Although the page seems to contradict itself somewhat. Early on they mention the Pontryagin index as a topological invariant in Yang Mills theories. But later on in the section on "various numbers of dimensions" they say that for SU(N) groups it's the 2nd Chern class that serves as the topological invariant, while for SO(N) groups it's the Pontryagin index.
The only reconciliation I can think of for that is that, from what I understand, the SU(2) subgroup is really the only thing that matters for all of the SU(N) groups when it comes to instantons, at least the normal kind. But SU(2) is isomorphic to SO(3) so maybe you can use the Pontryagin index for that reason. But then in that case, I don't know why the 2nd Chern class would be interesting.
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timeasmymeasure
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Date: 2010-07-12 02:17 pm (UTC)Although the page seems to contradict itself somewhat. Early on they mention the Pontryagin index as a topological invariant in Yang Mills theories. But later on in the section on "various numbers of dimensions" they say that for SU(N) groups it's the 2nd Chern class that serves as the topological invariant, while for SO(N) groups it's the Pontryagin index.
The only reconciliation I can think of for that is that, from what I understand, the SU(2) subgroup is really the only thing that matters for all of the SU(N) groups when it comes to instantons, at least the normal kind. But SU(2) is isomorphic to SO(3) so maybe you can use the Pontryagin index for that reason. But then in that case, I don't know why the 2nd Chern class would be interesting.