## The Verlinde algebra is twisted equivariant K-theory

Citations: | 41 - 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

### Citations

634 |
Field Theory and the Jones Polynomial, Commun.Math.Phys
- Witten
- 1989
(Show Context)
Citation Context ...where T is the circle group. This is the exponentiated Chern-Simons invariant. It is the integrand of the functional integral which defines the partition function (2.1) of quantum Chern-Simons theory =-=[W]-=-. Note that in defining the functional integral (over the universal family) we must extend the codomain of the ChernSimons invariant from T to C. Of course, that functional integral is a formal expres... |

235 |
Fusion rules and modular transformations in 2D conformal field theory
- Verlinde
- 1988
(Show Context)
Citation Context ...rate. 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 b... |

154 |
Categories and Cohomology Theories, Topology 13
- Segal
- 1974
(Show Context)
Citation Context ...aragraph 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 dimensi... |

79 |
Quantum invariants of knots and 3-manifolds, Walter de Gruyter
- Turaev
- 1994
(Show Context)
Citation Context ...a “modular tensor category,” or some close variation. In fact, there is a theorem that the entire three-dimensional topological quantum field theory can be reconstructed from this algebraic data (see =-=[T]-=-). In this context the Verlinde algebra is the Grothendieck group of the K-algebra E. In particular, it is an algebra over the Grothendieck group K • (pt) of K. Three-dimensional Chern-Simons theory i... |

62 |
Index theory for skew-adjoint Fredholm operators
- Atiyah, Patodi, et al.
- 1969
(Show Context)
Citation Context ...Fredholm operators Fred 0 (H) has the homotopy type of Z×BU, so is a classifying space for K 0 , and a suitable space of self-adjoint Fredholm operators Fred 1 (H) is a classifying space for K 1 (see =-=[AS]-=-). (Alternatively, we could take invertible operators of the form 1 + compact as a classifying space for K 1 .) Recall Bott periodicity which implies that K q depends only on the parity of q, so we ne... |

62 |
Higher algebraic structures and quantization
- Freed
(Show Context)
Citation Context ...ne, and these lines multiply suitably under composition of arrows. The K-module E consists of vector bundles over G suitably equivariant under this central extension. The precise statements appear in =-=[F2]-=-, but we failed to recognize the Grothendieck group as a twisted K-theory group. That belated realization led to the main theorem. In many cases the partition function (2.1) of a quantum field theory ... |

53 |
Graded Brauer groups and K-theory with local coecients Inst
- Donovan, Karoubi
- 1970
(Show Context)
Citation Context ...ious paragraph 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 di... |

52 | Quadratic functions in geometry, topology, and M-theory
- Hopkins, Singer
(Show Context)
Citation Context ...tial forms first appeared in differential geometry as Cheeger-Simons differential characters, and also goes by the name of smooth Deligne cohomology. A treatment in the spirit needed here is given in =-=[HS]-=-. The level λ ∈ H 4 (BG; Z) lifts uniquely to a differential cohomology class in ˇ H 4 (BG), and we fix a particular cocycle ˇ λ which represents it. Then if P → M is any principal G-bundle with conne... |

15 |
K-theory past and present
- Atiyah
(Show Context)
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’... |

8 | Extended Structures in Topological Quantum Field Theory
- Freed
- 1993
(Show Context)
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... |

1 |
Exponential isomorphisms for λ-rings
- Atiyah, Segal
- 1971
(Show Context)
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/... |