Abstract
We use methods of constructive field theory to generalize index theory to an infinite-dimensional setting. We study a family of Dirac operatorsQ on loop space. These operators arise in the context of supersymmetric nonlinear quantum field models with HamiltoniansH=Q 2. In these modelsQ is self-adjoint and Fredholm. A natural grading operator Γ exists such that ΓQ+QΓ=0. We studyQ +=P − QP +, whereP ±=1/2 (1±Γ) are the orthogonal projections onto the eigenspaces of Γ. We calculate the indexi(Q +) for Wess-Zumino models defined by a superpotentialV(ω). HereV is a polynomial of degreen≧2. We establish thati(Q +)=n−1=degδV. In particular, the field theory models have unbroken supersymmetry, and (forn≧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.
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Communicated by E. Lieb
Dedicated to Walter Thirring on his 60th birthday
Research supported in part by the National Science Foundation under Grant DMS/PHY-86-45122
Hertz Foundation Predoctoral Fellow
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Jaffe, A., Lesniewski, A. & Weitsman, J. Index of a family of Dirac operators on loop space. Commun.Math. Phys. 112, 75–88 (1987). https://doi.org/10.1007/BF01217680
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DOI: https://doi.org/10.1007/BF01217680