1,643 research outputs found

### 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

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