Abstract
We study the low energy behaviour of \( \mathcal{N}=\left( {2,2} \right) \) supersymmetric gauge theories in 1 + 1 dimensions, with orthogonal and symplectic gauge groups and matters in the fundamental representation. We observe supersymmetry breaking in super-Yang-Mills theory and in theories with small numbers of flavors. For larger numbers of flavors, we discover duality between regular theories with different gauge groups and matter contents, where regularity refers to absence of quantum Coulomb branch. The result is applied to study families of superconformal field theories that can be used for superstring compactifications, with corners corresponding to three-dimensional Calabi-Yau manifolds. This work is motivated by recent development in mathematics concerning equivalences of derived categories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
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].
E. Witten, θ Vacua in Two-dimensional Quantum Chromodynamics, Nuovo Cim. A 51 (1979) 325 [INSPIRE].
I. Affleck, M. Dine and N. Seiberg, Dynamical Supersymmetry Breaking in Supersymmetric QCD, Nucl. Phys. B 241 (1984) 493 [INSPIRE].
N. Seiberg, Exact results on the space of vacua of four-dimensional SUSY gauge theories, Phys. Rev. D 49 (1994) 6857 [hep-th/9402044] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
K.A. Intriligator and N. Seiberg, Duality, monopoles, dyons, confinement and oblique confinement in supersymmetric SO(N(c)) gauge theories, Nucl. Phys. B 444 (1995) 125 [hep-th/9503179] [INSPIRE].
K.A. Intriligator and P. Pouliot, Exact superpotentials, quantum vacua and duality in supersymmetric SP(N(c)) gauge theories, Phys. Lett. B 353 (1995) 471 [hep-th/9505006] [INSPIRE].
I. Affleck, J.A. Harvey and E. Witten, Instantons and (Super)Symmetry Breaking in (2+1)-Dimensions, Nucl. Phys. B 206 (1982) 413 [INSPIRE].
J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].
O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
A. Karch, Seiberg duality in three-dimensions, Phys. Lett. B 405 (1997) 79 [hep-th/9703172] [INSPIRE].
O. Aharony, IR duality in D = 3 N = 2 supersymmetric USp(2N(c)) and U(N(c)) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].
A. Kapustin, Seiberg-like duality in three dimensions for orthogonal gauge groups, arXiv:1104.0466 [INSPIRE].
A. Bondal and D. Orlov, Derived categories of coherent sheaves, Proceedings ICM. Vol. II, Beijing China (2002), Higher Education Press, Beijing China (2002), pg. 47 [math/0206295].
S. Hosono and H. Takagi, Mirror symmetry and projective geometry of Reye congruences I, arXiv:1101.2746 [INSPIRE].
E.A. Rødland, The Pfaffian Calabi-Yau, its Mirror and their link to the Grassmannian G(2, 7), Compositio Math. 122 (2000) 135 [math/9801092].
N. Seiberg, IR dynamics on branes and space-time geometry, Phys. Lett. B 384 (1996) 81 [hep-th/9606017] [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
E. Witten, Toroidal compactification without vector structure, JHEP 02 (1998) 006 [hep-th/9712028] [INSPIRE].
M. Reid, The complete intersection of two or more quadrics, Ph.D. Thesis, Trinity College, Cambridge U.K. (1972).
A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, alg-geom/9506012.
A. Kuznetsov, Derived Categories of Quadric Fibrations and Intersections of Quadrics, math/510670.
L.J. Dixon, P.H. Ginsparg and J.A. Harvey, \( \widehat{c}=1 \) superconformal field theory, Nucl. Phys. B 306 (1988) 470 [INSPIRE].
K.A. Intriligator and C. Vafa, Landau-Ginzburg orbifolds, Nucl. Phys. B 339 (1990) 95 [INSPIRE].
A. Hanany and K. Hori, Branes and N = 2 theories in two-dimensions, Nucl. Phys. B 513 (1998) 119 [hep-th/9707192] [INSPIRE].
P.H. Ginsparg, Applied conformal field theory, hep-th/9108028 [INSPIRE].
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].
A. Givental, Homological geometry and mirror symmetry, Proceedings of ICM 1994, Zürich Switzerland (1994), Birkhäuser, Basel Switzerland (1995), pg. 472.
K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
S.R. Coleman, More About the Massive Schwinger Model, Annals Phys. 101 (1976) 239 [INSPIRE].
H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton U.K. (1939).
E. Witten, On the conformal field theory of the Higgs branch, JHEP 07 (1997) 003 [hep-th/9707093] [INSPIRE].
O. Aharony and M. Berkooz, IR dynamics of D = 2, N = (4, 4) gauge theories and DLCQ of ’little string theories’, JHEP 10 (1999) 030 [hep-th/9909101] [INSPIRE].
E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, hep-th/9312104 [INSPIRE].
E. Witten, On the Landau-Ginzburg description of N = 2 minimal models, Int. J. Mod. Phys. A 9 (1994) 4783 [hep-th/9304026] [INSPIRE].
E. Silverstein and E. Witten, Global U(1) R symmetry and conformal invariance of (0, 2) models, Phys. Lett. B 328 (1994) 307 [hep-th/9403054] [INSPIRE].
K. Hori and A. Kapustin, Duality of the fermionic 2 − D black hole and N = 2 Liouville theory as mirror symmetry, JHEP 08 (2001) 045 [hep-th/0104202] [INSPIRE].
S. Cecotti and C. Vafa, On classification of N = 2 supersymmetric theories, Commun. Math. Phys. 158 (1993) 569 [hep-th/9211097] [INSPIRE].
M. Krawitz, FJRW rings and Landau-Ginzburg Mirror Symmetry, Ph.D. Thesis, University of Michigan, Ann Arbor U.S.A. (2010).
M. Herbst, K. Hori and D. Page, Phases Of N = 2 Theories In 1 + 1 Dimensions With Boundary, arXiv:0803.2045 [INSPIRE].
L. Borisov and A. Caldararu, The Pfaffian-Grassmannian derived equivalence, J. Alg. Geom. 18 (2009) 201 [math/0608404].
A. Kuznetsov, Homological projective duality for Grassmannians of lines, math/0610957.
A. Kuznetsov, Homological Projective Duality, Publ. Math. Inst. Hautes Études Sci. 105 (2007) 157 [math/0507292].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1104.2853
Rights and permissions
About this article
Cite this article
Hori, K. Duality in two-dimensional (2,2) supersymmetric non-Abelian gauge theories. J. High Energ. Phys. 2013, 121 (2013). https://doi.org/10.1007/JHEP10(2013)121
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2013)121