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I can't decipher the difference between large and small gauge transformations especially in its applications in physics.If perhaps one can engineer a simple physical theory that has such a transformation to describe its subtleties it would be great.

As usual I love really simple explanations as I am not any sort of expert. Diverse responses welcome. . . .

I don't know if I pose the question correctly, but please feel free to edit it.

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up vote 6 down vote accepted

The simplest example I can think of is to take as manifold the circle $M=S^1$, and as bundle the trivial circle bundle $X=S^1 \times S^1$, with bundle map $(s,t) \in X \mapsto s \in M$. A section $\sigma$ is written in this notation as $\sigma(s)=(s,f(s))$ for some continuous map $f \colon S^1 \to S^1$. Any continuous $f$ determines a section. The simplest example of a section $\sigma$ is to set $f$ constant. An example of a large gauge transformation is to replace our constant $f$ with the section $\tau$ given by $\tau(s)=(s,s)$. The replacement of $\sigma$ by $\tau$ (or of $\tau$ by $\sigma$) is large because $\tau$ wraps once around the circle while $\sigma$ wraps zero times around the circle, topological invariants of sections which ensure that replacement of one by another is large.

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If a configuration space is topologically non-trivial, one can distinguish between "small" gauge transformations which can be smoothly deformed to the identity, and "large" gauge transformations which cannot be smoothly deformed to the identity because they "wind" around the "handles" of the configuration space.

An important example of topologically non-trivial configuration space is a non-Abelian gauge theory. In the Abelian case the transverse gauge condition $\partial \cdot A=0$ is sufficient to remove the degeneracy. While in non-Abelian case, as Gribov showed, there are distinct transverse configurations $A\neq A^\prime$ such that $\partial \cdot A=\partial \cdot A^\prime=0$ and these configurations are connected with each other by "large" gauge transformations. This topological non-triviality of the configuration space has an important impact on QCD dynamics: (The Gribov problem and QCD dynamics, by N. Vandersickel and D. Zwanziger).

Chern-Simons theory provides another example where "large" gauge transformations have an important role: (Aspects of Chern-Simons Theory, by G. V. Dunne).

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