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SciPost Physics

Near-extremal limits of warped CFTs

Ankit Aggarwal, Alejandra Castro, Stéphane Detournay, Beatrix Mühlmann

Published Aug 9, 2023• 12 citations
Publisher

Abstract

Warped conformal field theories (WCFTs) are two-dimensional non-relativistic systems, with a chiral scaling and shift symmetry. We present a detailed derivation of the near-extremal limit for their torus partition function. This limit requires large values of the central charge, and is only consistent for non-unitary WCFT. We compare our analysis with previous studies of WCFT and its relation to a one-dimensional warped-Schwarzian theory. We discuss different ensembles of warped CFTs and contrast our results with analogous limits in two-dimensional CFTs.

DOI
10.21468/scipostphys.15.2.056

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Warped conformal field theories (WCFTs) are two-dimensional non-relativistic systems, with a chiral scaling and shift symmetry. We present a detailed derivation of the near-extremal limit for their torus partition function. This limit requires large values of the central charge, and is only consistent for non-unitary WCFT. We compare our analysis with previous studies of WCFT and its relation to a one-dimensional warped-Schwarzian theory. We discuss different ensembles of warped CFTs and contrast our results with analogous limits in two-dimensional CFTs.
Warped conformal field theories (WCFTs) are a novel class of non-relativistic theories. A simple, yet non-trivial, example of such theory is a massive Weyl fermion in (1 + 1)-dimensions, which we study in detail. We derive general properties of the spectrum and modular properties of partition functions of WCFTs. The periodic (Ramond) sector of this fermionic system is non-trivial, and we build two novel partition functions for this sector which have no counterpart in a CFT2. The thermodynamical properties of WCFTs are revisited in the canonical and micro-canonical ensemble.
Various aspects of warped conformal field theories (WCFTs) are studied including entanglement entropy on excited states, the Renyi entropy after a local quench, and out-of-time-order four-point functions. Assuming a large central charge and dominance of the vacuum block in the conformal block expansion, (i) we calculate the single-interval entanglement entropy on an excited state, matching previous finite temperature results by changing the ensemble; and (ii) we show that WCFTs are maximally chaotic, a result that is compatible with the existence of black holes in the holographic duals. Finally, we relax the aforementioned assumptions and study the time evolution of the Renyi entropy after a local quench. We find that the change in the Renyi entropy is topological, vanishing at early and late times, and nonvanishing in between only for charged states in spectrally-flowed WCFTs.
Abstract Various aspects of warped conformal field theories (WCFTs) are studied including entanglement entropy on excited states, the Rényi entropy after a local quench, and out-of-time-order four-point functions. Assuming a large central charge and dominance of the vacuum block in the conformal block expansion, (i) we calculate the single-interval entanglement entropy on an excited state, matching previous finite temperature results by changing the ensemble; and (ii) we show that WCFTs are maximally chaotic, a result that is compatible with the existence of black holes in the holographic duals. Finally, we relax the aforementioned assumptions and study the time evolution of the Rényi entropy after a local quench. We find that the change in the Rényi entropy is topological, vanishing at early and late times, and nonvanishing in between only for charged states in spectrally-flowed WCFTs.
We present a detailed discussion of entanglement entropy in (1+1)-dimensional Warped Conformal Field Theories (WCFTs). We implement the Rindler method to evaluate entanglement and Renyi entropies for a single interval and along the way we interpret our results in terms of twist field correlation functions. Holographically a WCFT can be described in terms of Lower Spin Gravity, a SL (2, ℝ) × U (1) Chern-Simons theory in three dimensions. We show how to obtain the universal field theory results for entanglement in a WCFT via holography. For the geometrical description of the theory we introduce the concept of geodesic and massive point particles in the warped geometry associated to Lower Spin Gravity. In the Chern-Simons description we evaluate the appropriate Wilson line that captures the dynamics of a massive particle.
We calculate the contributions of a general non-vacuum conformal family to Rényi entropy in two-dimensional conformal field theory (CFT). The primary operator of the conformal family can be either non-chiral or chiral, and we denote its scaling dimension by Δ . For the case of two short intervals on a complex plane, we expand the Rényi mutual information by the cross ratio x to order x 2 Δ +2 . For the case of one interval on a torus with low temperature, we expand the Rényi entropy by q =exp(−2π β /L), with β being the inverse temperature and L being the spatial period, to order q Δ +2 . To make the result meaningful, we require that the scaling dimension Δ cannot be too small. For two intervals on a complex plane we need Δ > 1, and for one interval on a torus we need Δ > 2. We work in the small Newton constant limit on the gravity side and so a large central charge limit on the CFT side, and find matches of gravity and CFT results.
A bstract Two-dimensional conformal field theories (CFTs) defined on non-orientable Riemann surfaces obey consistency Cardy conditions analogous to those in the orientable case. We revisit those conditions for irrational theories with central charge c > 1 in the context of two-point functions of primaries on the Real Projective plane $$ {\mathbbm{RP}}^2 $$ RP 2 and the partition function on the Klein bottle $$ {\mathbbm{K}}^2 $$ K 2 . Using the irrational versions of the Virasoro fusion and modular kernels we derive universal expressions for the non-orientable CFT data at large conformal dimension, assuming a gap in the spectrum of scalar primaries. In particular, we derive asymptotic formulas at finite central charge for the averaged Light-Light-Heavy product C LLH × Γ H of OPE coefficients with the $$ {\mathbbm{RP}}^2 $$ RP 2 one-point function normalizations, as well as for the parity-weighted density of heavy scalar primaries (or equivalently the density of heavy ΓH2 {\Gamma}_H^2 ΓH2​ Γ H 2 ). We discuss the gravitational interpretation of the results.