Gauge connections and Lie algebras? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T00:10:36Z http://mathoverflow.net/feeds/question/12248 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12248/gauge-connections-and-lie-algebras Gauge connections and Lie algebras? Matthew Dodelson 2010-01-18T22:55:20Z 2010-11-08T01:06:15Z <p>I'm probably missing something obvious, but I've been wondering what the motivation is for requiring the components <code>$A_\mu$</code> in a local trivialization of a gauge connection on a smooth principal <code>$G$</code>-bundle to lie in <code>$\mathfrak{g}$</code>, the Lie algebra of <code>$G$</code>. I can see that this gives a couple of nice properties; for example, in a local trivialization it ensures that under a gauge transformation <code>$A'_\mu=gA_\mu g^{-1}+g\partial_\mu g$</code> lies in <code>$\mathfrak{g}$</code>, and that the curvature form <code>$F=dA+A\wedge A$</code> lies in <code>$\mathfrak{g}$</code> (since <code>$\mathfrak{g}$</code> is closed under the Lie bracket). But is there a more intrinsic or geometric reason that <code>$A_\mu$</code> must be in <code>$\mathfrak{g}$</code>? Thanks.</p> http://mathoverflow.net/questions/12248/gauge-connections-and-lie-algebras/12255#12255 Answer by Steve Huntsman for Gauge connections and Lie algebras? Steve Huntsman 2010-01-18T23:20:59Z 2010-11-08T01:06:15Z <p>I'm not sure about the mathematical origins, but the original physical motivation was Yang and Mills's attempt to deal with the approximate SU(2)-symmetry of nucleons (protons and neutrons). The big step was (as I understand it) when Gell-Mann (and Ne'eman, independently at about the same time) realized that a diagram labeling experimentally observed particles was the weight diagram for SU(3). He made some <a href="http://backreaction.blogspot.com/2006_09_01_archive.html" rel="nofollow">predictions</a> at a conference:</p> <blockquote> <p>following the presentation on Strong interactions of strange particles by G. A. Snow, both Ne'eman and Gell-Mann raised their hands to ask for permission to speak. The chairman called Gell-Mann, who was the more eminent physicist of both, and Gell-Mann announced that "[...] we should look for the last particle called, say, Ω-, with S=-3, I=0. [Here, I is isospin.] At 1685 MeV it would be metastable and should decay by weak interaction [...]"</p> </blockquote> <p>and the rest was the <a href="http://en.wikipedia.org/wiki/Eightfold_Way_%28physics%29" rel="nofollow">eightfold way</a>.</p> <p>Of course, principal $G$-bundles and the connections on them had been around for quite some time before (Simons famously pointed this fact out to Yang later on).</p> http://mathoverflow.net/questions/12248/gauge-connections-and-lie-algebras/12257#12257 Answer by Deane Yang for Gauge connections and Lie algebras? Deane Yang 2010-01-18T23:31:55Z 2010-01-18T23:31:55Z <p>First, I never liked working with principal bundles; vector bundles seem easier and more natural to me. Second, I never like thinking about abstract principal $G$-bundles. I prefer fixing a representation of $G$ and viewing the principal $G$ bundle as a reduced frame bundle associated with a vector bundle.</p> <p>So let $E$ be a rank $k$ vector bundle and $F$ the bundle of arbitrary frames in $E$ (this is a principal $GL(k)$-bundle). Then $GL(k)$ acts on the right on $F$. Given a subgroup $G$ in $GL(k)$, let $F_G$ be a subbundle of $F$ such that if $f \in F_G$, then so is $f\cdot g$ for each $g \in G$.</p> <p>The primary example is $E = T_*M$ and $F_G$ is the bundle of orthonormal bases of the tangent space with respect to a Riemannian metric.</p> <p>What is the critical property we want a $G$-connection to satisfy? Well, any connection allows you to parallel translate an arbitrary frame $f \in F$ along a curve. We'd like the $G$-connection to be such that if $f \in F_G$, then the parallel translation remains in $F_G$. This leads to the right definition of a $G$-connection.</p> http://mathoverflow.net/questions/12248/gauge-connections-and-lie-algebras/12259#12259 Answer by José Figueroa-O'Farrill for Gauge connections and Lie algebras? José Figueroa-O'Farrill 2010-01-18T23:37:13Z 2010-03-18T15:13:30Z <p>As Mariano points out in his comment, this follows from the definition of a connection on a principle $G$-bundle $\pi: P \to M$.</p> <p>At every $p \in P$, the kernel of $\pi_* : T_pP \to T_{\pi(p)}M$ defines the <em>vertical subspace</em> of $T_pP$. Let's call it $V_p$. It is spanned by the fundamental vector fields of the $G$-action on $P$. Since this action is free, the fibres are principal homogeneous spaces and hence $V_p$ is isomorphic to the Lie algebra $\mathfrak{g}$. A connection (à la Ehresmann) is an equivariant choice of <em>horizontal subspace</em> $H_p$ complementary to $V_p$. Hence it can be defined as the kernel of a 1-form $\theta$ with values in the adjoint representation of $G$ (from equivariance of the horizontal subspace).</p> <p>The gauge field in your question is then the pullback via a local section of that connection 1-form. Hence locally it is a 1-form on $M$ with values in the Lie algebra $\mathfrak{g}$.</p> <p>So the reason the gauge field is $\mathfrak{g}$-valued is the equivariance of the of the connection (in the sense of Ehresmann).</p> <p>If you then ask why one imposes equivariance, one answer is that it is the natural condition in this context, but perhaps someone else has a more convincing reason.</p> http://mathoverflow.net/questions/12248/gauge-connections-and-lie-algebras/12857#12857 Answer by Orbicular for Gauge connections and Lie algebras? Orbicular 2010-01-24T18:45:19Z 2010-01-24T18:45:19Z <p>I always find it helpful to think about Cartan geometries first - they are less "abstract" than principal bundles and shed new light on things like Riemannian geometry.<br /> For a nice introduction see<br /> <a href="http://www.emis.de/journals/SIGMA/2009/080/sigma09-080.pdf" rel="nofollow">http://www.emis.de/journals/SIGMA/2009/080/sigma09-080.pdf</a><br /> (look for the hamster on page 4!) or the following nice book<br /> <a href="http://www.amazon.com/Differential-Geometry-Generalization-Erlangen-Mathematics/dp/0387947329" rel="nofollow">http://www.amazon.com/Differential-Geometry-Generalization-Erlangen-Mathematics/dp/0387947329</a></p>