Categories of massless D-branes and del Pezzo surfaces

  • Nicolas AddingtonEmail author
  • Paul S. Aspinwall


In analogy with the physical concept of a massless D-brane, we define a notion of “\( \mathbb{Q}\hbox{-}\mathrm{masslessness} \)” 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 \( \mathbb{Q}\hbox{-}\mathrm{massless} \) objects is a “fractional Calabi-Yau” category of graded matrix factorizations.


D-branes Differential and Algebraic Geometry 


  1. [1]
    M.R. Douglas, D-branes, categories and N = 1 supersymmetry, J. Math. Phys. 42 (2001) 2818 [hep-th/0011017] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. [2]
    P.S. Aspinwall and A.E. Lawrence, Derived categories and zero-brane stability, JHEP 08 (2001) 004 [hep-th/0104147] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [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. [4]
    M.R. Douglas, B. Fiol and C. Römelsberger, Stability and BPS branes, JHEP 09 (2005) 006 [hep-th/0002037] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    T. Bridgeland, Stability conditions on triangulated categories, Ann. Math. 166 (2007) 317 [math.AG/0212237].MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    A. Strominger, Massless black holes and conifolds in string theory, Nucl. Phys. B 451 (1995) 96 [hep-th/9504090] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    P.S. Aspinwall and M.R. Douglas, D-brane stability and monodromy, JHEP 05 (2002) 031 [hep-th/0110071] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    R.P. Horja, Derived category automorphisms from mirror symmetry, math.AG/0103231.
  10. [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].MathSciNetADSCrossRefGoogle Scholar
  11. [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).Google Scholar
  12. [12]
    R. Anno, Spherical functors, arXiv:0711.4409.
  13. [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. [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].MathSciNetzbMATHGoogle Scholar
  15. [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].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    T. Oda and H.S. Park, Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions, Tôhoku Math. J. 43 (1991) 375.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    D.A. Cox, The homogeneous coordinate ring of a toric variety, revised version, J. Algebraic Geom. 4 (1995) 17 [alg-geom/9210008] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  18. [18]
    D. Auroux, L. Katzarkov and D. Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, math.AG/0404281.
  19. [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].MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    I.M. Gelfand, M.M. Kapranov and A.V. Zelevinski, Discriminants, resultants and multidimensional determinants, Birkhäuser, Germany (1994).zbMATHCrossRefGoogle Scholar
  21. [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].MathSciNetADSCrossRefGoogle Scholar
  22. [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].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    I. Gel’fand, A. Zelevinskiǐand M. Kapranov, Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Algebra i Analiz 2 (1990) 1.Google Scholar
  24. [24]
    M. Herbst, K. Hori and D. Page, Phases of N = 2 theories in 1 + 1 dimensions with boundary, arXiv:0803.2045 [INSPIRE].
  25. [25]
    P.S. Aspinwall, D-branes on toric Calabi-Yau varieties, arXiv:0806.2612 [INSPIRE].
  26. [26]
    M. Ballard, D. Favero and L. Katzarkov, Variation of geometric invariant theory quotients and derived categories, arXiv:1203.6643 .
  27. [27]
    E. Segal, Equivalences between GIT quotients of Landau-Ginzburg B-models, Commun. Math. Phys. 304 (2011) 411 [arXiv:0910.5534] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  28. [28]
    P.S. Aspinwall and M.R. Plesser, Decompactifications and massless D-branes in hybrid models, JHEP 07 (2010) 078 [arXiv:0909.0252] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [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].MathSciNetzbMATHCrossRefGoogle Scholar
  30. [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].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    D. Halpern-Leistner and I. Shipman, Autoequivalences of derived categories via geometric invariant theory, arXiv:1303.5531.
  32. [32]
    E. Miller and B. Sturmfels, Combinatorial commutative algebra, Springer, U.S.A. (2005).Google Scholar
  33. [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].MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    N. Addington, New derived symmetries of some hyper-Kähler varieties, arXiv:1112.0487.
  35. [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].MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. [36]
    A.G. Kuznetsov, Derived categories of cubic and V 14 threefolds, Tr. Mat. Inst. Steklova 246 (2004)183 [math/0303037].Google Scholar
  37. [37]
    A. Canonaco and R.L. Karp, Derived autoequivalences and a weighted Beilinson resolution, J. Geom. Phys. 58 (2008) 743 [math/0610848].MathSciNetADSzbMATHCrossRefGoogle Scholar
  38. [38]
    P.S. Aspinwall, Some navigation rules for D-brane monodromy, J. Math. Phys. 42 (2001) 5534 [hep-th/0102198] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  39. [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].MathSciNetADSCrossRefGoogle Scholar
  40. [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].ADSCrossRefGoogle Scholar
  41. [41]
    P.S. Aspinwall and I.V. Melnikov, D-branes on vanishing del Pezzo surfaces, JHEP 12 (2004) 042 [hep-th/0405134] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    P.S. Aspinwall, Probing geometry with stability conditions, arXiv:0905.3137 [INSPIRE].
  43. [43]
    J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E 6 global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  44. [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].MathSciNetADSCrossRefGoogle Scholar
  45. [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].MathSciNetADSCrossRefGoogle Scholar
  46. [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].MathSciNetzbMATHGoogle Scholar
  47. [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].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    P.S. Green and T. Hübsch, Phase transitions among (many of ) Calabi-Yau compactifications, Phys. Rev. Lett. 61 (1988) 1163 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [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].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Department of MathematicsBox 90320, Duke UniversityDurhamU.S.A.

Personalised recommendations