scholarly journals The configuration space of gauge theory on open manifolds of bounded geometry

1997 ◽  
Vol 39 (1) ◽  
pp. 269-286 ◽  
Author(s):  
Jürgen Eichhorn ◽  
Gerd Heber
Keyword(s):  
Gauge Theory ◽  
Configuration Space ◽  
Bounded Geometry ◽  
Open Manifolds

GAUGE THEORY ON OPEN MANIFOLDS OF BOUNDED GEOMETRY

1992 ◽  
Vol 07 (17) ◽  
pp. 3927-3977 ◽  
Author(s):  
JÜRGEN EICHHORN
Keyword(s):  
Gauge Theory ◽  
Main Idea ◽  
Spectral Condition ◽  
Intersection Form ◽  
Bounded Geometry ◽  
Noncompact Manifolds ◽  
Open Manifolds ◽  
Stratified Space ◽  
Compact Case ◽  
Underlying Manifold

On compact manifolds (Mn, g) and for r>n/2+1 the configuration space [Formula: see text] is a well-defined object. [Formula: see text] is an affine space with a Sobolev space as vector space, and [Formula: see text] a Hilbert Lie group which acts smoothly and properly on [Formula: see text]. [Formula: see text] is a stratified space with Hilbert manifolds as strata. The existence problem has been solved for many interesting cases by Cliff Taubes and the description of the moduli space of instantons has been given by Donaldson. On noncompact manifolds none of the approaches of the compact case is further valid. We present here an intrinsic, self-consistent approach for gauge theory on open manifolds of bounded geometry up to order n/2+2. The main idea is to endow the space CP of gauge potentials and the gauge group with an intrinsic Sobolev topology. Bounded geometry of the underlying manifold and the considered connections provides all the Sobolev theorems which are needed to prove the existence of instantons if G=SU(2). We prove the existence of instantons if (M4, g) satisfies a certain spectral condition and has a positive definite L2 intersection form.

scholarly journals QUANTUM MECHANICS AS A SPONTANEOUSLY BROKEN GAUGE THEORY ON A U(1) GERBE

2008 ◽  
Vol 05 (02) ◽  
pp. 233-252 ◽  
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theory ◽  
Gauge Symmetry ◽  
Configuration Space ◽  
Fibre Bundle ◽  
Higgs Mechanism ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
Classical Action ◽  
Yang Mills

Any quantum-mechanical system possesses a U(1) gerbe naturally defined on configuration space. Acting on Feynman's kernel exp(iS/ħ), this U(1) symmetry allows one to arbitrarily pick the origin for the classical action S, on a point-by-point basis on configuration space. This is equivalent to the statement that quantum mechanics is a U(1) gauge theory. Unlike Yang–Mills theories, however, the geometry of this gauge symmetry is not given by a fibre bundle, but rather by a gerbe. Since this gauge symmetry is spontaneously broken, an analogue of the Higgs mechanism must be present. We prove that a Heisenberg-like noncommutativity for the space coordinates is responsible for the breaking. This allows to interpret the noncommutativity of space coordinates as a Higgs mechanism on the quantum-mechanical U(1) gerbe.

scholarly journals The BV formalism: Theory and application to a matrix model

2019 ◽  
Vol 31 (10) ◽  
pp. 1950035
Author(s):  
Roberta A. Iseppi
Keyword(s):  
Gauge Theory ◽  
Gauge Symmetry ◽  
General Solution ◽  
Master Equation ◽  
Configuration Space ◽  
Matrix Model ◽  
Gauge Theories ◽  
Brst Cohomology ◽  
Cohomology Complex ◽  
Theory Formula

We review the BV formalism in the context of [Formula: see text]-dimensional gauge theories. For a gauge theory [Formula: see text] with an affine configuration space [Formula: see text], we describe an algorithm to construct a corresponding extended theory [Formula: see text], obtained by introducing ghost and anti-ghost fields, with [Formula: see text] a solution of the classical master equation in [Formula: see text]. This construction is the first step to define the (gauge-fixed) BRST cohomology complex associated to [Formula: see text], which encodes many interesting information on the initial gauge theory [Formula: see text]. The second part of this article is devoted to the application of this method to a matrix model endowed with a [Formula: see text]-gauge symmetry, explicitly determining the corresponding [Formula: see text] and the general solution [Formula: see text] of the classical master equation for the model.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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