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Hori-mological projective duality. (English) Zbl 1427.14042
In the article under review, the authors establish two goals: 1) prove a version of homological projective duality (HPD) for Pfaffian varieties as proposed by A. Kuznetsov [Publ. Math., Inst. Hautes Étud. Sci. 105, 157–220 (2007; Zbl 1131.14017)] and 2) prove a duality between two quantum field theories called gauged linear sigma models (GLSM) as proposed by K. Hori [J. High Energy Phys. 2013, No. 10, Paper No. 121, 74 p. (2013; Zbl 1342.81635)].
For the first result the authors build on their previous work [N. Addington et al., Algebr. Geom. 2, No. 3, 332–364 (2015; Zbl 1322.14037)], [E. Segal and R. Thomas, J. Reine Angew. Math. 743, 245–259 (2018; Zbl 1454.14051)] and Kuznetzov’s own work on the homological projective duality of line Grassmannians (a special case of Pfaffian varieties). The version of HPD they prove is for a non-commutative crepant resolutions (as constructed by Š. Špenko and M. Van den Bergh [Invent. Math. 210, No. 1, 3–67 (2017; Zbl 1375.13007)]), the reason being that in general Pfaffian varieties are very singular, as opposed to the smooth case of Grassmannians treated by Kuznetzov.
The second result relies firstly on providing a mathematical, and algebro-geometric in nature, definition of B-branes for the gauged linear sigma model, which seems to be novel in the physics literature. Their definition of B-brane recovers the non-commutative crepant resolution of Pfaffian as an example of the general theory of Spenko and Van den Bergh. Finally, the duality of quantum field theories seems to be a consequence of HPD as proved in the first part.
The authors concentrated in the case of an odd-dimensional vector space and the stronger results are proved in this case, as the even-dimensional case doesn’t lend itself to a physical interpretation.
The article has an extensive introduction that explain the main construction used in the article and also include a sketch of the proof of the main duality result.

MSC:
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
16E35 Derived categories and associative algebras
16S38 Rings arising from noncommutative algebraic geometry
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References:
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