On a super-Virasoro group, a semigroup of annuli, and Gauss--Berezin integral operators

Technical Summary: On a super-Virasoro group, a semigroup of annuli, and Gauss–Berezin integral operators

Problem Statement
The paper addresses the extension of the theory of unitary representations of the Virasoro algebra to the realm of superalgebras and supergroups, specifically focusing on the Neveu–Schwarz (NS) Lie superalgebra. While the Virasoro algebra and its associated group of diffeomorphisms of the circle, $SDiff(S^1)$, are well-understood in the context of infinite-dimensional representation theory (including the construction of semigroups of annuli and their representations), the super-analogues present significant technical challenges.

The primary difficulties are twofold:

1. Functional-Analytic Nature: The representations involve unbounded operators in Hilbert spaces, requiring careful handling of domains, self-adjointness, and the multiplication of such operators.
2. Formal Super-Structure: The main objects of study—the semigroup of contact super-annuli and the semigroup of Gauss–Berezin operators—lie outside the standard formalism of super-mathematics (e.g., standard supermanifold theory). The author explicitly avoids assuming the reader has prior knowledge of Lie superalgebras, aiming instead for a treatment accessible to functional analysts familiar with classical Lie group representation theory.

The goal is to construct explicit representations of a semigroup of "super-annuli" (complex manifolds with boundary and contact structures) that extend the unitary highest weight representations of the Neveu–Schwarz supergroup.

Methodology
The author employs a specific approach to supergroups, following DeWitt and Rogers, by defining supergroups as groups over a fixed Grassmann algebra $A$ with a countable number of generators. The methodology proceeds through several distinct stages:

1. Grassmannization and Surplace Groups: The paper utilizes the Berezin–Kats construction to define the Grassmannization $g(A)$ of a Lie superalgebra $g$. It introduces "surplace groups" $G^{\circ\circ}(A)$, which are groups of actual infinitesimal transformations generated by elements of the form $1 + \lambda X$ (where $\lambda$ are nilpotent elements of $A$). This allows for a concrete realization of the Neveu–Schwarz supergroup $SCont(S^{1|1}_{\bullet}; A)$ as a group of contactomorphisms of the supercircle $S^{1|1}$.

2. Embedding into Orthosymplectic Structures: A key technical step is the embedding of the supergroup $SCont(S^{1|1}_{\bullet}; A)$ into an infinite-dimensional orthosymplectic supergroup $OSp(W; A)$. This is achieved by realizing the Lie superalgebra $cont(S^{1|1}_{\bullet})$ in terms of invariant differential operations on a space $W$ of functions and densities. This embedding is parameterized by complex numbers $\mu, \nu$, corresponding to the parameters in the operator definitions of the NS algebra.

3. Integration of Representations: The paper proves that unitary highest weight representations of the NS Lie superalgebra can be integrated to projective representations of the supergroup $SCont(S^{1|1}_{\bullet}; A)$. This involves establishing the self-adjointness of unbounded operators and handling the functional-analytic subtleties of the scale of Hilbert spaces defined by the operator $L_0$.

4. Semigroup of Super-Annuli and Potapov Transforms: The author constructs a semigroup $\Gamma_{\bullet}(A)$ of contact super-annuli. Elements of this semigroup are annuli equipped with contact structures and boundary parametrizations. The multiplication is defined by gluing annuli. The paper realizes these elements as affine Lagrangian super-linear relations in the orthosymplectic space $W[A]$. The Potapov transform is used to map these relations to operators.

5. Gauss–Berezin Operators: To represent the semigroup $\Gamma_{\bullet}(A)$, the paper introduces "Gauss–Berezin integral operators." These are super-hybrids of Gaussian operators (in bosonic Fock space) and Berezin operators (in fermionic Fock space). The kernel of such an operator is a formal series involving an exponential of a quadratic form in bosonic and fermionic variables.

Key Contributions and Results

• Explicit Description of the NS Supergroup: The paper provides explicit descriptions of the Neveu–Schwarz supergroup $NS(A)$ and its complexification, the semigroup of contact super-annuli $\Gamma_{\bullet}(A)$, in terms of contactomorphisms of the supercircle and gluing operations.
• Integration Theorem: It is proven that any unitary highest weight representation of the NS Lie superalgebra (with specific constraints on the central charge $c$ and conformal weight $h$) extends to a projective representation of the supergroup $SCont(S^{1|1}_{\bullet}; A)$.
• Semigroup Representation: The main result (Theorem 8.3 and Section 11) establishes that these representations can be further extended to the semigroup $\Gamma_{\bullet}(A)$. This is achieved by associating each element of the semigroup with a bounded Gauss–Berezin operator acting on the space of smooth vectors of the representation.
• Correspondence between Relations and Operators: A rigorous correspondence is established between affine Lagrangian super-linear relations (arising from the geometry of super-annuli) and Gauss–Berezin operators. Specifically, the product of operators corresponds to the product of the underlying Lagrangian relations, provided the relations are transversal.
• Handling Unboundedness: The paper successfully navigates the difficulties of unbounded operators by working within the space of smooth vectors ($H_{\infty}$) and utilizing the scale of Hilbert spaces defined by powers of $L_0$.

Significance and Claims
The paper claims to provide a "superization" of the work on the Virasoro algebra and the semigroup of annuli (specifically referencing works [41], [45], [62]). Its significance lies in:

1. Bridging Formalisms: It offers a framework for studying infinite-dimensional supergroups that is accessible to functional analysts, bypassing the heavy machinery of supermanifold theory while still capturing the essential algebraic and geometric structures.
2. New Operator Class: The introduction of Gauss–Berezin operators provides a concrete tool for handling the product of super-annuli in the context of representation theory, analogous to how Gaussian operators handle the Virasoro semigroup.
3. Completeness of the Picture: The work demonstrates that the deep structural results known for the Virasoro algebra (regarding highest weight representations and their extension to semigroups) have valid counterparts in the Neveu–Schwarz superalgebra context, despite the added complexity of fermionic variables and unboundedness.

The author modestly notes that the approach is specific to the Neveu–Schwarz case (chosen for technical correspondence with a preliminary paper) and that the definitions of supergroups used are one of several possible formalisms. The paper does not claim to solve all problems regarding infinite-dimensional supermanifolds but rather focuses on the specific functional-analytic and algebraic structures required to construct these representations.