Vortices and a TQFT for Lefschetz fibrations on 4–manifolds

Abstract

Adapting a construction of D Salamon involving the U.1/ vortex equations, we explore the properties of a Floer theory for 3-manifolds that fiber over S 1 which exhibits several parallels with monopole Floer homology, and in all likelihood coincides with it. The theory fits into a restricted analogue of a TQFT in which the cobordisms are required to be equipped with Lefschetz fibrations, and has connections to the dynamics of surface symplectomorphisms.57R57; 57R56, 53D401 Background and summary of resultsFor some time it has been known that two of the most important invariants of smooth closed 4-manifolds, the Donaldson and Seiberg-Witten invariants, can each be expressed in terms of .3 C 1/-dimensional topological quantum field theories (see Donaldson [6], Marcolli and Wang [27], Kronheimer and Mrowka [19]). In such a "TQFT," to each oriented 3-manifold Y (perhaps equipped with additional data, such as a spin c -structure), one associates canonically a group V .Y / satisfying, among several other conditions, the property that a cobordism X from Y 1 to Y 2 functorially induces a homomorphismIf X is a smooth closed oriented 4-manifold, divided into two pieces as X D X 1 [ Y X 2 with b C .X i / > 0, one views X 1 as a cobordism from the empty set ¿ to Y and X 2 as a cobordism from ¿ to Y (ie, Y with its orientation reversed). One has a natural identification V . Y / Š V .Y / , and the 4-dimensional invariant I X is obtained by a natural calculation in V .Y / involving the images of the maps F X 1 and F X 2 ; I X is independent of the choice of splitting of X into the two pieces X 1 and X 2 .In the presence of a symplectic structure ! on the spin c 4-manifold .X; s/, the famous work of C Taubes collected in [45] shows that the Seiberg-Witten invariant S W X .s/ agrees with a "Gromov invariant" Gr .X ;!/ .˛s/ which counts pseudoholomorphic submanifolds of X representing a homology class˛s corresponding to s . Kronheimer and Mrowka's work [19] (see [20] for a summary) lays the full foundations for the TQFT underlying S W X .s/, in which the role of the group V .Y / in the above description is played by HM.Y; s; Á/, where s is a spin c structure and Á 2 H 2 .Y I ‫/ޒ‬ is the

Symplectic Floer homology of area-preserving surface diffeomorphisms

“…Yi-Jen Lee [28; 29] has worked out a general bifurcation argument for what she calls Floer-type theories, including all of the necessary analysis. Michael Usher [50] has a nice summary of the invariance result this gives (which Lee conjectured in an earlier paper [27,Equation 3.2] but did not explicitly state as a Theorem in [28; 29]) and its algebraic aspects. We give a brief discussion of a simplified version sufficiently general for our needs.…”

“…This does not change the chain homotopy type of the chain complex (we need not worry about the sign as we are working with Z=2-coefficients, but the case with Z-coefficients is fine as well); see eg Usher [50,Section 3.3].…”

Symplectic Fibrations and the Abelian Vortex Equations

“…We note finally that the modules HF * (Y, π, r; w) fit into a field theory for Lefschetz fibrations over surfaces with boundary, which has been studied by M. Usher [16] and the author [12] (the latter extends the framework to a larger class of singular fibrations). This too is thought to be intimately related to Seiberg-Witten theory.…”

“…I am grateful to my Ph.D. supervisor, Simon Donaldson, for his ideas and advice. Thanks also to Michael Thaddeus for pointing out the Duistermaat-Heckman method, and to Michael Usher for telling me about his related work [16]. I acknowledge support from EPSRC Research Grant EP/C535995/1.…”

“…For discussion of the dependence on w we refer to [16]. Now, we can of course represent W by one of the forms v(Ω, τ, L) supplied by the theorem.…”