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A few Ricci-flat stacks as phases of exotic GLSM’s. (English) Zbl 1311.81184

Summary: In this Letter we follow up the recent work of Halverson, Kumar and Morrison on some exotic examples of gauged linear sigma models (GLSM’s). Specifically, they describe a set of \(\mathrm U(1) \times \mathbb{Z}_2\) GLSM’s with superpotentials that are quadratic in \(p\) fields rather than linear as is typically the case. These theories RG flow to sigma models on branched double covers, where the double cover is realized via a \(\mathbb{Z}_2\) gerbe. For that gerbe structure, and hence the double cover, the \(\mathbb{Z}_2\) factor in the gauge group is essential. In this Letter we propose an analogous geometric understanding of phases without that \(\mathbb{Z}_2\), in terms of Ricci-flat (but not Calabi-Yau) stacks which look like Fano manifolds with hypersurfaces of \(\mathbb{Z}_2\) orbifolds.

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
81T10 Model quantum field theories
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
32Q15 Kähler manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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