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Gauss--Berezin integral operators and spinors over supergroups $\mathrm{OSp}(2p|2q)$, and Lagrangian super-Grasmannians

Yuri Neretin

TL;DR

The work constructs an explicit spinor representation $\rho$ of the real orthosymplectic supergroup $OSp(2p|2q,\mathbb{R})$ via Gauss--Berezin integral operators and extends it to a complex domain to obtain a larger Olshanski-type semigroup. It then embeds this representation into a geometric framework on domains in the Lagrangian super-Grassmannian, where compositions of Lagrangian relations correspond to products of Gauss--Berezin operators, yielding a powerful bridge between super-geometry and operator theory. The paper develops a unified theory combining bosonic Gaussian operators and fermionic Berezin operators into Gauss--Berezin operators and establishes a canonical correspondence with contractive Lagrangian superlinear relations. This correspondence provides a concrete, diagrammatic interpretation of operator products as compositions of linear relations and offers a path toward studying unitary representations and super-Virasoro structures within a robust geometric–categorical setting.

Abstract

We obtain explicit formulas for the spinor representation $ρ$ of the real orthosymplectic supergroup $\mathrm{OSp}(2p|2q,\mathbb{R})$ by integral 'Gauss--Berezin' operators. Next, we extend $ρ$ to a complex domain and get a representation of a larger semigroup, which is a counterpart of Olshanski subsemigroups in semisimple Lie groups. Further, we show that $ρ$ can be extended to an operator-valued function on a certain domain in the Lagrangian super-Grassmannian (graphs of elements of the supergroup $\mathrm{OSp}(2p|2q,\mathbb{C})$ are Lagrangian super-subspaces) and show that this function is a 'representation' in the following sense: we consider Lagrangian subspaces as linear relations and composition of two Lagrangian relations in general position corresponds to a product of Gauss--Berezin operators

Gauss--Berezin integral operators and spinors over supergroups $\mathrm{OSp}(2p|2q)$, and Lagrangian super-Grasmannians

TL;DR

The work constructs an explicit spinor representation of the real orthosymplectic supergroup via Gauss--Berezin integral operators and extends it to a complex domain to obtain a larger Olshanski-type semigroup. It then embeds this representation into a geometric framework on domains in the Lagrangian super-Grassmannian, where compositions of Lagrangian relations correspond to products of Gauss--Berezin operators, yielding a powerful bridge between super-geometry and operator theory. The paper develops a unified theory combining bosonic Gaussian operators and fermionic Berezin operators into Gauss--Berezin operators and establishes a canonical correspondence with contractive Lagrangian superlinear relations. This correspondence provides a concrete, diagrammatic interpretation of operator products as compositions of linear relations and offers a path toward studying unitary representations and super-Virasoro structures within a robust geometric–categorical setting.

Abstract

We obtain explicit formulas for the spinor representation of the real orthosymplectic supergroup by integral 'Gauss--Berezin' operators. Next, we extend to a complex domain and get a representation of a larger semigroup, which is a counterpart of Olshanski subsemigroups in semisimple Lie groups. Further, we show that can be extended to an operator-valued function on a certain domain in the Lagrangian super-Grassmannian (graphs of elements of the supergroup are Lagrangian super-subspaces) and show that this function is a 'representation' in the following sense: we consider Lagrangian subspaces as linear relations and composition of two Lagrangian relations in general position corresponds to a product of Gauss--Berezin operators

Paper Structure

This paper contains 10 sections, 38 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. A survey of orthogonal spinors. Berezin operators and Lagrangian linear relations
  3. A survey of the oscillator representation. The category of Gaussian integral operators
  4. Gauss--Berezin integrals
  5. Gauss--Berezin integral operators
  6. Supergroups $\mathrm {OSp}(2p|2q)$
  7. Super-Grassmannians
  8. Superlinear relations
  9. Correspondence between Lagrangian superlinear relations and Gauss--Berezin operators
  10. Final remarks

Key Result

Proposition 2.1

The map $K\mapsto A_K$ is a one-to-one correspondence of the set of all polynomials $K(\xi,\overline\eta)$ and the set of all linear maps $\Lambda_q\to\Lambda_p$.

Theorems & Definitions (39)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.9
  • Theorem 2.10
  • Lemma 2.11
  • ...and 29 more