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March 21, 2021

Cosmic Strings in the Standard Model

Over at the n-Category Café, John Baez is making a big deal of the fact that the global form of the Standard Model gauge group is G=(SU(3)×SU(2)×U(1))/N G = (SU(3)\times SU(2)\times U(1))/N where NN is the ℤ 6\mathbb{Z}_6 subgroup of the center of G′=SU(3)×SU(2)×U(1)G'=SU(3)\times SU(2)\times U(1) generated by the element (e 2πi/3𝟙,−𝟙,e 2πi/6)\left(e^{2\pi i/3}\mathbb{1},-\mathbb{1},e^{2\pi i/6}\right).

The global form of the gauge group has various interesting topological effects. For instance, the fact that the center of the gauge group is Z(G)=U(1)Z(G)= U(1), rather than Z(G′)=U(1)×ℤ 6Z(G')=U(1)\times \mathbb{Z}_6, determines the global 1-form symmetry of the theory. It also determines the presence or absence of various topological defects (in particular, cosmic strings). I pointed this out, but a proper explanation deserved a post of its own.

None of this is new. I’m pretty sure I spent a sunny afternoon in the summer of 1982 on the terrace of Café Pamplona doing this calculation. (As any incoming graduate student should do, I spent many a sunny afternoon at a café doing this and similar calculations.)

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