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.

arXiv comment35+24 pages, v2: corrected a mistake in Sec 3, v3: minor errors fixed

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• Weighed average over Narain moduli space as $T\bar{T}$ deformation of CFT target space
• (Half) Wormholes under Irrelevant Deformation
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