Non Abelian T-duality in Gauged Linear Sigma Models

  • Nana Cabo Bizet
  • Aldo Martínez-Merino
  • Leopoldo A. Pando Zayas
  • Roberto Santos-Silva
Open Access
Regular Article - Theoretical Physics


Abelian T-duality in Gauged Linear Sigma Models (GLSM) forms the basis of the physical understanding of Mirror Symmetry as presented by Hori and Vafa. We consider an alternative formulation of Abelian T-duality on GLSM’s as a gauging of a global U(1) symmetry with the addition of appropriate Lagrange multipliers. For GLSMs with Abelian gauge groups and without superpotential we reproduce the dual models introduced by Hori and Vafa. We extend the construction to formulate non-Abelian T-duality on GLSMs with global non-Abelian symmetries. The equations of motion that lead to the dual model are obtained for a general group, they depend in general on semi-chiral superfields; for cases such as SU(2) they depend on twisted chiral superfields. We solve the equations of motion for an SU(2) gauged group with a choice of a particular Lie algebra direction of the vector superfield. This direction covers a non-Abelian sector that can be described by a family of Abelian dualities. The dual model Lagrangian depends on twisted chiral superfields and a twisted superpotential is generated. We explore some non-perturbative aspects by making an Ansatz for the instanton corrections in the dual theories. We verify that the effective potential for the U(1) field strength in a fixed configuration on the original theory matches the one of the dual theory. Imposing restrictions on the vector superfield, more general non-Abelian dual models are obtained. We analyze the dual models via the geometry of their susy vacua.


Duality in Gauge Field Theories Sigma Models String Duality Supersymmetric Gauge Theory 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    D.R. Morrison and M.R. Plesser, Towards mirror symmetry as duality for two-dimensional abelian gauge theories, Nucl. Phys. Proc. Suppl. 46 (1996) 177 [hep-th/9508107] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T duality, Nucl. Phys. B 479 (1996) 243 [hep-th/9606040] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
  4. [4]
    K. Hori et al., Mirror symmetry. Volume 1, Clay mathematics monographs, AMS, Providence U.S.A. (2003).Google Scholar
  5. [5]
    A. Giveon, M. Porrati and E. Rabinovici, Target space duality in string theory, Phys. Rept. 244 (1994) 77 [hep-th/9401139] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    X.C. de la Ossa and F. Quevedo, Duality symmetries from nonAbelian isometries in string theory, Nucl. Phys. B 403 (1993) 377 [hep-th/9210021] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    K. Sfetsos and D.C. Thompson, On non-abelian T-dual geometries with Ramond fluxes, Nucl. Phys. B 846 (2011) 21 [arXiv:1012.1320] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Y. Lozano, E. O Colgain, K. Sfetsos and D.C. Thompson, Non-abelian T-duality, Ramond Fields and Coset Geometries, JHEP 06 (2011) 106 [arXiv:1104.5196] [INSPIRE].
  9. [9]
    G. Itsios, Y. Lozano, J. Montero and C. Núñez, The AdS 5 non-Abelian T-dual of Klebanov-Witten as a \( \mathcal{N} \) = 1 linear quiver from M5-branes, JHEP 09 (2017) 038 [arXiv:1705.09661] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  10. [10]
    J. van Gorsel and S. Zacarías, A Type IIB Matrix Model via non-Abelian T-dualities, JHEP 12 (2017) 101 [arXiv:1711.03419] [INSPIRE].
  11. [11]
    K. Hori and D. Tong, Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N = (2,2) Theories, JHEP 05 (2007) 079 [hep-th/0609032] [INSPIRE].
  12. [12]
    A. Caldararu, J. Distler, S. Hellerman, T. Pantev and E. Sharpe, Non-birational twisted derived equivalences in abelian GLSMs, Commun. Math. Phys. 294 (2010) 605 [arXiv:0709.3855] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    K. Hori, Duality In Two-Dimensional (2, 2) Supersymmetric Non-Abelian Gauge Theories, JHEP 10 (2013) 121 [arXiv:1104.2853] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  14. [14]
    H. Jockers, V. Kumar, J.M. Lapan, D.R. Morrison and M. Romo, Nonabelian 2D Gauge Theories for Determinantal Calabi-Yau Varieties, JHEP 11 (2012) 166 [arXiv:1205.3192] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    S.J. Gates Jr., M.T. Grisaru, M. Roček and W. Siegel, Superspace Or One Thousand and One Lessons in Supersymmetry, Front. Phys. 58 (1983) 1 [hep-th/0108200] [INSPIRE].MATHGoogle Scholar
  16. [16]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    J. Bogaerts, A. Sevrin, S. van der Loo and S. Van Gils, Properties of semichiral superfields, Nucl. Phys. B 562 (1999) 277 [hep-th/9905141] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  18. [18]
    S.J. Gates Jr. and W. Merrell, D = 2 N = (2, 2) Semi Chiral Vector Multiplet, JHEP 10 (2007) 035 [arXiv:0705.3207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    S.J. Gates Jr., C.M. Hull and M. Roček, Twisted Multiplets and New Supersymmetric Nonlinear σ-models, Nucl. Phys. B 248 (1984) 157 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Giveon and M. Roček, On nonAbelian duality, Nucl. Phys. B 421 (1994) 173 [hep-th/9308154] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    M. Roček and E.P. Verlinde, Duality, quotients and currents, Nucl. Phys. B 373 (1992) 630 [hep-th/9110053] [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    J. Wess and J. Bagger, Supesymmetry and Supergravity, Princeton Series in Physics, Princeton University Press, Princeton U.S.A. (1992).Google Scholar
  23. [23]
    E.A. Rødland, The Pfaffian Calabi-Yau, its Mirror, and their Link to the Grassmannian G(2, 7), Compos. Math. 122 (2000) 135.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    A. Kanazawa, Pfaffian Calabi-Yau threefolds and mirror symmetry, Commun. Num. Theor. Phys. 6 (2012) 661 [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    A. Caldararu, J. Knapp and E. Sharpe, GLSM realizations of maps and intersections of Grassmannians and Pfaffians, arXiv:1711.00047 [INSPIRE].
  26. [26]
    S. Hosono and Y. Konishi, Higher genus Gromov-Witten invariants of the Grassmannian and the Pfaffian Calabi-Yau threefolds, Adv. Theor. Math. Phys. 13 (2009) 463 [arXiv:0704.2928] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Departamento de Física, División de Ciencias e IngenieríasUniversidad de GuanajuatoLeónMéxico
  2. 2.Mandelstam Institute for Theoretical Physics, School of Physics, and National Institute for Theoretical PhysicsUniversity of the WitwatersrandJohannesburgSouth Africa
  3. 3.Investigador Cátedra CONACyT, Facultad de Ciencias en Física y MatemáticasUniversidad Autónoma de ChiapasTuxtla GutiérrezMéxico
  4. 4.Leinweber Center for Theoretical Physics, Randall Laboratory of PhysicsThe University of MichiganAnn ArborU.S.A.
  5. 5.The Abdus Salam International Centre for Theoretical PhysicsTriesteItaly
  6. 6.Departamento de Ciencias Naturales y ExactasCUValles, Universidad de GuadalajaraAmecaMéxico

Personalised recommendations