Super theta functions and the Weil representation

A selective history of the Stone-von Neumann theorem

“…(This is not Kostant's definition but is basically equivalent to it, at least when G is connected-see [32,Corollary to Theorem 3.7]. There is no reasonable way to "exponentiate" the odd part of g.) For example (see [32,Remark 5.3.4] or [35]), there is a "super" version (H, h) of the Heisenberg group (for m bosonic and n fermionic degrees of freedom) with H the usual Heisenberg group of dimension 2m + 1, h 0 its Lie algebra, and h 1 a real vector space of dimension 2n with a (possibly indefinite) symmetric inner product B. 24 The bracket h 1 × h 1 → h 0 takes its values in the center z of h 0 and is given by [x, y] = B(x, y)z, where z is a basis element for z.…”

Noncommutative superspace and super Heisenberg group

“…Supersymmetric extensions of the Stone-von Neumann theorem were considered in [32,21]. It was shown in [21] that there exists a unique irreducible unitary sH(2n|2m) module up to isomorphism described as in the bosonic case. First we introduce the two-dimensional N = (2, 2) superspace spanned by (X µ , θ α ,θα).…”

“…So V = V0 ⊗ V1, where {q i , p i } is a basis for V0 and {ξ a , λ a } a basis for V1. The super Heisenberg algebra sh(V ) can be constructed as a central extension of the Abelian Lie superalgebra V by an even generator r [21]. We have an exact sequence…”

“…Heisenberg group [18,19,20] is constructed from the Heisenberg algebra by exponential mapping. If we include both of background NSNS and RR fields, it is expected that super Heisenberg group [21] would appear.…”