Parity and the modular bootstrap

Abstract

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.

arXiv comment21 pages, 3 figures, v2 reference and equation added, v3 minor edits and figure 2 improved

References in corpus (45)

• Symmetries and Strings in Field Theory and Gravity
• Conformal Four Point Functions and the Operator Product Expansion
• Conformal Partial Waves and the Operator Product Expansion
• Quantum Gravity Partition Functions in Three Dimensions
• Theta, Time Reversal, and Temperature
• Fermion Path Integrals And Topological Phases
• Universal Spectrum of 2d Conformal Field Theory in the Large c Limit
• Deconfined quantum critical points: symmetries and dualities
• Elliptic Genera of Symmetric Products and Second Quantized Strings
• A Universal Inequality for CFT and Quantum Gravity
• Time-Reversal Breaking in QCD${}_4$, Walls, and Dualities in 2+1 Dimensions
• Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
• Comments on Global Symmetries, Anomalies, and Duality in (2+1)d
• Intrinsic and emergent anomalies at deconfined critical points
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Poincare Series, 3D Gravity and CFT Spectroscopy
• Constraints on 2d CFT partition functions
• On the modular structure of the genus-one Type II superstring low energy expansion
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
• A Cardy Formula for Three-Point Coefficients: How the Black Hole Got its Spots
• Anomaly matching for phase diagram of massless ${\mathbb{Z}}_N$-QCD
• Universal Bounds on Charged States in 2d CFT and 3d Gravity
• A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance
• A Symmetry Breaking Scenario for QCD${}_3$
• Vacuum structure of bifundamental gauge theories at finite topological angles
• Circle compactification and 't Hooft anomaly
• Generalized global symmetries and holography
• Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\infty )$ non-linear $\sigma$-models
• Time-Reversal Symmetry, Anomalies, and Dualities in (2+1)$d$
• From 4d Yang-Mills to 2d ${\mathbb{CP}}^{N-1}$ model: IR problem and confinement at weak coupling
• Global inconsistency, 't Hooft anomaly, and level crossing in quantum mechanics
• Modular forms and a generalized Cardy formula in higher dimensions
• Universality of Sparse $d>2$ Conformal Field Theory at Large $N$
• Constraints on Flavored 2d CFT Partition Functions
• Charged structure constants from modularity
• Witten Diagrams for Torus Conformal Blocks
• Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus
• Modular Constraints on Conformal Field Theories with Currents
• Modular invariance on $S^1\times S^3$ and circle fibrations
• A conformal block Farey tail
• Black hole microstates in AdS
• Black holes from CFT: Universality of correlators at large c
• Holography on Non-Orientable Surfaces
• Detachable circles and temperature-inversion dualities for CFT${}_d$
• Emergent gravity from Eguchi-Kawai reduction

Cited by in corpus (24)

• Anomalies and Bounds on Charged Operators
• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• Sphere Packing and Quantum Gravity
• Bootstrapping Non-Invertible Symmetries
• Fast Conformal Bootstrap and Constraints on 3d Gravity
• Tauberian-Cardy formula with spin
• Einstein gravity from ANEC correlators
• Phases of scrambling in eigenstates
• ${\mathbb{Z}}_N$ Symmetries, Anomalies, and the Modular Bootstrap
• Solutions of modular bootstrap constraints from quantum codes
• Modular Constraints on Superconformal Field Theories
• Lessons from the Ramond sector
• Quantum stabilizer codes, lattices, and CFTs
• Classical Codes and Chiral CFTs at Higher Genus
• Discreteness and Integrality in Conformal Field Theory
• The Holographic Landscape of Symmetric Product Orbifolds
• Fake Z
• Siegel Paramodular Forms and Sparseness in AdS${}_3$/CFT${}_2$
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Conformal Bootstrap Deformations
• Light-state Dominance from the Conformal Bootstrap
• Aspects of the S transformation Bootstrap
• Chiral Modular Bootstrap
• Crossing, Modular Averages and $N\leftrightarrow k$ in WZW Models