Conformal perturbation theory and higher spin entanglement entropy on the torus

TL;DR

<3-5 sentences>We investigate a free fermion CFT in 1+1 dimensions perturbed by holomorphic spin-3 currents on a torus (circle of length $L$ at finite temperature $\beta^{-1}$), with the perturbation parameterized by a chemical potential $\mu$. Using bosonization, we compute the order $\mu^2$ corrections to the thermal partition function and to single-interval Renyi and entanglement entropies, carefully accounting for both quantum fluctuations and winding modes; the results are expressed in terms of weight-six quasi-modular forms for thermal data and quasi-elliptic functions for Renyi entropies, with a consistent integration prescription on the torus. We compare two integration prescriptions, showing they differ by a finite spin-4 counterterm and that one yields holomorphic, modular-covariant results while the other can be related to a deformed partition function by ${\widehat E}_2(\tau,\bar\tau)$. The spin-3 deformation results have a universal cylinder (high-temperature) limit that matches known ${\cal W}_\infty$ predictions, while winding-mode contributions are crucial for obtaining the correct finite-size entropies, including the correct $\Delta \to L$ limit where the Renyi entropy reproduces the thermal entropy. Our analysis also clarifies connections to the large-$N$ Yang-Mills genus expansion on the torus and illuminates how higher-spin hair in hs$[\lambda]$ theories manifests in finite-size entropies of the dual CFTs.>

Abstract

We study the free fermion theory in 1+1 dimensions deformed by chemical potentials for holomorphic, conserved currents at finite temperature and on a spatial circle. For a spin-three chemical potential μ, the deformation is related at high temperatures to a higher spin black hole in hs[0] theory on AdS_3 spacetime. We calculate the order μ^2 corrections to the single interval Renyi and entanglement entropies on the torus using the bosonized formulation. A consistent result, satisfying all checks, emerges upon carefully accounting for both perturbative and winding mode contributions in the bosonized language. The order μ^2 corrections involve integrals that are finite but potentially sensitive to contact term singularities. We propose and apply a prescription for defining such integrals which matches the Hamiltonian picture and passes several non-trivial checks for both thermal corrections and the Renyi entropies at this order. The thermal corrections are given by a weight six quasi-modular form, whilst the Renyi entropies are controlled by quasi-elliptic functions of the interval length with modular weight six. We also point out the well known connection between the perturbative expansion of the partition function in powers of the spin-three chemical potential and the Gross-Taylor genus expansion of large-N Yang-Mills theory on the torus. We note the absence of winding mode contributions in this connection, which suggests qualitatively different entanglement entropies for the two systems.

Figures (1)

• Figure 1: Left: Using prescription-1, integration along the A-cycle is performed first, avoiding any poles along the integration path. The integral along ${\cal C}$ is finite and independent of any smooth deformations of the path. The integral along ${\cal C}$ and ${\cal C}'$, differ by any non-zero residue around a pole (red dot). Right: Using prescription-2, we apply Stokes' theorem to the integral over the torus after excising disks $D_2$ centred around singular contact points (in red).