TFT construction of RCFT correlators I: Partition functions

By Jürgen Fuchs, Ingo Runkel, Christoph Schweigert. 2002

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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.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/b1e2b93c899af013bade86e71c4ead6348ce14b6

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Citations in the corpus

Quantum picturalism (2009, methodology): “Related graphical methods have been around for a bit more than a decade now in mathematical physics and pure mathematics, for example in [24, 71, 72, 73, 74] and references therein.”
On finite symmetries and their gauging in two dimensions (2017, background): “The crucial idea of [18, 8] is to take this as the definition of gauging in the generalized sense.”
Nonsemisimple Fusion Algebras and the Verlinde Formula (2003, methodology): “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].”
Anyon condensation and tensor categories (2013, methodology): “e algebras L∨ M(idM) ≃Z(A) and A, together with an algebraic homomorphism Z(A) →A, form a so-called Cardy algebra [Ko], a notion which classiﬁes open-closed rational conformal ﬁeld theories (see also [FRS1][Fj][KR2]). This connection between anyon condensation and rational CFT is not accidental and ”
The virtue of defects in 4D gauge theories and 2D CFTs (2010, background): “[9, 10, 11]) in a 2d CFT are operators supported on closed curves characterized by the property that they commute with both the holomorphic and antiholomorphic energy-momentum tensor of the CFT.”
More Morphisms between Bundle Gerbes (2007, unknown): “... modules and Jandl structures in terms of morphisms between bundle gerbes is one step to understand the relation between two approaches to two-dimensional conformal field theories: on the one hand the Lagrangian approach with a metric and a bundle gerbe G being the relevant structure [GR02] and o”
Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center (2005, background): “accidental,” we hope that a more systemat ic derivation can be given. In fact, the task to place the structures encountered in the study of nonsemisimpleVerlinde algebras into the categorical context [32, 33, 34, 35] was already formulated in [15]. With the quantum-group counterpart of nonsemisimple”