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Chern-Simons Invariants from Ensemble Averages

Meer Ashwinkumar, Matthew Dodelson, Abhiram Kidambi, Jacob M. Leedom, Masahito Yamazaki

TL;DR

This work shows that ensemble averaging over Narain-type CFT moduli spaces defined by an indefinite lattice with quadratic form $Q$ yields a holographic dual described by a three-dimensional Abelian Chern-Simons theory with a $K$-matrix given by $Q$, and the averaged partition function becomes a Siegel-Eisenstein series expressed as a sum over lens-space CS invariants. For even $Q$ the bulk dual is purely bosonic, while odd $Q$ requires spin-Chern-Simons theory, with spin-structure dependent invariants precisely matching the spin-refined Eisenstein sums. The paper also demonstrates a pre-average holographic dual via Maxwell-Chern-Simons theory, where boundary wavefunctions reproduce the CFT theta functions for each moduli point, and discusses higher genus generalizations and the role of the Siegel-Weil theorem in linking moduli averages to bulk topological data. The results illuminate how ensemble averages naturally encode bulk sums over geometries and connect topological field theory, modular forms, and holography, offering a framework to study holography with ensembles and possibly shedding light on quantum gravity in lower dimensions.

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.

Chern-Simons Invariants from Ensemble Averages

TL;DR

This work shows that ensemble averaging over Narain-type CFT moduli spaces defined by an indefinite lattice with quadratic form yields a holographic dual described by a three-dimensional Abelian Chern-Simons theory with a -matrix given by , and the averaged partition function becomes a Siegel-Eisenstein series expressed as a sum over lens-space CS invariants. For even the bulk dual is purely bosonic, while odd requires spin-Chern-Simons theory, with spin-structure dependent invariants precisely matching the spin-refined Eisenstein sums. The paper also demonstrates a pre-average holographic dual via Maxwell-Chern-Simons theory, where boundary wavefunctions reproduce the CFT theta functions for each moduli point, and discusses higher genus generalizations and the role of the Siegel-Weil theorem in linking moduli averages to bulk topological data. The results illuminate how ensemble averages naturally encode bulk sums over geometries and connect topological field theory, modular forms, and holography, offering a framework to study holography with ensembles and possibly shedding light on quantum gravity in lower dimensions.

Abstract

We discuss ensemble averages of two-dimensional conformal field theories associated with an arbitrary indefinite lattice with integral quadratic form . We provide evidence that the holographic dual after the ensemble average is the three-dimensional Abelian Chern-Simons theory with kinetic term determined by . 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.

Paper Structure

This paper contains 18 sections, 126 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ensemble Average of Bosonic CFTs
  3. Lattices with Even Quadratic Forms
  4. CFT Partition Function
  5. Ensemble Average and the Siegel-Weil Theorem
  6. Bulk Interpretation
  7. Positive Definite Case
  8. Higher Genus
  9. Ensemble Average of Fermionic CFTs
  10. Review of Spin Chern-Simons Theory
  11. Partition Functions with Spin Structure
  12. Ensemble Average
  13. Spin Chern-Simons Invariants
  14. Holographic Dual before Ensemble Average
  15. Discussion
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Change of spin structures under the mapping class group transformations $T, S$. Three even spin structures $(0,0), (0,1), (1,0)$ make a triplet, while the odd spin structure $(1,1)$ is a singlet.