Exoflops in two dimensions

  • Paul S. AspinwallEmail author
Open Access
Regular Article - Theoretical Physics


An exoflop occurs in the gauged linear σ-model by varying the Kähler form so that a subspace appears to shrink to a point and then reemerge “outside” the original manifold. This occurs for K3 surfaces where a rational curve is “flopped” from inside to outside the K3 surface. We see that whether a rational curve contracts to an orbifold phase or an exoflop depends on whether this curve is a line or conic. We study how the D-brane category of the smooth K3 surface is described by the exoflop and, in particular, find the location of a massless D-brane in the exoflop limit. We relate exoflops to noncommutative resolutions.


D-branes Superstring Vacua Conformal Field Models in String Theory 


Open Access

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  1. [1]
    E. Witten, Phases of N = 2 theories in two dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    P.S. Aspinwall and M.R. Plesser, Decompactifications and massless D-branes in hybrid models, JHEP 07 (2010) 078 [arXiv:0909.0252] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. Bertolini, I.V. Melnikov and M.R. Plesser, Hybrid conformal field theories, JHEP 05 (2014) 043 [arXiv:1307.7063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    B.R. Greene, D.R. Morrison and C. Vafa, A geometric realization of confinement, Nucl. Phys. B 481 (1996) 513 [hep-th/9608039] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    N. Addington and P.S. Aspinwall, Categories of massless D-branes and del Pezzo surfaces, JHEP 07 (2013) 176 [arXiv:1305.5767] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
  7. [7]
    I.M. Gelfand, M.M. Kapranov and A.V. Zelevinski, Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston U.S.A. (1994).CrossRefzbMATHGoogle Scholar
  8. [8]
    P.S. Aspinwall, B.R. Greene and D.R. Morrison, Measuring small distances in N = 2 sigma models, Nucl. Phys. B 420 (1994) 184 [hep-th/9311042] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    P.S. Aspinwall, Enhanced gauge symmetries and K3 surfaces, Phys. Lett. B 357 (1995) 329 [hep-th/9507012] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    P. Candelas, X. De La Ossa, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two-parameter models (I), Nucl. Phys. B 416 (1994) 481 [hep-th/9308083] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    P.S. Aspinwall, D-branes on Calabi-Yau manifolds, in J.M. Maldacena ed., Progress in string theory. TASI 2003 lecture notes, World Scientific (2005), pp. 1-152 [hep-th/0403166] [INSPIRE].
  12. [12]
    R. Rouquier, Categorification of sl2 and braid groups, in Trends in representation theory of algebras and related topics, Contemp. Math. 406 (2006) 137, American Mathematical Society, Providence U.S.A. (2006).Google Scholar
  13. [13]
    R. Anno, Spherical functors, arXiv:0711.4409.
  14. [14]
    N. Addington, New derived symmetries of some hyperkähler varieties, arXiv:1112.0487.
  15. [15]
    R.P. Horja, Derived category automorphisms from mirror symmetry, Duke Math. J. 127 (2005) 1 [math.AG/0103231] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    P.S. Aspinwall and M.R. Douglas, D-brane stability and monodromy, JHEP 05 (2002) 031 [hep-th/0110071] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980) 35.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, arithmetic, and geometry: in honor of Yu.I. Manin, vol. II, Progr. Math. 270 (2009) 503, Birkhäuser, Boston U.S.A. (2009) [math.AG/0503632].
  20. [20]
    M. Herbst, K. Hori and D. Page, Phases of Open image in new window theories in 1+1 dimensions with boundary, arXiv:0803.2045 [INSPIRE].
  21. [21]
    P.S. Aspinwall, Topological D-branes and commutative algebra, Commun. Num. Theor. Phys. 3 (2009) 445 [hep-th/0703279] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    E. Segal, Equivalences between GIT quotients of Landau-Ginzburg B-models, Commun. Math. Phys. 304 (2011) 411 [arXiv:0910.5534] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    L.L. Avramov and D.R. Grayson, Resulutions and cohomology over complete intersections, in D. Eisenbud et al. eds., Computations in algebraic geometry with Macaulay 2, Algorithms and Computations in Mathematics 8, pp. 131-178, Springer-Verlag (2001).Google Scholar
  24. [24]
    D. Halpern-Leistner and I. Shipman, Autoequivalences of derived categories via geometric invariant theory, arXiv:1303.5531.
  25. [25]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    M. Van den Bergh, Non-commutative crepant resolutions, in The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo 2002, Springer (2004), pp. 749-770 [math.RA/0211064].
  27. [27]
    P.S. Aspinwall and D.R. Morrison, Quivers from matrix factorizations, Commun. Math. Phys. 313 (2012) 607 [arXiv:1005.1042] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    A.C. Avram, M. Kreuzer, M. Mandelberg and H. Skarke, The web of Calabi-Yau hypersurfaces in toric varieties, Nucl. Phys. B 505 (1997) 625 [hep-th/9703003] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUnited States

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