Parity and the modular bootstrap

We consider unitary, modular invariant, two-dimensional CFTs which are invariant under the parity transformation $P$. Combining $P$ with modular inversion $S$ leads to a continuous family of fixed points of the $SP$ transformation. A particular subset of this locus of fixed points exists along the line of positive left- and right-moving temperatures satisfying $\beta_L \beta_R = 4\pi^2$. We use this fixed locus to prove a conjecture of Hartman, Keller, and Stoica that the free energy of a large-$c$ CFT$_2$ with a suitably sparse low-lying spectrum matches that of AdS$_3$ gravity at all temperatures and all angular potentials. We also use the fixed locus to generalize the modular bootstrap equations, obtaining novel constraints on the operator spectrum and providing a new proof of the statement that the twist gap is smaller than $(c-1)/12$ when $c>1$. At large $c$ we show that the operator dimension of the first excited primary lies in a region in the $(h,\overline{h})$-plane that is significantly smaller than $h+\overline{h}<c/6$. Our results for the free energy and constraints on the operator spectrum extend to theories without parity symmetry through the construction of an auxiliary parity-invariant partition function.


Background
The modular bootstrap, first considered by Cardy [50], is a "medium-temperature" expansion (about β = 2π) of CFT 2 observables. The basic idea is to impose invariance of the partition function under the S transformation Invariance under S implies the vanishing of certain derivatives of Z at the fixed point of this transformation (τ, τ ) = (i, −i).
In [51], Hellerman rediscovered the modular bootstrap and used it to prove an upper bound on the scaling dimension of the first excited Virasoro primary ∆ (1) . Hellerman's bound states that ∆ (1) < c/6 at large c. In the language of AdS 3 gravity, c/6 is the dimension of the lightest BTZ black hole which dominates the canonical ensemble, at β = 2π − 0 + . However, BTZ black holes are known to exist all the way down to ∆ = c/12, suggesting it should be possible to improve the upper bound on ∆ (1) to c/12. So far, any such attempt has involved going to higher order in the expansion around the fixed point, e.g. [52,53]. While this strategy shows improvement at finite c, none have successfully improved the bound at asymptotically large c. New tools may be necessary.
A possible generalization of the modular bootstrap is to use fixed points of other transformations of the full modular group SL(2, Z). This means supplementing S invariance with T invariance Z(τ, τ ) = Z(τ + 1, τ + 1). For example, reference [54] considered the transformation ST . 1 However, individual terms in the partition function evaluated at such fixed points are not positive. This precludes applying ordinary bootstrap techniques around these SL(2, Z) fixed points.
In this paper, we make some progress by constructing an auxiliary, parity-invariant partition function that is amenable to standard bootstrap techniques.

Structure of the paper
In section 2, we show that combining parity symmetry P with modular inversion S leads to a continuous family of fixed points of the SP transformation. This continuous family of fixed points contains points with arbitrary β ∈ R + . This generalizes the S fixed point, which sits at β = 2π. The partition function along this fixed locus is manifestly non-negative, allowing us to employ standard bootstrap techniques.
In section 4, we generalize the modular bootstrap equations of [51] to the case of SPinvariant partition functions. These equations can be evaluated at any generic fixed point  2. We also give a new proof of the statement that the twist gap is at most (c − 1)/12 when c > 1.
Section 5 contains brief remarks on related topics. Kinematic definitions are collected in appendix A.

Modular invariance plus parity
The parity transformation acts on the partition function as follows (see appendix A for definitions) We remind the reader that the partition function is a function of independent complex variables τ and τ . Let us justify the action of parity (2.1). Physically, parity maps a state with energy E and spin J to a state with energy E and spin −J. Thus, on the section τ = τ * , parity changes the sign of the angular potential. In particular, the partition function is invariant under P if and only if the density of states ρ(E, J) satisfies ρ(E, J) = ρ(E, −J).
Equation (2.1) is the natural holomorphic upliftment of this to C 2 .
From (1.1) and (2.1), we see that the combination SP transforms the partition function The fixed locus of this transformation is τ τ = 1, which is a complex curve (and hence a real 2-surface) inside C 2 .
Since P and S both square to the identity and commute with one another, SP is the only nontrivial transformation incorporating both. Altogether, S, P , and T can be shown to generate the following general transformations:

Slices of the SP fixed locus
We now consider various slices of the SP fixed locus, i.e. the curve τ τ = 1. First, let us parametrize the partition function in terms of β and θ. Using the definitions in (A.1) and (2.2), SP invariance implies that which has a fixed locus along the curve β 2 + θ 2 = 4π 2 . Restricting to real β and θ along this locus, we have access to temperatures β ≤ 2π, which includes the infinite-temperature limit β → 0. We remark that the partition function for real (β, θ) is generically complex, but is real if and only if the theory is parity invariant. However, the individual terms in the partition function are not positive definite.
If we instead take β L , β R as independent variables, the statement of SP invariance becomes (again using (A.1) and (2.2)) with fixed locus β L β R = 4π 2 . From (A.2), we see that restricting to real β L and β R along this slice allows us to access temperatures with β ≥ 2π, including the zero-temperature limit β → ∞. We can reach this limit either by taking β L → ∞ or β R → ∞. The partition function is manifestly real and positive along the real (β L , β R ) slice: . This will be of crucial importance later. The only point along the curve β L β R = 4π 2 that is also an SL(2, Z) fixed point is ). This is because T acts as an imaginary shift of the left and right Altogether, by considering either the real (β L , β R ) slice or the real (β, θ) slice, we have access to fixed points at any temperature β ∈ R + . Henceforth, we restrict our attention to the slice with β L and β R both real.

CFT derivation of AdS free energy
HKS [32] proved two important results for the free energy of CFT 2 at finite temperature and zero angular potential. First, they showed that if the central charge c is large, the free energy at all temperatures is completely fixed by the spectrum of light states (∆ < c/12) only. Second, assuming that the light spectrum is sparse (ρ(∆) e 2π∆ ) in addition to large c, they were able to reproduce the free energy of Einstein gravity in AdS 3 .
HKS also provided evidence that a slightly stronger sparseness condition on the low-lying spectrum fixes the free energy at nonzero angular potential. Ultimately however, they were unable to prove this statement, leaving it as an open conjecture. In this section we will give a proof of their conjecture.

Proof of the HKS conjecture
Let be a small positive number. Let us define heavy states to be the states which have h > c/24 + and h > c/24 + , The remaining set of states is denoted by H c , which contains states with low twist. Here the superscript c is a mnemonic for set complement. Let Z H denote the partition function restricted to the states in H, and Z H c the partition function restricted to the states in H c .
For brevity we will denote by Z the partition function evaluated at (β L , β R ), and by Z the partition function evaluated at (β L , β R ). The SP invariance of the partition function implies Using the fact that the SP transformed temperatures are smaller, we can immediately get an upper bound on Z H : Here the first inequality is a direct consequence of (3.4), the second equality uses SP invariance (3.2), and the third simply uses the positivity of Z H c . Combining (3.4) and (3.5) we find: What the last inequality achieves is an upper bound on Z H , the contribution of the heavy states to the partition function, in terms of the contribution of the non-heavy states. This immediately implies that we can also place an upper bound on the full partition function: The first inequality comes from positivity of Z H and the second one follows from writing Z = Z H + Z H c and using (3.6). An identical analysis for the case β L β R < 4π 2 bounds Z by Z H c . Notice that the upper bound goes to infinity as ε → 1, which occurs as β L β R → 4π 2 .
Thus, as long as we keep both β L and β R fixed (with β L β R = 4π 2 ) as we scale c → ∞, we The log Z H c and log Z H c pieces are O(c) due to the contribution of the vacuum. This shows that the free energy is dominated by the non-heavy states at large c. To match the bulk AdS 3 gravity answer, we want log Z H c to be given solely by the vacuum contribution up to O(1) corrections for β L β R > 4π 2 (and similarly for log Z H c ). This will be assured if we assume that ρ(h, h) Here, means that one is allowed to have polynomial prefactors. In turn this condition is guaranteed if Thus, we have shown that (3.10) implies the AdS 3 free energy (3.9) in a large-c theory. The reverse implication can also be proven as long as we modify the bound for states that do not We will omit the details here, but the essence of the argument is the same: bound the density of states around any such state so that it does not contribute at the same order as the vacuum answer.
Remarkably, we can now show the results in this section hold if we relax the requirement that the CFT be parity invariant. Starting with a possibly non parity symmetric Z(β L , β R ), construct the parity invariant (and still modular invariant) object If ρ satisfies (3.10), ρ also satisfies (3.10) since Thus Z(β L , β R ) is given by (3.9). Next note that log Z vac ≤ log Z ≤ log Z. Since (3.9) says that log Z ≈ log Z vac , we see that log Z is both upper and lower bounded by the same quantity, and hence must be equal to it.

Cardy formula for ρ(E L , E R )
Like in [32], the above universal form of the free energy (3.9) allows us to conclude that the Cardy formula for the density of states extends beyond the regime of validity of Cardy's saddle point derivation. We can get the microcanonical density of states by inverse-Laplace transforming (3.9): where γ > 0. We can do the integral by saddle point. If √ E L E R > c/24, this saddle point is at a value of β L and β R such that β L β R < 4π 2 . We thus get the Cardy formula Note that this formula is valid for all √ E L E R > c/24, and not just for values of E L , E R that are much bigger than c. This gives a CFT derivation of the Bekenstein-Hawking entropy for all rotating BTZ black holes which dominate the canonical ensemble.
Orthogonally to the methods spelled out in this section, reproducing the AdS 3 free energy at finite temperature and angular potential can be achieved by assuming a particular center symmetry breaking pattern [39].

The parity-modular bootstrap
Now we turn to deriving constraints on the operator spectrum.

New bootstrap constraints
Let us work in the real (β L , β R ) plane. Recall that invariance under S implies that Z(β L , β R ) = Z 4π 2 /β L , 4π 2 /β R . The fixed point of this transformation is at β L = β R = 2π. This implies This was used in [51] to derive an upper bound on the dimension ∆ (1) of the lightest primary in a modular invariant 2d CFT.
Generalizing this to SP invariant theories, we get that along the fixed locus β L β R = 4π 2 the following derivatives vanish: Interestingly the presence of the fixed locus gives us a new set of functional relations which contain derivatives with N L + N R even. At β L = β R = 2π, the two terms in (4.2) are separately zero for N L + N R odd, in agreement with (4.1).
Let us examine the implications of (4.1) and (4.2) at low order. For (N L , N R ) = (1, 0), we find We can also get the behavior of E L along the fixed line as The inequality is strict for states with E L = E R , which exist in any CFT spectrum. Let us now sum this inequality over the spectrum. The left-hand-side is E L − E R . Using parity symmetry, the right-hand-side is E R − E L . Thus, Now (4.3) says that on the fixed line, E L = −4π 2 E R /β 2 L . So for (4.4) to be true, E R must be strictly negative for temperatures on the fixed line with β L < β R . In particular, E R ∈ [−c/24, 0) after imposing unitarity. It is important that E R cannot be zero, which follows from the strictness of the inequality in (4.4). Taking the limit β L → 0 on the fixed line, we see that for some e L ∈ R + . (4.5) Interestingly, this is the high-temperature scaling of energy with temperature in CFT, which we have shown holds in the purely left-moving sector, even though β = (β L + β R )/2 → ∞.
One can derive further results about expectation values at higher orders, but these apply only to parity-invariant theories. We now turn to constraints on the spectrum of operators which will also apply to parity non-invariant theories.

Bounds on the spectrum of operators
Assuming c > 1, the partition function can be expressed as a sum over Virasoro characters, which can be written down explicitly in terms of the Dedekind eta function. The vacuum Virasoro character is (4.6) and the character for a non-conserved Virasoro primary with weights (h, h) is Let us consider the following specific linear combination of derivatives from (4.2):   Via numerical work, reference [53] improved the bound on ∆ (1) for c of order 100 to ∆ (1) c/9. However, at asymptotically large c, the best result to compare to is still ∆ (1) < c/6 obtained in [51]. Keeping in mind our qualitative goals, we have not tried to exhaustively apply the machinery of linear programming. We leave it for future work to see if one can do better by exhaustively searching for an optimal functional about points other than β L = β R = 2π.

Bounds on the twist gap
It is known that all unitary, modular-invariant CFT 2 's with c > 1 and no conserved currents have a sequence of Virasoro primaries with twist accumulating to (c − 1)/12 [53,55]. In this section we will derive a simpler statement: there must be a Virasoro primary with twist smaller than (c − 1)/12 + for every > 0. Recall that twist equals 2 min(h, h).
Let us consider the following linear combination of derivatives from (4.2): 1) , (4.10) Here we pick the constant α 1 to be as in (4.11) and we pick α 2 to be such that the functional vanishes on the vacuum (4.12). The constant q can be any real number greater than or equal to two. A sample plot of the regions in the (h, h) plane where the functional is negative is given in in Figure 3. The blue shaded regions are where the functional is negative, and thus we must have a state that is in the blue region and has h and h positive. Note that the existence of a continuous fixed locus and the even order derivative constraints were both crucial here. We needed to go to the limit β L → ∞ along the fixed locus and include the (2, 0) derivative to derive this result.

Time reversal
One might wonder about additional discrete symmetries other than parity, and if they provide additional constraints. One natural candidate is time reversal symmetry. However time The parity transformation is usually defined as flipping the signs of all the spatial coordinates. When the number of spatial dimensions is even, this transformation has determinant one, and thus belongs to the part of the orthogonal group that is connected to the identity. It was emphasized in [56] that a more uniform treatment of even and odd dimensions is possible if we instead work with the reflection transformation R, which is defined to flip only one of the spatial coordinates. The determinant of this transformation is always −1. Importantly, in Euclidean signature, the 'time' coordinate is not special, so 'time reversal' is also simply a reflection, which is why time reversal symmetry does not provide us with new constraints.

Fixed locus in four-point function crossing equation
Let us consider the four point correlation function of identical scalar operators. Usually crossing is stated as the mapping (z, z) → (1 − z, 1 − z) of the cross ratios, which has the unique fixed point (z, z) = (1/2, 1/2). Parity acts as (z, z) → (z, z). The combination of crossing and parity thus acts as (z, z) → (1 − z, 1 − z), so we see that parity again extends the fixed point to a fixed locus z + z = 1.
In dimensions bigger than two, the conformal block is automatically symmetric under z ↔ z. For example, one can see this explicitly in D = 4, where the conformal block contribution of a primary with spin and dimension ∆ is [57,58] . More abstractly, we use conformal invariance to place three points at 0, (1, 0, . . . , 0) and ∞, and the fourth point in the 1 − 2 plane. In dimensions bigger than two, the symmetry under exchanging z and z is automatic because of the existence of a rotation in the 2 − 3 plane that maps z to z. 2 Thus, the fixed point (z, z) = (1/2, 1/2) is always enhanced to the fixed line z + z = 1 in dimensions bigger than two.
In two dimensions, there is no third dimension that we can use to rotate z to z, and therefore the z ↔ z symmetry is not automatic. This is expected given the many examples of perfectly good CFT 2 's which have left-right asymmetry. In more detail, the 2D global conformal block of a primary with quantum numbers (h, h) is (again considering identical external scalars) Note the manifest asymmetry with respect to interchange of z and z. To get something parity symmetric, we have to add the contribution of the parity-transformed state with the same OPE coefficient. The four point function in a parity invariant CFT 2 , therefore has the following conformal block decomposition: Now one can see explicitly that G(z, z) = G(z, z). Thus, combining crossing and parity, we conclude that there is a fixed line z + z = 1.

Theories with an internal U (1) symmetry
Consider a relativistic CFT 2 with an internal U (1) symmetry. Let us note immediately that not every theory with a U (1) symmetry admits an extra 'charge conjugation' symmetry.
Now we consider an SP transformation to see if we have a fixed line in the presence of these chemical potentials: The fixed locus is given by τ τ = 1 and zτ = z. Notice that the latter constraint also makes the anomalous prefactor vanish. In terms of real parameters τ = τ 1 + iτ 2 and z = z 1 + iz 2 , we have a two-dimensional fixed plane in R 4 given by , τ 2 = 2z 1 z 2 z 2 1 + z 2 2 . (5.8) In analogy with the unflavored case considered in this paper, we expect that this fixed locus can be used to improve the results of [59,60,64] for CFT 2 's with a U (1) symmetry.

Acknowledgments
We would like to thank Nathan Benjamin, Scott Collier,

A Definitions
The grand canonical ensemble is parameterized by the inverse-temperature β and angular potential θ. In two-dimensional CFTs they are often packaged into 2πτ . . = iβ L . . = θ + iβ , 2πτ . . = −iβ R . . = θ − iβ . (A.1) As is customary, we will treat τ and τ as a priori independent complex variables. When trading variables, this then translates into treating (β L , β R ) or (β, θ) as independent sets of complex variables. Given these definitions it is easy to see that Introduce the variables q and q via q . . = exp (2πiτ ), q . . = exp (−2πiτ ) .
The partition function is defined as a function from C 2 to C: where c and c are the central charges for the holomorphic and the anti-holomorphic Virasoro algebras. One can show the partition function is modular invariant with respect to SL(2, Z) transformations on τ and τ , without restricting to the section τ = τ * : see [32] for an argument using the Weierstrass preparation theorem.
States are labeled by a pair of non-negative real numbers (h, h), which are the eigenvalues of the L 0 and L 0 . The conformal dimension ∆ and spin s of a state are then defined as The twist of a state is defined as In the literature, states are also frequently parametrized either by their left-and right-moving energies, defined as: or by their total energy E and angular momentum J, defined as: In the main text, we will only consider theories with c = c.