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25
Holonomy and parallel transport for abelian gerbes, Preprint math.DG/0007053
"... In this paper we establish a onetoone correspondence between S 1gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higherorder analogue of the familiar equivalence ..."
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Cited by 54 (10 self)
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In this paper we establish a onetoone correspondence between S 1gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higherorder analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with presently working as a postdoc at the University of Nottingham, UK 1 group S 1 on a simply connected manifold M is a group morphism from the thin second homotopy group to S 1, satisfying a smoothness condition, where a homotopy between maps from [0,1] 2 to M is thin when its derivative is of rank ≤ 2. For the nonsimply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the nonabelian case via the theory of double Lie groupoids. 1
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
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Cited by 52 (6 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = FVect, where F is a field. An object X ∈ A with twosided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
"... ..."
Statesum construction of twodimensional openclosed TQFTs
 In preparation
"... We present a state sum construction of twodimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional twodimensional cobordisms to those of openclosed cobordisms, i.e. smoo ..."
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Cited by 11 (5 self)
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We present a state sum construction of twodimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional twodimensional cobordisms to those of openclosed cobordisms, i.e. smooth compact oriented 2manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a twodimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an openclosed TQFT with a finite set of Dbranes using the example of the groupoid algebra of a finite groupoid.
On Galois extensions of braided tensor categories, mapping class group representations and simple current extensions
 In preparation
"... We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed Gcategories, recently introduced for the purposes of 3manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non ..."
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Cited by 10 (4 self)
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We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed Gcategories, recently introduced for the purposes of 3manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G nontrivial objects of grade g exist in C ⋊ S. 1
Representations of algebraic quantum groups and reconstruction theorems for tensor categories
"... We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the TannakaKrein reconstruction problem. We show that every concrete semisimple tensor ∗category with conjugates is equivalent to the category of finite dimensional nondegene ..."
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Cited by 8 (4 self)
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We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the TannakaKrein reconstruction problem. We show that every concrete semisimple tensor ∗category with conjugates is equivalent to the category of finite dimensional nondegenerate ∗representations of a discrete algebraic quantum group. Working in the selfdual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and Rmatrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical TannakaKrein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well known general result concerning discrete multiplier Hopf ∗algebras. 1
Conformal Orbifold Theories and Braided Crossed GCategories
, 2004
"... The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G−LocA of twisted representations. This category is a braided crossed Gcategory in the sense of Turaev [60]. Its degree zero subcategor ..."
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Cited by 7 (2 self)
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The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G−LocA of twisted representations. This category is a braided crossed Gcategory in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of Gmanifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ❀ modular category ❀ 3manifold invariant. Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep AG of the orbifold theory AG. We prove the equivalence Rep AG ≃ (G−LocA) G, which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep AG. In the opposite direction we have G−LocA ≃ Rep AG ⋊ S, where S ⊂ Rep AG is the full
Topological quantum field theories from compact Lie groups. arXiv:0905.0731
, 2009
"... Let G be a compact Lie group and BG a classifying space for G. Then a class in H n 1 BG; Z leads to an ndimensional topological quantum field theory (TQFT), at least for n�1,2,3. The theory for n�1is trivial, but we include it for completeness. The theory for n�2has some infinities if G is not a fi ..."
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Cited by 5 (0 self)
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Let G be a compact Lie group and BG a classifying space for G. Then a class in H n 1 BG; Z leads to an ndimensional topological quantum field theory (TQFT), at least for n�1,2,3. The theory for n�1is trivial, but we include it for completeness. The theory for n�2has some infinities if G is not a finite group; it is a topological limit of 2dimensional YangMills theory. The most direct analog for n�3 is an L 2 version of the topological quantum field theory based on the classical ChernSimons invariant, which is only partially defined. The TQFT constructed by Witten and ReshetikhinTuraev which goes by the name ‘ChernSimons theory ’ (sometimes ‘holomorphic ChernSimons theory ’ to distinguish it from the L 2 theory) is completely finite. The theories we construct here are extended, or multitiered, TQFTs which go all the way down to points. For the n�3 ChernSimons theory, which we term a ‘0123 theory ’ to emphasize the extension down to points, we only treat the cases where G is finite or G is a torus, the latter being one of the main novelties in this paper. In other words, for toral theories we provide an answer to the longstanding question: What does ChernSimons theory attach to a point? The answer is a bit subtle as ChernSimons is an anomalous field theory of oriented manifolds. 1 This framing anomaly was already flagged in Witten’s seminal paper [Wi]. Here we interpret the anomaly as an invertible