Non-chiral 2d CFT with integer energy levels

The partition function of 2d conformal field theory is a modular invariant function. It is known that the partition function of a holomorphic CFT whose central charge is a multiple of 24 is a polynomial in the Klein function. In this paper, by using the medium temperature expansion we show that every modular invariant partition function can be mapped to a holomorphic partition function whose structure can be determined similarly. We use this map to study partition function of CFTs with half-integer left and right conformal weights. We show that the corresponding left and right central charges are necessarily multiples of 4. Furthermore, the degree of degeneracy of high-energy levels can be uniquely determined in terms of the degeneracy in the low energy states.


Introduction
An important question in conformal field theory (CFT) is to what extent a theory can be identified in terms of its constraints and symmetries. The bootstrap hypothesis [1,2,3] is based on the crossing symmetry. Recently in 4d CFT the crossing symmetry has been used to obtain an upper bound on the weights of the fields that appear in the operator product expansion of scalar operators [4]- [8] and a lower bound on the stress tensor central charge [9,10]. Similarly an upper bound on the scaling dimension of the first scalar operator appearing in the OPE of two quasi-primary scalar operators has been obtained in two dimensions [4].
In two dimensions, the infinite dimensional group of the conformal symmetry makes the bootstrap project more efficient. Furthermore, the partition function of a 2d CFT should be invariant under modular transformations. The modular group PSL(2, Z) is the disconnected diffeomorphism group of the torus where τ = τ 1 + iτ 2 is the complex structure taking value in the upper half plane (τ 2 ≥ 0) and τ = τ 1 − iτ 2 . The generators of the modular group are T : (τ,τ ) → (τ + 1,τ + 1), Invariance under T -transformation (henceforth T -invariance) constrains the spin of states and the difference between the left and right central charges of the conformal field theory. S-invariance constrains the density of states and the spectrum of the theory. In [11] the S-invariance of partition function has been used to estimate the density of states in the saddle-point approximation for a unitary CFT. It is seen that the density of states at conformal dimension h grows exponentially with the square root of h [12]. The Cardy formula is a key ingredient in the AdS 3 /CFT 2 correspondence; it reproduces the Bekenstein-Hawking entropy of the BTZ-black holes [13]. 1 S-invariance has been also used to compute the 'logarithmic correction' [12] and 'the beyond the logarithmic corrections' to the Cardy formula [15]. In [16] it is shown that in theories with sparse light spectrum and large central charge, the Cardy formula also works for energies greater than the central charge.
Recently, the modular invariance of partition function has been used in order to obtain an upper bound on conformal dimensions of the primary fields. In [17] for holomorphically factorizable models whose left and right central charges are multiples of 24, an upper bound on the lowest primary fields has been obtained, ∆ ≤ min This upper bound is saturated in extremal CFTs [18,19]. Extremal CFT's are promising holographic duals to the pure gravity with negative cosmological constant [17,20]. The vacuum state corresponds to the AdS space, and the primary fields above the vacuum correspond to the BTZ black hole. Modular invariance is enough to determine the partition function of an extremal CFT. For c = 24 an extremal CFT is known and its uniqueness has been conjectured [21,22]. The holomorphic and anti-holomorphic parts of the partition function are modular functions. A modular function can be written in terms of a polynomial in the Klein function J [23]. While for the other values of the central charge the partition functions are known it is not clear whether such CFTs exist [24]. In general, for CFT's in which there is no chiral algebra beyond the Virasoro algebra and c ≫ 1, the following upper bound on the lowest primary operator has been obtained [25]- [29] ∆ ≤ c tot 12 In fact for asymptotically large central charge this inequality is valid for ∆ n with n ≤ e πc 12 [30,31]. The upper bound (4) can be computed by using the medium temperature expansion. This method uses the S-invariance of the partition function at the self-dual point τ = −τ = i [25]. Considering a small neighborhood of in the limit s → 0, one obtains an infinite set of constraints on the partition function: Combining the constraints that can be obtained by different selections of (N L , N R ) leads to certain universal constraints on the spectrum [25], [30]- [33].
In this work we use the medium temperature expansion method in a different manner. We note that Eq. (6) indicates that for any (smooth) odd function f (x, This observation leads to an interesting result: corresponding to every S-invariant non-chiral partition function Z(τ,τ ), there exist an S-invariant chiral function Z(τ ). The map Z(τ,τ ) → Z(τ ) can be interpreted as the chiralization of the partition function. The chiral function corresponding to the non-chiral partition function can be easily obtain by insertingτ = −τ in Z(τ,τ ). That is, This equation implies that Z(τ ) can be obtained by analytic continuation of the 'canonical' partition function to the complex β-plane. The behavior of the chiral function Z(τ ) under T transformation depends on the spectrum of the main theory.
Focusing on a special class of CFTs whose primary operators have half integer scaling dimensions (henceforth HI-CFT), we show that the corresponding chiral partition function is an eigen-function of T whose eigen-value is e −iπc tot 12 Since Z(τ ) is by construction S-invariant, the identity (ST ) 3 = 1 implies that c tot ∈ 8 Z. Thus, in such theories c L and c R are inevitably multiples of 4. We show that the corresponding chiral partition function Z(τ ) can be determined in terms of 1 + [ k 3 ] positive integers.
Since the degree of degeneracy of levels in Z(τ,τ ) and Z(τ ) are equivalent (as can be inferred from Eq.(9)), Eq. (11) implies that the degree of degeneracy of high-energy levels in Z(τ,τ ) can be uniquely determined in terms of the degeneracy in the low energy states. The organization of the paper is as follows. In sections 2 we review the effect of two constraints on the partition function. One of them is the T -invariance and the other one is the simple fact that partition function should be real-valued. In section 3 we study the S-invariance of partition function and use the medium temperature expansion to obtain the chiralization map. Sections 4 and 5 are devoted to the HI-CFT's. We study the chiral partition function Z(τ ) in section 4, and identify a subclass of HI-CFT partition functions in terms of free-fermions in section 5. Some technical details are relegated to the appendices. Our main results are summarized in section 6.

Constraints on the spin values
Consider a two dimensional unitary CFT on a circle of length 2π. The partition function of the theory at temperature 1 β and chemical potential µ c is as follows in which µ := µ c β, H is the Hamiltonian and P is the momentum on the compact spatial direction. The eigenvalues of H and P are ∆ − ctot 24 and j − c dif 24 respectively [34].
where c L and c R are the left and right central charges. This partition function can be interpreted as a CFT partition function on a torus whose complex structure is given by In this picture, the conformal weights are given by and the partition function can be written as follows. 2 in which Henceforward we assume the following: • The partition function is invariant under modular transformation; • The spectrum contains the identity operator h =h = 0; • The density of states ρ(h,h) are positive integer numbers; • The partition function is real.
In the following, we show that T -invariance indicates that the spin j ∈ Z and c dif ∈ 24Z. Furthermore we show that since the partition function is real-valued, at each energy level, the number of states with spin j and −j + c dif 12 are equivalent.

T -invariance of partition function
Therefore, T -invariance of the partition function requires that For µ = 0 Eq. (19) gives The summands in (20) are non-negative. Consequently j − c dif 24 is necessarily an integer. The vacuum state (j = 0) enforces that c dif ∈ 24Z. Therefore, j ∈ Z. From Eq. (19) one verifies that these conditions are also sufficient.

Partition function is real-valued
The imaginary part of the partition function (12) is zero.
where J ∆ ⊂ [−∆, ∆] denotes the set of spins of states with energy ∆. From the T -invariance we already know that j − c dif 24 ∈ Z. Using the orthogonality of sin j − c dif 24 µ (as a function of µ) in Eq. (22) Assuming the ordering ∆ 1 < ∆ 2 < · · · , Eq.(23) reads By considering the β → ∞ limit one verifies that Using Eq. (25) in Eq.(24), the same argument implies that ρ( Since j − c dif 24 is the momentum eigen-value, we conclude that Corollary 2.1. A 2d CFT whose partition function is real-valued and T -invariant is parity even.

Invariance of partition function under S-transformation
S-invariance of the partition function, implies that [25], 3 in which At the self dual point τ = ω = i, andτ =ω = −i, this condition readŝ whereD L = τ ∂ ∂τ andD R =τ ∂ ∂τ are respectively the left and the right dilatation operators. For a holomorphic test function F (τ ) Eq. (30) implies that for any (smooth) odd function Using where u L = ie x L and u R = −ie ix R . 4 An immediate result of the identities (33) and (34) is where u L and u R are two independent C parameters taking value in the upper half-plane and in the lower half-plane respectively.

Chiralization of the partition Function
Consider the case u L = −u R = τ and define 4 Since the growth of ρ(h,h) in Eq. (16) is controlled by the Cardy formula, the partition function is convergent if the imaginary parts of u L and u R are positive and negative respectively. We assume that Z(u L , u R ) gives a biholomorphic analytic continuation of Z(β, µ) to complex u L in the upper half-plane and u R in the lower half-plane.
Eq.(16) (for q =q) gives where 5ρ From (36) we learn that the function Z(τ ) is invariant under S-transformation. In summary, We call the map ch : the chiralization map and Z(τ ) the ch-image of Z(τ,τ ). Table 1 shows some example of the known partition function and the corresponding ch-images.

CFT's with half-integer conformal weights
In this section we investigate a family of CFT's in which ∆ ∈ Z. Since j ∈ Z, the corresponding conformal weights are half-integers. Hence we call such a CFT an HI-CFT. In the following Z(τ,τ ) and Z(τ ) denote the partition function of an HI-CFT and the corresponding ch-image respectively. From Eq.(38) one verifies that Since Z(τ ) is S-invariant, using the identity (T S) 3 = 1 one obtains Consequently, From the T -invariance of Z(τ,τ ) we have learned that c dif ∈ 24 Z. Therefore, Corollary 4.1. For an HI-CFT Now we are ready to obtain the basis for Z(τ ). Let's start with c L , c R ∈ 12N. In that case c tot ∈ 24Z and Z(τ ) defined by Eq.(38) is a well-defined modular invariant meromorphic function in the upper half plane. Therefore it can be given as a polynomial in the Klein function J [23], The Klein function can be written in terms of the Jacobi Theta functions θ i (τ ) (i = 2, 3, 4) and the Dedekind function η(τ ).
where j(τ ) := 1 2 In the following we show that for c tot ∈ 8N, Z(τ ) can be written in terms of a polynomial in j.
in the upper half τ -plane. Then b. It is a polynomial in j.
Proof. T 3 -invariance is obvious. Eq.(47) and Eq.(48) imply that there exist {a (r−1) } such that Therefore, The order of the poles of f (r) {a (r) }, τ and f (r−1) {a (r−1) }, τ are r and r − 1 respectively. The Sinvariance of f (r) {a (r) }, τ , j and J imply that f (r−1) {a (r−1) }, τ is also S-invariant. By iteration one obtains where The function f (−1) {a (−1) }, τ 3 is modular invariant. It has no pole in the upper half plane and is zero at τ = i∞. Thus it is zero in the upper half plane.
Corollary 4.3. The ch-image of the HI-CFT partition function with total central charge c tot = 8k, has an expansion in terms of j as follows The degeneracy of the vacuum state is given by n 0 . In the following we assume that n 0 = 1. Eq.(54) shows that Z(τ ), and consequently the number of states with energy ∆ i.e.ρ(∆) can be uniquely determined if c tot and the integers n r , or equivalently, the low-energy (i.e. ∆ ≤ k 3 ) density of states are given. 6 Finally, consider an HI-CFT whose ch-image Z(τ ) is extremal, i.e. Z(τ ) = q −k/3 [1 + O(q)]. In that case, the coefficients n r can be uniquely determined in terms of the central charge. Furthermore, the scaling dimension of the first primary field after identity is ∆ 1 = ctot 24 + 1, which is in agreement with the upper bound given in Eq.(4).

AdS/CFT correspondence
It is known that the Cardy formula reproduces the Bekenstein-Hawking entropy at ∆ ≫ 1. In [17] it has been observed that for k ∈ 3 N, 7 the number of primary fields is given by the Cardy formulâ Therefore it is natural to assume that the primary fields correspond to the micro-states of the BTZ black hole. In Table 2 the Fourier expansion of the ch-image is given for c tot = 8, 16, 24. The coefficients of the expansions determine the density of stateρ(∆) which equals the number of states with energy ∆ and spin j ∈ [−∆, ∆]. For k = 1, 2, 3 the first high energy state (i.e. ∆ = 1 + k 3 ) has weight ∆ = 1, 1, 2 respectively. It is an interesting observation that the corresponding number of states can be estimated by the Cardy formula.

A basis for HI-CFT partition function
In the previous section we have observed that the ch-image of the HI-CFT partition function is a polynomial in j. Motivated by the fact that j is the ch-image of 1 2 3 i=1 θ i η 8 in this section we study a class of HI-CFT's whose partition functions can be given as a polynomial in θ i η and θ i η .
central charge ch-Image of partition function The functions θ i η have the following Fourier expansion. where The S-transformation of the Dedekind function η and the Theta functions are as follows.
T -transformation of these functions is given by, Since an HI-CFT only contains primary fields with half integer scaling dimension, from Eqs.(56-62) one infers that the corresponding partition function is a polynomial in x, y and z defined as follows.
x := ϑ 2 η where A(q), B(q) and C(q) are polynomials in q with positive integer coefficients. The functions x, y and z are not independent. They are related through the standard relations between the Theta functions and Dedekind function.
x − y + z = 0, (66) By using Eq.(67) and the transformation rules T : x → e 2iπ/3 x, one can show that the most general modular covariant combination of x, y, z can be written as follows.
S-invariance of the partition function implies that Z(τ,τ ) should be an even function in R − a,b,c,d . In Appendix A we show that R − a,b,c,d R − a ′ ,b ′ ,c ′ ,d ′ is a linear combination of R + a,b,c,d . Hence, we concentrate on polynomials in R + a,b,c,d and drop the + sign for simplicity. Noting that one can use Eq.(66) to show that These identities together with Eq.(67) result in the following recurrence relations.
Eqs.(80)-(83) show that every R a,b,c,d is a polynomial in j,j, and To show it, we first consider the chiral function Noting that R 0,0 = 6, R 1,0 = 0, one verifies that j is the single generator of R c,d .
Since the HI-CFT partition function is modular invariant, we investigate the invariance of R under T transformation. h is modular invariant. j and k are eigen-functions of T with eigen-values e respectively. In order to determine g i , (i = 0 · · · 5), we write them as follows where F Therefore, every HI-CFT partition function can be written as Noting that the ch-image of h is j and the ch-image of k equals -48, 8 one easily verifies that the ch-image of Z(τ,τ ) is a function of j in agreement with corollary 4.3.
which corresponds to 8 right-handed and 8 left-handed fermions. The corresponding ch-image is • c tot = 16. In this case the partition function is not unique.
The ch-image of Z(τ,τ ) is j 2 (independent of a and b). The factor 1 a+b indicates that there is a single vacuum state. The coefficients a an b should be determined in such a way that the density of states are positive integers. By inspecting the first few terms in the Fourier expansion of Z(τ,τ ), one can obtain the following necessary condition.
in which m and n are nonnegative integers. This gives where m ′ := m 8 and n ′ := n 8 . Using Eq.(107) in Eq.(105) one obtains For 7 ≤ m ′ ≤ 31 the energy densities are obviously positive integers. We have not been able to exclude the partition functions corresponding to 0 ≤ m ′ ≤ 6. Therefore, we are optimistic that there should be 32 different HI-CFT's with c tot = 16.

Summary
In this work we have studied modular invariant partition functions of unitary CFT's whose conformal weights are half-integers, hence HI-CFT's. By using the medium temperature expansion we have obtained a chiralization map which maps every S-invariant non-chiral partition function to an Sinvariant chiral partition function. We have used the chiralization map to show that the left and right central charges of an HI-CFT are multiples of 4. Furthermore, we have shown that the partition function after chiralization can be written as a polynomial in j = J 1/3 , where J is the Klein function.
In this way we have realized that the degree of degeneracy of the high energy levels ∆ > c L +c R

24
can be uniquely determined in terms 1 + c L +c R 24 integers corresponding to the degeneracy in the low energy states.
We have identified a class of HI-CFT's whose partition functions can be given in terms of the Jacobi Theta function θ i and the Dedekind function η. In Eq.(102) we have given the most general form of such partition functions.
A S-invariant combinations of R a,b,c,d The multiplication rule for R − a,b,c,d can be obtained as follows.
We have separated the above terms in 5 combinations. We show that each combination is an R + . It is clear that these terms have the following structure.
In order to proceed we need to classify different orderings of a, b, c and (a ′ , b ′ , c ′ ). In general, there are nine of them as follows We first consider cases 1, 4 and 7.