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<p>We study weakly-coupled descriptions/channel decompositions of the 4d <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 2 theories of class <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">S</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{S} \)</EquationSource> </InlineEquation> of type <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="fraktur">su</mi> <mfenced close=")" open="("> <mi>N</mi> </mfenced> </math></EquationSource> <EquationSource Format="TEX">\( \mathfrak{su}(N) \)</EquationSource> </InlineEquation>, from the perspective of the 3d <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 4 mirror duals of their circle compactifications. This is a delicate problem when the channel decomposition produces pathological, or bad, 4d configurations that correspond to spheres with non-maximal punctures. The star-shaped quivers, describing the 3d mirrors associated with such bad 4d configurations, are bad 3d <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 4 theories. Leveraging recent results regarding 3d bad theories, we identify a new and interesting family of bad theories, which we coin <i>broken</i> theories, that naturally arise in this context. Using these broken theories, we develop a systematic and analytic method that determines the generically non-Lagrangian matter sectors and the weakly-coupled gauge groups in such channel decompositions. We understand these weakly-coupled descriptions as emerging dynamically via Higgs mechanisms triggered by operators acquiring vacuum expectation values.</p>

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Breaking bad theories of class \( \mathcal{S} \)

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月8日

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  • Riccardo Comi,
  • Sebastiano Garavaglia,
  • Simone Giacomelli,
  • Sara Pasquetti,
  • Palash Singh

摘要

We study weakly-coupled descriptions/channel decompositions of the 4d N \( \mathcal{N} \) = 2 theories of class S \( \mathcal{S} \) of type su N \( \mathfrak{su}(N) \) , from the perspective of the 3d N \( \mathcal{N} \) = 4 mirror duals of their circle compactifications. This is a delicate problem when the channel decomposition produces pathological, or bad, 4d configurations that correspond to spheres with non-maximal punctures. The star-shaped quivers, describing the 3d mirrors associated with such bad 4d configurations, are bad 3d N \( \mathcal{N} \) = 4 theories. Leveraging recent results regarding 3d bad theories, we identify a new and interesting family of bad theories, which we coin broken theories, that naturally arise in this context. Using these broken theories, we develop a systematic and analytic method that determines the generically non-Lagrangian matter sectors and the weakly-coupled gauge groups in such channel decompositions. We understand these weakly-coupled descriptions as emerging dynamically via Higgs mechanisms triggered by operators acquiring vacuum expectation values.