Elliptic Genera and Quantum Field Theory

It is shown that in elliptic cohomology as recently formulated in the mathematical literature the supercharge of the supersymmetric nonlinear sigma model plays a role similar to the role of the Dirac operator in X-theory. This leads to several insights concerning both elliptic cohomology and string theory. Some of the relevant calculations have been done previously by Schellekens and Warner in a different context. If M is a spin manifold of dimension n, we can consider the Dirac operator i]/>, acting on a field ψa which is a section of the spinor bundle S. More generally, if R is any representation of the structure group Spin(n) of the tangent bundle, we can consider the Dirac operator acting on a field ψΛii α, and i being respectively a spinor index and an index labeling the representation R; in mathematical terms, ψ is a section of S® TR, TR being the Spin(n) bundle associated with the representation R of Spin (n). In [1], an infinite series of representations Rb i = 0,1,2,... was singled out. The first few were