Journal of High Energy Physics

, 2017:106 | Cite as

Brane brick models in the mirror

  • Sebastián Franco
  • Sangmin Lee
  • Rak-Kyeong Seong
  • Cumrun Vafa
Open Access
Regular Article - Theoretical Physics


Brane brick models are Type IIA brane configurations that encode the 2d \( \mathcal{N}=\left(0,2\right) \) gauge theories on the worldvolume of D1-branes probing toric Calabi-Yau 4-folds. We use mirror symmetry to improve our understanding of this correspondence and to provide a systematic approach for constructing brane brick models starting from geometry. The mirror configuration consists of D5-branes wrapping 4-spheres and the gauge theory is determined by how they intersect. We also explain how 2d (0, 2) triality is realized in terms of geometric transitions in the mirror geometry. Mirror symmetry leads to a geometric unification of dualities in different dimensions, where the order of duality is n − 1 for a Calabi-Yau n-fold. This makes us conjecture the existence of a quadrality symmetry in 0d. Finally, we comment on how the M-theory lift of brane brick models connects to the classification of 2d (0, 2) theories in terms of 4-manifolds.


Brane Dynamics in Gauge Theories D-branes Supersymmetric gauge theory 


Open Access

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  1. [1]
    S. Kachru and E. Silverstein, 4D conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998) 4855 [hep-th/9802183] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    I.R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    D.R. Morrison and M.R. Plesser, Nonspherical horizons. 1, Adv. Theor. Math. Phys. 3 (1999) 1 [hep-th/9810201] [INSPIRE].
  4. [4]
    C. Beasley, B.R. Greene, C.I. Lazaroiu and M.R. Plesser, D3-branes on partial resolutions of Abelian quotient singularities of Calabi-Yau threefolds, Nucl. Phys. B 566 (2000) 599 [hep-th/9907186] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    B. Feng, A. Hanany and Y.-H. He, D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 [hep-th/0003085] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S. Benvenuti, S. Franco, A. Hanany, D. Martelli and J. Sparks, An infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals, JHEP 06 (2005) 064 [hep-th/0411264] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht, Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    F. Cachazo, B. Fiol, K.A. Intriligator, S. Katz and C. Vafa, A geometric unification of dualities, Nucl. Phys. B 628 (2002) 3 [hep-th/0110028] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    S. Franco, D. Ghim, S. Lee, R.-K. Seong and D. Yokoyama, 2d (0, 2) quiver gauge theories and D-branes, JHEP 09 (2015) 072 [arXiv:1506.03818] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  10. [10]
    S. Franco, S. Lee and R.-K. Seong, Brane brick models, toric Calabi-Yau 4-folds and 2d (0, 2) quivers, JHEP 02 (2016) 047 [arXiv:1510.01744] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Franco, S. Lee and R.-K. Seong, Brane brick models and 2d (0, 2) triality, JHEP 05 (2016)020 [arXiv:1602.01834] [INSPIRE].
  12. [12]
    A. Gadde, S. Gukov and P. Putrov, (0, 2) trialities, JHEP 03 (2014) 076 [arXiv:1310.0818] [INSPIRE].
  13. [13]
    S. Franco, S. Lee, R.-K. Seong and C. Vafa, work in progress.Google Scholar
  14. [14]
    A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T duality, Nucl. Phys. B 479 (1996) 243 [hep-th/9606040] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    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].
  16. [16]
    N. Seiberg, Electric-magnetic duality in supersymmetric non-Abelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    H. Garcia-Compean and A.M. Uranga, Brane box realization of chiral gauge theories in two-dimensions, Nucl. Phys. B 539 (1999) 329 [hep-th/9806177] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D. Kutasov and J. Lin, (0, 2) dynamics from four dimensions, Phys. Rev. D 89 (2014) 085025 [arXiv:1310.6032] [INSPIRE].
  20. [20]
    B. Feng, Y.-H. He, K.D. Kennaway and C. Vafa, Dimer models from mirror symmetry and quivering amoebae, Adv. Theor. Math. Phys. 12 (2008) 489 [hep-th/0511287] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M. Futaki and K. Ueda, Tropical coamoeba and torus-equivariant homological mirror symmetry for the projective space, Commun. Math. Phys. 332 (2014) 53 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
  23. [23]
    K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [INSPIRE].
  24. [24]
    S. Franco, A. Hanany, K.D. Kennaway, D. Vegh and B. Wecht, Brane dimers and quiver gauge theories, JHEP 01 (2006) 096 [hep-th/0504110] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    B. Feng, A. Hanany and Y.-H. He, Phase structure of D-brane gauge theories and toric duality, JHEP 08 (2001) 040 [hep-th/0104259] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    B. Feng, S. Franco, A. Hanany and Y.-H. He, Symmetries of toric duality, JHEP 12 (2002) 076 [hep-th/0205144] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford graduate texts in mathematics, Oxford University Press, Oxford U.K., (2007).Google Scholar
  28. [28]
    S. Elitzur, A. Giveon and D. Kutasov, Branes and N = 1 duality in string theory, Phys. Lett. B 400 (1997) 269 [hep-th/9702014] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    A. Giveon and D. Kutasov, Brane dynamics and gauge theory, Rev. Mod. Phys. 71 (1999) 983 [hep-th/9802067] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Cvetič, G.W. Gibbons, H. Lü and C.N. Pope, Ricci flat metrics, harmonic forms and brane resolutions, Commun. Math. Phys. 232 (2003) 457 [hep-th/0012011] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    B. Feng, A. Hanany, Y.H. He and A. Iqbal, Quiver theories, soliton spectra and Picard-Lefschetz transformations, JHEP 02 (2003) 056 [hep-th/0206152] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    J.P. Gauntlett, N. Kim and D. Waldram, M5-branes wrapped on supersymmetric cycles, Phys. Rev. D 63 (2001) 126001 [hep-th/0012195] [INSPIRE].ADSGoogle Scholar
  33. [33]
    J.P. Gauntlett and N. Kim, M5-branes wrapped on supersymmetric cycles. 2, Phys. Rev. D 65 (2002) 086003 [hep-th/0109039] [INSPIRE].
  34. [34]
    F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
  36. [36]
    E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    M. Berkooz, M.R. Douglas and R.G. Leigh, Branes intersecting at angles, Nucl. Phys. B 480 (1996) 265 [hep-th/9606139] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Sebastián Franco
    • 1
    • 2
  • Sangmin Lee
    • 3
    • 4
    • 5
  • Rak-Kyeong Seong
    • 6
  • Cumrun Vafa
    • 7
  1. 1.Physics DepartmentThe City College of the CUNYNew YorkU.S.A.
  2. 2.The Graduate School and University CenterThe City University of New YorkNew YorkU.S.A.
  3. 3.Center for Theoretical PhysicsSeoul National UniversitySeoulKorea
  4. 4.Department of Physics and AstronomySeoul National UniversitySeoulKorea
  5. 5.College of Liberal StudiesSeoul National UniversitySeoulKorea
  6. 6.School of PhysicsKorea Institute for Advanced StudySeoulKorea
  7. 7.Jefferson Physical LaboratoryHarvard UniversityCambridgeU.S.A.

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