Abstract
We consider a set of toric Calabi–Yau varieties which arise as deformations of the small resolutions of type A surface singularities. By careful analysis of the heuristics of B-brane transport in the associated gauged linear sigma models, we predict the existence of a mixed braid group action on the derived category of each variety, and then prove that this action does indeed exist. This generalizes the braid group action found by Seidel and Thomas for the undeformed resolutions. We also show that the actions for different deformations are related, in a way that is predicted by the physical heuristics.
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Adolphson A.: Hypergeometric functions and rings generated by monomials. Duke Math. Journal 73(2), 269–290 (1994)
Anno, R.: Spherical functors. arXiv:0711.4409
Anno, R., Logvinenko, T.: Spherical DG-functors. arXiv:1309.5035
Ballard, M., Favero, D., Katzarkov, L.: Variation of geometric invariant theory quotients and derived categories. arXiv:1203.6643
Bartocci, C., Bruzzo, U., Hernández Ruipérez, D.: Fourier–Mukai and Nahm transforms in geometry and mathematical physics. Progress in Mathematics, no. 276, Birkhäuser (2009)
Bezrukavnikov, R., Riche, S.: Affine braid group actions on derived categories of Springer resolutions. Annales scientifiques de l’École normale supérieure, 45, fascicule 4, pp. 535–599 (2012). arXiv:1101.3702
Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties. arXiv:alg-geom/9506012
Cautis, S., Kamnitzer, J.: Braiding via geometric Lie algebra actions. Compos. Math. 148(2), 464–506 (2012). arXiv:1001.0619
Chan, K., Pomerleano, D., Ueda, K.: Lagrangian torus fibrations and homological mirror symmetry for the conifold. arXiv:1305.0968
Coates, T., Corti, A., Iritani, H., Tseng, H.-H.: Computing genus-zero twisted Gromov-Witten invariants. Duke Math. J. 147(3), 377–438 (2009). arXiv:math/0702234
Donovan, W., Segal, E.: Window shifts, flop equivalences and Grassmannian twists. Compos. Math. 150(6), 942–978 (2014). arXiv:1206.0219
Fulton W.: Introduction to toric varieties. Annals of Mathematics Studies no. 131. Princeton University Press, Princeton (1993)
Herbst, M., Hori, K., Page, David, P.: Phases of \({\mathcal{N}=2}\) theories in 1 + 1 dimensions with boundary. arXiv:0803.2045
Halpern-Leistner, D.: The derived category of a GIT quotient. arXiv:1203.0276
Halpern-Leistner, D., Shipman, I.: Autoequivalences of derived categories via geometric invariant theory. arXiv:1303.5531
Hori, K.: Duality in two-dimensional (2,2) supersymmetric non-abelian gauge theories. J. High Energy Phys. 1310, 121 (2013). arXiv:1104.2853
Hori, K., Vafa, C.: Mirror symmetry. arXiv:hep-th/0002222
Huybrechts D.: Fourier–Mukai transforms in algebraic geometry. Oxford University Press, Oxford (2007)
Huybrechts, D., Thomas, R.P.: \({\mathbb{P}}\) -objects and autoequivalences of derived categories. Math. Res. Lett. 13, 87–98 (2006). arXiv:math/0507040
Kawamata, Y.: D-equivalence and K-equivalence. J. Diff. Geom. 61(1), 147–171 (2002). arXiv:math/0205287
Lipman, J.: Notes on derived functors and Grothendieck duality. In: Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Math. no. 1960, pp. 1–259.Springer, New York, (2009). http://www.math.purdue.edu/~lipman/Duality.pdf
Reid, M.: Young person’s guide to canonical singularities. Algebraic geometry, Bowdoin (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, 345–414 (1987) (MR0927963)
Segal, E.: Equivalences between GIT quotients of Landau–Ginzburg B-models. Commun. Math. Phys. 304(2), 411–432 (2011). arXiv:0910.5534
Seidel, P., Thomas, R.P.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108, 37–108 (2001). arXiv:math/0001043
Slodowy, P.: Simple singularities and simple algebraic groups. Lecture Notes in Math. no. 815, Springer, New York (1980)
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Communicated by H. Ooguri
W.D. is grateful for the support of EPSRC grant EP/G007632/1.
E.S. is grateful for the support of an Imperial College Junior Research Fellowship.
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Donovan, W., Segal, E. Mixed Braid Group Actions From Deformations of Surface Singularities. Commun. Math. Phys. 335, 497–543 (2015). https://doi.org/10.1007/s00220-014-2226-3
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DOI: https://doi.org/10.1007/s00220-014-2226-3