Superbosonisation, Riesz superdistributions, and highest weight modules

Abstract

Superbosonisation, introduced by Littelmann-Sommers-Zirnbauer, is a generalisation of bosonisation, with applications in Random Matrix Theory and Condensed Matter Physics. In this survey, we link the superbosonisation identity to Representation Theory and Harmonic Analysis and explain two new proofs, one via the Laplace transform and one based on a multiplicity freeness statement.

1 Appendix

Supergeometry We summarise some basic definitions from supergeometry.

Definition 2 A C-superspace is a pair X=(X₀ O _x) where X ₀ is a topological space and O _x is a sheaf of supercommutative superalgebras over C with local stalks. A morphism f : X →Y of C-superspaces is a pair (f ₀,f ^#) comprising a continuous map f ₀ : X ₀ →Y ₀ and a sheaf map ${f^\# }:f_0^{ - 1}{O_Y} \to {O_X}$, which is local in the sense that ${f^\# }({m_Y}{,_{f0(x)}}) \subseteq mx,x$ for any x, where m _X,x is the maximal ideal of O _X,x.

Global sections f ∈ Γ(O _x) of O _X are called superfunctions. Due to the locality condition, the value $f(x): = f + {m_X}_{,x} \in {O_{X,x}}/{m_{X,x}}$ is defined for any x. Open subspaces of a C-superspace X are given by (U,O _X|_U), for any open subset U ⊆ X ₀.

We consider two types of model spaces.

Definition 3 For a complex super-vector space V, we define H denotes the sheaf of holomorphic functions. The space $({V_{\bar 0}},Ov)$ is called the complex affine superspace associated with V, and denoted by V.

If instead, V is a cs vector space (see Definition 1), then we define the sheaf $Ov: = C_{{V_{\bar 0}}}^\infty \otimes \wedge {({V_{\bar 1}})^*}$ denotes the sheaf of complex-valued smooth func¬tions. The space $({V_{\bar 0}}.{O_V})$ is called the cs affine superspace associated with V, and denoted by V. (The cs terminology is due to J. Bernstein.)

In turn, this gives two flavours of supermanifolds.

Definition 4 Let X be a C-superspace, whose underlying topological space X ₀ is Hausdorff, and which admits a cover by open subspaces isomorphic to open sub-spaces of some complex resp. cs affine superspace V, where V may vary. Then X is called a complex supermanifold resp. a cs manifold.

Complex supermanifolds and cs manifolds form full subcategories of the category of C-superspaces that admit finite products. The assignment sending the complex affine superspace V to the cs affine superspace obtained by forgetting the complex structure on Vō extends to a product-preserving cs-ification functor from complex supermanifolds to cs manifolds; the cs-ification of X is denoted by X _cs.

This point of view is also espoused by Witten in recent work [28].

Supergroups and supergroup pairs We give some basic definitions on supergroups. Details can be found in [6,8,22].

Definition 5 A complex Lie supergroup (resp. a cs Lie supergroup) is a group object in the category of complex supermanifolds (resp. cs manifolds). A morphism of (complex or cs) Lie supergroups is a morphism of group objects in the category of complex supermanifolds (resp. cs manifolds). The cs-ification functor maps complex Lie supergroups to cs Lie supergroups and morphisms of complex Lie supergroups to morphisms of cs Lie supergroups.

Definition 6 A complex (resp. cs) supergroup pair (g,G₀) is given by a complex (resp. real) Lie group G ₀ and a complex Lie superalgebra g, together with a morphism Ad : G₀→ Aut(g) of complex (resp. real) Lie groups such that ${g_{\bar 0}}$ is the Lie algebra of G ₀ (resp. its complexification), Ad extends the adjoint action of G ₀ on ${g_{\bar 0}}$, and [·, ·] extends d Ad. A morphism of supergroup pairs (d∅, ∅₀) consists of a morphism ∅₀ of complex (resp. real) Lie groups and a morphism d<j) of Lie superalgebras that is ∅₀-equivariant for the Ad-actions, such that d∅ extends d(∅₀).

The following is well-known, cf. [6,19,20].

Proposition 11 There is an equivalence of the categories of complex (resp. cs) Lie supergroups and of complex (resp. cs) supergroup pairs. It maps any Lie supergroup to the pair consisting of its Lie superalgebra and its underlying Lie group.

Definition 7 A closed embedding of (complex resp. cs) Lie supergroups is called a closed (complex resp. cs) subsupergroup. A closed supergroup subpair (h, H ₀) ⊆ (g, G ₀) consists of a Lie subsuperalgebra h ⊆ g and a closed subgroup H ₀ ⊆ G ₀, such that ${g_{\bar 0}}$ is a supergroup pair. Given a complex Lie supergroup G, a cs form of G is a closed subsupergroup H of G _cs such that in the supergroup pairs (h,H ₀) and (g,G ₀) of H resp. G, one has h = g. In this case, H ₀ is a real form of G ₀.

If G is a complex Lie supergroup with associated supergroup pair (g,G ₀), then (g, H ₀), for a closed subgroup H ₀ ⊆ G ₀, is the supergroup pair of a cs form of G if and only if H ₀ is a real form of G ₀, or equivalently, if (g, H ₀) is a cs supergroup pair. To define a cs form H of G, it thus suffices to specify a real form H ₀ ⊆ G ₀.

Points If C is any category, and X is an object of C, then an S-valued point (where S is another object of C) is defined to be a morphism x : S → X. Suggestively, one writes x ∈ _S X in this case, and denotes the set of all x ∈ _S X by X(S).

For any morphism f : X → Y, one may define a set-map f _S: X(S) → Y(S) by

$$fs(x): = f(x): = f \circ x \in sY,{\kern 1pt} x \in sX.$$

Taking the generic point x = id _X ∈ _X X, the values f (x) completely determine f. The following statement is known as Yoneda’s Lemma: For any collection of maps f s : X (S) → Y (S), there exists a morphism f : X → Y such that f _S(x) = f (x) for all x ∈ _S X if and only if f _T(x(t)) = f _S(x)(t), for all t: T → S. Here, x(t) are called specialisations of x, so the condition states that (f _S) is invariant under specialisation.

Thus, the Yoneda embedding functor X ↦ X(—) from C to [C ^op ,Sets] is fully faithful. It preserves products, so if C admits finite products, it induces a fully faith¬ful embedding of the category of group objects in C into the category [C ^op, Grp] of group-valued functors. In other words, an object X of C is a group object if and only if for any S, X(S) admits a group law that is invariant under specialisation.

Berezin integrals Let X be a cs manifold and Ber _X to the Berezinian sheaf. The sheaf of Berezinian densities |Ber|_X is the twist by the orientation sheaf. Given local coordinates (x ^a) = (x, ξ,) on U, one may consider the distinguished basis |D ( x ^a)| of the module of Berezinian densities |Ber|_X[22].

A retraction is a morphism r : X → X ₀, left inverse to the canonical embedding j : X0 → X. A system of coordinates (x, ξ) of X is called adapted to r if x=r^#(x₀). Given an adapted system, we write ω = |D(x,ξ,)|f and f _I ∈ Γ(O _X0) where dim X = *|q. Then one defines ∫ _{X/X 0} ω) := |dx ₀ |f _{1,….,q}, which is an ordinary density on X ₀. This quantity only depends on r, and not on the choice of an adapted system of coordinates. If this density is absolutely integrable on X ₀, then we say that ft) is absolutely integrable with respect to r, and define ∫ _X ω) := ∫ _{X 0} ∫ _{X/X 0} ω >. Unless supp ω) is compact, this quantity depends heavily on r.

Acknowledgements This research was funded by the grants no. DFG ZI 513/2-1 and SFB TR/12, provided by Deutsche Forschungsgemeinschaft (DFG). We wish to thank Martin Zirnbauer for extensive discussions, detailed comments, and for bringing this topic to our attention. We thank INdAM for its hospitality. The first named author wishes to thank Jacques Faraut for his interest, and the second named author wishes to thank Bent Ørsted for some useful comments.