Summary: The book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The main goal is to introduce the concept of vertex algebra in a coordinate-independent way, and to define the spaces of conformal blocks attached to an arbitrary vertex algebra and a smooth algebraic curve, possibly equipped with some extra geometric data. From this point of view vertex algebras appear as the algebraic objects that encode the local geometric structure of various moduli spaces associated with algebraic curves. The book bridges the gap between the algebraic (formal power series) approach and the more abstract geometric theory of Beilinson and Drinfeld, and features several previously unpublished results. \normalsize
\small Summary: We establish a general link between integrable systems in algebraic geometry (expressed as Jacobian flows on spectral curves) and soliton equations (expressed as evolution equations on flat connections). Our main result is a natural isomorphism between a moduli space of spectral data and a moduli space of differential data, each equipped with an infinite collection of commuting flows. Our isomorphism between the two sides provides a geometric description of the entire phase space of the hierarchy. \normalsize
Summary: We construct a natural quantization of the complement of the zero-section in the cotangent bundle to a Riemann surface, associated to a projective structure on the surface.
Summary: We study relations between two fundamental constructions associated to vector bundles on a curve: the generalized theta function on the moduli space, and the Szeg\"o kernel function on the square of the curve. Two types of relations are demonstrated: the determinant of the Szeg\"o kernel is equated with the pullback of the theta function to the square of the curve, and the expansion of the Szeg\"o kernel along the diagonal describes the logarithmic derivatives of theta along moduli of curves and bundles.
Summary: We investigate the space of extended connections on a vector bundle, which are simultaneous generalizations of projective structures on a curve and of connections on a bundle. Of particular interest is a quadratic map from extended connections to projective structures, which is a deformation of the quadratic part of the Hitchin map and related to isomonodromic deformations. Using theta functions, this produces the Klein and Wirtinger maps, new rational maps from moduli of bundles to projective connections and kernels. In the case of line bundles, these maps relate to several classical constructions with theta functions. We prove finiteness statements for these maps, providing low--dimensional parametrizations of the Jacobian.
We study interactions between the categories of $\D$--modules on smooth and singular varieties. We show that for a large class of singular varieties $Y$, the category of $\D_Y$--modules is equivalent to the category of stratifications, which are sheaves equipped with infinitesimal parallel transport. It follows that $\D$--modules are unaffected by a class of bijective maps, the {\em cuspidal quotients. In particular when $Y$ has a smooth bijective normalization $X$, we obtain a Morita equivalence of $\D_Y$ and $\D_X$ (generalizing results of Smith--Stafford for complex curves and Berest-Etingof-Ginzburg for the rational Cherednik algebras). We also use this equivalence to enlarge the category of induced $\D$--modules on a smooth variety $X$ by collecting induced $\D$--modules on varying cuspidal quotients. The resulting {\em cusp--induced $\D_X$--modules possess the good properties of induced $\D$--modules (in particular a Riemann--Hilbert description) as well as, in the case of curves, a simple characterization. In particular we refine recent results of several authors describing ideals in the Weyl algebra, which play a central role in integrable systems.
Summary: We study the variation of the moduli spaces of bundles over varying curves, using localization of representations of vertex algebras. The Sugawara construction, a basic vertex algebraic feature of loop algebras, is used to construct the heat equations (KZ equations) for nonabelian theta functions in their most general form. Moreover, the flexibility of the technique allows us to describe various limits of the heat operators, including the quadratic part of the Beilinson-Drinfeld quantization of the Hitchin system. In particular we obtain an algebraic Hamiltonian description of the equations of isomonodromic deformation of meromorphic connections on algebraic curves, deforming the quadratic part of the Hitchin system. Moreover the general isomonodromic deformations are exhibited as classical limits of generalized KZ equations.
Summary: We establish a new route from algebraic geometry to the KP hierarchy through noncommutative geometry. The moduli space of our solutions to KP (the differential solitons) may be described as Hilbert schemes of points on noncommutative surfaces, moduli of projective $\D$--modules on smooth curves, or compactified Jacobians on cuspidal curves. In low genus, the differential solitons provide complete and simple solutions to some of the more widely studied puzzles of integrable systems: the ``bispectral phenomenon'' of rational solitons (genus zero) and the behavior of the poles of KP solitons according to a finite--dimensional many--body system. When the curve has higher genus, the differential solitons open an intriguing new chapter by providing the first continuous families of solutions of the KP equation not covered by the Krichever construction.
Summary: We introduce and study a factorization (or ``vertex manifold'') structure on the adelic Grassmannian, a ubiquitous moduli space of $\D$--bundles on a curve. This factorization structure is used to construct continuous families of new vertex algebras, as well as to give a geometric construction of the $\mathcal W_{1+\infty$ vertex algebra, which is the symmetry algebra of the KP equations.
Summary: We lay the foundations for a theory of $\W$--geometry, underlying $\W_n$--vertex algebras and generalizing the relation of the Virasoro algebra with Teichm\"uller theory. We show how the fields of the $\W$--algebras on an algebraic curve may be described as $n$--dimensional projective structures on a canonical ``thickening'' of the curve. Relations with Hitchin's higher Teichm\"uller spaces are developed.
Summary: It is shown that the Drinfeld--Sokolov hierarchies (generalized KdV equations) are embedded in a Hamiltonian way inside of a ``universal'' Hitchin system, generalizing a result of Donagi-Markman. Relations with the geometry of the affine Springer fibration are explored.
Summary: We investigate the properties of affine opers on general curves. In particular we complete the space of opers using affine opers with singularities, and explore the relations with the Drinfeld compactification of the moduli of $B$--bundles using Pl\"ucker data.