Detecting Standard Model Gauge Group from Generalized Fractional Quantum Hall Effect

The Standard Model of particle physics stands as one of the most profound and successful frameworks for describing the fundamental workings of nature. The global form of the Standard Model gauge group, however, remains an open question: it can be $\left(SU(3)_C\times SU(2)_W\times U(1)_Y\right)/\Gamma$ with $\Gamma=1,\mathbb{Z}_2,\mathbb{Z}_3$ or $\mathbb{Z}_6$. The work introduces the fractional topological transport coefficient $\xi$ involving the $U(1)$ B-L symmetry and the $U(1)$ one-form magnetic symmetry of the renormalizable Standard Model, and show that it distinguishes the global form of the Standard Model gauge group. The gauge group is fully determined for specific values of $\xi$, which also depends on the choice of action of the B-L symmetry on the Standard Model known as a fractionalization class. This transport coefficient can be measured in a contact term for the two-point function of the B-L symmetry current with the magnetic one-form symmetry current of the Standard Model. This parallels topological transport in the Fractional Quantum Hall Effect, with quarks and the B-L symmetry playing the role of anyons and the $U(1)$ electromagnetic global symmetry respectively.