Flavor Symmetries and Automatic Enhancement in the 6d Supergravity Swampland

We argue for the quantum-gravitational inconsistency of certain 6D <math display="inline"><mrow><mi mathvariant="script">N</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math> supergravity theories, whose anomaly-free gauge algebra <math display="inline"><mrow><mi mathvariant="fraktur">g</mi></mrow></math> and hypermultiplet spectrum <math display="inline"><mi>M</mi></math> were observed by Raghuram et al. [J. High Energy Phys. 07 (2021) 048] to be realizable only as part of a larger gauge sector <math display="inline"><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="fraktur">g</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊃</mo><mi mathvariant="fraktur">g</mi><mo>,</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊃</mo><mi>M</mi><mo stretchy="false">)</mo></mrow></math> in F-theory. To detach any reference to a string theoretic method of construction, we utilize flavor symmetries to provide compelling reasons why the vast majority of such <math display="inline"><mrow><mo stretchy="false">(</mo><mi mathvariant="fraktur">g</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow></math> theories are not compatible with quantum gravity constraints, and how the “automatic enhancement” to <math display="inline"><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="fraktur">g</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>′</mo></mrow></msup><mo stretchy="false">)</mo></mrow></math> remedies this. In the first class of models, with <math display="inline"><mrow><msup><mrow><mi mathvariant="fraktur">g</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mi mathvariant="fraktur">g</mi><mo stretchy="false">⊕</mo><mi mathvariant="fraktur">h</mi></mrow></math>, we show that there exists an unbroken flavor symmetry <math display="inline"><mi mathvariant="fraktur">h</mi></math> acting on the matter <math display="inline"><mi>M</mi></math>, which, if ungauged, would violate the no-global-symmetries hypothesis. This argument also applies to 1-form center symmetries, which govern the gauge group topology and massive states in representations different from those of massless states. In a second class, we find that <math display="inline"><mi mathvariant="fraktur">g</mi></math> is incompatible with the flavor symmetry of certain supersymmetric strings that must exist by the completeness hypothesis.