Symplectic geometry of Frobenius structures

Abstract

The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in two-dimensional topological field theories "coupled to gravity at the tree level". The purpose of these notes is to reiterate and expand the viewpoint, outlined in the paper [7] of T. Coates and the author, which recasts this concept in terms of linear symplectic geometry and exposes the role of the twisted loop group L (2) GL N of hidden symmetries.We try to keep the text introductory and non-technical. In particular, we supply details of some simple results from the axiomatic theory, including a several-line proof of the genus 0 Virasoro constraints not mentioned elsewhere, but merely quote and refer to the literature for a number of less trivial applications, such as the quantum Hirzebruch-Riemann-Roch theorem in the theory of cobordism-valued Gromov-Witten invariants. The latter is our joint work in progress with Tom Coates, and we would like to thank him for numerous discussions of the subject.The author is also thankful to Claus Hertling, Yuri Manin and Matilde Marcolli for the invitation to the workshop on Frobenius structures held at MPIM Bonn in Summer 2002.Gravitational descendents. In Witten's formulation of topological gravity [43] one is concerned with certain "correlators" called gravitational descendents. The totality of genus 0 gravitational descendents is organized as the set of Taylor coefficients of a single formal function F of a sequence of vector variables t = (t 0 , t 1 , t 2 , ...). The vectors t i are elements of a finite-dimensional vector space 1 which we will denote H. One assumes that H is equipped with a symmetric nondegenerate bilinear form (·, ·) and with a distinguished non-zero element 1. The axioms are formulated as certain partial differential equations on F : an infinite set of Topological Recursion Relations (TRR), the String Equation (SE) and the Dilaton Equation (DE). To state the axioms using the following coordinate notation: introduce a basis {φ α } in H such thatand write ∂ α,k for the partial derivative ∂/∂t α k . Then the axioms read:Research is partially supported by NSF Grant DMS-0072658. 1 or super-space, but we will systematically ignore the signs that may come out of this possibility.

Gromov–Witten Invariants of Toric Fibrations

“…As it was mentioned in 1.1, the dependence of the genus-0 descendant potential on Novikov's variables is governed by divisor equations. They can be stated in terms of the cone L B as follows [3,7]. Novikov's variables Q i represent degrees (of holomorphic curves) which form a basis in H 2 (B, Q).…”

“…Following [7,3], one associates to F M a Lagrangian cone L M in a symplectic loop space (H, Ω). Let H denote the cohomology space H * (M, Q), (·, ·) the Poincare pairing on H, and 1 ∈ H the unit element.…”

“…Here being overruled means that each tangent space T to L M is tangent to L M exactly along the subspace zT (see [3,7]). This property is invariant under the action of the twisted loop group L (2) GL(H).…”

Quantum Cobordisms and Formal Group Laws

“…Moreover, the tangent spaces T τ of L are tangent to L exactly along zT τ . All these facts, properly generalized, remain true in genus-zero Gromov-Witten theory with a non-trivial target space X (see [9]). …”

“…We will mostly consider here only the simplest example, elucidating one of the key aspects of the theory. We refer the reader to [9] for a more comprehensive survey of the subject and to [5] for all further details. Consider M 0,n , n ≥ 3, the Deligne-Mumford compactification of the moduli space of configurations of n distinct ordered points on the Riemann sphere CP 1 .…”