Morse field theory

Abstract

In this paper we define and study the moduli space of metric-graph-flows in a manifold $M$. This is a space of smooth maps from a finite graph to $M$, which, when restricted to each edge, is a gradient flow line of a smooth (and generically Morse) function on $M$. Using the model of Gromov- Witten theory, with this moduli space replacing the space of stable holomorphic curves in a symplectic manifold, we obtain invariants, which are (co)homology operations in $M$. The invariants obtained in this setting are classical cohomology operations such as cup product, Steenrod squares, and Stiefel- Whitney classes. We show that these operations satisfy invariance and gluing properties that fit together to give the structure of a topological quantum field theory. By considering equivariant operations with respect to the action of the automorphism group of the graph, the field theory has more structure. It is analogous to a homological conformal field theory. In particular we show that classical relations such as the Adem relations and Cartan formulae are consequences of these field theoretic properties. These operations are defined and studied using two different methods. First, we use algebraic topological techniques to define appropriate virtual fundamental classes of these moduli spaces. This allows us to define the operations via the corresponding intersection numbers of the moduli space. Secondly, we use geometric and analytic techniques to study the smoothness and compactness properties of these moduli spaces. This will allow us to define these operations on the level of Morse-Smale chain complexes, by appropriately counting metric-graph-flows with particular boundary conditions.