Consistent Orientation of Moduli Spaces

In a series of papers by Freed, Hopkins, and Teleman (2003, 2005, 2007a) the relationship between positive energy representations of the loop group of a compact Lie group G and the twisted equivariant K-theory K ^τ+dimG _G (G) was developed. Here G acts on itself by conjugation. The loop group representations depend on a choice of ‘level’, and the twisting τ is derived from the level. For all levels the main theorem is an isomorphism of abelian groups, and for special transgressed levels it is an isomorphism of rings: the fusion ring of the loop group andK ^τ+dimG _G (G) as a ring. For G connected with π₁G torsionfree, it has been proven that the ring K ^τ+dimG _G (G) is a quotient of the representation ring of G and can be calculated explicitly. In these cases it agrees with the fusion ring of the corresponding centrally extended loop group. This chapter explicates the multiplication on the twisted equivariant K-theory for an arbitrary compact Lie group G. It constructs a Frobenius ring structure on K ^τ+dimG _G (G). This is best expressed in the language of topological quantum field theory: a two-dimensional topological quantum field theory (TQFT) is constructed over the integers in which the abelian group attached to the circle is K ^τ+dimG _G (G).