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In physics, the fields in (pre-)quantum field theory whose dynamics is described by the theory may in general depend on certain fixed structure which also has an interpretation of a field, but one that is fixed and not regarded as having dynamics in the given model (in theoretical physics). These are often called background fields.
In particuar a sigma-model quantum field theory describes the propagation of a brane on some target space $X$. Typically the dynamics is controled by a gauge field on $X$ under which the brane is charged. This is the background gauge field of the $\sigma$-model. The term “background” alludes to the fact that this background gauge field is fixed and only serves to influence the dynamics of the brane coupled to it. More generally, one could also study the quantum dynamics of the possible background fields themselves. This would be given by the corresponding gauge theory on $X$.
A formalization of the notion is discussed at field in Definition – Physical fields.
The electrically charged relativistic particle is described by a sigma-model with background gauge field the electromagnetic field on spacetime.
The string of string theory may be described by a sigma-model with background gauge field the Kalb-Ramond field.
Chern-Simons theory may be understood as the sigma-model of a membrane charged under a background field that is a Chern-Simons circle 3-bundle.
background field