Building upon Dyson's fundamental 1962 article known in random-matrix theory as the threefold way, we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important physical examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds.The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V , a space of endomorphisms in End(V ⊕ V * ), a bilinear form on V ⊕ V * which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V * . Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way.This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.
We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group G C and that with respect to a compatible maximal compact subgroup U of G C the action on Z is Hamiltonian. There is a corresponding gradient map µ p : X → p * where g = k ⊕ p is a Cartan decomposition of g. We obtain a Morse-like function η p := µ p 2 on X. Associated with critical points of η p are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds S β of X which are called pre-strata. In cases where µ p is proper, the pre-strata form a decomposition of X and in cases where X is compact they are the strata of a Morse-type stratification of X. Our results are generalizations of results of Kirwan obtained in the case where G = U C and X = Z is compact.
Abstract. We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna's slice theorem as well as a version of the Hilbert-Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.
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