Research Proposal

My interests concern the geometrization of the complicated and fascinating algebraic structures, that have been introduced to mathematics from physics. I propose two projects involving the investigation and application of new geometric structures, which I uncovered in soliton theory and conformal field theory. My dissertation work clarifies and expands the role of algebraic geometry in the study of soliton equations. In particular, I discover how geometric objects known as formal spectral curves control a wide variety of mechanical systems. My first proposed project brings together singularity theory and Lie theory in a novel way, in a thorough investigation of the ubiquitous role of spectral curves in integrable systems and representation theory. My second proposed project involves a far-reaching generalization of Teichmuller theory, a fundamental topic in the differential and algebraic geometry of surfaces. String theorists discovered that conformal field theory provides a completely new viewpoint on the classical geometry of surfaces, involving infinite-dimensional representation theory. Their work suggests that there should exist a huge collection of ``higher'' versions of Teichmuller theory, describing strange new symmetry algebras on surfaces. I have developed [BF2] a detailed program to uncover the geometric meaning of these W-symmetries, and thus rigorously construct the conjectured higher Teichmuller spaces. I expect these ``jet moduli spaces'' to display many of the diverse features of ordinary Teichmuller theory, as well as to provide new insights into the classical theory.

Spectral Curves and Loop Groups.

Algebraic Geometry of Integrable Systems.

The characteristic feature of integrable mechanical systems is the presence of tori in the phase space, along which the flows are linearized. A crucial development in the study of infinite-dimensional integrable systems, or soliton equations, was the discovery that the phase spaces often contain many such tori that are in fact Jacobian varieties, associated with algebraic curves known as spectral curves. Thus one can start from an algebraic curve with some extra structure, and construct a special ``algebro-geometric'' solution to a soliton equation. (See [SW].) In my thesis [BF1] a simple, direct and general link is established between integrable systems in algebraic geometry (the study of spectral curves and their Jacobians) and soliton equations (usually expressed as flows on spaces of flat connections). It is derived from an abstract group theoretic construction, when applied to loop groups. This leads to the geometric understanding of a much broader range of systems than the classical approach, including the generalized Drinfeld-Sokolov systems [DS] (special cases of which are the famous Korteweg-deVries hierarchy, the nonlinear Schrodinger hierarchy and many others). The great flexibility of the technique suggests that it can be applied in interesting new settings, such as difference equations and equations in Frobenius (in characteristic p).

New Curves, New Solutions.

Perhaps the most interesting new feature of this work, however, is the appearance of formal spectral curves. In the classical construction of algebro-geometric solutions, one starts from a compact spectral curve [DM]. In the new approach, these global curves are replaced with small, local pieces of curves, arising as branched covers. Applying the general construction to these one obtains the entire phase space of the corresponding soliton equation, instead of the special locus obtained classically. Moreover, utilizing the Jacobian varieties for these curves [CC], we can express the totality of the flows of the system as straight-line flows on Jacobians. The different topologies (as branched covers) of smooth spectral curves correspond to the different generalized Drinfeld-Sokolov systems (many of which had no geometric construction classically). If, however, the spectral curve is allowed to be singular, continuous new families of integrable systems of soliton type are obtained. It would be very interesting to understand the relation between the behavior of the integrable system and the geometry of the corresponding singularity. The classical algebro-geometric solutions of soliton equations may be explicitly written down in terms of theta functions of Jacobians. In the formal setting, tau functions on formal Jacobians replace theta functions. Thus it should be possible to understand directly tau-function formulas for solutions from formal geometry. These formulas could then be related to the explicit calculations of soliton equations from loop groups that were the origin of my work (see also [FF]).

Singular Curves and Representation Theory.

The discovery of the role of formal curves and their Jacobians in integrable systems suggests a broad range of questions about the interplay between this geometry and the structure of loop groups. In terms of representation theory, spectral curves arise from Cartan subalgebras of the loop algebra. The topological invariant of the curve provides a rough classification of these subalgebras. However, their fine classification leads directly to the study of singularities, a connection which has not been studied in detail. Suppose Lg is the loop algebra of a finite dimensional Lie algebra g. Cartan subalgebras of Lg may be described as maps from the (formal) punctured disc into the moduli space M of Cartan subalgebras of g. Such a map can be extended to the puncture if we compactify M, and the study of these extensions amounts to the fine classification of the Cartans in Lg. Thus the study of spectral curves is closely related to that of compactifications of M, or equivalently, of degenerations of Cartans in g. There are two natural compactifications of M, in a Grassmannian and in a Hilbert scheme. It would be very useful to obtain a basic understanding of both and of the relation between them. This is a question of independent interest in finite-dimensional Lie theory.

Hitchin System and Loop Grassmannians.

The understanding of the collection of formal spectral curves, or equivalently of Cartans of the loop group, should have a wide range of uses in representation theory. The Hitchin integrable system, which relates moduli spaces of bundles with Jacobians of spectral curves, has a formal analog. This analog is intimately related with the loop versions of the adjoint quotient, nilpotent orbits and Springer theory from finite-dimensional Lie theory. This set of questions arose from conversations with Victor Ginzburg. My first application in this direction [BZ] is the discovery of a fascinating structure in fixed point sets on loop Grassmannians (introduced in [KL]). Suppose p is an element of the loop algebra, whose centralizer is a Cartan subalgebra a, corresponding to a spectral curve Sigma. Then the formal group of a acts on the fixed point set of p in the loop Grassmannian, and I calculate the orbits to be in one to one correspondence with the isomorphism classes of spectral curves of the same topological type as Sigma. Moreover, the orbit associated with a given spectral curve is formally isomorphic to the Jacobian of that curve. However, these orbits are finite dimensional varieties, while the Jacobian is an infinite-dimensional formal scheme. This seeming contradiction in fact implies that the fixed point set carries an intriguing, highly non-reduced structure as a scheme: it is covered with infinite-dimensional nilpotent ``fuzz''. I believe a more detailed knowledge of the geometry of the associated spectral curves and their singularities will provide a complete description of these extraordinary objects.

Higher Uniformization and Conformal Field Theory.

Aspects of Teichmuller Theory.

One of the most interesting aspects of the study of Riemann surfaces is the theory of Teichmuller spaces. These are the spaces of deformations of complex structures on a fixed topological surface. A remarkable feature of Teichmuller theory is the interplay between many different branches of mathematics. In algebraic geometry, Teichmuller spaces arise as moduli spaces of (marked) algebraic curves. In differential geometry, they arise as spaces of uniformizations of a surface by hyperbolic metrics. Such a metric is equivalent to a covering of the surface by hyperbolic space, which is realized as the upper half plane in C. More generally, a projective structure on a surface is an atlas of local isomorphisms from the surface into the projective line, with transition functions given by Mobius transformations (that is, from the action of SL_2(C) on P^1). The uniformization theorem states that every complex structure on a surface admits a unique refinement to a projective structure, with transitions in SL_2(R), thus giving rise to a hyperbolic metric. This also provides an interpretation of Teichmuller space in terms of representations of the fundamental group into SL_2(R). An entirely new viewpoint on Teichmuller (or moduli) spaces came out of string theory, in particular conformal field theory. It was discovered that the infinitesimal deformations of algebraic curves are ``controlled'' by the Virasoro algebra. This implies that the moduli spaces are in some sense homogeneous spaces for this infinite-dimensional Lie algebra. It follows that representation theory of the Virasoro algebra naturally lives on Teichmuller space [BS]. In physics, this means that for every conformal field theory the conformal blocks form a natural sheaf on Teichmuller space. This has led to new insights about classical geometric problems.

Towards Jet Moduli Spaces.

The physics point of view on Teichmuller space has extraordinary conjectural implications. The Virasoro algebra is the SL2 case of a family of (vertex operator) algebras, associated with any simple Lie group G. These so-called W algebras should be thought of as symmetry algebras for enhanced conformal field theories. Thus one is led to ask: are there enhanced moduli spaces, one for every G, on which conformal blocks of field theories with W-symmetry (and thus, representation theory of W-algebras) naturally live? In other words, are there natural geometries of which W-algebras are the symmetries? If so, which of the varied facets of Teichmuller theory can be generalized to the new setting?

Opers and Thickenings.

I believe the answer lies in the geometry of opers. Opers are a class of flat connections on G-bundles on Riemann surfaces, first introduced in the study of soliton equations by Drinfeld and Sokolov [DS], and extensively investigated in the context of the geometric Langlands program by Beilinson and Drinfeld [BD1]. When G is a classical group, opers may be identified with certain differential operators. W-algebras arise in soliton theory from Hamiltonian structures on the space of opers. The relation to uniformization theory comes from the identification between opers when G=SL_2, and projective structures. So the first step to understanding higher Teichmuller spaces is to identify a relation between G-opers in general and some sort of uniformization. I have recently completed this first step [BF2], by introducing for any Riemann surface X a canonical formal thickening of X, closely related to jet bundles. The crucial property of this thickening is that opers on X correspond precisely to ``horizontal'' flag structures on the thickening. A flag structure on a manifold is an atlas of local isomorphisms into the flag manifold G/B of G, with transition functions given by the G-action on G/B. Thus for G=SL_2, a flag structure is precisely a projective structure. (For general G, a horizontality condition - Griffiths transversality - must be imposed on the flag structure.) This identifies the geometric content of opers as G-uniformizations of the (thickened) surface. The next step is the introduction of the W (or ``jet'') moduli spaces as formal deformation spaces of the thickened Riemann surface. Once this is rigorously defined, the construction of conformal blocks for W-algebras is straightforward: representations of W-algebras are expressed in terms of operator-valued opers, and we know the intrinsic meaning of opers on the thickened surfaces. This connection of jet moduli with representation theory and conformal field theory can be expected to have interesting implications in both directions, generalizing work of Beilinson, Schechtman and Ginzburg [BS,BG,G2].

Higher Teichmuller Theory and Applications.

The jet moduli spaces are the higher analogs of the algebro-geometric theory of Teichmuller space, and are only expected to exist as formal thickenings of the ordinary moduli spaces. One might hope, however, for an analytic theory which provides a global complex manifold and analogs of the various other aspects of Teichmuller theory. (For example there are simple analogs of quadratic differentials and affine structures, and of the relation of the latter with projective structures.) I propose to construct a global version of the jet moduli spaces, as a space measuring the obstruction for a flat connection to be an oper. Given a flat connection (with some technical hypotheses) I construct analytically a thickening of the curve, which for opers produces the jet thickening above. The map sending a flat connection to this thickening defines the space of thickenings, my candidate for the jet Teichmuller space, as a quotient of the space of flat connections. This map appears to be a nonlinear version of the famous Hitchin integrable system, which appears from linearizing the map along the classical Teichmuller space.

The Uniformization Conjecture.

An informal conjecture in the physics literature asserts that the W-moduli space (once defined) should be isomorphic to Hitchin's higher Teichmuller space [H], defined in terms of representations of the fundamental group of the Riemann surface into the split real form of G. In our language, this is the analog of the uniformization theorem: every ``thickened curve'' in the jet Teichmuller space admits a unique real flag structure. The proposed construction above also indicates how one could attempt to prove this: a representation of the fundamental group into G_R produces a flat connection, which now gets mapped to a point in the jet Teichmuller space. So we now have a map, between spaces of the same dimension, and it remains to analyze it in detail to show it is an isomorphism.

Opers and the Geometric Langlands Program.

Finally, a vital motivation for this program is the pivotal role played by opers in the geometric Langlands correspondence [BD2,G1,F] which (conjecturally) relates flat connections on a curve with automorphic sheaves on a moduli space of bundles on the curve. Beilinson and Drinfeld have constructed this correspondence for flat connections which are opers. In order to understand connections which aren't opers, one needs a handle on the obstruction for a connection to be an oper. In other words, one wants to relate the different singular oper structures carried by a connection. I hope that the interpretation of the jet moduli space as an obstruction space sheds light on the problem. In the case of SL2, my results demonstrate that a bundle which is not an oper gives rise to an oper on another complex surface. Thus Teichmuller theory comes in for the first time into the Langlands program. I hope to use this technique to provide a new approach to the geometric Langlands correspondence, for rank two bundles. Thus the perspective provided by W-symmetry has potential applications even in the simplest case of SL2, where the theory reduces to ordinary Teichmuller theory.

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