Superunitary Representations of Heisenberg Supergroups

Numerous Lie supergroups do not admit superunitary representations except the trivial one, e.g., Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of superunitary representation, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrodinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible superunitary representations and serve as ground to the main result of this paper: a generalized Stone-von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrodinger-like representations to metaplectic supergroups, also fit into this definition of superunitary representations.Comment: 59 pages. v2: section 6 (Proof of Stone-von Neumann theorem) and section 7 (super unitary dual) were corrected and rewritte

Deformation Quantization for Heisenberg Supergroup

We construct a non-formal deformation machinery for the actions of the Heisenberg supergroup analogue to the one developed by M. Rieffel for the actions of R^d. However, the method used here differs from Rieffel's one: we obtain a Universal Deformation Formula for the actions of R^{m|n} as a byproduct of Weyl ordered Kirillov's orbit method adapted to the graded setting. To do so, we have to introduce the notion of C*-superalgebra, which is compatible with the deformation, and which can be seen as corresponding to noncommutative superspaces. We also use this construction to interpret the renormalizability of a noncommutative Quantum Field Theory.Comment: 49 page

Resurgent Deformation Quantisation

We construct a version of the complex Heisenberg algebra based on the idea of endless analytic continuation. In particular, we exhibit an integral formula for the product of resurgent operators with algebraic singularities. This algebra would be large enough to capture quantum effects that escape ordinary formal deformation quantisation.Comment: 28 pages, v2: published versio

Non-formal star-exponential on contracted one-sheeted hyperboloids

In this paper, we exhibit the non-formal star-exponential of the Lie group SL(2,R) realized geometrically on the curvature contraction of its one-sheeted hyperboloid orbits endowed with its natural non-formal star-product. It is done by a direct resolution of the defining equation of the star-exponential and produces an expression with Bessel functions. This yields a continuous group homomorphism from SL(2,R) into the von Neumann algebra of multipliers of the Hilbert algebra underlied by this natural star-product. As an application, we prove a new identity on Bessel functions.Comment: 33 pages, 3 figures, v2: changes in section 3, references adde

Symmetries of noncommutative scalar field theory

We investigate symmetries of the scalar field theory with harmonic term on the Moyal space with euclidean scalar product and general symplectic form. The classical action is invariant under the orthogonal group if this group acts also on the symplectic structure. We find that the invariance under the orthogonal group can be restored also at the quantum level by restricting the symplectic structures to a particular orbit.Comment: 12 pages, revised versio