Index of a family of Dirac operators on loop space

We use methods of constructive field theory to generalize index theory to an infinite-dimensional setting. We study a family of Dirac operators Q on loop space. These operators arise in the context of supersymmetric nonlinear quantum field models with Hamiltonians H=Q2. In these models Q is self-adjoint and Fredholm. A natural grading operator Γ exists such that ΓQ+QΓ=0. We study Q+=P-QP+, where P±=1/2(1 ± Γ) are the orthogonal projections onto the eigenspaces of Γ. We calculate the index i(Q+) for Wess-Zumino models defined by a superpotential V(φ). Here V is a polynomial of degree n ≥ 2. We establish that i(Q+)=n-1=degδV. In particular, the field theory models have unbroken supersymmetry, and (for n ≥ 3) they have degenerate vacua. We believe that this is the first index theorem for a Dirac operator that couples infinitely many degrees of freedom. (orig.)