Symplectic Singularities, Color Confinement, and the Quantum Dirac Sheaf

Abstract

A singularity $\mathbb{C}^{2r}/G$, with $G$ a split symplectic reflection group, may or may not be crepant. Then the total space $\mathscr{X}$ of the Donagi-Witten integrable system is crepant for some 4d $\mathcal{N}=2$ 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 $\mathcal{N}=2$ SCFT carries a Frobenius algebra $\mathcal{R}$, the quantum cohomology ring of $\mathscr{X}$ (defined via mirror symmetry), and $\mathscr{X}$ is crepant iff its central Witten index $\dim\mathcal{R}$ is equal to its Euler number $\chi(\mathscr{X})$. When the SCFT has a Lagrangian, $\mathcal{R}$ 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 $\mathcal{R}$: a relevant quantum datum is the Dirac sheaf $\mathscr{L}$ which refines Dirac charge quantization. We get several other results of independent interest, and we fully classify all special geometries of $\bigstar$-type in rank $r>6$.