Twisted equivariant K -theory and the Verlinde algebra

Abstract

As we outlined in the introduction, the role of K-theory in physics started when topological invariants of D-branes in string theory took values in K-groups. When the D-brane has a nontrivial background B-field, the corresponding D-brane invariants take values in the twisted K-groups where the B-field is a three-form defining the twisting in the K-group. Just as there is a G-equivariant K-theory, there is also a G-equivariant twisted K-theory with a definition in terms of Fredholm operators on a Hilbert space having suitable G-action.One such G-space is G=Ad(G) with the adjoint action of G on itself, and in this case, it is sometimes possible to determine the twisted G-equivariant K-theory of G. Throughout this chapter, G is a compact, simply connected, simple real Lie group. In some cases the assertions hold more generally as for the isomorphism R(G) → KG(*) which is true for a general compact group. When one considers either the loop group on G or the associated Kac-Moody Lie algebra to Lie(G), the classical Lie algebra of G, there is a representation theory for a given level or central charge c. The representations of central charge c define a fusion algebra Verc(G), called the Verlinde algebra which is a quotient ring of R(G) for simply connected compact Lie groups G which are simple. The main assertion in this chapter is that this Verlinde fusion algebra is related to twisted K-theory of Ad(G). The isomorphism R(G) → KG(*) has a quotient which is an isomorphism giving a commutative square A figure is presented. © Springer-Verlag Berlin Heidelberg 2008.