28 research outputs found
Bott periodicity for symmetric ground states of gapped free-fermion systems
Building on the symmetry classification of disordered fermions, we give a
proof of the proposal by Kitaev, and others, for a "Bott clock" topological
classification of free-fermion ground states of gapped systems with symmetries.
Our approach differs from previous ones in that (i) we work in the standard
framework of Hermitian quantum mechanics over the complex numbers, (ii) we
directly formulate a mathematical model for ground states rather than
spectrally flattened Hamiltonians, and (iii) we use homotopy-theoretic tools
rather than K-theory. Key to our proof is a natural transformation that squares
to the standard Bott map and relates the ground state of a d-dimensional system
in symmetry class s to the ground state of a (d+1)-dimensional system in
symmetry class s+1. This relation gives a new vantage point on topological
insulators and superconductors.Comment: 55 pages; one figure added; corrections in Section 8; proofs in
Section 6 expande
Chevalley's restriction theorem for reductive symmetric superpairs
Let (g,k) be a reductive symmetric superpair of even type, i.e. so that there
exists an even Cartan subspace a in p. The restriction map S(p^*)^k->S(a^*)^W
where W=W(g_0:a) is the Weyl group, is injective. We determine its image
explicitly.
In particular, our theorem applies to the case of a symmetric superpair of
group type, i.e. (k+k,k) with the flip involution where k is a classical Lie
superalgebra with a non-degenerate invariant even form (equivalently, a
finite-dimensional contragredient Lie superalgebra). Thus, we obtain a new
proof of the generalisation of Chevalley's restriction theorem due to Sergeev
and Kac, Gorelik.
For general symmetric superpairs, the invariants exhibit a new and surprising
behaviour. We illustrate this phenomenon by a detailed discussion in the
example g=C(q+1)=osp(2|2q,C), endowed with a special involution. In this case,
the invariant algebra defines a singular algebraic curve.Comment: 35 pages; v4: revised submission to J.Alg., accepted for publication
under the proviso of revisio