Skip to main content
SpringerLink
Log in

Categories of massless D-branes and del Pezzo surfaces

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M.R. Douglas, D-branes, categories and N = 1 supersymmetry, J. Math. Phys. 42 (2001) 2818 [hep-th/0011017] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. P.S. Aspinwall and A.E. Lawrence, Derived categories and zero-brane stability, JHEP 08 (2001) 004 [hep-th/0104147] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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, JHEP 09 (2005) 006 [hep-th/0002037] [INSPIRE].

    Article  ADS  Google Scholar 

  5. T. Bridgeland, Stability conditions on triangulated categories, Ann. Math. 166 (2007) 317 [math.AG/0212237].

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Strominger, Massless black holes and conifolds in string theory, Nucl. Phys. B 451 (1995) 96 [hep-th/9504090] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  7. P.S. Aspinwall and M.R. Douglas, D-brane stability and monodromy, JHEP 05 (2002) 031 [hep-th/0110071] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. R.P. Horja, Derived category automorphisms from mirror symmetry, math.AG/0103231.

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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).

  12. R. Anno, Spherical functors, arXiv:0711.4409.

  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].

    MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  16. T. Oda and H.S. Park, Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions, Tôhoku Math. J. 43 (1991) 375.

    Article  MathSciNet  MATH  Google Scholar 

  17. D.A. Cox, The homogeneous coordinate ring of a toric variety, revised version, J. Algebraic Geom. 4 (1995) 17 [alg-geom/9210008] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  18. D. Auroux, L. Katzarkov and D. Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, math.AG/0404281.

  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].

    Article  MathSciNet  MATH  Google Scholar 

  20. I.M. Gelfand, M.M. Kapranov and A.V. Zelevinski, Discriminants, resultants and multidimensional determinants, Birkhäuser, Germany (1994).

    Book  MATH  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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. 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].

  26. M. Ballard, D. Favero and L. Katzarkov, Variation of geometric invariant theory quotients and derived categories, arXiv:1203.6643 .

  27. E. Segal, Equivalences between GIT quotients of Landau-Ginzburg B-models, Commun. Math. Phys. 304 (2011) 411 [arXiv:0910.5534] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  28. P.S. Aspinwall and M.R. Plesser, Decompactifications and massless D-branes in hybrid models, JHEP 07 (2010) 078 [arXiv:0909.0252] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  31. D. Halpern-Leistner and I. Shipman, Autoequivalences of derived categories via geometric invariant theory, arXiv:1303.5531.

  32. E. Miller and B. Sturmfels, Combinatorial commutative algebra, Springer, U.S.A. (2005).

    Google Scholar 

  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].

    Article  MathSciNet  MATH  Google Scholar 

  34. N. Addington, New derived symmetries of some hyper-Kähler varieties, arXiv:1112.0487.

  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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. A.G. Kuznetsov, Derived categories of cubic and V 14 threefolds, Tr. Mat. Inst. Steklova 246 (2004)183 [math/0303037].

    Google Scholar 

  37. A. Canonaco and R.L. Karp, Derived autoequivalences and a weighted Beilinson resolution, J. Geom. Phys. 58 (2008) 743 [math/0610848].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. P.S. Aspinwall, Some navigation rules for D-brane monodromy, J. Math. Phys. 42 (2001) 5534 [hep-th/0102198] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  41. P.S. Aspinwall and I.V. Melnikov, D-branes on vanishing del Pezzo surfaces, JHEP 12 (2004) 042 [hep-th/0405134] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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 E 6 global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    MathSciNet  MATH  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  48. P.S. Green and T. Hübsch, Phase transitions among (many of ) Calabi-Yau compactifications, Phys. Rev. Lett. 61 (1988) 1163 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Box 90320, Duke University, Durham, NC, 27708-0320, U.S.A.

    Nicolas Addington & Paul S. Aspinwall

Authors
  1. Nicolas Addington
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul S. Aspinwall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Addington.

Additional information

ArXiv ePrint: 1305.5767

Rights and permissions

Reprints and permissions

About this article

Cite this article

Addington, N., Aspinwall, P.S. Categories of massless D-branes and del Pezzo surfaces. J. High Energ. Phys. 2013, 176 (2013). https://doi.org/10.1007/JHEP07(2013)176

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2013)176

Keywords

Search

Navigation