<p>We uncover an infinite class of novel zero-form non-invertible symmetries in a broad family of four-dimensional models, studied years ago by Gaillard and Zumino (GZ), which includes several extended supergravities as particular subcases. The GZ models consist of abelian gauge fields coupled to a neutral sector, typically including a set of scalars, whose equations of motion are classically invariant under a continuous group <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">G</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{G} \)</EquationSource> </InlineEquation> acting on the electric and magnetic field strengths via symplectic transformations. The standard lore holds that, at the quantum level, these symmetries are broken to an integral subgroup <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <msub> <mi mathvariant="script">G</mi> <mi>ℤ</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">\( {\mathcal{G}}_{\mathbb{Z}} \)</EquationSource> </InlineEquation>. We show that, in fact, a much larger subgroup <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <msub> <mi mathvariant="script">G</mi> <mi>ℚ</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">\( {\mathcal{G}}_{\mathbb{Q}} \)</EquationSource> </InlineEquation> survives, albeit through non-invertible topological defects. We explicitly construct these defects and compute some of their fusion rules. As illustrative examples, we consider the axion-dilaton-Maxwell model and the bosonic sector of a class of <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 2 supergravities of the kind that appear in type II Calabi-Yau compactifications. Finally, we comment on how (part of) these non-invertible zero-form symmetries can be broken by gauging the <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math display="inline"> <msub> <mi mathvariant="script">G</mi> <mi>ℤ</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">\( {\mathcal{G}}_{\mathbb{Z}} \)</EquationSource> </InlineEquation> subgroup of invertible symmetries.</p>

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Gaillard-Zumino non-invertible symmetries

  • Fabio Apruzzi,
  • Luca Martucci

摘要

We uncover an infinite class of novel zero-form non-invertible symmetries in a broad family of four-dimensional models, studied years ago by Gaillard and Zumino (GZ), which includes several extended supergravities as particular subcases. The GZ models consist of abelian gauge fields coupled to a neutral sector, typically including a set of scalars, whose equations of motion are classically invariant under a continuous group G \( \mathcal{G} \) acting on the electric and magnetic field strengths via symplectic transformations. The standard lore holds that, at the quantum level, these symmetries are broken to an integral subgroup G \( {\mathcal{G}}_{\mathbb{Z}} \) . We show that, in fact, a much larger subgroup G \( {\mathcal{G}}_{\mathbb{Q}} \) survives, albeit through non-invertible topological defects. We explicitly construct these defects and compute some of their fusion rules. As illustrative examples, we consider the axion-dilaton-Maxwell model and the bosonic sector of a class of N \( \mathcal{N} \) = 2 supergravities of the kind that appear in type II Calabi-Yau compactifications. Finally, we comment on how (part of) these non-invertible zero-form symmetries can be broken by gauging the G \( {\mathcal{G}}_{\mathbb{Z}} \) subgroup of invertible symmetries.