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.
Bibliography:
- [BD1] A. Beilinson and V. Drinfeld. Opers. Preprint,
1995.
- [BD2] A. Beilinson and V. Drinfeld. Quantization of
Hitchin's Integrable System and Hecke Eigensheaves. Preprint, 1998.
- [BG] A. Beilinson and V. Ginzburg. Infinitesimal structure
of moduli spaces of G-bundles. Int. Math. Res. Not. 4 (63-74) 1992.
- [BS] A. Beilinson and V. Schechtman. Determinant bundles
and Virasoro algebras. Comm. Math. Phys. 118 (651-701) 1988.
- [BZ] D. Ben-Zvi. Formal Prym varieties and the affine
Springer fibration. Preprint, 1999.
- [BF1] D. Ben-Zvi and E. Frenkel. Moduli spaces, flat
connections and integrable systems. Preprint, 1998.
- [BF2] D. Ben-Zvi and E. Frenkel. Jet Moduli. Preprint,
1998.
- [CC] C. Contou-Carrere. Jacobienne locale, groupe de
bivecteurs de Witt universel, et symbole
modere. C. R. Acad. Sci. Paris, serie I 318 (743-746) 1994.
- [DM] R. Donagi and E. Markman. Spectral
covers, algebraically completely integrable Hamiltonian systems, and moduli of
bundles. Integrable systems and quantum groups, B. Dubrovin et al. (eds.)
LNM 1620, 1995.
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integrability of soliton equations. Invent. Math. 120 (379-408) 1995.
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duality and Bethe ansatz. Proc. Int. Cong. Math. Phys. 1994.
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and Langlands' Duality. alg-geom 9511007.
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spaces. Prog. Math. 129 (231-266) 1995. (Birkhauser)
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affine flag manifolds. Israel J. of Math. 62 No. 1 (129-168) 1988.
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