Holomorphic curves in symplectic manifolds and integrable systems

Abstract

Award: DMS-0707150

Principal Investigator: Todor Milanov

The long term goal of these projects is to understand the

topology of moduli spaces of holomorphic curves in symplectic

manifolds in terms of the theory of integrable systems. The

first project aims to construct an integrable hierarchy which

governs the topology of the moduli spaces of degree d stable

holomorphic maps from a genus-g, nodal Riemann surface to a

variety X which is assumed to be a toric complete intersection.

The topological information is encoded in a formal power series

called the total descendant potential of the toric variety, and

the goal is to show that this potential is a solution to an

integrable hierarchy. The second project in this research

program is dedicated to an interaction of integrable systems with

symplectic field theory of a compact Kaehler manifold Y,

developing a potential related to the total descendant potential

described above. Relationships are expected to emerge between

Gromov-Witten invariants for Y and for a projective line bundle

over Y, with the relationships described via transformations of

integrable systems.

Symplectic geometry is the structure underlying the Hamiltonian

formalism of classical mechanics, in which the behavior of a

mechanical system is determined by an energy-like function.

Geometric spaces carrying such structures can be high-dimensional

and structurally complicated, and much recent work in the area is

devoted to exploring symplectic structures through associated

spaces of well-imbedded two-dimensional subsurfaces of them.

Two-dimensional surfaces have been studied for more than 150

years and our detailed understanding of them leads to constraints

upon the associated invariants of high-dimensional symplectic

manifolds. The principal investigator's work explores new

constraints of this kind that take the form of well-behaved

differential equations, the integrable systems of the proposal

title.