Hecke-Langlands Duality and Witten's Gravitational Moonshine

TL;DR

This paper explores a duality between conformal blocks of 2D rational CFT and Hecke eigenfields, connecting various mathematical structures and applying these ideas to quantum gravity and geometric Langlands duality.

Contribution

It introduces a novel dual description of 2D rational CFT conformal blocks using Hecke operators and eigensheaves, linking multiple mathematical frameworks.

Findings

Reconstruction of partition functions and characters via Hecke operators

Application to Galois number fields, lattices, and minimal models

Construction of 3D quantum gravity as AdS/CFT dual of Monster vertex algebra

Abstract

We show that there is a dual description of conformal blocks of $d=2$ rational CFT in terms of Hecke eigenfields and eigensheaves. In particular, partition functions, conformal characters and lattice theta functions may be reconstructed from the action of Hecke operators. This method can be applied to: 1) rings of integers of Galois number fields equipped with the trace (or anti-trace) form; 2) root lattices of affine Kac-Moody algebras and WZW-models; 3) minimal models of Belavin-Polyakov-Zamolodchikov and related $d=2$ spin-chain/lattice models; 4) vertex algebras of Leech and Niemeier lattices and others. We also use the original Witten's idea to construct the 3-dimensional quantum gravity as the AdS/CFT-dual of $c=24$ Monster vertex algebra of Frenkel-Lepowsky-Meurman. Concerning the geometric Langlands duality, we use results of Beilinson-Drinfeld, Frenkel-Ben-Zvi, Gukov-Kapustin-Witten and many others (cf. references).