400 research outputs found
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
Indefinite Morse 2-functions; broken fibrations and generalizations
A Morse 2-function is a generic smooth map from a smooth manifold to a
surface. In the absence of definite folds (in which case we say that the Morse
2-function is indefinite), these are natural generalizations of broken
(Lefschetz) fibrations. We prove existence and uniqueness results for
indefinite Morse 2-functions mapping to arbitrary compact, oriented surfaces.
"Uniqueness" means there is a set of moves which are sufficient to go between
two homotopic indefinite Morse 2-functions while remaining indefinite
throughout. We extend the existence and uniqueness results to indefinite, Morse
2-functions with connected fibers.Comment: 74 pages, 41 figures; further errors corrected, some exposition
added, other exposition improved, following referee's comment
Four-dimensional symplectic cobordisms containing three-handles
We construct four-dimensional symplectic cobordisms between contact
three-manifolds generalizing an example of Eliashberg. One key feature is that
any handlebody decomposition of one of these cobordisms must involve
three-handles. The other key feature is that these cobordisms contain chains of
symplectically embedded two-spheres of square zero. This, together with
standard gauge theory, is used to show that any contact three-manifold of
non-zero torsion (in the sense of Giroux) cannot be strongly symplectically
fillable. John Etnyre pointed out to the author that the same argument together
with compactness results for pseudo-holomorphic curves implies that any contact
three-manifold of non-zero torsion satisfies the Weinstein conjecture. We also
get examples of weakly symplectically fillable contact three-manifolds which
are (strongly) symplectically cobordant to overtwisted contact three-manifolds,
shedding new light on the structure of the set of contact three-manifolds
equipped with the strong symplectic cobordism partial order.Comment: This is the version published by Geometry & Topology on 28 October
200
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4-Manifolds With Inequivalent Symplectic Forms and 3-Manifolds With Inequivalent Fibrations
We exhibit a closed, simply connected 4-manifold carrying two
symplectic structures whose first Chern classes in lie in disjoint orbits of the diffeomorphism group of . Consequently, the moduli space of symplectic forms on is disconnected. The example is in turn based on a 3-manifold . The symplectic structures on come from a pair of fibrations whose Euler classes lie in disjoint orbits for the action of on .Mathematic
Small coupling limit and multiple solutions to the Dirichlet Problem for Yang Mills connections in 4 dimensions - Part I
In this paper (Part I) and its sequels (Part II and Part III), we analyze the
structure of the space of solutions to the epsilon-Dirichlet problem for the
Yang-Mills equations on the 4-dimensional disk, for small values of the
coupling constant epsilon. These are in one-to-one correspondence with
solutions to the Dirichlet problem for the Yang Mills equations, for small
boundary data. We prove the existence of multiple solutions, and, in
particular, non minimal ones, and establish a Morse Theory for this non-compact
variational problem. In part I, we describe the problem, state the main
theorems and do the first part of the proof. This consists in transforming the
problem into a finite dimensional problem, by seeking solutions that are
approximated by the connected sum of a minimal solution with an instanton, plus
a correction term due to the boundary. An auxiliary equation is introduced that
allows us to solve the problem orthogonally to the tangent space to the space
of approximate solutions. In Part II, the finite dimensional problem is solved
via the Ljusternik-Schirelman theory, and the existence proofs are completed.
In Part III, we prove that the space of gauge equivalence classes of Sobolev
connections with prescribed boundary value is a smooth manifold, as well as
some technical lemmas used in Part I. The methods employed still work when the
4-dimensional disk is replaced by a more general compact manifold with
boundary, and SU(2) is replaced by any compact Lie group
Saddle point solutions in Yang-Mills-dilaton theory
The coupling of a dilaton to the -Yang-Mills field leads to
interesting non-perturbative static spherically symmetric solutions which are
studied by mixed analitical and numerical methods. In the abelian sector of the
theory there are finite-energy magnetic and electric monopole solutions which
saturate the Bogomol'nyi bound. In the nonabelian sector there exist a
countable family of globally regular solutions which are purely magnetic but
have zero Yang-Mills magnetic charge. Their discrete spectrum of energies is
bounded from above by the energy of the abelian magnetic monopole with unit
magnetic charge. The stability analysis demonstrates that the solutions are
saddle points of the energy functional with increasing number of unstable
modes. The existence and instability of these solutions are "explained" by the
Morse-theory argument recently proposed by Sudarsky and Wald.Comment: 11 page
Monopole--Antimonopole Chains
We present new static axially symmetric solutions of SU(2) Yang-Mills-Higgs
theory, representing chains of monopoles and antimonopoles in static
equilibrium. They correspond to saddlepoints of the energy functional and exist
both in the topologically trivial sector and in the sector with topological
charge one.Comment: 9 pages, 2 figure
Enumerative geometry of Calabi-Yau 4-folds
Gromov-Witten theory is used to define an enumerative geometry of curves in
Calabi-Yau 4-folds. The main technique is to find exact solutions to moving
multiple cover integrals. The resulting invariants are analogous to the BPS
counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold
invariants to be integers and expect a sheaf theoretic explanation.
Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including
the sextic Calabi-Yau in CP5, are also studied. A complete solution of the
Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic
anomaly equation.Comment: 44 page
On field theory quantization around instantons
With the perspective of looking for experimentally detectable physical
applications of the so-called topological embedding, a procedure recently
proposed by the author for quantizing a field theory around a non-discrete
space of classical minima (instantons, for example), the physical implications
are discussed in a ``theoretical'' framework, the ideas are collected in a
simple logical scheme and the topological version of the Ginzburg-Landau theory
of superconductivity is solved in the intermediate situation between type I and
type II superconductors.Comment: 27 pages, 5 figures, LaTe
Notes on bordered Floer homology
This is a survey of bordered Heegaard Floer homology, an extension of the
Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is
placed on how bordered Heegaard Floer homology can be used for computations.Comment: 73 pages, 29 figures. Based on lectures at the Contact and Symplectic
Topology Summer School in Budapest, July 2012. v2: Fixed many small typo
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