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I've a question about the definition of 'large' gauge transformations. There are two competing definitions:

  1. small gauge transformations are equal to the identity at spatial infinity while large gauge transformations are not

  2. small gauge transformations are continuously connected to the identity while large gauge transformations are not

It seems to me that these definitions are not equivalent. For example consider a SU(2) gauge transformation on 3-space: $$\Omega_n(\bf{x})=\exp(i\omega(|\bf x|) ~\sigma_i \hat x^i/2)=\cos(\omega/2)+i\sin(\omega/2)\sigma_i \hat x^i$$ where $\omega(\cdot)$ is any monotonic function such that $\omega(0)=0$ and $\omega(\infty)=4\pi n$ (this is taken from (2.37) of http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html).

This gauge transformation is not continuously connected to the identity (in fact it has winding number $n$); but this gauge transformation is the identity at spatial infinity. So is $\Omega_n(x)$ a small or large gauge transformation?

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  • $\begingroup$ Any more information about your second statement? Because in the first definition, for a gauge transformation to go smoothly to the identity at infinity it needs to be continuously connected to the identity as well. It can't suddenly make a discrete jump when considering spatial infinity. So there might be something missing where you found the second statement. $\endgroup$
    – Guliano
    Commented Jan 14, 2023 at 15:57
  • $\begingroup$ @JulianDeV. I found the second statement on p23 of classe.cornell.edu/~pt267/files/documents/A_instanton.pdf $\endgroup$
    – dennis
    Commented Jan 14, 2023 at 16:03
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    $\begingroup$ There are at least two different things that are sloppily called "large gauge transformations", see this answer of mine $\endgroup$
    – ACuriousMind ♦
    Commented Jan 15, 2023 at 11:31
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    $\begingroup$ As far as I'm aware the two concepts of Large Gauge Transformation you present are different and it's just unfortunate that they get the same name. The first refers to asymptotic symmetries, which are local transformations that turn out to be physical because of their behavior at infinity, instead of mere redundancies. For more details on why these exist see my answer here physics.stackexchange.com/questions/719053/…. $\endgroup$
    – Gold
    Commented Jan 15, 2023 at 14:09
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    $\begingroup$ Well there are various such transformations in various theories and what the charge means certainly depends on the specific case. In electrodynamics, the conservation of the large gauge charge in the scattering problem means "charge conservation at every angle" and likewise in gravity the conservation of supertranslation charge means "energy conservation at every angle". In the quantum theory conservation of the charges implies in soft theorems, an extremely important result for holography in asymptotically flat spacetimes. $\endgroup$
    – Gold
    Commented Jan 15, 2023 at 14:34

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