TFT construction of RCFT correlators I: partition functions

Abstract

We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of Moore--Seiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are A-modules, and (generalised) defect lines are A-A-bimodules. The relation with three-dimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in three-manifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIM-rep properties. We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore--Seiberg data.

Nonsemisimple Fusion Algebras and the Verlinde Formula

“…For instance, the state spaces of the three-dimensional TFT are the spaces of chiral blocks of the CFT, and the modular S matrix (or, to be precise, the symmetric matrix that diagonalizes the fusion rules) is, up to normalization, the invariant of the Hopf link in the three-dimensional TFT. Also, a full (nonchiral) CFT based on a given chiral CFT corresponds to a certain Frobenius algebra in the category C, and the correlation functions of the full CFT can be determined by combining methods from three-dimensional TFT and from noncommutative algebra in monoidal categories [27,28]. In the nonrational case, C is no longer modular, in particular not semisimple, but in any case it should still be an additive braided monoidal category.…”

Torus structure on graphs and twisted partition functions for minimal and affine models

“…Dimensions of the 11 blocks d i are equal to (6,10,14,18,20,20,20,18,14,10,6). Dimension of the 12 blocks d x are equal to (6,8,6,10,14,10,10,14,10,20,28,20).…”

“…Dimension of the 12 blocks d x are equal to (6,8,6,10,14,10,10,14,10,20,28,20). The quadratic sum rule reads:…”

“…There are several possible theories associated to this value of central charge, one is the diagonal theory associated to the pair (A 4 , A 5 ), another is the one we are considering here. There are 15 toric matrices (also graph matrices in this case) of type W A 4 (λ, 00) = W A 4 (λ) with λ labelling the vertices of the generalized Dynkin diagram, λ A 4 = {(00), (30), (03), (11), (22), (10), (40), (21), (02), (13), (20), (12), (04), (01), (31)} and 24 of type W E 5 (σ, 00) = W E 5 (σ) with σ = ρ × ν labels 27 Compatibility of weights of SU (2) versus SU (3) is only meaningful modulo integers. …”

A geometric construction for permutation equivariant categories from modular functors

“…All manifolds are smooth oriented manifolds and all maps are smooth and orientation preserving. We freely use the graphical calculus for morphisms in ribbon categories for which we refer to [JS91,FRS02].…”