Conformal Correlation Functions, Frobenius Algebras and Triangulations

TL;DR

This paper generalizes 2D rational conformal field theory using Frobenius algebras in modular tensor categories, providing a triangulation-based correlator construction that ensures modular invariance and factorization.

Contribution

It introduces a new formulation of 2D rational CFT via Frobenius algebras in modular categories, extending lattice TFT methods to more general surfaces and boundary conditions.

Findings

Constructed correlators that are modular invariant

Established boundary conditions as algebra representations

Derived modular invariants and NIM-reps from partition functions

Abstract

We formulate two-dimensional rational conformal field theory as a natural generalization of two-dimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the Moore-Seiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of T-duality. We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIM-reps of the fusion rules, respectively.