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Non-birational twisted derived equivalences in abelian GLSMs. (English) Zbl 1231.14035
The authors argue that A. Kuznetsov’s homological projective duality [Publ. Math., Inst. Hautes Étud. Sci. 105, 157–220 (2007; Zbl 1131.14017)] can be used to explain dualities between non-birational Kähler phases of gauged linear sigma models (GLSMs).
Physicists expected in the past that different geometric Kähler phases of the same GLSM are birational one to another. However, the geometry at the Landau-Ginzburg point of the GLSM for the complete intersection of four quadrics in $${\mathbb P}^7$$ is a branched double cover of $${\mathbb P}^3$$ with a singular octic branch locus. Such a double cover is not birational to the complete intersection of quadrics in $${\mathbb P}^7$$.
In the first half of this paper, this example is studied in detail. Physical arguments are provided which support the view of the authors that the double octic should be replaced by a non-commutative space (in the sense of Kontsevich), represented by the derived category of coherent sheaves which are modules over the sheaf of even parts of Clifford algebras over $${\mathbb P}^3$$.
The authors also analyse GLSMs for other complete intersections of quadrics and of higher degree hypersurfaces in projective spaces; some of these examples involve non-Calabi-Yau 3-folds. Based on these examples it is conjectured that Kuznetsov’s homological projective duality can be applied to GLSMs in general.

##### MSC:
 14J81 Relationships between surfaces, higher-dimensional varieties, and physics 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J33 Mirror symmetry (algebro-geometric aspects) 32L81 Applications of holomorphic fiber spaces to the sciences 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81R60 Noncommutative geometry in quantum theory
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