Flat Connections and Non-Local Conserved Quantities in Irrational Conformal Field Theory

TL;DR

This paper reformulates the Ward identities of irrational conformal field theory as a linear system with flat connections, revealing new non-local conserved quantities and providing solutions for various correlators.

Contribution

It introduces a novel reformulation of ICFT Ward identities using flat connections and identifies new non-local conserved quantities, with explicit solutions for key correlators.

Findings

Reformulation of Ward identities as flat connection systems

Identification of new non-local conserved quantities

Explicit solutions for coset and affine-Sugawara correlators

Abstract

Irrational conformal field theory (ICFT) includes rational conformal field theory as a small subspace, and the affine-Virasoro Ward identities describe the biconformal correlators of ICFT. We reformulate the Ward identities as an equivalent linear partial differential system with flat connections and new non-local conserved quantities. As examples of the formulation, we solve the system of flat connections for the coset correlators, the correlators of the affine-Sugawara nests and the high-level $n$-point correlators of ICFT.