nLab equivariant connection

Idea

An equivariant connection is a connection on a bundle $\nabla$ over a space $X$ with action by a group $H$ which is equipped with $H$-equivariant structure, hence equivalently – in the language of higher differential geometry of smooth groupoids – an extension of a connection $\nabla \;\colon\; X \longrightarrow \mathbf{B}G_{conn}$ to a connection $\nabla_{equ}$ on the action groupoid $X//H$:

$$\array{ X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}G_{conn} \\ \downarrow & \nearrow_{\mathrlap{\nabla_{equ}}} \\ X//H } \,.$$

Examples

• reduced phase space

Related concepts

• equivariant bundle

• equivariant cohomology