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Sheaves of the Virasoro and Neveu-Schwarz algebras. (English) Zbl 0645.17008

\(K\)-theory, arithmetic and geometry, Semin., Moscow Univ. 1984-86, Lect. Notes Math. 1289, 52-66 (1987).
This article is a first step towards realizing the representation theory of those infinite-dimensional Lie algebras that arise in (super) string theory in terms of sheaves over moduli space. Formal pseudodifferential operators are defined over a ring with derivation \((A,\partial)\) and then localized in terms of sheaves over a Riemann surface. The introduction of a non-commutative analogue of the residue allows them to write down a generalization of the Virasoro algebra with central term proportional to the residue. Specializing to \(A=\mathbb{C}[t,t^{-1}]\) and \(\partial =\partial_t\) recovers the usual Virasoro with central term dependent on ; \(\operatorname{Res} H^{dR}(A,\partial)\simeq \mathbb{C}c\). When \(\Omega^*_A\) is the algebra of forms over \(A\), the ‘tangent space’ is given by \(T_A=\Omega_A^{-1}\) and there is an exact sequence (for each \(n)\) \(0\to H\to V_n\to T_A\to 0\).
\(V_n\) has a natural structure of Lie algebra and this is a sequence of Lie algebra homomorphisms: \(H\), the de Rham cohomology of \(A\), belongs to the centre of \(V_n\) which is therefore a central extension of \(T_A\) (explicitly given). Localizing this sequence in the sense of families of Riemann surfaces gives rise to an exact sequence \[ 0\to \mathfrak{O}_{\mathfrak{M}}\to R^1\pi_* V_n \to T\mathfrak{M}\to 0 \] of sheaves over moduli space \(\mathfrak{M}\). It is conjectured that \(R^1\pi_* V_n\) can be canonically identified with the sheaf of differential operators of order \(\leq 1\) on the Mumford sheaf \(\lambda_{n/2}\). This work is immediately generalized to Neveu-Schwarz superalgebras.
[For the entire collection see Zbl 0621.00010.]
Reviewer: A. A. Beĭlinson

MSC:

17B65 Infinite-dimensional Lie (super)algebras
17B68 Virasoro and related algebras
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Citations:

Zbl 0621.00010