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The phrase large gauge transformation has two related uses, so the allowed gauge transformations and boundary conditions must be specified.

In the topological sense, a large gauge transformation lies outside the identity component of the allowed transformation group. For example, on a spatial circle of circumference $L$, the U(1) transformationshave winding numberThis integer cannot change during a homotopy through periodic functions. Hence $n\ne 0$ cannot be deformed to the identity, although $g_n(0)=g_n(L)=1$. Similarly, based SU(2) transformations on compactified three-space can have nonzero winding even while tending to the identity at infinity. ^[ref]

In discussions of asymptotic electromagnetism, transformations with nonvanishing limiting gauge parameters at infinity are also called large. After choosing the boundary phase space, their generators can carry nonzero boundary charges and yield a Ward identity relating scattering amplitudes. This usage concerns asymptotic action on physical data; a boundary value alone does not characterize the disconnected components of a general gauge-map group.

For an Abelian gauge transformation, the local electromagnetic field tensor is unchanged. Global winding or an asymptotic charge can nevertheless distinguish its action from a gauge transformation regarded as a pure redundancy.

Applying $ϵ=b_i(t)x^i$ to $A_{\mu }=0$ givesIts field strength vanishes, so it is a large gauge transformation of the background. To arise as the zero-momentum limit of a physical transverse perturbation, $b_i(t)$ must obey the zero-momentum vector equation of motion. In the variables used in the question this iswith a constant growing solution and a decaying solution proportional to ${\int }^{t}d{t}^{\prime }/a^3(t^{\prime })$. This condition makes the large gauge mode an adiabatic mode that can be continued to small nonzero momentum.

A theta vacuum is a coherent state $|\theta ⟩={\sum }_n{e}^{in\theta }|n⟩$ formed from gauge vacua of different winding numbers. Large gauge transformations shift $n$ and act on this state by a phase. The Yang-Mills theta term gives the corresponding weighting of spacetime topological sectors.