Skip to main content
SpringerLink
Log in

Ground states of duality-twisted sigma-models with K 3 target space

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

Abstract

We analyze the ground states of a two-dimensional sigma-model whose target space is an elliptically fibered K 3, with the sigma-model compactified on S 1 with boundary conditions twisted by a duality symmetry. We show that the Witten index receives contributions from two kinds of states: (i) those that can be mapped to cohomology with coefficients in a certain line bundle over the target space, and (ii) states whose wavefunctions are localized at singular fibers. We also discuss the orbifold limit and possible connections with geometric quantization of the target space.

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. A. Kumar and C. Vafa, U manifolds, Phys. Lett. B 396 (1997) 85 [hep-th/9611007] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. J.T. Liu and R. Minasian, U-branes and T 3 fibrations, Nucl. Phys. B 510 (1998) 538 [hep-th/9707125] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. A. Dabholkar and J.A. Harvey, String islands, JHEP 02 (1999) 006 [hep-th/9809122] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. O.J. Ganor, U duality twists and possible phase transitions in (2 + 1)-dimensions supergravity, Nucl. Phys. B 549 (1999) 145 [hep-th/9812024] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. S. Hellerman, J. McGreevy and B. Williams, Geometric constructions of nongeometric string theories, JHEP 01 (2004) 024 [hep-th/0208174] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. A. Dabholkar and C. Hull, Duality twists, orbifolds and fluxes, JHEP 09 (2003) 054 [hep-th/0210209] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  7. A. Flournoy, B. Wecht and B. Williams, Constructing nongeometric vacua in string theory, Nucl. Phys. B 706 (2005) 127 [hep-th/0404217] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. C. Hull, A Geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. M. Cvetič, T. Liu and M.B. Schulz, Twisting K3 × T 2 orbifolds, JHEP 09 (2007) 092 [hep-th/0701204] [INSPIRE].

    Article  ADS  Google Scholar 

  10. S. Kawai and Y. Sugawara, Mirrorfolds with K3 fibrations, JHEP 02 (2008) 065 [arXiv:0711.1045] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. O.J. Ganor and Y.P. Hong, Selfduality and Chern-Simons Theory, arXiv:0812.1213 [INSPIRE].

  12. O.J. Ganor, Y.P. Hong and H. Tan, Ground States of S-duality Twisted N = 4 Super Yang-Mills Theory, JHEP 03 (2011) 099 [arXiv:1007.3749] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  13. Y. Terashima and M. Yamazaki, SL(2, R) Chern-Simons, Liouville and Gauge Theory on Duality Walls, JHEP 08 (2011) 135 [arXiv:1103.5748] [INSPIRE].

    Article  ADS  Google Scholar 

  14. O.J. Ganor, Y.P. Hong, R. Markov and H.S. Tan, Static Charges in the Low-Energy Theory of the S-duality Twist, JHEP 04 (2012) 041 [arXiv:1201.2679] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. J.A. Harvey, G.W. Moore and A. Strominger, Reducing S duality to T duality, Phys. Rev. D 52 (1995) 7161 [hep-th/9501022] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  16. M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Topological reduction of 4 − D SYM to 2−D σ-models,Nucl. Phys. B 448 (1995) 166[hep-th/9501096][INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  18. E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].

  20. K. Hori et al., Mirror symmetry, Clay mathematics monographs 1, AMS (2003) [INSPIRE].

  21. B. Zumino, Supersymmetry and Kähler Manifolds, Phys. Lett. B 87 (1979) 203 [INSPIRE].

    Article  ADS  Google Scholar 

  22. E. Witten, Mirror manifolds and topological field theory, hep-th/9112056 [INSPIRE].

  23. B.R. Greene, A.D. Shapere, C. Vafa and S.-T. Yau, Stringy Cosmic Strings and Noncompact Calabi-Yau Manifolds, Nucl. Phys. B 337 (1990) 1 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  25. N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].

  26. N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T duality, Nucl. Phys. B 479 (1996) 243 [hep-th/9606040] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. P.S. Aspinwall and D.R. Morrison, String theory on K3 surfaces, hep-th/9404151 [INSPIRE].

  29. N. Seiberg, Observations on the Moduli Space of Superconformal Field Theories, Nucl. Phys. B 303 (1988) 286 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  30. R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience (1994) [ISBN-10: 0471050598, ISBN-13: 978-0471050599].

  32. D.R. Morrison, The Geometry underlying mirror symmetry, alg-geom/9608006 [INSPIRE].

  33. C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez and J.M. Muñoz Porras, Mirror symmetry on elliptic K3 surfaces via Fourier-Mukai transform, Commun. Math. Phys. 195 (1998) 79 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  34. N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  35. N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. N. Nekrasov and A.S. Schwarz, Instantons on noncommutative R 4 and (2, 0) superconformal six-dimensional theory, Commun. Math. Phys. 198 (1998) 689 [hep-th/9802068] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. M. Berkooz, Nonlocal field theories and the noncommutative torus, Phys. Lett. B 430 (1998) 237 [hep-th/9802069] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  38. E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. M.R. Douglas, Branes within branes, in proceedings of NATO Advanced Study Institute on Strings, Branes and Dualities, pg. 267-275, Cargèse, France, 26 May-14 June 1997, hep-th/9512077 [INSPIRE].

  40. P.S. Aspinwall, K3 surfaces and string duality, in S.T. Yau ed., Differential geometry inspired by string theory, pg. 1-95 [hep-th/9611137] [INSPIRE].

  41. P.S. Aspinwall, Enhanced gauge symmetries and K3 surfaces, Phys. Lett. B 357 (1995) 329 [hep-th/9507012] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  42. M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory. Vol. 1: Introduction, Cambridge Monographs On Mathematical Physics, Cambridge University Press, Cambridge, U.K. (1987).

  43. J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string Cambridge University Press, Cambridge, U.K. (1998).

  44. D.-E. Diaconescu, M.R. Douglas and J. Gomis, Fractional branes and wrapped branes, JHEP 02 (1998) 013 [hep-th/9712230] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  45. D.-E. Diaconescu and J. Gomis, Fractional branes and boundary states in orbifold theories, JHEP 10 (2000) 001 [hep-th/9906242] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. A. Connes, M.R. Douglas and A.S. Schwarz, Noncommutative geometry and matrix theory: Compactification on tori, JHEP 02 (1998) 003 [hep-th/9711162] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. M.R. Douglas and C.M. Hull, D-branes and the noncommutative torus, JHEP 02 (1998) 008 [hep-th/9711165] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  48. N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  49. A.S. Cattaneo and G. Felder, A Path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212 (2000) 591 [math/9902090] [INSPIRE].

  50. R. Blumenhagen, L. Görlich, B. Körs and D. Lüst, Asymmetric orbifolds, noncommutative geometry and type-I string vacua, Nucl. Phys. B 582 (2000) 44 [hep-th/0003024] [INSPIRE].

    Article  ADS  Google Scholar 

  51. N.J. Hitchin, The Selfduality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59 [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  52. S. Axelrod, S. Della Pietra and E. Witten, Geometric quantization of Chern-Simons gauge theory, J. Diff. Geom. 33 (1991) 787 [INSPIRE].

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of California, Berkeley, CA, 94720, U.S.A

    Ori J. Ganor, Sharon Jue & Shannon McCurdy

Authors
  1. Ori J. Ganor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sharon Jue
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shannon McCurdy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ori J. Ganor.

Additional information

ArXiv ePrint: 1211.4179

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ganor, O.J., Jue, S. & McCurdy, S. Ground states of duality-twisted sigma-models with K 3 target space. J. High Energ. Phys. 2013, 17 (2013). https://doi.org/10.1007/JHEP02(2013)017

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP02(2013)017

Keywords

Search

Navigation