We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings f:X→P(V) of Fano manifolds X. More precisely, the category of B-branes of such hybrid models corresponds to the HPD category of the embedding f. B-branes on these hybrid models can be seen as global matrix factorizations over some compact space B or, equivalently, as the derived category of the sheaf of A-modules on B, where A is an A_∞ algebra. This latter interpretation corresponds to a noncommutative resolution of B. We compute explicitly the algebra A by several methods, for some specific class of hybrid models, and find that in general it takes the form of a smash product of an A_∞ algebra with a cyclic group. Then we apply our results to the HPD of f corresponding to a Veronese embedding of projective space and the projective embedding of Fano complete intersections in P^n.
Sibasish BanerjeeWeyertal 86-90, Department of Mathematics, University of Cologne, 50679, Cologne, GermanyPietro LonghiInstitute for Theoretical Physics, ETH Zurich, 8093, Zurich, SwitzerlandMauricio Andrés Romo JorqueraYau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
Algebraic GeometryarXiv subject: High Energy Physics - Theory (hep-th)mathscidoc:2207.45001
We study BPS states of 5d N=1 SU(2) Yang-Mills theory on S^1×R^4. Geometric engineering relates these to enumerative invariants for the local Hirzebruch surface F_0. We illustrate computations of Vafa-Witten invariants via exponential networks, verifying fiber-base symmetry of the spectrum at certain points in moduli space, and matching with mirror descriptions based on quivers and exceptional collections. Albeit infinite, parts of the spectrum organize in families described by simple algebraic equations. Varying the radius of the M-theory circle interpolates smoothly with the spectrum of 4d N=2 Seiberg-Witten theory, recovering spectral networks in the limit.
Richard EagerKavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, Japan; Mathematical Institute, Heidelberg University, MATHEMATIKON, Im Neuenheim Feld 205, 69120 Heidelberg, GermanyKentaro HoriKavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, JapanJohanna KnappInstitute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8-10, 1040 Vienna, AustriaMauricio Andrés Romo JorqueraKavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, Japan; School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
Complex Variables and Complex AnalysisAlgebraic Geometrymathscidoc:2207.08002
Chinese Annals of Mathematics, Series B, 38, (4), 901–912, 2017.7
The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model. The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rødland. A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.