# Weak isospin and types of weak charge

My understanding is that QCD has three color charges that are conserved as a result of global SU(3) invariance. What about SU(2) weak? Does it have two types of charges? What I'm getting at is:

U(1) --> 1 type of charge

SU(2) --> ?

SU(3) --> 3 types of charge

Does SU(2) have two types? If not, what is the relation between SU(N) invariance and the number of charge types?

Idea: Maybe both I and I_3 (weak isospin and its third component) are conserved before electroweak symmetry breaking? Is that true? If so, then that would answer my question.

• Actually each of the symmetry generators commutes with the Hamiltonian and gives a conserved charge. For an SU(N) group the number of generators is $N^2 - 1$. The total number of generators in the SM is $1 + (2^2 - 1) + (3^2 -1) = 12$. – Michael Brown Mar 14 '13 at 13:41
• @Michael Brown, I am aware that local invariance of SU(N) leads to N^2-1 gauge fields, however I think this is a different question from what I am asking. SU(3) has 8 gauge fields (quantized becomes 8 gluons), but 3 colors. 3 conserved charges, due to global invariance of SU(3), not local. – user1247 Mar 14 '13 at 13:54
• Look up Noether's theorem again. There is a conserved current $j_\mu^A$ for every generator $T^A$ of the group, i.e. the currents live in the adjoint rep. Therefore there are $N^2 - 1$ conserved charges. The number of components in the fundamental rep is a different matter. For SU(N) that is $N$. – Michael Brown Mar 14 '13 at 14:45
• @Michael Brown, I thought that color charge was conserved, and that SU(3) implied three such colors. Am I mistaken that the "3" colors are related to the "3" in SU(3), and that their conservation is related to Noether's theorem? Any help would be appreciated. – user1247 Mar 14 '13 at 15:24
• The "3" is the same 3 as in the SU(3). There are indeed three colours. I didn't say gluons are conserved. But there are 8 conserved quantities in an SU(3) theory, which are the generators of the symmetry group. If it helps you can think of these formally as colour-anticolour pairs (look up 't Hooft double line formalism) since the charges live in the adjoint rep which is isomorphic to fundamental $\otimes$ antifundamental. There is always a conserved quantity associated to any continuous symmetry via Noether's theorem. # of symmetries = # of conserved quantities. – Michael Brown Mar 14 '13 at 16:39