Bootstrapping holographic warped CFTs or: how I learned to stop worrying and tolerate negative norms

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 Compere-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.

33 pages, 8 figures; v2: appendix and references added, matches published version

References in corpus (10)

• Semi-classical central charge in topologically massive gravity
• Boundary conditions for spacelike and timelike warped AdS_3 spaces in topologically massive gravity
• Poincare Series, 3D Gravity and CFT Spectroscopy
• Warped Conformal Field Theory as Lower Spin Gravity
• Microscopic Theory of Black Hole Superradiance
• Constraints on Flavored 2d CFT Partition Functions
• Tweaking one-loop determinants in AdS${}_3$
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• Modular Bootstrap Revisited
• Generating spaces of modular forms with $\eta$-quotients

Cited by in corpus (26)

• A note on the complex SYK model and warped CFTs
• Towards the generalized gravitational entropy for spacetimes with non-Lorentz invariant duals
• Warped Information and Entanglement Islands in AdS/WCFT
• Warped Symmetries of the Kerr Black Hole
• Modular Hamiltonians in flat holography and (W)AdS/WCFT
• Entanglement and chaos in warped conformal field theories
• Warped Schwarzian theory
• 2d Galilean Field Theories with Anisotropic Scaling
• On Galilean conformal bootstrap
• Complexity of warped conformal field theory
• Reflected Entropy in AdS${}_3$/WCFT
• Holographic entanglement entropy in AdS${}_3$$/$WCFT
• New boundary conditions for (extended) ${\mathrm{AdS}}_3$ supergravity
• Structure Constants from Modularity in Warped CFT
• Rényi Mutual Information in Holographic Warped CFTs
• Supersymmetric Warped Conformal Field Theory
• Near-Extremal Limits of Warped CFTs
• New Chiral Gravity
• Near-Extremal Limits of Warped Black Holes
• Supersymmetric Galilean conformal blocks
• Note on Bulk Reconstruction in AdS${}_3$/WCFT${}_2$
• Warped Conformal Symmetries of the Accelerating Kerr Black Hole
• Generalized Gibbs Ensemble of 2D CFTs with U(1) Charge from the AGT Correspondence
• Modular matrices of 2d Carrollian and warped CFTs
• Boundary Conditions for Warped AdS${}_3$ in Quadratic Ensemble
• Warped CFT duals of the Plebański-Demiański family of solutions