nLab
instanton sector

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

\infty-Chern-Weil theory

Contents

Definition

In general

Given a gauge field configuration modeled by a GG-principal connection, its instanton sector or charge sector is the equivalence class of the underlying principal bundle.

In SU(n)SU(n)-Yang-Mills theory

Notably for Yang-Mills theory on a 4-dimensional spacetime and with a gauge group the special unitary group G=SU(n)G = SU(n), GG-principal bundles PP are entirely classified by their second Chern class c 2(P)c_2(P) and hence the value c 2(P)H 4(X,)c_2(P) \in H^4(X, \mathbb{Z}) is the instanton sector. Given the GG-principal connection of the gauge field the image in de Rham cohomology of this class may be expressed by the integration of differential forms [ XF ,F ]H dR 4(X)[\int_{X} \langle F_\nabla , \F_\nabla \rangle] \in H_{dR}^4(X), where F F_{\nabla} is the curvature and ,\langle -,-\rangle the invariant polynomial which corresponds to c 2c_2 under the Chern-Weil homomorphism.

gauge field: models and components

physicsdifferential geometrydifferential cohomology
gauge fieldconnection on a bundlecocycle in differential cohomology
instanton/charge sectorprincipal bundlecocycle in underlying cohomology
gauge potentiallocal connection differential formlocal connection differential form
field strengthcurvatureunderlying cocycle in de Rham cohomology
gauge transformationequivalencecoboundary
minimal couplingcovariant derivativetwisted cohomology
BRST complexLie algebroid of moduli stackLie algebroid of moduli stack
extended Lagrangianuniversal Chern-Simons n-bundleuniversal characteristic map

Revised on June 10, 2013 15:20:36 by Urs Schreiber (89.204.137.122)