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Published on Jul 30, 2012
Andre Henriques, University of Utrecht, Netherlands
The idea of extending quantum field theories to manifolds of lower dimension was first proposed by Dan Freed in the nineties. In the case of conformal eld theory (CFT), we are talking of an extension of the Atiyah-Segal axioms where one replaces the bordism category of Riemann surfaces by a suitable bordism 2-category, whose objects are points, whose morphism are 1-manifolds, and whose 2-morphisms are pieces of Riemann surface. There is a beautiful classication of full (rational) CFT due to Fuchs, Runkel and Schweigert. the classication say roughly the following. Fix a chiral algebra A (= vertex algebra). Then the set of full CFT whose left and right chiral algebras agree with A is classied by Frobenius algebras internal to Rep(A). A famous example to which one can successfully apply this is the case when the chiral algebra A is ane su(2): in that case, the Frobenius algebras in Rep(A) are classied by An, Dn, E6, E7, E8, and so are the corresponding CFTs.
Recently, Kapustin and Saulina gave a conceptual interpretation of the FRS classication in terms of 3-dimentional Chern-Simons theory with defects. Those defects are also given by Frobenius algebra object in Rep(A). Inspired by the proposal of Kapustin and Saulina, we will (partially) construct the three-tier CFT associated to a Frobenius algebra object.