nLab flux

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Differential cohomology

differential cohomology

Contents

Definition

In gauge theory of higher abelian Yang-Mills theory type, where field configurations on some manifold $X$ are circle n-bundles with connection $\nabla$, the magnetic flux of the field configuration through a $(n+1)$-dimensional closed manifold $\Sigma \hookrightarrow X$ is

$\Phi_\sigma(\nabla) = \int_\Sigma F_\nabla \,,$

where $F_\nabla$ is the curvature $(n+1)$-form.

Properties

If $\Sigma$ is contractible in $X$, hence if there is a $\hat \Sigma \hookrightarrow X$ such whose boundary is $\partial \hat \Sigma = \Sigma$ then this is the (higher) magnetic charge enclosed by $\Sigma$.

Revised on January 11, 2013 00:28:31 by Urs Schreiber (82.113.98.146)