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Supersymmetry and localization. (English) Zbl 0873.58003

The aim of the present paper is to derive very general localization formulae in the framework of supergeometry. Namely, the authors consider an integral over a finite dimensional (super)manifold \(M\), where the integrand is invariant under the action of an odd vector field \(Q\). They formulate sufficient conditions on \(M\) and \(Q\) under which the integral localizes onto the zero locus of the number part of \(Q\).
One of the possible ways to apply the theorems of the present paper is based on the use of the Batalin-Vilkovisky formalism where the calculation of physical quantities reduces to the calculation of integrals of functions invariant with respect to an odd vector field. In this paper the authors do not consider concrete examples of the application of their results. They only mention that their results can be used, for instance, to calculate integrals of \(OSp(n|m)\)-invariant functions and obtain the statement about reduction in the Parisi-Sourlas model.
Reviewer: V.Abramov (Tartu)

MSC:

58A50 Supermanifolds and graded manifolds
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References:

[1] Batalin, I., Vilkovisky, G.: Gauge algebra and quantization. Phys. Lett.102B, 27 (1981)
[2] Berezin, F.A.: Introduction to superanalysis. Dordrecht-Boston: D. Reidel Pub. Co., 1987 · Zbl 0659.58001
[3] Berline, N., Getzler E., Vergne, M.: Heat Kernels and Dirac Operators. Berlin: Springer Verlag, 1991 · Zbl 1037.58015
[4] Blau, M., Thompson, G.: Localization and Diagonalization. A Review of Functional Integral Techniques for Low-Dimensional Gauge Theories and Topological Field Theories. hep-th/9501075
[5] Duistermaat, J.J., Heckman, G.J.: On the variation in the cohomology of the symplectic form of the reduced phase space. Inv. Math.69, 259 (1982); and ibid72, 153 (1983) · Zbl 0503.58015 · doi:10.1007/BF01399506
[6] Hirsch, M.W.: Differential Topology. New York, Heidelberg, Berlin, Springer-Verlag, 1976 · Zbl 0356.57001
[7] Parisi, G., Sourlas, N.: Supersymmetric field theories and stochastic differential equations. Nucl. Phys.B206, 321 (1982) · Zbl 0968.81547 · doi:10.1016/0550-3213(82)90538-7
[8] Polyakov, A.M.: Cargese Lecture. (1995)
[9] Schwarz, A.S.: Semiclassical Approximation in Batalin-Vilkovisky Formalism. Commun. Math. Phys.158, 373 (1993) · Zbl 0855.58005 · doi:10.1007/BF02108080
[10] Schwarz, A.S.: Superanalogs of Symplectic and Contact Geometry and their Applications to Quantum Field Theory. MSRI preprint 055-94, to be published in Berezin memorial volume
[11] Witten, E.: Two Dimensional Gauge Theories Revisited. J. Geom. Phys.9, 303 (1992); T. Karki, A.J. Niemi.: Duistermaat-Heckman Theorem and Integrable Models. UUITP-02-94 (1993) · Zbl 0768.53042 · doi:10.1016/0393-0440(92)90034-X
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