We study topological defect lines in two-dimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting onedimensional phase boundaries can be used to extract symmetries and order-disorder dualities of the CFT. The case of central charge c = 4/5, for which there are two inequivalent world sheet phases corresponding to the tetra-critical Ising model and the critical three-states

Abstract. This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2-categories, and Cat-categories. The latter two are exactly the same (except that strictly speaking a Cat-category should have small hom-categories, but that need not concern us here). The first two are nominally different — the 2-categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2-categories. Nonetheless, the theories of bicategories, 2-categories, and Catcategories have rather different flavours.

Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2-functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topological gauge theory, the Dijkgraaf-Witten model. To give this definition rigorously, we first define a bicategory of cobordisms between cobordisms. We also give some explicit description of a higher-categorical version of Vect, denoted 2Vect, a bicategory of 2-vector spaces. Along the way, we prove several results showing how to construct 2-vector spaces of Vect-valued presheaves on certain kinds of groupoids. In particular, we use the case when these are groupoids whose objects are connections, and whose morphisms are gauge transformations, on the manifolds on which the extended TQFT is to be defined. On cobordisms between these manifolds, we show how a construction of “pullback and pushforward ” of presheaves gives both the morphisms and 2-morphisms in 2Vect for the extended TQFT, and that these

There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended ” functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra “of observables”, the latter uses n-functors which assign to each patch a “propagator of states”. In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2-dimensional extended Minkowskian QFT 2-functor (”parallel surface transport”) naturally yields a local net. This is obtained by postcomposing the propagation 2-functor with an operation that mimics the passage from the Schrödinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general pseudo-

Fuchs, Runkel, Schweigert et al. (“FRS”) have developed a detailed formalism for studying CFT in terms of (Wilson-)graphs decorated in modular tensor categories. Here we discuss aspects of how the Poincaré dual of their prescription can be understood in terms of locally trivialized 2-functors from surface elements to bimodules, similar to how Stolz and Teichner describe enriched elliptic objects. Even though this might begin to look like a paper, the following are

Abstract. This paper describes a relationship between essentially finite groupoids and 2-vector spaces. In particular, we show to construct 2-vector spaces of Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation— a weak functor—from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail. It has applications in constructing quantum field theories, among others. 1.

2.2. Soergel bimodules 6

We study topological defect lines in two-dimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting onedimensional phase boundaries can be used to extract symmetries and order-disorder dualities of the CFT. The case of central charge c = 4/5, for which there are two inequivalent world sheet phases corresponding to the tetra-critical Ising model and the critical three-states Potts model, is treated as an illustrative example. Contents