Three Dimensional Pure Gravity and Generalized Hecke Operators

We investigate the 3d pure gravity in AdS space and its dual boundary conformal field theory. We study the modularity of the holomorphically factorizable partition function. We show that the chiral and anti-chiral partition functions are modular covariant, the primary fields are integer, and the central charges are the integer multiple of 8, $c=8\mathbb{N}$. For dual pure gravity, this leads to the constraint on the symmetric group of the corresponding Chern-Simon theory. The symmetry group of the theory should be $SO(2,1)\times SO(2,1)$ and its three-fold diagonal cover. We introduce the generalized Hecke operators which map the modular covariant functions to the modular covariant functions. Using this operator, we obtain the partition function of the extremal CFT for $c=8\mathbb{N}$, and the corresponding microcanonical entropy. We show there are some subleading terms in the entropy beyond the logarithmic correction to the Bekenstein-Hawking entropy.


Intoduction
In three dimensional classical Einstein gravity, every two solutions are equivalent and there is no propagating degrees of freedom. In 3d gravity with negative cosmological constant (AdS gravity), the existence of the BTZ black hole [1,2] makes the theory more interesting to consider this theory as a toy model to understand the higher dimensional gravity [3]- [10].
Three dimensional AdS gravity has asymptotic Virasoro symmetry. During the quantization of this theory, Brown-Henneaux showed that the theory has left and right moving Virasoro algebras which are part of the structure of the conformal field theory [14]. The corresponding central charge is c = 3l 2g (where l is the length of the AdS space). This shows the existence of the boundary conformal field theory. This duality is an example of the AdS/CFT correspondence, which is a correspondence between a bulk gravity, and boundary CFT in the lower dimension [11]- [13].
Using the AdS/CFT dictionary, one can obtain useful informations about the bulk theory by studying the boundary CFT. Solvability of the 3d gravity and the AdS/CFT correspondence, makes this theory more powerful to reveal some fundamental aspects of the quantum gravity [16]- [18].
We investigate the minimal theory of 3d gravity in this paper. The 3d pure gravity is dual to the holomorphically factorizable extremal 2d CFT on the boundary, where there are some constraints on the central charges of the CFT [19]. These constraints on the dual CFT come from the equivalence between 3d Einstein gravity and Chern-Simons gauge theory [20,21].
In three dimension, the Einstein-Hilbert action with negative cosmological constant can be expressed in terms of the gauge invariant action as follows: where the gauge field A is built from SO(2, 1) gauge field w(spin connection), and verbin e. The gauge group of this action is SO (2,2). The gauge group SO(2, 2) is locally equivalent to SO(2, 1) × SO (2,1). Therefore, in the oriented space-time the action can be written as: = k L I L + k R I R .
where k L and k R , are Chern-Simons couplings. The allowed values of the k L and k R , can be calculated from the quantization of the Chern-Simons couplings. Since the fundamental group of the gauge group SO(2, 1) × SO(2, 1) is U(1) × U(1), every diagonal covering groups of this group can be considered as the gauge group of the Chern-Simons theory. For an n-fold diagonal cover of SO(2, 1) × SO(2, 1), the quantization condition of the k L and k R is obtained as follows Using the AdS/CFT dictionary, equation (2) shows the partition function of the dual CFT should be holomorphically factorizable: The quantization condition (3) leads to the following quantization condition on the left and right central charges: Quantization condition (4) is equivalent to the T invariant constraint of the partition function of the dual CFT [19]. The purity of the 3d gravity for the dual CFT means that the primary fields of low dimensions should come from the identity. Therefore, the scaling dimension of the lowest primary fields excluding identity should be k + 1. This class of CFTs is called extremal CFT [22,23]. The vacuum state of the extremal CFT corresponds to the AdS space and the other primary fields correspond to the BTZ black hole. Now, we are ready to solve the pure gravity. In order to achieve this goal, we need to find the dual CFT. First step toward obtaining the partition function of the extremal CFT is determining the gauge group of the Chern-Simons gravity. As (3) shows, the simplest gauge group is SO(2, 1) × SO(2, 1). For this gauge group, k L and k R are integer numbers and the dual extremal CFT satisfies all of the constraints. The left and right central charges are integer multiple of 24 and the chiral and anti-chiral partition functions are modular invariant.
Modular invariance is a powerful tool that reveals interesting aspects of the conformal field theory [24]- [29]. Modularity determines the partition function precisely. Every modular function can be expressed in terms of the polynomial of the Klein J function: For c L = 24, 48, and 72, the partition function of the extremal CFT is calculated as follows: The closed-form of the extremal partition function is derived using the Hecke operators The Hecke operators map the modular functions to the modular functions and for f (τ ) = q −1 +O(q); Therefore, the partition function of external CFT is obtained as follows [34] 1 : Where, a −r are the coefficients of the low states of the vacuum. For c L = 24, there is an extremal CFT which is unique [30,31], while for the other values of the central charges it is not clear whether such CFT exists [32].
In [19], Witten considered the SO(2, 1) × SO(2, 1) gauge group, now the question arise: "Does any covering group of the gauge group SO(2, 1) × SO(2, 1) exists which satisfies all constraints on the dual CFT?" 1 One can also write the closed-form of the partition function in terms of the unique modular function J m (τ ) which only has an order-m pole at q = 0 [35]: where p(m) is the partition number.
In order to address this question, we study the holomorphically factorizable CFT in this paper. We show that modular invariance of the holomorphically factorizable CFT is necessary and sufficient condition for deriving the authorized values of the covering group. In section 2, we study the modularity of the holomorphically factorizable partition functions and we calculate the authorized covering group. We show that the chiral and the anti-chiral partition functions are modular covariant (i.e. they are modular invariant up to an overall phase). Moreover, we introduce the bases for the modular covariant functions. Since the chiral and the anti-chiral partition functions are not modular invariant, we can not use the Hecke operators. In section 3, we introduce the generalized Hecke operators which map modular covariant functions into the modular covariant functions. We also investigate its Fourier expansion. In section 4, we show that the authorized gauge group for the Chern-Simons, is SO(2, 1)×SO(2, 1) and its three-fold cover. In this section, we study the dual CFT for the case where the gauge group is three-fold diagonal cover of the group SO(2, 1) × SO(2, 1). We obtain the partition function and the microcanonical entropy of the theory.

Partition Function
The partition function of the unitary 2d CFT in the upper half plane where, c L and c R are the left and right central charges and ρ(h,h) is the density of the state. The holomorphically factorizable partition functions can be written as the multiplications of the chiral and anti-chiral partition functions as follows: where the chiral partition function Z(τ ), and the anti-chiral partition functionZ(τ ) are defined as follows

Modularity of Partition Function
The modular covariance of the chiral and anti-chiral partition functions is the necessary and sufficient condition for the modular invariance of the partition function Z(τ,τ ). Modular covariant means that the chiral and anti-chiral partition functions take an overall phase under the S and T transformations. Modular covariance of Z(τ ) under the S transformation demands: The identity S 2 = 1, shows the phase β should be π or 2π. For β = π, we called the corresponding partition function Z − (τ ). The Fourier expansion of Z − (τ ) under S transformation at the self dual point τ = i yields: Using (17) for β = π, and (18) show that Z − (τ = i) is equal to zero. In Eq. (18) all phases are positive, so some of the density of states should be negative. Therefore, Z − (τ ) is not a physical partition function. Covariance of Z(τ ) under T transformation requires: Plugging (15) into (19), for τ r = 0 yields The summands (20) and (21) are non-negative. Therefore, Using (ST ) 3 = 1 and invariance of Z(τ ) under S transformation one can obtain: Consequently; Similarly, forZ(τ ) we have:h For m L , m R / ∈ 3N modular invariance of the partition function Z(τ,τ ), enforces that m L = m R = k. In this case, the partition function Z(τ,τ ) automatically becomes real. For m L , m R ∈ 3N, if we put the weak condition of the reality of the partition function, this constrains leads to the equality of the right and left central charges.

The Basis for the Modular Covariant Functions
In this section we derive the basis for Z(τ ). For m L integer multiple of three, i.e. c L ∈ 24Z, Z(τ ) is modular invariant. Therefore, it is a polynomial in terms of the Klein function J(τ ) [34]: with some coefficients h r . The Klein function has the following Fourier expansion: where, j has expansion in terms of the Jacobi Theta functions and Eta function as follows In order to obtain the bases for Z(τ ), we use the lemma in [33].
on the upper half τ -plane, is a polynomial in j.
Proof. From Eq. (29) and Eq.(30) we conclude that there exist Therefore, Since The order of the pole of f (r) {a (r) }, τ and f (r−1) {a (r−1) }, τ are r and r − 1 respectively. By iteration one can obtain where Since the function is modular invariant. It has no pole on the upper half plane and is zero at τ = i∞. Thus, it is zero on the upper half plane.
Corollary 2.2. The chiral partition function is a polynomial in function j as follows:

The Hecke Operators
Let us define the subgroup of the modular group with S and T 3 generators, which we call this group Γ 3 .
The modular covariant function f 3 (τ ) with the Fourier expansion: is invariant under the group Γ 3 . The Hecke operators are linear operators which map modular form space M k , onto itself and are defined as follows: The Hecke operators map the Modular functions, onto the modular functions.
In this section, we generalize the definition of the Hecke operators, 3-Hecke operators for the group Γ 3 , which map the modular covariant functions onto the modular covariant functions.
We called the operators T then, T n f 3 (τ ) has the Fourier expansion: where Proof. By putting the Fourier expansion of the function f 3 (τ ) (40) into (39) we have The last sum in (43) is zero for d ∤ 3m − 1 and is equal to d for d | 3m − 1: Since, d | 3m − 1; writing 3m − 1 = pd and replacing n d with d (because d | n) yeilds n d e 2πiτ dp 3 .
The last term in the sum has the form q ( pd 3 ). For all terms which pd 3 is constant pd 3 = m − 1 3 one can obtain:

The Order n Transformations
For positive integer n, the order n transformation Γ(n), is defined as follows where a, b, c, and d are integers. The Γ(1) = Γ transformations correspond to the modular transformations. The transformations A 1 and A 2 in Γ(n) are called equivalent if there exist a modular transformation V ∈ Γ, such that It is clear that the relation ∼ is an equivalence relation. So, the transformations Γ(n) can be divided into the equivalence classes. Two element of Γ(n) are in the same class, if and only if, they are equivalent.

Lemma 3.3. For every equivalence class of Γ(n), there is a triangular representation A 3 :
where n = 3p + i(i = 1, 2) and p ∈ N.
Proof. As shown in [34], in every equivalence class of Γ(n) there is a representation of triangular form For A 1 and A 2 (two equivalent elements in Γ(n)), there is V = 1 q 0 1 ∈ Γ Such that In order to prove this theorem, it it is necessary to show that b 2 is multiple of three.
Since a 1 d 1 = n and n = 3N, so d 1 can not be multiple integer of three and takes 3s + 1 or 3s + 2 values. For fixed value of d, b 1 takes 3r, 3r + 1 and 3r + 2 values. By substituting these values to (52), one can show b 2 can be multiple of three (by choosing appropriate values of q). where, First, we show if (54) holds, then A 1 ∼ A 2 . For some integer q, if we consider V as follows where q = 3q ′ , then, such that The above equality shows r = 0 (since a 1 = 0). From ps − qr = 1, we can conclude ps = 1, so p = s = 1 or p = q = −1. Let us consider p = s = 1 (for the other case we replace V by −V ). By equating the entries in the above equation, we have a 1 = a 2 , d 1 = d 2 , and 3b 2 = 3b 1 + qd.
Since, a 1 d 1 = n and n is not integer multiple of three, therefore, q = 3q ′ : Lemma 3.5. For A 1 ∈ Γ(n), V 1 ∈ Γ there exists transformation A 2 ∈ Γ(n) and V 1 ∈ Γ such that Proof. det(A 1 V 1 ) = detA 1 detV 1 = n, so A 1 V 1 ∈ Γ(n). According to lemma 3.3, there exists A 2 ∈ Γ(n) and V 2 ∈ Γ such that Now, by using (63) for V 1 = S = 0 −1 1 0 transformation, we derive the elements of A 2 and V 2 in terms of the element of A 1 . By equating the entries in (63), we have From (64), we recognize that V 2 has two independent entries δ 2 and γ 2 . Since; n = a 1 d 1 = 3N , the second and the forth equation in (64) show δ 2 and α 2 are multiples of three.
We already know S and T = 1 1 0 1 transformations are generators of the modular group and each elements of the modular group can be written in the below form Since V 2 has two independent entries, one can write it as follows From (66) we have For the case where n is not multiple of three, from (64) and (67) we conclude n 1 and n 2 are multiple of three. Proof. Since d|n one can rewrite the 3-Hecke operator as follow where A is an element of Γ(n): From (70) we have: Using lemma (3.5), we have As we showed earlier, n 1 and n 2 are multiple of three, therefore: Substituting (72) to (70) yields to: (73) shows that the 3-Hecke operators are invariant under S transformation. The Fourier expansion (41), shows that the 3-Hecke operators are modular covariant under T transformation and are invariant under T 3 transformation. Since every elements of the group of Γ 3 are built from the multiplication of the S and T 3 generators, we conclude that the 3-Hecke operators map the Γ 3 invariant functions f 3 (τ ) into the Γ 3 invariant functions.

Three Dimensional Gravity
Our focus in this section is solving the pure quantum gravity in the sense of finding the dual boundary CFT. As it is shown in [19], the dual CFT is extremal which means that the lowest dimension of the primary fields excluding the identity, is k + 1 for c = 24k, and the partition function should be holomorphically factorizable. The authorized values of the left and right central charges can be obtained from the symmetry group of the Chern-Simons gauge theory. The symmetry group can be the group SO(2, 1) × SO(2, 1) and its nth diagonal cover: where, k L and k R are the Chern-Simons couplings. From the AdS/CFT dictionary, the corresponding central charges are obtained as follows Therefore, the first step to solve the quantum gravity is determining the covering group. The holomorphically factorizable and modular invariance of the partition function is necesary and sufficient condition for determining the symmetry group.
In section (2) we show for the modular invariant holomorphically factorizable partition function, the scaling dimensions of the chiral and anti-chiral partition functions should be an integer number and the left and right central charges are integer multiples of eight: From (74) and (76), we conclude that the authorized values of n should be 1 or 3. So, the symmetry group of the Chern-Simons gauge theory should be SO(2, 1) × SO(2, 1) and its three-fold diagonal cover.
For pure gravity with the asymptotic space-time AdS 3 , the vacuum state is the trivial state where its chiral partition function is obtained as follows The vacuum state corresponds to the Anti de Sitter Space, classically. Since, the vacuum partition function (77), is not modular covariant, there should be other states in the theory. This is in the agreement with the existence of the BTZ black hole in the theory. The mass and the angular momentum of the classical BTZ black hole in terms of the Virasoro generators L 0 andL 0 are obtained as follows and the entropy is From (79) and (80), we conclude that L 0 1. Hence, the full partition function has the following form: The modular covariant constraint determines the partition function uniquely. For k L , k R ∈ Z, the chiral and anti-chiral partition functions are modular functions. In [19], the partition function and the entropy are investigated. In this section we study the case where k L , k R ∈ Z 3 . We showed in section (2), modular invariant of Z(τ,τ ) demands c L = c R = 8k, k ∈ Z.
From corollary (2.2), the chiral partition function is obtained as follows where the n r coefficients are determined from the fact that the density of the low dimensional state, should be equal to the density of the corresponding low dimensional state of the vacuum. For k = 1 to k = 11, the chiral partition function is obtained in [36]. Here are some examples: The closed-form of the partition function is obtained by using the generalize 3-Hecke operator (39). The Fourier expansion (40) and (41) shows for and The 3-Hecke operators have the following expansion and Therefore, for c = 8k the chiral partition function is obtained as follows where i = 1 is for k = odd and i = 2 corresponds with k = even. The a r coefficients, are the low state density of the vacuum: In order to determine the entropy, let us write the partition function as: Using (91) and (41), the b k,m coefficients are obtained as follows where c 1 (m), and c 2 (m) are the j and j 2 Fourier expansion's coefficients, respectively. The c i coefficients are obtained as follows ( up to the exponentially suppressed terms) [37], [38]: and a r = P (r) − P (r − 1).
The partition numbers P (r) are obtained from Peterson-Rademacher expansion: where, Kl(a,b;k) is the Kloosterman sum. In (94), the leading term in the large km limit is d = 1: Now, we compare the first and the second terms in the Eq.(98). Using (95) and the asymptotic behavior of the Bessel Function: we have In the large k and m, where m k is constant, the leading terms in the entropy comes from r = k. Therefore, in the semiclassical limit the entropy is obtained as follows S(k, m) = ln b k,m = ln kc i k(m − 1 3 ) + · · · (102) where the first term is Bekenstein-Hawking entropy and other terms determine the logarithmic corrections [39]- [43]. In (102), there are some subleading terms which are corresponding to r < k in (94). As we showed in (100), these terms are important for large values of m−1/3 k , which means the size of the BTZ black hole is in the order of AdS scale.

Summary
In this study we have investigated the 3d quantum gravity and its corresponding CFT. The equivalence between 3d gravity and Chern-Simons gauge theory, shows that the boundary CFT should be extremal and the partition function should be holomorphically factorizable and the left and right central charges are (24k L , 24k R ). The values of the gauge couplings have been obtained from the gauge group of the Chern-Simons theory.
The gauge group of the Chern-Simons theory can be SO(2, 1) × SO(2, 1) and its n-fold diagonal cover. From modular invariance of the holomorphically factorizable partition function, we have shown, the authorized values of the gauge group should be SO(2, 1) × SO(2, 1) group and its 3-fold gauge group. We also have shown that, the chiral and antichiral partition functions are modular covariant, and the scaling dimension of the primary fields are integer numbers and (c L = 24k L , c R = 24k R ) for k L , k R ∈ Z 3 . We have obtained the bases for the chiral partition functions in-terms of the Klein function and j = J 1 3 . Furthermore, we have introduced the generalized Hecke operators (3-Hecke operators) which map the modular covariant functions to modular covariant functions. We have studied the 3d pure gravity and corresponding boundary CFT for the case where the gauge group of the Chern-Simons theory is 3-fold diagonal cover (i.e. left and right central charges are integer multiple of eight ). Using the 3-Hecke operators we have obtained the closed-form for the partition function of the extremal CFT and the microcanonical entropy. We showed the microcanonical entropy is equal to the Bekenstein-Hawking entropy, the logarithmic corrections and some subleading terms which, are important when the size of the BTZ black hole is of the order of the AdS scale.