Skip to main content
SpringerLink
Log in

Absence of sign problem in two-dimensional \( \mathcal{N} = \left( {2,2} \right) \) super Yang-Mills on lattice

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

Abstract

We show that \( \mathcal{N} = \left( {2,2} \right) \) SU(N) super Yang-Mills theory on lattice does not have sign problem in the continuum limit, that is, under the phase-quenched simulation phase of the determinant localizes to 1 and hence the phase-quench approximation becomes exact. Among several formulations, we study models by Cohen-Kaplan-Katz-Unsal (CKKU) and by Sugino. We confirm that the sign problem is absent in both models and that they converge to the identical continuum limit without fine tuning. We provide a simple explanation why previous works by other authors, which claim an existence of the sign problem, do not capture the continuum physics.

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. S.P. Martin, A supersymmetry primer, hep-ph/9709356 [SPIRES].

  2. T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  3. N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A large-N reduced model as superstring, Nucl. Phys. B 498 (1997) 467 [hep-th/9612115] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  4. L. Motl, Proposals on nonperturbative superstring interactions, hep-th/9701025 [SPIRES].

  5. R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Matrix string theory, Nucl. Phys. B 500 (1997) 43 [hep-th/9703030] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  6. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  7. N. Itzhaki, J.M. Maldacena, J. Sonnenschein and S. Yankielowicz, Supergravity and the large-N limit of theories with sixteen supercharges, Phys. Rev. D 58 (1998) 046004 [hep-th/9802042] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  8. A.G. Cohen, D.B. Kaplan, E. Katz and M. Ünsal, Supersymmetry on a Euclidean spacetime lattice. I: a target theory with four supercharges, JHEP 08 (2003) 024 [hep-lat/0302017] [SPIRES].

    Article  ADS  Google Scholar 

  9. F. Sugino, Super Yang-Mills theories on the two-dimensional lattice with exact supersymmetry, JHEP 03 (2004) 067 [hep-lat/0401017] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  10. A.G. Cohen, D.B. Kaplan, E. Katz and M. Unsal, Supersymmetry on a Euclidean spacetime lattice. II: target theories with eight supercharges, JHEP 12 (2003) 031 [hep-lat/0307012] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  11. S. Catterall, A geometrical approach to \( \mathcal{N} = 2 \) super Yang-Mills theory on the two dimensional lattice, JHEP 11 (2004) 006 [hep-lat/0410052] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. H. Suzuki and Y. Taniguchi, Two-dimensional \( \mathcal{N} = \left( {2,2} \right) \) super Yang-Mills theory on the lattice via dimensional reduction, JHEP 10 (2005) 082 [hep-lat/0507019] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  13. D.B. Kaplan and M. Ünsal, A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges, JHEP 09 (2005) 042 [hep-lat/0503039] [SPIRES].

    Article  ADS  Google Scholar 

  14. A. D’Adda, I. Kanamori, N. Kawamoto and K. Nagata, Exact extended supersymmetry on a lattice: twisted \( \mathcal{N} = 2 \) super Yang-Mills in two dimensions, Phys. Lett. B 633 (2006) 645 [hep-lat/0507029] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  15. S. Catterall, D.B. Kaplan and M. Ünsal, Exact lattice supersymmetry, Phys. Rept. 484 (2009) 71 [arXiv:0903.4881] [SPIRES].

    Article  ADS  Google Scholar 

  16. J. Giedt, R. Brower, S. Catterall, G.T. Fleming and P. Vranas, Lattice super-Yang-Mills using domain wall fermions in the chiral limit, Phys. Rev. D 79 (2009) 025015 [arXiv:0810.5746] [SPIRES].

    ADS  Google Scholar 

  17. M.G. Endres, Dynamical simulation of \( \mathcal{N} = 1 \) supersymmetric Yang-Mills theory with domain wall fermions, Phys. Rev. D 79 (2009) 094503 [arXiv:0902.4267] [SPIRES].

    ADS  Google Scholar 

  18. K. Demmouche et al., Simulation of 4d \( \mathcal{N} = 1 \) supersymmetric Yang-Mills theory with Symanzik improved gauge action and stout smearing, Eur. Phys. J. C 69 (2010) 147 [arXiv:1003.2073] [SPIRES].

    Article  ADS  Google Scholar 

  19. J.M. Maldacena, M.M. Sheikh-Jabbari and M. Van Raamsdonk, Transverse fivebranes in matrix theory, JHEP 01 (2003) 038 [hep-th/0211139] [SPIRES].

    Article  ADS  Google Scholar 

  20. M. Hanada, S. Matsuura and F. Sugino, Two-dimensional lattice for four-dimensional \( \mathcal{N} = 4 \) supersymmetric Yang-Mills, arXiv:1004.5513 [SPIRES].

  21. M. Hanada, A proposal of a fine tuning free formulation of 4d \( \mathcal{N} = 4 \) super Yang-Mills, JHEP 11 (2010) 112 [arXiv:1009.0901] [SPIRES].

    Article  ADS  Google Scholar 

  22. T. Ishii, G. Ishiki, S. Shimasaki and A. Tsuchiya, \( \mathcal{N} = 4 \) super Yang-Mills from the plane wave matrix model, Phys. Rev. D 78 (2008) 106001 [arXiv:0807.2352] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  23. G. Ishiki, S.-W. Kim, J. Nishimura and A. Tsuchiya, Deconfinement phase transition in \( \mathcal{N} = 4 \) super Yang-Mills theory on R × S 3 from supersymmetric matrix quantum mechanics, Phys. Rev. Lett. 102 (2009) 111601 [arXiv:0810.2884] [SPIRES].

    Article  ADS  Google Scholar 

  24. G. Ishiki, S.-W. Kim, J. Nishimura and A. Tsuchiya, Testing a novel large-N reduction for \( \mathcal{N} = 4 \) super Yang-Mills theory on R × S 3, JHEP 09 (2009) 029 [arXiv:0907.1488] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  25. M. Hanada, L. Mannelli and Y. Matsuo, Large-N reduced models of supersymmetric quiver, Chern-Simons gauge theories and ABJM, JHEP 11 (2009) 087 [arXiv:0907.4937] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  26. G. Ishiki, S. Shimasaki and A. Tsuchiya, A novel large-N reduction on S 3 : demonstration in Chern-Simons theory, Nucl. Phys. B 834 (2010) 423 [arXiv:1001.4917] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  27. M. Hanada, J. Nishimura and S. Takeuchi, Non-lattice simulation for supersymmetric gauge theories in one dimension, Phys. Rev. Lett. 99 (2007) 161602 [arXiv:0706.1647] [SPIRES].

    Article  ADS  Google Scholar 

  28. K.N. Anagnostopoulos, M. Hanada, J. Nishimura and S. Takeuchi, Monte-Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature, Phys. Rev. Lett. 100 (2008) 021601 [arXiv:0707.4454] [SPIRES].

    Article  ADS  Google Scholar 

  29. M. Hanada, A. Miwa, J. Nishimura and S. Takeuchi, Schwarzschild radius from Monte-Carlo calculation of the Wilson loop in supersymmetric matrix quantum mechanics, Phys. Rev. Lett. 102 (2009) 181602 [arXiv:0811.2081] [SPIRES].

    Article  ADS  Google Scholar 

  30. M. Hanada, Y. Hyakutake, J. Nishimura and S. Takeuchi, Higher derivative corrections to black hole thermodynamics from supersymmetric matrix quantum mechanics, Phys. Rev. Lett. 102 (2009) 191602 [arXiv:0811.3102] [SPIRES].

    Article  ADS  Google Scholar 

  31. M. Hanada, J. Nishimura, Y. Sekino and T. Yoneya, Monte-Carlo studies of matrix theory correlation functions, Phys. Rev. Lett. 104 (2010) 151601 [arXiv:0911.1623] [SPIRES].

    Article  ADS  Google Scholar 

  32. M. Hanada, J. Nishimura, Y. Sekino and T. Yoneya, Non-perturbative studies of matrix theory correlation functions and the gauge-gravity correspondence, in preparation.

  33. S. Catterall and T. Wiseman, Towards lattice simulation of the gauge theory duals to black holes and hot strings, JHEP 12 (2007) 104 [arXiv:0706.3518] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  34. S. Catterall and T. Wiseman, Black hole thermodynamics from simulations of lattice Yang-Mills theory, Phys. Rev. D 78 (2008) 041502 [arXiv:0803.4273] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  35. S. Catterall and T. Wiseman, Extracting black hole physics from the lattice, JHEP 04 (2010) 077 [arXiv:0909.4947] [SPIRES].

    Article  ADS  Google Scholar 

  36. M. Hanada and I. Kanamori, Lattice study of two-dimensional \( \mathcal{N} = \left( {2,2} \right) \) super Yang-Mills at large-N , Phys. Rev. D 80 (2009) 065014 [arXiv:0907.4966] [SPIRES].

    ADS  Google Scholar 

  37. S. Catterall, A. Joseph and T. Wiseman, Thermal phases of D1-branes on a circle from lattice super Yang-Mills, JHEP 12 (2010) 022 [arXiv:1008.4964] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  38. S. Catterall, A. Joseph and T. Wiseman, Gauge theory duals of black hole-black string transitions of gravitational theories on a circle, arXiv:1009.0529 [SPIRES].

  39. W. Krauth, H. Nicolai and M. Staudacher, Monte-Carlo approach to M-theory, Phys. Lett. B 431 (1998) 31 [hep-th/9803117] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  40. I. Kanamori, A method for measuring the Witten index using lattice simulation, Nucl. Phys. B 841 (2010) 426 [arXiv:1006.2468] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  41. J. Ambjørn, K.N. Anagnostopoulos, W. Bietenholz, T. Hotta and J. Nishimura, Large-N dynamics of dimensionally reduced 4D SU(N) super Yang-Mills theory, JHEP 07 (2000) 013 [hep-th/0003208] [SPIRES].

    Article  ADS  Google Scholar 

  42. M. Hanada, L. Mannelli and Y. Matsuo, Four-dimensional \( \mathcal{N} = 1 \) super Yang-Mills from matrix model, Phys. Rev. D 80 (2009) 125001 [arXiv:0905.2995] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  43. H. Suzuki, Two-dimensional \( \mathcal{N} = \left( {2,2} \right) \) super Yang-Mills theory on computer, JHEP 09 (2007) 052 [arXiv:0706.1392] [SPIRES].

    Article  ADS  Google Scholar 

  44. I. Kanamori, Vacuum energy of two-dimensional \( \mathcal{N} = \left( {2,2} \right) \) super Yang-Mills theory, Phys. Rev. D 79 (2009) 115015 [arXiv:0902.2876] [SPIRES].

    ADS  Google Scholar 

  45. T. Azeyanagi, M. Hanada, T. Hirata and H. Shimada, On the shape of a D-brane bound state and its topology change, JHEP 03 (2009) 121 [arXiv:0901.4073] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  46. I. Kanamori, H. Suzuki and F. Sugino, Euclidean lattice simulation for the dynamical supersymmetry breaking, Phys. Rev. D 77 (2008) 091502 [arXiv:0711.2099] [SPIRES].

    ADS  Google Scholar 

  47. I. Kanamori, F. Sugino and H. Suzuki, Observing dynamical supersymmetry breaking with euclidean lattice simulations, Prog. Theor. Phys. 119 (2008) 797 [arXiv:0711.2132] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  48. I. Kanamori and H. Suzuki, Some physics of the two-dimensional \( \mathcal{N} = \left( {2,2} \right) \) supersymmetric Yang-Mills theory: lattice Monte-Carlo study, Phys. Lett. B 672 (2009) 307 [arXiv:0811.2851] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  49. J. Giedt, Non-positive fermion determinants in lattice supersymmetry, Nucl. Phys. B 668 (2003) 138 [hep-lat/0304006] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  50. S. Catterall, First results from simulations of supersymmetric lattices, JHEP 01 (2009) 040 [arXiv:0811.1203] [SPIRES].

    Article  ADS  Google Scholar 

  51. M. Ünsal, Compact gauge fields for supersymmetric lattices, JHEP 11 (2005) 013 [hep-lat/0504016] [SPIRES].

    Article  Google Scholar 

  52. R. Gregory and R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70 (1993) 2837 [hep-th/9301052] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  53. O. Aharony, J. Marsano, S. Minwalla and T. Wiseman, Black hole-black string phase transitions in thermal 1+ 1-dimensional supersymmetric Yang-Mills theory on a circle, Class. Quant. Grav. 21 (2004) 5169 [hep-th/0406210] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  54. I. Kanamori and H. Suzuki, Restoration of supersymmetry on the lattice: two-dimensional \( \mathcal{N} = \left( {2,2} \right) \) supersymmetric Yang-Mills theory, Nucl. Phys. B 811 (2009) 420 [arXiv:0809.2856] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  55. S. Catterall, a private communication.

  56. J. Giedt, Deconstruction, 2D lattice super-Yang-Mills and the dynamical lattice spacing, hep-lat/0405021 [SPIRES].

  57. D. Kadoh and H. Suzuki, SUSY WT identity in a lattice formulation of 2D \( \mathcal{N} = \left( {2,2} \right) \) SYM, Phys. Lett. B 682 (2010) 466 [arXiv:0908.2274] [SPIRES].

    MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Washington, Seattle, WA, 98195-1560, U.S.A.

    Masanori Hanada

  2. Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot, 76100, Israel

    Masanori Hanada

  3. Theoretical Physics Laboratory, RIKEN Nishina Center, Wako, Saitama, 351-0198, Japan

    Masanori Hanada & Issaku Kanamori

  4. Institut für Theoretische Physik, Universität Regensburg, D-93040, Regensburg, Germany

    Issaku Kanamori

  5. INFN sezione di Torino, and Dipartimento di Fisica Teorica, Università di Torino, Via P. Giuria 1, 10125, Torino, Italy

    Issaku Kanamori

Authors
  1. Masanori Hanada
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Issaku Kanamori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Issaku Kanamori.

Additional information

ArXiv ePrint: 1010.2948v3

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hanada, M., Kanamori, I. Absence of sign problem in two-dimensional \( \mathcal{N} = \left( {2,2} \right) \) super Yang-Mills on lattice. J. High Energ. Phys. 2011, 58 (2011). https://doi.org/10.1007/JHEP01(2011)058

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP01(2011)058

Keywords

Search

Navigation