Abelian conformal field theory and \(N=2\) supercurves (Q1911567)

Abelian conformal field theory, that is the theory of free fermions over a compact Riemann surface, has been studied, since the late 1980s, from several points of view. The first rigorous mathematical foundation of abelian conformal field theory was provided by the work of \textit{N. Kawamoto}, \textit{Y. Namikawa}, \textit{A. Tsuchiya} and \textit{Y. Yamada} [Commun. Math. Phys. 116, No. 2, 247-308 (1988; Zbl 0648.35080)], Y. Shimizu and K. Ueno. Their approach uses the framework of Sato's universal Grassmannian manifold, the Krichever correspondence between data on Riemann surfaces and soliton equations, complex cobordism theory, formal groups, and other well-developed theories. In his recent paper ``On conformal field theory'' [in: Vector bundles in algebraic geometry, Proc. 1993 Durham Symp., Lond. Math. Soc. Lect. Note Ser. 208, 283-345 (1995; Zbl 0846.17027)] \textit{K. Ueno} proposed another approach to abelian conformal field theory. Namely, taking the vertex operator algebra constructed from the Heisenberg algebra as a gauge group, and applying the basic constructions in the nonabelian conformal theory to this situation, the relationship between conformal blocks and theta functions of higher level gives another mathematically rigorous description of the physicists' operator formalism in the fermionic theory. The aim of the present paper is to give a geometric interpretation of one of Ueno's results stated in this context. More precisely, Ueno had described the spaces of vacua (conformal blocks) by imposing an extra gauge condition expressed by vertex operators of even level \(M\) among the Fock spaces. His main result, in this regard, consisted then in establishing an isomorphism between the space of conformal blocks and the space of \(M\)th-order theta functions on the Jacobian of the underlying pointed stable curve \(C\). In the paper under review, the author points out that Ueno's result can be extended to the case of level \(M = 1\), and he gives another description of the space of conformal blocks in this particular odd-level case. The basic framework used here is, on the one hand, the Beilinson-Bernstein theory of localizations for representations of certain infinite-dimensional Lie algebras associated with special ``dressed'' moduli spaces and, on the other hand, a version of Skornyakov's theory of the \(\pi\)-Picard group of locally free sheaves on supercurves (for \(N=1\) or \(N=2)\) with symmetry group \(\pi\). The respective general theories of Beilinson-Bernstein [cf. \textit{A. A. Beilinson} and \textit{V. V. Schechtman}, Commun. Math. Phys. 118, No. 4, 651-701 (1988; Zbl 0665.17010)] and of Skornyakov-Manin [cf. \textit{Yu. I. Manin}, Topics in non-commutative geometry (1991; Zbl 0724.17007)] are tailored to the particular context of abelian conformal field theory, and this program forms the main body of the paper. The central results are then the following theorems: (1) The \(\pi\)-Picard functor for a proper smooth supercurve of dimension \(N = 1\) or \(N = 2\) is representable by a smooth superscheme, the so-called \(\pi\)-Picard variety. (2) The space of (Ueno's) conformal blocks equals a fiber of the Beilinson-Bernstein localization of the Fock representation on the \(\pi\)-Picard scheme of the respective supercurve. (3) In the case of level \(M = 1\), the space of conformal blocks is canonically isomorphic to the space of theta functions on the Jacobian of the underlying pointed stable curve \(C\). As for related and further results, the author refers to two forthcoming papers entitled ``Moduli of \(N = 1\) stable superconformal curves and abelian conformal field theory'' and ``Moduli of \(N = 2\) superconformal curves''.