<p>We explore (−1)-form symmetries within the framework of geometric engineering in M-theory. By constructing the Symmetry Topological Field Theory (SymTFT) for selected 5d <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_25790_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 1, 4d <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_25790_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 2 and 4d <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_25790_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 1 theories, we formalize the geometric origin of these symmetries and compute the mixed anomaly polynomials involving (−1)-form and higher- form symmetries. Our findings consistently reveal both discrete and continuous (−1)-form symmetries, aligning with established field theory results, while also uncovering new (−1)-form symmetry factors and structural insights. In particular, we study the SymTFT of 4d <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_25790_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 1 theories from M-theory on a class of spaces with <i>G</i><sub>2</sub> holonomy, and obtain properties such as modified instanton sums and 4-group structures observed in other 4d gauge theories. Additionally, we systematically construct symmetry operators for continuous abelian symmetries, refining existing proposals, and providing an M-theory origin for them.</p>

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(1)-form symmetries from M-theory and SymTFTs

  • Marwan Najjar,
  • Leonardo Santilli,
  • Yi-Nan Wang

摘要

We explore (−1)-form symmetries within the framework of geometric engineering in M-theory. By constructing the Symmetry Topological Field Theory (SymTFT) for selected 5d N \( \mathcal{N} \) = 1, 4d N \( \mathcal{N} \) = 2 and 4d N \( \mathcal{N} \) = 1 theories, we formalize the geometric origin of these symmetries and compute the mixed anomaly polynomials involving (−1)-form and higher- form symmetries. Our findings consistently reveal both discrete and continuous (−1)-form symmetries, aligning with established field theory results, while also uncovering new (−1)-form symmetry factors and structural insights. In particular, we study the SymTFT of 4d N \( \mathcal{N} \) = 1 theories from M-theory on a class of spaces with G2 holonomy, and obtain properties such as modified instanton sums and 4-group structures observed in other 4d gauge theories. Additionally, we systematically construct symmetry operators for continuous abelian symmetries, refining existing proposals, and providing an M-theory origin for them.