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Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 138 (14 self)
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For a copy with the handdrawn figures please email
Algebras and Hopf algebras IN BRAIDED CATEGORIES
, 1995
"... This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise superalgebras and superHopf algebras, as well as colourLie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras i ..."
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Cited by 84 (13 self)
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This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise superalgebras and superHopf algebras, as well as colourLie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied, the notion of ‘braidedcommutative ’ or ‘braidedcocommutative ’ Hopf algebras (braided groups) is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group Aut (C) is given. The theory has important implications for the theory of quasitriangular Hopf algebras (quantum groups). It also includes important examples such as the degenerate Sklyanin algebra and the quantum plane.
HigherDimensional Algebra I: Braided Monoidal 2Categories
 Adv. Math
, 1996
"... We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their relevance to 4d TQFTs and 2tangles. Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give lon ..."
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Cited by 53 (9 self)
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We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their relevance to 4d TQFTs and 2tangles. Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2category Z(C) as the `center' of a semistrict monoidal category C, in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category. As a corollary this yields a strictification theorem for braided monoidal 2categories. 1 Introduction This is the first of a series of articles developing the program introduced in the paper `HigherDimensional Algebra and Topological Quantum Field Theory' [1], henceforth referred to as `HDA'. This program consists of generalizing algebraic concep...
The Verlinde algebra is twisted equivariant Ktheory
"... Ktheory in various forms has recently received much attention in 10dimensional superstring theory. Our raised consciousness about twisted Ktheory led to the serendipitous discovery that it enters in a different way into 3dimensional topological field theories, in particular ChernSimons theory. ..."
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Cited by 43 (4 self)
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Ktheory in various forms has recently received much attention in 10dimensional superstring theory. Our raised consciousness about twisted Ktheory led to the serendipitous discovery that it enters in a different way into 3dimensional topological field theories, in particular ChernSimons theory. Namely, as the title of the paper reports, the Verlinde algebra is a certain twisted Ktheory group. This assertion, and its proof, is joint work with Michael Hopkins and Constantin Teleman. The general theorem and proof will be presented elsewhere [FHT]; our goal here is to explain some background, demonstrate the theorem in a simple nontrivial case, and motivate it through the connection with topological field theory. From a mathematical point of view the Verlinde algebra is defined in the theory of loop groups. Let G be a compact Lie group. There is a version of the theorem for any compact group G, but here for the most part we focus on connected, simply connected, and simple groups—G = SU2 is the simplest example. In this case a central extension of the free loop group LG is determined by the level, which is a positive integer k. There is a finite set of equivalence classes of positive energy representations of this central extension; let Vk(G) denote the free abelian group they generate. One of the influences of 2dimensional conformal field theory on the theory of loop groups is the construction of an algebra structure on Vk(G), the fusion product. This is the Verlinde algebra [V]. Let G act on itself by conjugation. Then with our assumptions the equivariant cohomology group H3 G (G) is free of rank one. Let h(G) be the dual Coxeter number of G, and define ζ(k) ∈ H3 G (G) to be k + h(G) times a generator. We will see that elements of H3 may be used to twist Ktheory, and so elements of equivariant H 3 twist equivariant Ktheory. Theorem (FreedHopkinsTeleman). There is an isomorphism of algebras
Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 43 (13 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
Twisted Ktheory and Loop groups
 Proceedings of the International Congress of Mathematicians, Vol. III (Beijing
"... Abstract. Twisted Ktheory has received much attention recently in both mathematics and physics. We describe some models of twisted Ktheory, both topological and geometric. Then we state a theorem which relates representations of loop groups to twisted equivariant Ktheory. This is joint work with ..."
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Cited by 24 (2 self)
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Abstract. Twisted Ktheory has received much attention recently in both mathematics and physics. We describe some models of twisted Ktheory, both topological and geometric. Then we state a theorem which relates representations of loop groups to twisted equivariant Ktheory. This is joint work with Michael Hopkins and Constantin Teleman. The loop group of a compact Lie group G is the space of smooth maps S 1 → G with multiplication defined pointwise. Loop groups have been around in topology for quite some time [Bo], and in the 1980s were extensively studied from the point of view of representation theory [Ka], [PS]. In part this was driven by the relationship to conformal field theory. The interesting representations of loop groups are projective, and with fixed projective cocycle τ there is a finite number of irreducible representations up to isomorphism. Considerations from conformal field theory [V] led to a ring structure on the abelian group R τ (G) they generate, at least for transgressed twistings. This is the Verlinde ring. For G simply connected R τ (G) is a quotient of the representation ring of G, but that is not true in general. At about this time Witten [W] introduced a threedimensional topological quantum field theory in which the Verlinde ring plays an important role. Eventually it was understood that the fundamental object in that theory is a “modular tensor category ” whose Grothendieck group is the Verlinde ring. Typically it is a category of representations of a loop group or quantum group. For the special case of a finite group G the topological field theory is specified by a certain
Bundle Gerbes for ChernSimons and WESSZUMINOWITTEN THEORIES
, 2005
"... We develop the theory of ChernSimons bundle 2gerbes and multiplicative bundle gerbes associated to any principal Gbundle with connection and a class in H4 (BG, Z) for a compact semisimple Lie group G. The ChernSimons bundle 2gerbe realises differential geometrically the CheegerSimons invarian ..."
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Cited by 23 (8 self)
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We develop the theory of ChernSimons bundle 2gerbes and multiplicative bundle gerbes associated to any principal Gbundle with connection and a class in H4 (BG, Z) for a compact semisimple Lie group G. The ChernSimons bundle 2gerbe realises differential geometrically the CheegerSimons invariant. We apply these notions to refine the DijkgraafWitten correspondence between three dimensional ChernSimons functionals and WessZuminoWitten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H 4 (BG, Z) to H3 (G, Z). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for nonsimply connected Lie groups. The implications for WessZuminoWitten models are also discussed.
An introduction to ncategories
 In 7th Conference on Category Theory and Computer Science
, 1997
"... ..."
Integral transforms and Drinfeld centers in derived algebraic geometry
"... Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derive ..."
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Cited by 20 (4 self)
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Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derived stacks which we call stacks with air. The class of stacks with air includes in particular all quasicompact, separated derived schemes and (in characteristic zero) all quotients of quasiprojective or smooth derived schemes by affine algebraic groups, and is closed under derived fiber products. We show that the (enriched) derived categories of quasicoherent sheaves on stacks with air behave well under algebraic and geometric operations. Namely, we identify the derived category of a fiber product with the tensor product of the derived categories of the factors. We also identify functors between derived categories of sheaves with integral transforms (providing a generalization of a theorem of Toën [To1] for ordinary schemes over a ring). As a first application, for a stack Y with air, we calculate the Drinfeld center (or synonymously,