Categorical symmetry of the standard model from gravitational anomaly

In the Standard Model, some combination of the baryon <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="bold">B</a:mi></a:math> and lepton <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi mathvariant="bold">L</d:mi></d:math> number symmetry is free of mixed anomalies with strong and electroweak <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>s</g:mi><g:mi>u</g:mi><g:mo stretchy="false">(</g:mo><g:mn>3</g:mn><g:mo stretchy="false">)</g:mo><g:mo>×</g:mo><g:mi>s</g:mi><g:mi>u</g:mi><g:mo stretchy="false">(</g:mo><g:mn>2</g:mn><g:mo stretchy="false">)</g:mo><g:mo>×</g:mo><g:mi>u</g:mi><g:mo stretchy="false">(</g:mo><g:mn>1</g:mn><g:msub><g:mo stretchy="false">)</g:mo><g:mover accent="true"><g:mi>Y</g:mi><g:mo stretchy="false">˜</g:mo></g:mover></g:msub></g:math> 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 <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>n</q:mi><q:msub><q:mi>ν</q:mi><q:mi>R</q:mi></q:msub></q:msub></q:math> is not equal to the family number <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>N</s:mi><s:mi>f</s:mi></s:msub></s:math>. Thus, the invertible <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mi mathvariant="bold">B</u:mi><u:mo>−</u:mo><u:mi mathvariant="bold">L</u:mi></u:mrow></u:math> symmetry current conservation can be violated quantum mechanically by gravitational backgrounds such as gravitational instantons. Specifically, we show that a noninvertible categorical <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mrow><y:mi mathvariant="bold">B</y:mi><y:mo>−</y:mo><y:mi mathvariant="bold">L</y:mi></y:mrow></y:math> 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 <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi mathvariant="double-struck">Z</cb:mi></cb:math> and <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:msub><fb:mi mathvariant="double-struck">Z</fb:mi><fb:mn>16</fb:mn></fb:msub></fb:math>, 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 with a framing anomaly corresponding to a 2D rational conformal field theory with an appropriate rational chiral central charge, and the 4D <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msubsup><ib:mi mathvariant="double-struck">Z</ib:mi><ib:mn>4</ib:mn><ib:mrow><ib:mi>TF</ib:mi></ib:mrow></ib:msubsup></ib:math>-time-reversal symmetric topological superconductor with 3D boundary topological order. Published by the American Physical Society 2024