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Categories of massless D-branes and del Pezzo surfaces. (English) Zbl 1342.83313

Summary: In analogy with the physical concept of a massless D-brane, we define a notion of “Q-masslessness” for objects in the derived category. This is defined in terms of monodromy around singularities in the stringy Kähler moduli space and is relatively easy to study using “spherical functors”. We consider several examples in which del Pezzo surfaces and other rational surfaces in Calabi-Yau threefolds are contracted. For precisely the del Pezzo surfaces that can be written as hypersurfaces in weighted \(\mathbb{P}^3\), the category of \(Q\)-massless objects is a “fractional Calabi-Yau” category of graded matrix factorizations.

MSC:

83E30 String and superstring theories in gravitational theory
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J81 Relationships between surfaces, higher-dimensional varieties, and physics
14J26 Rational and ruled surfaces
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References:

[1] M.R. Douglas, D-branes, categories and N = 1 supersymmetry, J. Math. Phys.42 (2001) 2818 [hep-th/0011017] [INSPIRE]. · Zbl 1036.81027
[2] P.S. Aspinwall and A.E. Lawrence, Derived categories and zero-brane stability, JHEP08 (2001) 004 [hep-th/0104147] [INSPIRE].
[3] P.S. Aspinwall, D-branes on Calabi-Yau manifolds, in Progress in String Theory. TASI 2003 lecture notes, J.M. Maldacena, World Scientific, Singapore (2005), hep-th/0403166 [INSPIRE].
[4] M.R. Douglas, B. Fiol and C. Römelsberger, Stability and BPS branes, JHEP09 (2005) 006 [hep-th/0002037] [INSPIRE].
[5] T. Bridgeland, Stability conditions on triangulated categories, Ann. Math.166 (2007) 317 [math.AG/0212237]. · Zbl 1137.18008
[6] A. Strominger, Massless black holes and conifolds in string theory, Nucl. Phys.B 451 (1995) 96 [hep-th/9504090] [INSPIRE]. · Zbl 0925.83071
[7] P.S. Aspinwall and M.R. Douglas, D-brane stability and monodromy, JHEP05 (2002) 031 [hep-th/0110071] [INSPIRE].
[8] E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys.B 403 (1993) 159 [hep-th/9301042] [INSPIRE]. · Zbl 0910.14020
[9] R.P. Horja, Derived category automorphisms from mirror symmetry, math.AG/0103231. · Zbl 1075.18006
[10] W. Lerche, P. Mayr and N. Warner, Noncritical strings, Del Pezzo singularities and Seiberg-Witten curves, Nucl. Phys.B 499 (1997) 125 [hep-th/9612085] [INSPIRE]. · Zbl 0934.81036
[11] R. Rouquier, Categorification of \(s{l_2}\) and braid groups, in Trends in representation theory of algebras and related topics, Contemporary Mathematics volume 406, American Mathematical Society, U.S.A. (2006). · Zbl 1162.20301
[12] R. Anno, Spherical functors, arXiv:0711.4409. · Zbl 1374.14015
[13] D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, arithmetic, and geometry: in honor of Yu.I. Manin. Volume II, Y. Tschinkel and Y. Zarhin, Progress in Mathematics volume 270, Birkäuser, Boston Inc., Boston, U.S.A. (2009), math.AG/0506347.
[14] V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom.3 (1994) 493 [alg-geom/9310003] [INSPIRE]. · Zbl 0829.14023
[15] P.S. Aspinwall and B.R. Greene, On the geometric interpretation of N = 2 superconformal theories, Nucl. Phys.B 437 (1995) 205 [hep-th/9409110] [INSPIRE]. · Zbl 1052.32502
[16] T. Oda and H.S. Park, Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions, Tôhoku Math. J.43 (1991) 375. · Zbl 0782.52006
[17] D.A. Cox, The homogeneous coordinate ring of a toric variety, revised version, J. Algebraic Geom.4 (1995) 17 [alg-geom/9210008] [INSPIRE]. · Zbl 0846.14032
[18] D. Auroux, L. Katzarkov and D. Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, math.AG/0404281. · Zbl 1175.14030
[19] L.A. Borisov, L. Chen and G.G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc.18 (2005) 193 [math/0309229]. · Zbl 1178.14057
[20] I.M. Gelfand, M.M. Kapranov and A.V. Zelevinski, Discriminants, resultants and multidimensional determinants, Birkhäuser, Germany (1994). · Zbl 0827.14036
[21] P.S. Aspinwall, B.R. Greene and D.R. Morrison, Calabi-Yau moduli space, mirror manifolds and space-time topology change in string theory, Nucl. Phys.B 416 (1994) 414 [hep-th/9309097] [INSPIRE]. · Zbl 0899.32006
[22] D.R. Morrison and M.R. Plesser, Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nucl. Phys.B 440 (1995) 279 [hep-th/9412236] [INSPIRE]. · Zbl 0908.14014
[23] I. Gel’fand, A. Zelevinskiǐand M. Kapranov, Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Algebra i Analiz2 (1990) 1.
[24] M. Herbst, K. Hori and D. Page, Phases of N = 2 theories in 1 + 1 dimensions with boundary, arXiv:0803.2045 [INSPIRE].
[25] P.S. Aspinwall, D-branes on toric Calabi-Yau varieties, arXiv:0806.2612 [INSPIRE]. · Zbl 1162.81033
[26] M. Ballard, D. Favero and L. Katzarkov, Variation of geometric invariant theory quotients and derived categories, arXiv:1203.6643 . · Zbl 1400.14048
[27] E. Segal, Equivalences between GIT quotients of Landau-Ginzburg B-models, Commun. Math. Phys.304 (2011) 411 [arXiv:0910.5534] [INSPIRE]. · Zbl 1216.81122
[28] P.S. Aspinwall and M.R. Plesser, Decompactifications and massless D-branes in hybrid models, JHEP07 (2010) 078 [arXiv:0909.0252] [INSPIRE]. · Zbl 1290.81095
[29] M. Herbst and J. Walcher, On the unipotence of autoequivalences of toric complete intersection Calabi-Yau categories, Math. Ann.353 (2012) 783 [arXiv:0911.4595]. · Zbl 1248.14022
[30] A.C. Avram, P. Candelas, D. Jancic and M. Mandelberg, On the connectedness of moduli spaces of Calabi-Yau manifolds, Nucl. Phys.B 465 (1996) 458 [hep-th/9511230] [INSPIRE]. · Zbl 0896.14027
[31] D. Halpern-Leistner and I. Shipman, Autoequivalences of derived categories via geometric invariant theory, arXiv:1303.5531. · Zbl 1371.14023
[32] E. Miller and B. Sturmfels, Combinatorial commutative algebra, Springer, U.S.A. (2005). · Zbl 1090.13001
[33] P. Seidel and R.P. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J.108 (2001) 37 [math.AG/0001043] [INSPIRE]. · Zbl 1092.14025
[34] N. Addington, New derived symmetries of some hyper-Kähler varieties, arXiv:1112.0487. · Zbl 1372.14009
[35] P.S. Aspinwall, R.L. Karp and R.P. Horja, Massless D-branes on Calabi-Yau threefolds and monodromy, Commun. Math. Phys.259 (2005) 45 [hep-th/0209161] [INSPIRE]. · Zbl 1082.81069
[36] A.G. Kuznetsov, Derived categories of cubic and V14threefolds, Tr. Mat. Inst. Steklova246 (2004)183 [math/0303037]. · Zbl 1107.14028
[37] A. Canonaco and R.L. Karp, Derived autoequivalences and a weighted Beilinson resolution, J. Geom. Phys.58 (2008) 743 [math/0610848]. · Zbl 1149.18007
[38] P.S. Aspinwall, Some navigation rules for D-brane monodromy, J. Math. Phys.42 (2001) 5534 [hep-th/0102198] [INSPIRE]. · Zbl 1019.81050
[39] P. Candelas, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. 2, Nucl. Phys.B 429 (1994) 626 [hep-th/9403187] [INSPIRE]. · Zbl 1020.32506
[40] P. Candelas, X. De La Ossa, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. 1, Nucl. Phys.B 416 (1994) 481 [hep-th/9308083] [INSPIRE]. · Zbl 0899.14017
[41] P.S. Aspinwall and I.V. Melnikov, D-branes on vanishing del Pezzo surfaces, JHEP12 (2004) 042 [hep-th/0405134] [INSPIRE].
[42] P.S. Aspinwall, Probing geometry with stability conditions, arXiv:0905.3137 [INSPIRE].
[43] J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E6global symmetry, Nucl. Phys.B 482 (1996) 142 [hep-th/9608047] [INSPIRE]. · Zbl 0925.81309
[44] D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2, Nucl. Phys.B 476 (1996) 437 [hep-th/9603161] [INSPIRE]. · Zbl 0925.14007
[45] D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys.B 483 (1997) 229 [hep-th/9609070] [INSPIRE]. · Zbl 0925.81228
[46] P.S. Aspinwall, S.H. Katz and D.R. Morrison, Lie groups, Calabi-Yau threefolds and F-theory, Adv. Theor. Math. Phys.4 (2000) 95 [hep-th/0002012] [INSPIRE]. · Zbl 0992.81060
[47] S. Kachru and C. Vafa, Exact results for N = 2 compactifications of heterotic strings, Nucl. Phys.B 450 (1995) 69 [hep-th/9505105] [INSPIRE]. · Zbl 0957.14509
[48] P.S. Green and T. Hübsch, Phase transitions among (many of ) Calabi-Yau compactifications, Phys. Rev. Lett.61 (1988) 1163 [INSPIRE].
[49] B.R. Greene, D.R. Morrison and C. Vafa, A geometric realization of confinement, Nucl. Phys.B 481 (1996) 513 [hep-th/9608039] [INSPIRE]. · Zbl 0925.32006
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