nLab
flux

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

$${\Phi }_{\sigma }(\nabla )={\int }_{\Sigma }{F}_{\nabla },$$\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 $\Sigma$.

Related concepts

• conserved current, charge

• electric charge, magnetic charge

• flux

• flux compactification