Harmonic analysis of 2d CFT partition functions

Abstract

We apply the theory of harmonic analysis on the fundamental domain of $SL(2,\mathbb{Z})$ to partition functions of two-dimensional conformal field theories. We decompose the partition function of $c$ free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space $\mathbb{H}/SL(2,\mathbb{Z})$, and of target space moduli space $O(c,c;\mathbb{Z})\backslash O(c,c;\mathbb{R})/O(c)\times O(c)$. This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to AdS${}_3$ gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies.

35+24 pages, v2: corrected a mistake in Sec 3, v3: minor errors fixed

References in corpus (12)

• More on Supersymmetric and 2d Analogs of the SYK Model
• Conformal Bootstrap in Mellin Space
• Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions
• Poincare Series, 3D Gravity and CFT Spectroscopy
• Eikonal Methods in AdS/CFT: Regge Theory and Multi-Reggeon Exchange
• Fast Conformal Bootstrap and Constraints on 3d Gravity
• A Crossing-Symmetric OPE Inversion Formula
• Wormholes without averaging
• Adding Flavor to the Narain Ensemble
• In search of conformal theories
• Poincaré Series, 3d Gravity and Averages of Rational CFT
• Averaging over moduli in deformed WZW models

Cited by in corpus (15)

• Semiclassical 3D gravity as an average of large-c CFTs
• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings
• Scalar Modular Bootstrap and Zeros of the Riemann Zeta Function
• Casimir Energy and Modularity in Higher-dimensional Conformal Field Theories
• A note on ensemble holography for rational tori
• Bootstrapping Closed Hyperbolic Surfaces
• $T\overline{T}$-deformed modular forms
• On Classification of Fermionic Rational Conformal Field Theories
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Integrating three-loop modular graph functions and transcendentality of string amplitudes
• Weighed average over Narain moduli space as $T\bar{T}$ deformation of CFT target space
• (Half) Wormholes under Irrelevant Deformation
• Rademacher expansion of modular integrals
• Modular linear differential equations for four-point sphere conformal blocks