Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 525Now, let X = G be a compact Lie group and, for simplicity, we shallassume that it is simply connected, though the theory works in the generalcase. We consider G as G−space, the group acting on itself by conjugation.Since H 3 (G, Z) ∼ = Z we can construct twisted K−theories for eachinteger k. Moreover, we can also do this equivariantly, thus obtainingAbelian groups KG,k ∗ (G). These will all be R(G)−modules.Now, the group multiplication map µ : G × G → G is compatible withconjugation and so is a G−map. In addition, to the pull back µ ∗ , we canalso consider the push–forward µ ∗ . This depends on Poincaré duality forK−theory and it works also, when appropriately formulated, in the presentcontext.If dim(G) is even, this gives us a commutative multiplication onKG,k 0 (G), while for dim(G) odd, our multiplication is on K1 G,k(G). In eithercase we get a ring.The claim of [Freed (2001); Freed et. al. (2003)] is that this ring (accordingto the parity of dim(G) is naturally isomorphic to the Verlindealgebra of G at level k − h (where h is the Coxeter number). The Verlindealgebra is a key tool in certain quantum field theories and it has been muchstudied by physicists, topologists, group theorists and algebraic geometers.The K−theory approach is totally new and much more direct than mostother ways.The Verlinde algebra is defined in the theory of loop groups. Let G bea compact Lie group. There is a version of the Theorem for any compactgroup G, but here for the most part we focus on connected, simply connected,and simple groups—G = SU 2 is the simplest example. In this casea 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 classesof positive energy representations of this central extension; let V k (G) denotethe free Abelian group they generate. One of the influences of 2Dconformal field theory on the theory of loop groups is the construction ofan algebra structure on V k (G), the fusion product. This is the Verlindealgebra [Verlinde (1988)].More precisely, let G act on itself by conjugation. Then with our assumptionsthe equivariant cohomology group HG 3 (G) is free of rank one. Leth(G) be the dual Coxeter number of G, and define ζ(k) ∈ HG 3 (G) to bek + h(G) times a generator. We will see that elements of H 3 may be usedto twist K−theory, and so elements of equivariant H 3 twist equivariantparameterizing a family of related mathematical objects such as schemes or topologicalspaces, especially when the members of these families have nontrivial automorphisms.