Superbosonization of invariant matrix ensembles
Lie Theory
March 11, 2008 11:00 AM to 12:00 PM
Speakers:
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Abstract: |
Superbosonization is a new variant of the method of commuting and anticommuting variables. The object of departure of the supersymmetry method is the Fourier transform of the probability measure of the given ensemble of disordered Hamiltonians. This Fourier transform is evaluated on a supermatrix built from commuting and anti-commuting variables and thus becomes a superfunction; more precisely a function f, which is defined on a complex vector space V0 and takes values in the exterior algebra V 1 of another complex vector space V1. If the probability measure is invariant under a group K, so that the function f is equivariant with respect to K acting on V0 and V1, then a standard result from invariant theory tells us that f can be viewed as the pullback of a superfunction F defined on the quotient of V = V0 V1 by the group K. The heart of the superbosonization method is a formula which reduces the integral of f to an integral of the lifted function F .
In the talk will be concerned with the algebra AG(V ) of G-equivariant holomorphic functions f : V0 ! (V 1 ), v 7! f(v) = g.f (g−1v) (g 2 G), for the classical Lie groups G = GLn(C),On(C), and Spn(C), and the vector spaces V0 = Hom(Cn,Cp) Hom(Cp,Cn), V1 = Hom(Cn,Cq) Hom(Cq,Cn).
Our strategy is to lift f 2 AG(V ) to another algebra A(W) of holomorphic functions F : W0 ! (W 1 ), using a surjective homomorphism A(W) ! AG(V ). The aim is then to show a statement of reduction the superbosonization formula transferring the Berezin superintegral of f 2 AG(V ) to such an integral of F 2 A(W).
References
[1] P. Littelmann, H.-J. Sommers, M.R. Zirnbauer, Superbosonization of invariant matrix ensembles, preprint arXiv:0707.2929v1 [math-ph] (2007). |
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