Abstract
We use modular invariance to derive constraints on the spectrum of warped conformal field theories (WCFTs) — nonrelativistic quantum field theories described by a chiral Virasoro and U(1) Kac-Moody algebra. We focus on holographic WCFTs and interpret our results in the simplest holographic set up: three dimensional gravity with Compère-Song-Strominger boundary conditions. Holographic WCFTs feature a negative U(1) level that is responsible for negative norm descendant states. Despite the violation of unitarity we show that the modular bootstrap is still viable provided the (Virasoro-Kac-Moody) primaries carry positive norm. In particular, we show that holographic WCFTs must feature either primary states with negative norm or states with imaginary U(1) charge, the latter of which have a natural holographic interpretation. For large central charge and arbitrary level, we show that the first excited primary state in any WCFT satisfies the Hellerman bound. Moreover, when the level is positive we point out that known bounds for CFTs with internal U(1) symmetries readily apply to unitary WCFTs.
Highlights
We focus on holographic warped conformal field theories (WCFTs) and interpret our results in the simplest holographic set up: three dimensional gravity with Compere-Song-Strominger boundary conditions
In the Hellerman bound for holographic WCFTs (4.26) different characters lead to O(10−3) corrections, the latter of which are small due to the small number of τ derivatives taken in the crossing equation
We found that despite the mild violation of unitarity, the modular bootstrap is still feasible in theories with negative norm descendant states
Summary
We consider three-dimensional Einstein gravity with CSS boundary conditions and an asymptotic Virasoro-Kac-Moody algebra. We begin by recalling that in asymptotically AdS3 spacetimes with Brown-Henneaux boundary conditions the boundary metric is flat and nondynamical [1] This boundary condition is necessary, not sufficient, to recover the two copies of the Virasoro algebra that characterize the dual conformal field theory.. Where x± = t ± φ and B (x+) = ∂+B(x+) This boundary condition, along with other boundary conditions on the subleading components of the metric, lead to an asymptotic symmetry group described by a left-moving Virasoro and U(1) Kac-Moody algebras [8]. These are the symmetries of a WCFT and one finds that at the boundary the VirasoroKac-Moody symmetries act as x+ → f (x+), x− → x− + g(x+),.
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