P. Putrov

J. Wang

We demonstrate that the Standard Model's baryon minus lepton symmetry defect can become categorical by absorbing the gravitational anomaly.

In the Standard Model, the strong and electroweak forces are free of gauge anomalies and the difference between baryon number (B) and lepton number (L) is conserved. This B − L symmetry is, however, broken by gravity, a phenomenon hypothesised to give rise to leptogenesis in the early universe. We show that a new noninvertible symmetry survives under gravity and its defects encompass Chern–Simons gravity theory.

In the Standard Model, some combination of the baryon B and lepton L number symmetry is free of mixed anomalies with strong and electroweak su(3)×su(2)×u(1)Ỹ  gauge forces. However, it can still suffer from a mixed gravitational anomaly, hypothetically pertinent to leptogenesis in the very early universe. This happens when the total "sterile right-handed" neutrino number nνR is not equal to the family number Nf. Thus the invertible B−L symmetry current conservation can be violated quantum mechanically by gravitational backgrounds such as gravitational instantons. In specific, we show that a noninvertible categorical B−L generalized symmetry still survives in gravitational backgrounds. In general, we propose a construction of noninvertible symmetry charge operators as topological defects derived from invertible anomalous symmetries that suffer from mixed gravitational anomalies. Examples include the perturbative local and nonperturbative global anomalies classified by ℤ and ℤ16 respectively. For this construction, we utilize the anomaly inflow bulk-boundary correspondence, the 4d Pontryagin class and the gravitational Chern-Simons 3-form, the 3d Witten-Reshetikhin-Turaev-type topological quantum field theory corresponding to a 2d rational conformal field theory with an appropriate rational chiral central charge, and the 4d ℤTF4-time-reversal symmetric topological superconductor with 3d boundary topological order.

Categorical symmetry

Categorical symmetry of the standard model from gravitational anomaly