Deformations of the osp(n|2)-action on the superspace of symbols of differential operators on R1|n
Abstract
We study formal deformations of the natural osp(n|2)-action, n≥ 3, on the superspace Scnd=⨁k≥ 0Fcnd−k/2 of symbols of linear differential operators on weighted densities over R1|n. Starting from the first cohomology space computed in \cite{10}, we compute the cup-product Hd1∨ Hd1→ Hd2 which carries the quadratic obstructions. The answer is governed by the osp(n|2)-invariant operators Ak=η1⋯ηn∂xk−1: the two cocycles hk and h̃k spanning the off-diagonal part of Hd1 are exactly the two derivatives of the coboundary of Ak with respect to the two weights. Consequently, all the products of two off-diagonal classes and all the products of two diagonal classes vanish, and the whole obstruction is carried, for each k, by a single non-trivial 2-cocycle Ωk. If 2d∉N the space Hd1∨Hd1 is identically zero, so every infinitesimal deformation is integrable. If 2d=m∈N we obtain exactly m quadratic integrability conditions, τ2−n−k(tk−t̃k)+τkt̃k=0, 1≤ k≤ m, and we prove that they are also sufficient: no condition of order ≥ 3 occurs and the versal deformation is of degree one in the parameters. In particular every integrable formal deformation is equivalent to its infinitesimal part.
Related
- Stable Symmetric Series, Differential Operators, and Jack Deformations
- Hyponormal Toeplitz operators with operator valued normal symbols
- Commuting Toeplitz operators with harmonic symbols on the Fock space
- The spectral structure of Toeplitz operators with polynomial symbols on the harmonic Bergman space
- Zero products of Toeplitz operators with harmonic symbols on the Bergman space of the unit ball
BibTeX
@misc{basdouri2026deformations,
title = {{Deformations of the $\mathfrakosp(n|2)$-action on the superspace of symbols of differential operators on $\R^1|n$}},
author = {Imed Basdouri and Mabrouk Ben Ammar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02426},
eprint = {2609.02426},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02426},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Deformations of the osp(n|2)-action on the superspace of symbols of differential operators on R^1|n
AU - Imed Basdouri
AU - Mabrouk Ben Ammar
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02426
UR - https://arxiv.org/abs/2609.02426
AB - We study formal deformations of the natural osp(n|2)-action, n≥ 3, on the superspace Sc^n_d=⨁_k≥ 0Fc^n_d−k/2 of symbols of linear differential operators on weighted densities over R^1|n. Starting from the first cohomology space computed in \cite{10}, we compute the cup-product Hd^1∨ Hd^1→ Hd^2 which carries the quadratic obstructions. The answer is governed by the osp(n|2)-invariant operators A_k=η_1⋯η_n∂_x^k−1: the two cocycles h_k and h̃_k spanning the off-diagonal part of Hd^1 are exactly the two derivatives of the coboundary of A_k with respect to the two weights. Consequently, all the products of two off-diagonal classes and all the products of two diagonal classes vanish, and the whole obstruction is carried, for each k, by a single non-trivial 2-cocycle Ω_k. If 2d∉N the space Hd^1∨Hd^1 is identically zero, so every infinitesimal deformation is integrable. If 2d=m∈N we obtain exactly m quadratic integrability conditions, τ_2−n−k(t_k−t̃_k)+τ_kt̃_k=0, 1≤ k≤ m, and we prove that they are also sufficient: no condition of order ≥ 3 occurs and the versal deformation is of degree one in the parameters. In particular every integrable formal deformation is equivalent to its infinitesimal part.
ER -