Conformal Correlation Functions, Frobenius Algebras and Triangulations

TL;DR

This work reframes two-dimensional rational conformal field theory as a natural extension of lattice TFT by embedding chiral data in a modular tensor category ${\mathcal C}$ and introducing a symmetric special Frobenius algebra object $A\in{\mathcal C}$. Morita equivalence of such algebras guarantees that different but related $A$ yield the same full CFT, enabling a geometric construction of all correlators from a triangulated world sheet and linking boundary conditions to $A$-modules. The authors derive modular-invariant torus partition functions and NIM-reps for annuli from a 3D TFT perspective and provide a general, triangulation-independent prescription for all correlators, including bulk and boundary fields, with factorization guaranteed by Frobenius and associativity properties. They also show how T-duality arises via Morita equivalence in orbifolded chiral data, and outline a broader program of classification and reconstruction of full RCFTs from chiral data. The framework promises a unified, algebraic route to RCFTs and their dualities, with potential extensions to unorientable surfaces and nonrational theories.

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.