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<p>A singularity ℂ<sup>2<i>r</i></sup>/<i>G</i>, with <i>G</i> a split symplectic reflection group, may or may not be crepant. Then the total space 𝒳 of the Donagi-Witten integrable system is crepant for some 4d <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> <mo>=</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N}=2 \)</EquationSource> </InlineEquation> SCFT and non-crepant for others. Which physical mechanism controls the (dis)crepancy? Surprisingly, it is the detailed physics of color confinement (and its generalizations for non-Lagrangian QFT). A 4d <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> <mo>=</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N}=2 \)</EquationSource> </InlineEquation> SCFT carries a Frobenius algebra ℛ, the quantum cohomology ring of 𝒳 (defined via mirror symmetry), and 𝒳 is crepant iff its <i>central Witten index</i> dim ℛ is equal to its Euler number <i>χ</i>(𝒳). When the SCFT has a Lagrangian, ℛ is fixed by compatibility with confinement, and physics may require a discrepancy to be present. The quantum cohomology depends on quantum-geometric data, and a classical Seiberg-Witten geometry may have several inequivalent ℛ: a relevant quantum datum is the <i>Dirac sheaf</i> ℒ which refines Dirac charge quantization.</p><p>We get several other results of independent interest, and we fully classify all special geometries of ⋆-type in rank r &gt; 6.</p>

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Symplectic singularities, color confinement, and the quantum Dirac sheaf

  • 期刊论文

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  • 发表日期：

    2026年5月14日

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  • Sergio Cecotti

摘要

A singularity ℂ2r/G, with G a split symplectic reflection group, may or may not be crepant. Then the total space 𝒳 of the Donagi-Witten integrable system is crepant for some 4d N = 2 \( \mathcal{N}=2 \) SCFT and non-crepant for others. Which physical mechanism controls the (dis)crepancy? Surprisingly, it is the detailed physics of color confinement (and its generalizations for non-Lagrangian QFT). A 4d N = 2 \( \mathcal{N}=2 \) SCFT carries a Frobenius algebra ℛ, the quantum cohomology ring of 𝒳 (defined via mirror symmetry), and 𝒳 is crepant iff its central Witten index dim ℛ is equal to its Euler number χ(𝒳). When the SCFT has a Lagrangian, ℛ is fixed by compatibility with confinement, and physics may require a discrepancy to be present. The quantum cohomology depends on quantum-geometric data, and a classical Seiberg-Witten geometry may have several inequivalent ℛ: a relevant quantum datum is the Dirac sheaf ℒ which refines Dirac charge quantization.

We get several other results of independent interest, and we fully classify all special geometries of ⋆-type in rank r > 6.