, Heat kernels on the AdS2 cone and logarithmic corrections to extremal black hole entropy, http://dx.doi.org/10.1007/JHEP03(2014)043

Web End =JHEP 03 (2014) 043 [arXiv:1311.6286] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1311.6286

Web End =INSPIRE ].

[39] R.K. Gupta, S. Lal and S. Thakur, Logarithmic corrections to extremal black hole entropy in N = 2, 4 and 8 supergravity, arXiv:1402.2441 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1402.2441

Web End =INSPIRE ].

[40] N. Banerjee, S. Banerjee, R.K. Gupta, I. Mandal and A. Sen, Supersymmetry, localization and quantum entropy function, http://dx.doi.org/10.1007/JHEP02(2010)091

Web End =JHEP 02 (2010) 091 [arXiv:0905.2686] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0905.2686

Web End =INSPIRE ].

[41] D.V. Vassilevich, Heat kernel expansion: users manual, http://dx.doi.org/10.1016/j.physrep.2003.09.002

Web End =Phys. Rept. 388 (2003) 279 [http://arxiv.org/abs/hep-th/0306138

Web End =hep-th/0306138 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0306138

Web End =INSPIRE ].

[42] S. Bhattacharyya, A. Grassi, M. Mario and A. Sen, A one-loop test of quantum supergravity, http://dx.doi.org/10.1088/0264-9381/31/1/015012

Web End =Class. Quant. Grav. 31 (2014) 015012 [arXiv:1210.6057] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1210.6057

Web End =INSPIRE ].

[43] R. Camporesi and A. Higuchi, Spectral functions and zeta functions in hyperbolic spaces,http://dx.doi.org/10.1063/1.530850

Web End =J. Math. Phys. 35 (1994) 4217 [http://inspirehep.net/search?p=find+J+J.Math.Phys.,35,4217

Web End =INSPIRE ].

20

JHEP11(2014)002

[44] R. Camporesi and A. Higuchi, On the eigen functions of the Dirac operator on spheres and real hyperbolic spaces, http://dx.doi.org/10.1016/0393-0440(95)00042-9

Web End =J. Geom. Phys. 20 (1996) 1 [http://arxiv.org/abs/gr-qc/9505009

Web End =gr-qc/9505009 ] [http://inspirehep.net/search?p=find+EPRINT+gr-qc/9505009

Web End =INSPIRE ].

[45] R. Camporesi and A. Higuchi, The plancherel measure for p-forms in real hyperbolic spaces,J. Geom. Phys. 15 (1994) 57.

[46] M.R. Gaberdiel, R. Gopakumar and A. Saha, Quantum W -symmetry in AdS3,

http://dx.doi.org/10.1007/JHEP02(2011)004

Web End =JHEP 02 (2011) 004 [arXiv:1009.6087] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1009.6087

Web End =INSPIRE ].

[47] R. Gopakumar, R.K. Gupta and S. Lal, The heat kernel on AdS, http://dx.doi.org/10.1007/JHEP11(2011)010

Web End =JHEP 11 (2011) 010 [arXiv:1103.3627] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.3627

Web End =INSPIRE ].

[48] S. Lang, Sl2(R), Graduate Texts in Mathematics volume 105, Springer, Germany (1998).

JHEP11(2014)002

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