Grafting Seiberg-Witten monopoles

We demonstrate that the operation of taking disjoint unions of J-holomorphic curves (and thus obtaining new J-holomorphic curves) has a Seiberg-Witten counterpart. The main theorem asserts that, given two solutions (A_i, psi_i), i=0,1 of the Seiberg-Witten equations for the Spin^c-structure W^+_{E_i}= E_i direct sum (E_i tensor K^{-1}) (with certain restrictions), there is a solution (A, psi) of the Seiberg-Witten equations for the Spin^c-structure W_E with E= E_0 tensor E_1, obtained by `grafting' the two solutions (A_i, psi_i).Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-7.abs.htm

Seiberg-Witten invariants and pseudo-holomorphic subvarieties for self-dual, harmonic 2-forms

A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 has a non-trivial, closed, self-dual 2-form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set, the symplectic form and the metric define an almost complex structure; and the latter can be used to define pseudo-holomorphic submanifolds and subvarieties. The main theorem in this paper asserts that if the 4-manifold has a non zero Seiberg-Witten invariant, then the zero set of any given self-dual harmonic 2-form is the boundary of a pseudo-holomorphic subvariety in its complement.Comment: 44 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper8.abs.htm

Critical-exponent Sobolev norms and the slice theorem for the quotient space of connections

The use of certain critical-exponent Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and similar non-linear elliptic partial differential equations. Indeed, the estimates one can obtain using these critical-exponent norms appear to be the best possible when one needs to bound the norm of a Green's operator for a Laplacian, depending on a connection varying in a non-compact family, in terms of minimal data such as the first positive eigenvalue of the Laplacian or the L^2 norm of the curvature of the connection. Following Taubes, we describe a collection of critical-exponent Sobolev norms and general Green's operator estimates depending only on first positive eigenvalues or the L^2 norm of the connection's curvature. Such estimates are particularly useful in the gluing construction of solutions to non-linear partial differential equations depending on a degenerating parameter, such as the approximate, reference solution in the anti-self-dual or PU(2) monopole equations. We apply them here to prove an optimal slice theorem for the quotient space of connections. The result is optimal in the sense that if a point [A] in the quotient space is known to be just L^2_1-close enough to a reference point [A_0], then the connection A can be placed in Coulomb gauge relative to the connection A_0, with all constants depending at most on the first positive eigenvalue of the covariant Laplacian defined by A_0 and the L^2 norm of the curvature of A_0. In this paper we shall for simplicity only consider connections over four-dimensional manifolds, but the methods and results can adapted to manifolds of arbitrary dimension to prove slice theorems which apply when the reference connection is allowed to degenerate.Comment: LaTeX 2e, 43 pages. Estimates in section 5.2 strengthene

Existence of a New Instanton in Constrained Yang-Mills-Higgs Theory

Our goal is to discover possible new 4-dimensional euclidean solutions (instantons) in fundamental SU(2) Yang-Mills-Higgs theory, with a constraint added to prevent collapse of the scale. We show that, most likely, there exists one particular new constrained instanton (\Istar) with vanishing Pontryagin index. This is based on a topological argument that involves the construction of a non-contractible loop of 4-dimensional configurations with a certain upperbound on the action, which we establish numerically. We expect \Istar to be the lowest action non-trivial solution in the vacuum sector of the theory. There also exists a related static, but unstable, solution, the new sphaleron \Sstar. Possible applications of \Istar to the electroweak interactions include the asymptotics of perturbation theory and the high-energy behaviour of the total cross-section.Comment: 32 pages, Latex, NIKHEF-H/93-02 (March 1993), postscript file including 10 figures available by anonymous ftp from nikhefh.nikhef.n

Seiberg-Witten-Floer Homology and Gluing Formulae

This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion \spinc structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.Comment: 63 pages, LaTe

Lagrangians for the Gopakumar-Vafa conjecture

This article explains how to construct immersed Lagrangian submanifolds in C^2 that are asymptotic at large distance from the origin to a given braid in the 3-sphere. The self-intersections of the Lagrangians are related to the crossings of the braid. These Lagrangians are then used to construct immersed Lagrangians in the vector bundle O(-1) oplus O(-1) over the Riemann sphere which are asymptotic at large distance from the zero section to braids.Comment: This is the version published by Geometry & Topology Monographs on 22 April 200