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The degree of the Nicolai map in supersymmetric quantum mechanics. (English) Zbl 0658.28008

In a previous paper [J. Funct. Anal. 68, 388-403 (1986)], the author has introduced, for certain smooth maps on an abstract Wiener space, a measure theoretic or probabilistic notion of degree, based on the attached Gaussian measure, which is beyond the sweep of the Leray- Schauder degree. The paper under review gives an application of this theory with suitable modifications to the nonlinear map A: \(C^{\alpha}\to {\mathcal D}'\) between the spaces on the circle S, defined by \[ (Aw)(t)=dw(t)/dt+(\nabla V)(w(t)),\quad t\in S. \] Here V is a real-valued \(C^{\infty}\) function on \({\mathbb{R}}^ m\) satisfying a certain growth condition. \(C^{\alpha}\equiv C^{\alpha}(S,{\mathbb{R}}^ m),\quad 0<\alpha <1/2,\) is the Banach space of the Hölder continuous loops w: \(S\to {\mathbb{R}}^ m\), and \({\mathcal D}'\equiv {\mathcal D}'(S,{\mathbb{R}}^ m)\), the space of distributions. There are the Ornstein-Uhlenbeck measure \(d\lambda\) on \(C^{\alpha}\) and the white noise measure \(d\kappa\) on the image space \(C^{\alpha -1}\) of the linear map \(\Lambda \equiv d/dt+1: C^{\alpha}\to {\mathcal D}'.\) The main result is to impart a meaning to the degree deg A for the map A: There exists an integer deg A for which \[ \int_{C^{\alpha}}A^*(\phi d\kappa)=(\deg A)\int_{C^{\alpha - 1}}\phi d\kappa \] for every \(\phi \in L^{\infty}(C^{\alpha - 1},d\kappa)\). Here \(A^*(\phi d\kappa)\) is the pullback of the signed measure \(\phi\) \(d\kappa\), as considered as a volume form on \(C^{\alpha -1}\), by the map A. As a corollary it is shown that \(\kappa\)-a.e. \(\xi \in C^{\alpha -1}\) is a regular value of A, and that, for \(\kappa\)-a.e. \(\xi \in C^{\alpha -1}\), the set \(A^{(-1)}(\xi)\) is finite and \[ \deg A=\sum_{A(w)=\xi}sgn(\nabla_ wA)\equiv \sum_{A(w)=\xi}sgn \det_ 2[\Lambda^{-1}(d/dt+\nabla^ 2_ wV)], \] where \(\det_ 2\) denotes the Carleman-Fredholm determinant. It is further shown that deg A equals the Leray-Schauder degree of the map \(\nabla V: {\mathbb{R}}^ m\to {\mathbb{R}}^ m\), if \(\nabla V\) is proper. The map A is what is called a Nicolai map [H. Nicolai, Nucl. Phys. B 176, 419-428 (1980)] for a certain supersymmetric quantum mechanical model. \(A^*(d\kappa)\) is then formally the path integral for its corresponding Hamiltonian \(H=D^ 2\) acting on the \(L^ 2\) differential forms on \({\mathbb{R}}^ m\), where \(D=d_ V+d^*_ V\) with \(d_ V=e^{-V}\cdot d\cdot e^ V.\) The paper also visualizes this point to show: If \(f_ j\in C_ 0^{\infty}({\mathbb{R}}^ m),\) \(1\leq j\leq n\), then \[ \int_{C^{\alpha}}f_ 1(w(t_ 1))...f_ n(w(t_ n))A^*(d\kappa) \]
\[ =Str[e^{-t_ 1D^ 2/2}f_ 1e^{-(t_ 2-t_ 1)D^ 2/2}...f_ ne^{-(1-t_ n)D^ 2/2}], \] with \(0<t_ 1<...<t_ n<1\), where Str denotes the supertrace. In particular, it is seen that \(\int_{C^{\alpha}}A^*(d\kappa)=Str e^{-D^ 2/2},\) and hence deg A equals the index of D.
Reviewer: T.Ichinose

MSC:

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46G12 Measures and integration on abstract linear spaces
81T60 Supersymmetric field theories in quantum mechanics
46G15 Functional analytic lifting theory
81S40 Path integrals in quantum mechanics
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References:

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