I'm reading Chapter 10.4 on the 't Hooft-Polyakov monopoles in Ryder's Quantum Field Theory.
On page 412 he explains why magnetic monopoles cannot appear in the Weinberg-Salam model.
I'm I right by saying that he shows that the electromagnetic gauge group U(1)em is not compactly embedded into the U(1)×U(1) subgroup of SU(2)×U(1)?
He then immediately concludes that the first fundamental group of the unbroken symmetry, which is H=U(1)em, π1(H) must be trivial or doesn't exists. Could someone refer me why?
Comment: I know that in the SU(2)×U(1) ones must consider the second homotopy group from S2 to the orbit G/H=SU(2)×U(1)/U(1), where H is the isotropy group of a vacuum state, after symmetry breaking. But the second homotopy group of a quotient can be related through a exact series to the kernel of the map from π1(H) into π1(G).
What I do not understand is by which theorem for H having a non-compact covering group π1(H) must be trivial or non-existing (???)?
This post imported from StackExchange Physics at 2014-05-04 11:10 (UCT), posted by SE-user Anne O'Nyme