## The Verlinde algebra is twisted equivariant K-theory

Citations: | 40 - 4 self |

### BibTeX

@MISC{Freed_theverlinde,

author = {Daniel S. Freed and Vk(g K},

title = {The Verlinde algebra is twisted equivariant K-theory},

year = {}

}

### OpenURL

### Abstract

K-theory in various forms has recently received much attention in 10-dimensional superstring theory. Our raised consciousness about twisted K-theory led to the serendipitous discovery that it enters in a different way into 3-dimensional topological field theories, in particular Chern-Simons theory. Namely, as the title of the paper reports, the Verlinde algebra is a certain twisted K-theory 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 2-dimensional 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 K-theory, and so elements of equivariant H 3 twist equivariant K-theory. Theorem (Freed-Hopkins-Teleman). There is an isomorphism of algebras

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136 |
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Citation Context ...sented by cocycles gij with values in the space of lines, in other words by complex line bundles Lij → Uij which satisfy a cocycle condition. This is the data often given to define a gerbe. Following =-=[A]-=-, we present a model of twisted K-theory in terms of operators on an infinite dimensional separable complex Hilbert space H. Let PGL(H) be the projective general linear group of Hilbert space. Kuiper’... |

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Citation Context ... α.) The complex vector space of these vectors is then functorially attached to the closed manifold Y : (2.2) Y n−1 ↦−→ E(Y ) C-module. The standard story ends here. But, as many people observed (see =-=[F1]-=-, for example) it is beneficial to go further and consider hypersurfaces S ⊂ Y which express Y as the union of submanifolds Y1,Y2 with boundary. Then the locality property leads us by analogy to defin... |

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Citation Context ...aph we used the units in integral cohomology, the group Z/2Z. For complex K-theory there is a richer group of units2 (1.3) GL1(K) ∼ Z/2Z × CP ∞ × BSU. 2 This is a consequence of results in [DK], [S], =-=[ASe]-=-, [AP]. 3In our problem the last factor doesn’t enter and all the interest is in the first two, which we denote GL1(K) ′. As a first approximation, view K as the category of all finite dimensional Z/... |