General super imprimitivity theorem: Mackey induction and systems of imprimitivity for super homogeneous spaces with odd directions

The rigorous theory of unitary representations of super Lie groups built by Carmeli, Cassinelli, Toigo and Varadarajan establishes a super analogue of Mackey's inducing construction and imprimitivity theorem, but only for so-called special sub super Lie groups, where the odd part of the subalgebra equals the odd part of the ambient algebra and the homogeneous space is purely even. The open problem is to extend the induced-representation construction and the super imprimitivity theorem to general super homogeneous spaces with nontrivial odd directions, and to base them on quasi-invariant rather than invariant measures, thereby completing Mackey's machinery for super Lie groups in full generality.

Background

A super Lie group is a symmetry object whose infinitesimal algebra is a Lie superalgebra $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$, an anticommutative algebra with an even part $\mathfrak g_0$ (an ordinary Lie algebra) and an odd part $\mathfrak g_1$, where the bracket of two odd elements is symmetric and lands in $\mathfrak g_0$. By a theorem of Koszul and others, the category of super Lie groups is equivalent to the category of super Harish-Chandra pairs $(G_0,\mathfrak g)$, consisting of a classical Lie group $G_0$ integrating $\mathfrak g_0$ together with the action of $G_0$ on $\mathfrak g$; this is the framework used in the source paper. A unitary representation of a super Lie group $(G_0,\mathfrak g)$ is a pair $(\pi_0,\rho)$: an ordinary unitary representation $\pi_0$ of $G_0$ on a super Hilbert space (a $\mathbb Z_2$-graded Hilbert space with a graded symmetry operator), together with odd operators $\rho(X)$, $X\in\mathfrak g_1$, on the smooth vectors satisfying the super-commutation rules, with $-i\rho(X)$ essentially self-adjoint. The odd operators are typically unbounded, and a basic result of the source paper is that the commutation rules force them to be well behaved, making the definition essentially unique. In classical representation theory, Mackey's little-group method classifies representations of groups with a normal abelian subgroup, and its abstract core is the imprimitivity theorem: systems of imprimitivity (a unitary representation together with a projection-valued measure on a homogeneous space $G_0/H_0$ satisfying a covariance condition) are exactly the representations induced from unitary representations of the subgroup $H_0$; the inducing construction is an equivalence of categories and is transitive. The source paper (Commun. Math. Phys. 263 (2006) 217-258) extends this to the super setting and uses it to classify the irreducible unitary representations of super semidirect products, in particular the super Poincare groups in arbitrary dimension and number $N$ of supersymmetries, recovering the physics classification of super multiplets. However, its Theorem 2 is proved under two restrictions that the authors explicitly flag: first, the sub super Lie group $(H_0,\mathfrak h)$ must be special, i.e. $\mathfrak h_1=\mathfrak g_1$, so that the super homogeneous space on which the system of imprimitivity lives is a purely even manifold (this is what makes the super Poincare applications work, but it excludes every subgroup whose odd part is a proper subspace of the ambient odd part); the authors call this restriction 'severe'. Second, the homogeneous space $G_0/H_0$ is assumed to carry an invariant measure, whereas the classical theory needs only a quasi-invariant measure (obtained, for example, through Weil's rho-function construction). Later frameworks for unitary supergroup representations, such as the C*-crossed-product and direct-integral theory of Neeb and Salmasian, build on this foundation but do not supply the missing general imprimitivity theorem, and subsequent applications (for instance to supersymmetric quantum fields in the quantum-probability literature) still invoke only the special case. Within the same paper's classification applications, two further items are left open and remain unaddressed in the later literature: the determination, in general spacetime dimension $D$ and number of supersymmetries $N$, of the restriction $\kappa'_p$ of the spin representation to the product of the even little group $L_0^p$ and the automorphism ($R$-)group acting on the odd part (only the $D=4$ formulas, due to Ferrara, Savoy and Zumino, are written out), and the removal of the regularity assumption that the relevant $L_0$-orbits in $T_0^*$ are locally closed.

Problem Statement

Let $(G_0,\mathfrak g)$ be a super Lie group, viewed as a super Harish-Chandra pair, and let $(H_0,\mathfrak h)$ be a closed sub super Lie group with $\mathfrak h_0\subset\mathfrak g_0$ and $\mathfrak h_1\subseteq\mathfrak g_1$ arbitrary, in particular with no assumption that $\mathfrak h_1=\mathfrak g_1$, so that the associated super homogeneous space $X=G/H$ is in general a genuinely non-even superspace whose structure sheaf has odd fiber directions. Solve the following, generalizing Theorems and constructions of Carmeli-Cassinelli-Toigo-Varadarajan (Commun. Math. Phys. 263 (2006) 217-258) beyond their special-subgroup hypothesis: (1) Construct, for every unitary representation $(\sigma,\rho_\sigma)$ of $(H_0,\mathfrak h)$ on a super Hilbert space, the induced unitary representation of $(G_0,\mathfrak g)$ acting on a suitable graded space of sections over $X$, handling the analytic difficulty that the odd operators are in general unbounded and must satisfy the super-commutation rules essentially in the self-adjoint sense. (2) Define the appropriate notion of a super system of imprimitivity for $(G_0,\mathfrak g)$ based on $X$, i.e. the correct super analogue of a covariant projection-valued spectral measure over the even quotient $G_0/H_0$ together with the compatible odd operator action. (3) Prove the super imprimitivity theorem in this generality: the inducing construction of (1) is an equivalence of categories between the category of unitary representations of $(H_0,\mathfrak h)$ and the category of super systems of imprimitivity of $(G_0,\mathfrak g)$ based on $X$; and establish it under only a quasi-invariance hypothesis on the measure class of $G_0/H_0$ (as in the classical rho-function theory), removing the invariant-measure assumption. The general case (arbitrary $\mathfrak h_1\subseteq\mathfrak g_1$, $X$ a genuinely non-even superspace, quasi-invariant measure class on $G_0/H_0$) is what has to be established; proving only the special-subgroup, purely even, or semidirect-product case does not settle it.

References

1. C. Carmeli, G. Cassinelli, A. Toigo, V. S. Varadarajan, 'Unitary Representations of Super Lie Groups and Applications to the Classification and Multiplet Structure of Super Particles', Communications in Mathematical Physics 263 (2006), no. 1, 217-258. DOI: 10.1007/s00220-005-1452-0; arXiv:hep-th/0501061, https://arxiv.org/abs/hep-th/0501061, https://doi.org/10.1007/s00220-005-1452-0
2. Karl-Hermann Neeb, Hadi Salmasian, 'Crossed product algebras and direct integral decomposition for Lie supergroups', Pacific Journal of Mathematics 282 (2016), no. 1, 213-232. DOI: 10.2140/pjm.2016.282.213; arXiv:1506.01558, https://arxiv.org/abs/1506.01558, https://doi.org/10.2140/pjm.2016.282.213
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