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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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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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