Chern-Simons Invariants from Ensemble Averages

TL;DR

This paper explores the relationship between ensemble-averaged 2D conformal field theories linked to indefinite lattices and their holographic duals, revealing connections to Abelian Chern-Simons theories and modular forms.

Contribution

It provides a novel framework connecting ensemble averages of conformal field theories to three-dimensional Chern-Simons theories, including spin variants, and elucidates their modular properties.

Findings

Partition functions as modular forms

Duality with Abelian Chern-Simons theories

Identification of phenomena in spin Chern-Simons cases

Abstract

We discuss ensemble averages of two-dimensional conformal field theories associated with an arbitrary indefinite lattice with integral quadratic form $Q$. We provide evidence that the holographic dual after the ensemble average is the three-dimensional Abelian Chern-Simons theory with kinetic term determined by $Q$. The resulting partition function can be written as a modular form, expressed as a sum over the partition functions of Chern-Simons theories on lens spaces. For odd lattices, the dual bulk theory is a spin Chern-Simons theory, and we identify several novel phenomena in this case. We also discuss the holographic duality prior to averaging in terms of Maxwell-Chern-Simons theories.