gaugeequivalent

Gauge-equivalent is a term used in differential geometry and gauge theory to describe a relationship between connections that differ only by a gauge transformation. Let P be a principal G-bundle over a manifold M, and let A be a connection on P. A smooth map g: M -> G, called a gauge transformation, acts on A by A^g = g A g^{-1} + g d g^{-1} in a local trivialization. Two connections A and A^g are gauge equivalent if such a g exists. This relation partitions the space of connections into gauge orbits, and the quotient space of connections modulo gauge transformations is the gauge moduli space.

The curvature F of a connection, which measures its field strength, transforms covariantly under gauge transformations: F^g = g F g^{-1}. Consequently, gauge-equivalent connections have physically indistinguishable curvature up to the adjoint action, and many gauge-invariant quantities—such as the trace of powers of F, Wilson loops, and characteristic classes—are preserved.

In electromagnetism, a simple abelian case with G = U(1), the gauge transformation A -> A + dχ leads to an equivalent potential A^χ, while the observable field F = dA remains unchanged (F^χ = F). In nonabelian gauge theories, such as Yang–Mills, the same principle holds with the more general transformation law, making gauge equivalence a central organizing principle.

Applications include the study of moduli spaces of connections, instantons and monopoles in mathematical physics, and the formulation of physical theories where redundant degrees of freedom are factored out to reveal true physical configurations.