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\(\mathbb{C}^2/\mathbb{Z}_{n}\) Fractional Branes and Monodromy

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Abstract

We construct geometric representatives for the \(\mathbb{C}^{2}/\mathbb{Z}_{n}\) fractional branes in terms of branes wrapping certain exceptional cycles of the resolution. In the process we use large radius and conifold-type monodromies, and also check some of the orbifold quantum symmetries. We find the explicit Seiberg-duality which connects our fractional branes to the ones given by the McKay correspondence. We also comment on the Harvey-Moore BPS algebras.

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References

  1. Douglas M.R. (2001). D-branes, categories and N =  1 supersymmetry. J. Math. Phys. 42: 2818–2843

    Article  ADS  MathSciNet  Google Scholar 

  2. Aspinwall P.S. and Lawrence A.E. (2001). Derived categories and zero-brane stability. JHEP 08: 004

    Article  ADS  MathSciNet  Google Scholar 

  3. Aspinwall P.S. and Douglas M.R. (2002). D-brane stability and monodromy. JHEP 05: 031

    Article  ADS  MathSciNet  Google Scholar 

  4. Douglas, M.R., Moore, G.W.: D-branes, Quivers, and ALE Instantons. http://arxiv.org/list/ hep-th/9603167, 1996

  5. McKay, J.: Graphs, singularities, and finite groups. In: The Santa Cruz Conference on Finite Groups. Proc. Sympos. Pure Math. 37. Providence, R.I.: Amer. Math. Soc. 1980, pp 183–186

  6. Bridgeland T., King A. and Reid M. (2001). Mukai implies McKay: The McKay correspondence as an equivalence of derived categories. J. Amer. Math. Soc. 14(3): 535–554

    Article  MathSciNet  Google Scholar 

  7. Witten E. (1998). D-branes and K-theory. JHEP 12: 019

    Article  ADS  MathSciNet  Google Scholar 

  8. Diaconescu D.-E., Douglas M.R. and Gomis J. (1998). Fractional branes and wrapped branes. JHEP 02: 013

    Article  ADS  MathSciNet  Google Scholar 

  9. Aspinwall P.S. (2003). A point’s point of view of stringy geometry. JHEP 01: 002

    Article  ADS  MathSciNet  Google Scholar 

  10. Diaconescu D.-E. and Gomis J. (2000). Fractional branes and boundary states in orbifold theories. JHEP 10: 001

    Article  ADS  MathSciNet  Google Scholar 

  11. Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. In: Recent Trends in String Theory. River Edge, NJ: World Scientific, 2004, pp 1–152

  12. Karp, R.L.: On the \(\mathbb{C}^{n}/\mathbb{Z}_{m}\) fractional branes. http://arxiv.org/list/hep-th/0602165, 2006

  13. Berenstein, D., Herzog, C.P., Ouyang, P., Pinansky, S.: Supersymmetry breaking from a Calabi-Yau singularity. JHEP 0509, 084 (2005)

  14. Verlinde, H., Wijnholt, M.: Building the standard model on a D3-brane. http://arxiv.org/list/hep-th/0508089, 2005

  15. Cox, D.A., Katz, S.: Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs 68, Providence, RI: Amer. Math. Soc. 1999

  16. Witten E. (1993). Phases of N =  2 theories in two dimensions. Nucl. Phys. B 403: 159–222

    Article  ADS  MathSciNet  Google Scholar 

  17. Aspinwall P.S., Greene B.R. and Morrison D.R. (1994). Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory. Nucl. Phys. B 416: 414–480

    Article  ADS  MathSciNet  Google Scholar 

  18. Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants, and multidimensional determinants. Boston, MA: Birkhäuser, 1994

  19. Morrison D.R. and Plesser M.R. (1995). Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties. Nucl. Phys. B  440: 279–354

    Article  ADS  MathSciNet  Google Scholar 

  20. Cox D.A. (1995). The homogeneous coordinate ring of a toric variety. J. Alg. Geom. 4(1): 17–50

    Google Scholar 

  21. Aspinwall P.S. (2001). Some navigation rules for D-brane monodromy. J. Math. Phys. 42: 5534–5552

    Article  ADS  MathSciNet  Google Scholar 

  22. Fulton, W.: Introduction to toric varieties. Ann. Math. Studies 131, Princeton, NJ: Princeton University Press, 1993

  23. Horja R.P. (2005). Derived category automorphisms from mirror symmetry. Duke Math. J. 127(1): 1–34

    Article  MathSciNet  Google Scholar 

  24. Orlov D.O. (1997). Equivalences of derived categories and K3 surfaces. J. Math. Sci. (New York) 84(5): 1361–1381

    Article  MathSciNet  Google Scholar 

  25. Kawamata Y. (2004). Equivalences of derived categories of sheaves on smooth stacks. Amer. J. Math. 126(5): 1057–1083

    MathSciNet  Google Scholar 

  26. Strominger A. (1995). Massless black holes and conifolds in string theory. Nucl. Phys. B  451: 96–108

    Article  ADS  MathSciNet  Google Scholar 

  27. Aspinwall P.S., Karp R.L. and Horja R.P. (2005). Massless D-branes on Calabi-Yau threefolds and monodromy. Commun. Math. Phys. 259: 45–69

    Article  ADS  MathSciNet  Google Scholar 

  28. Distler, J., Jockers, H., Park, H.-j.: D-brane monodromies, derived categories and boundary linear sigma models. http://arxiv.org/list/hep-th/0206242, 2002

  29. Seidel P. and Thomas R. (2001). Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1): 37–108

    Article  MathSciNet  Google Scholar 

  30. Greene B.R. and Kanter Y. (1997). Small volumes in compactified string theory. Nucl. Phys. B 497: 127–145

    Article  ADS  MathSciNet  Google Scholar 

  31. De la Ossa X., Florea B. and Skarke H. (2002). D-branes on noncompact Calabi-Yau manifolds: K-theory and monodromy. Nucl. Phys. B 644: 170–200

    Article  ADS  MathSciNet  Google Scholar 

  32. Beilinson A.A. (1978). Funkt. Anal. i Pril. 12(3): 68–69

    MathSciNet  Google Scholar 

  33. Barth, W., Peters, C., Van de Ven, A., Hulek, K.: Compact complex surfaces. 2nd Enlarged ed., Ergebnisse der Mathematik 4, Berlin: Springer-Verlag, 2004

  34. Sprinberg G.G. and Verdier J.-L. (1983). Construction géométrique de la correspondance de McKay. Ann. Sci. École Norm. Sup. (4) 16(3): 409–449

    Google Scholar 

  35. Kapranov M. and Vasserot E. (2000). Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316(3): 565–576

    Article  MathSciNet  Google Scholar 

  36. Skarke H. (2001). Non-perturbative gauge groups and local mirror symmetry.. JHEP 11: 013

    Article  ADS  MathSciNet  Google Scholar 

  37. Seiberg N. (1995). Electric-magnetic duality in supersymmetric nonAbelian gauge theories. Nucl. Phys. B  435: 129–146

    Article  ADS  MathSciNet  Google Scholar 

  38. Argyres, P.C., Plesser, M.R., Seiberg, N.: The Moduli Space of N=2 SUSY QCD and Duality in N=1 SUSY QCD. Nucl. Phys. B471, 159–194 (1996)

    Google Scholar 

  39. Berenstein, D., Douglas, M.R.: Seiberg duality for quiver gauge theories. http://arxiv.org/list/hep-th/0207027, 2002

  40. Robles-Llana, D., Rocek, M.: Quivers, quotients, and duality. http://arxiv.org/list/hep-th/0405230, 2004

  41. Douglas M.R., Fiol B. and Romelsberger C. (2005). The spectrum of BPS branes on a noncompact Calabi-Yau. JHEP 09: 057

    Article  ADS  MathSciNet  Google Scholar 

  42. Hartshorne R: Algebraic geometry. GTM, No. 52, New York: Springer-Verlag, 1977

  43. Kawamata Y. (2005). Log crepant birational maps and derived categories. J. Math. Sci. Univ. Tokyo 12(2): 211–231

    MathSciNet  Google Scholar 

  44. Herzog C.P. and Karp R.L. (2006). Exceptional collections and D-branes probing toric singularities. JHEP 02: 061

    Article  ADS  MathSciNet  Google Scholar 

  45. Borisov L.A., Chen L. and Smith G.G. (2005). The orbifold Chow ring of toric Deligne-Mumford stacks. J. Amer. Math. Soc. 18(1): 193–215

    Article  MathSciNet  Google Scholar 

  46. Auroux, D., Katzarkov. L., and Orlov, D.: Mirror symmetry for weighted projective planes and their noncommutative deformations. http://arxiv.org/list/math.AG/0404281, 2004

  47. Canonaco, A.: The Beilinson complex and canonical rings of irregular surfaces. Mem. Amer. Math. Soc. 862, Providence, RI: Amer. Math. Soc., 2006

  48. Rudakov, A.N., et al.,: Helices and vector bundles. London Math. Soc. Lecture Note Ser. 148, Cambridge: Cambridge Univ. Press, 1990

  49. Katz S., Pantev T. and Sharpe E. (2003). D-branes, orbifolds, and Ext groups. Nucl. Phys. B  673: 263–300

    Article  ADS  MathSciNet  Google Scholar 

  50. Toen B. (1999). Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford.. K-Theory 18(1): 33–76

    Article  MathSciNet  Google Scholar 

  51. Harvey J.A. and Moore G.W. (1996). Algebras, BPS States and Strings. Nucl. Phys. B  463: 315–368

    Article  ADS  MathSciNet  Google Scholar 

  52. Harvey J.A. and Moore G.W. (1998). On the algebras of BPS states. Commun. Math. Phys. 197: 489–519

    Article  ADS  MathSciNet  Google Scholar 

  53. Minasian R. and Moore G.W. (1997). K-theory and Ramond-Ramond charge. JHEP 11: 002

    Article  ADS  MathSciNet  Google Scholar 

  54. Ringel C.M. (1990). Hall algebras and quantum groups. Invent. Math. 101(3): 583–591

    Article  ADS  MathSciNet  Google Scholar 

  55. Kapranov M. (1998). Heisenberg doubles and derived categories. J. Algebra 202(2): 712–744

    Article  MathSciNet  Google Scholar 

  56. Bridgeland T. (2006) Stability conditions on a non-compact Calabi-Yau threefold. Commun. Math. Phys. 266, 715–733

    Article  ADS  MathSciNet  Google Scholar 

  57. Seiberg N. and Witten E. (1994). Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426: 19–52

    Article  ADS  MathSciNet  Google Scholar 

  58. Aspinwall P.S. and Karp R.L. (2003). Solitons in Seiberg-Witten theory and D-branes in the derived category. JHEP 04: 049

    Article  ADS  MathSciNet  Google Scholar 

  59. Burban, I., Schiffmann, O.: On the Hall algebra of an elliptic curve, I. http://arxiv.org/list/math.AG/ 0505148, 2005

  60. Toen, B.: Derived Hall Algebras. http://arxiv.org/list/math.QA/0501343, 2005

  61. Katz S. and Sharpe E. (2003). D-branes open string vertex operators and Ext groups. Adv. Theor. Math. Phys. 6: 979–1030

    MathSciNet  Google Scholar 

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  1. Department of Physics, Rutgers University, Piscataway, NJ, 08854-8019, USA

    Robert L. Karp

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Communicated by M.R. Douglas

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Karp, R.L. \(\mathbb{C}^2/\mathbb{Z}_{n}\) Fractional Branes and Monodromy. Commun. Math. Phys. 270, 163–196 (2007). https://doi.org/10.1007/s00220-006-0162-6

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