Integration of Grassmann variables over invariant functions on flat superspaces

Abstract

We study integration over functions on superspaces. These functions are invariant under a transformation which maps the whole superspace onto the part of the superspace which only comprises purely commuting variables. We get a compact expression for the differential operator with respect to the commuting variables which results from Berezin integration over all Grassmann variables. Also, we derive Cauchy--like integral theorems for invariant functions on supervectors and symmetric supermatrices. This extends theorems partly derived by other authors. As an physical application, we calculate the generating function of the one--point correlation function in random matrix theory. Furthermore, we give another derivation of supermatrix Bessel--functions for U(k_1/k_2).

31 pages

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• Dirac Spectrum of the Wilson Dirac Operator for QCD with Two Colors
• Random matrices with row constraints and eigenvalue distributions of graph Laplacians
• Ren-integrable and ren-symmetric integrable systems
• Orthosymplectically invariant functions in superspace
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Supersymmetry Method for Chiral Random Matrix Theory with Arbitrary Rotation Invariant Weights
• Invariant integration on orthosymplectic and unitary supergroups
• On the Efetov-Wegner terms by diagonalizing a Hermitian supermatrix
• Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance
• A use of geometric calculus to reduce Berezin integral to the limit of a Riemann sum
• Asymptotic Coincidence of the Statistics for Degenerate and Non-Degenerate Correlated Real Wishart Ensembles
• Universality Emerging in a Universality: Derivation of the Ericson Transition in Stochastic Quantum Scattering and Experimental Validation