Abstract
We derive general BPS boundary conditions in two-dimensional \( \mathcal{N} \) = (2, 2) supersymmetric gauge theories. We analyze the solutions of these boundary conditions, and in particular those that allow the bulk fields to have poles at the boundary. We also present the brane configurations for the half- and quarter-BPS boundary conditions of the \( \mathcal{N} \) = (2, 2) supersymmetric gauge theories in terms of branes in Type IIA string theory. We find that both A-type and B-type brane configurations are lifted to M-theory as a system of M2-branes ending on an M5-brane wrapped on a product of a holomorphic curve in ℂ2 with a special Lagrangian 3-cycle in ℂ3.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Gaiotto and E. Witten, Supersymmetric boundary conditions in N = 4 super Yang-Mills theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
A. Hashimoto, P. Ouyang and M. Yamazaki, Boundaries and defects of N = 4 SYM with 4 supercharges. Part I: boundary/junction conditions, JHEP 10 (2014) 107 [arXiv:1404.5527] [INSPIRE].
A. Hashimoto, P. Ouyang and M. Yamazaki, Boundaries and defects of N = 4 SYM with 4 supercharges. Part II: brane constructions and 3d N = 2 field theories, JHEP 10 (2014) 108 [arXiv:1406.5501] [INSPIRE].
D. Gaiotto, S-duality of boundary conditions and the geometric Langlands program, Proc. Symp. Pure Math. 98 (2018) 139 [arXiv:1609.09030] [INSPIRE].
T. Dimofte and D. Gaiotto, An E7 surprise, JHEP 10 (2012) 129 [arXiv:1209.1404] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Walls, lines, and spectral dualities in 3d gauge theories, JHEP 05 (2014) 047 [arXiv:1302.0015] [INSPIRE].
T. Okazaki and S. Yamaguchi, Supersymmetric boundary conditions in three-dimensional N = 2 theories, Phys. Rev. D 87 (2013) 125005 [arXiv:1302.6593] [INSPIRE].
F. Aprile and V. Niarchos, N = 2 supersymmetric field theories on 3-manifolds with A-type boundaries, JHEP 07 (2016) 126 [arXiv:1604.01561] [INSPIRE].
Y. Yoshida and K. Sugiyama, Localization of three-dimensional N = 2 supersymmetric theories on S1 × D2, PTEP 2020 (2020) 113B02 [arXiv:1409.6713] [INSPIRE].
T. Dimofte, D. Gaiotto and N.M. Paquette, Dual boundary conditions in 3d SCFT’s, JHEP 05 (2018) 060 [arXiv:1712.07654] [INSPIRE].
I. Brunner, J. Schulz and A. Tabler, Boundaries and supercurrent multiplets in 3D Landau-Ginzburg models, JHEP 06 (2019) 046 [arXiv:1904.07258] [INSPIRE].
K. Costello, T. Dimofte and D. Gaiotto, Boundary chiral algebras and holomorphic twists, arXiv:2005.00083 [INSPIRE].
M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 gauge theory, JHEP 10 (2016) 108 [arXiv:1603.08382] [INSPIRE].
H.-J. Chung and T. Okazaki, (2, 2) and (0, 4) supersymmetric boundary conditions in 3d N = 4 theories and type IIB branes, Phys. Rev. D 96 (2017) 086005 [arXiv:1608.05363] [INSPIRE].
K. Costello and D. Gaiotto, Vertex operator algebras and 3d N = 4 gauge theories, JHEP 05 (2019) 018 [arXiv:1804.06460] [INSPIRE].
A. Hanany and T. Okazaki, (0, 4) brane box models, JHEP 03 (2019) 027 [arXiv:1811.09117] [INSPIRE].
T. Okazaki, Abelian dualities of N = (0, 4) boundary conditions, JHEP 08 (2019) 170 [arXiv:1905.07425] [INSPIRE].
D.S. Berman, M.J. Perry, E. Sezgin and D.C. Thompson, Boundary conditions for interacting membranes, JHEP 04 (2010) 025 [arXiv:0912.3504] [INSPIRE].
K. Hosomichi and S. Lee, Self-dual strings and 2D SYM, JHEP 01 (2015) 076 [arXiv:1406.1802] [INSPIRE].
T. Okazaki and D.J. Smith, Topological M-strings and supergroup Wess-Zumino-Witten models, Phys. Rev. D 94 (2016) 065016 [arXiv:1512.06646] [INSPIRE].
D. Gaiotto and H.-C. Kim, Duality walls and defects in 5d N = 1 theories, JHEP 01 (2017) 019 [arXiv:1506.03871] [INSPIRE].
D.V. Belyaev and P. van Nieuwenhuizen, Rigid supersymmetry with boundaries, JHEP 04 (2008) 008 [arXiv:0801.2377] [INSPIRE].
D.S. Berman and D.C. Thompson, Membranes with a boundary, Nucl. Phys. B 820 (2009) 503 [arXiv:0904.0241] [INSPIRE].
M. Faizal and D.J. Smith, Supersymmetric Chern-Simons theory in presence of a boundary, Phys. Rev. D 85 (2012) 105007 [arXiv:1112.6070] [INSPIRE].
M. Faizal, Y. Luo, D.J. Smith, M.-C. Tan and Q. Zhao, Gauge and supersymmetry invariance of N = 2 boundary Chern-Simons theory, Nucl. Phys. B 914 (2017) 577 [arXiv:1601.05429] [INSPIRE].
H. Ooguri, Y. Oz and Z. Yin, D-branes on Calabi-Yau spaces and their mirrors, Nucl. Phys. B 477 (1996) 407 [hep-th/9606112] [INSPIRE].
S. Govindarajan, T. Jayaraman and T. Sarkar, World sheet approaches to D-branes on supersymmetric cycles, Nucl. Phys. B 580 (2000) 519 [hep-th/9907131] [INSPIRE].
K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [INSPIRE].
S. Govindarajan, T. Jayaraman and T. Sarkar, On D-branes from gauged linear sigma models, Nucl. Phys. B 593 (2001) 155 [hep-th/0007075] [INSPIRE].
K. Hori, Linear models of supersymmetric D-branes, in KIAS annual international conference on symplectic geometry and mirror symmetry, (2000), pg. 111 [hep-th/0012179] [INSPIRE].
M. Kontsevich, Homological algebra of mirror symmetry, alg-geom/9411018 [INSPIRE].
D. Gaiotto, G.W. Moore and E. Witten, Algebra of the infrared: string field theoretic structures in massive N = (2, 2) field theory in two dimensions, arXiv:1506.04087 [INSPIRE].
D. Gaiotto, G.W. Moore and E. Witten, An introduction to the web-based formalism, arXiv:1506.04086 [INSPIRE].
E. Witten, Mirror manifolds and topological field theory, AMS/IP Stud. Adv. Math. 9 (1998) 121 [hep-th/9112056] [INSPIRE].
M.R. Douglas, D-branes, categories and N = 1 supersymmetry, J. Math. Phys. 42 (2001) 2818 [hep-th/0011017] [INSPIRE].
P.S. Aspinwall and A.E. Lawrence, Derived categories and zero-brane stability, JHEP 08 (2001) 004 [hep-th/0104147] [INSPIRE].
E.R. Sharpe, D-branes, derived categories, and Grothendieck groups, Nucl. Phys. B 561 (1999) 433 [hep-th/9902116] [INSPIRE].
W. Lerche, P. Mayr and J. Walcher, A new kind of McKay correspondence from non-Abelian gauge theories, hep-th/0103114 [INSPIRE].
S. Hellerman and J. McGreevy, Linear sigma model toolshed for D-brane physics, JHEP 10 (2001) 002 [hep-th/0104100] [INSPIRE].
S.H. Katz and E. Sharpe, D-branes, open string vertex operators, and Ext groups, Adv. Theor. Math. Phys. 6 (2003) 979 [hep-th/0208104] [INSPIRE].
E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].
A. Kapustin and D. Orlov, Remarks on A branes, mirror symmetry, and the Fukaya category, J. Geom. Phys. 48 (2003) 84 [hep-th/0109098] [INSPIRE].
M. Herbst, K. Hori and D. Page, Phases of N = 2 theories in 1 + 1 dimensions with boundary, arXiv:0803.2045 [INSPIRE].
D. Honda and T. Okuda, Exact results for boundaries and domain walls in 2d supersymmetric theories, JHEP 09 (2015) 140 [arXiv:1308.2217] [INSPIRE].
K. Hori and M. Romo, Exact results in two-dimensional (2, 2) supersymmetric gauge theories with boundary, arXiv:1308.2438 [INSPIRE].
E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
R. Mazzeo and E. Witten, The Nahm pole boundary condition, arXiv:1311.3167 [INSPIRE].
W. Nahm, A simple formalism for the BPS monopole, Phys. Lett. B 90 (1980) 413 [INSPIRE].
A. Hanany and K. Hori, Branes and N = 2 theories in two-dimensions, Nucl. Phys. B 513 (1998) 119 [hep-th/9707192] [INSPIRE].
D.-E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys. B 503 (1997) 220 [hep-th/9608163] [INSPIRE].
D. Tsimpis, Nahm equations and boundary conditions, Phys. Lett. B 433 (1998) 287 [hep-th/9804081] [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].
K. Hori, Duality in two-dimensional (2, 2) supersymmetric non-Abelian gauge theories, JHEP 10 (2013) 121 [arXiv:1104.2853] [INSPIRE].
N.P. Warner, Supersymmetry in boundary integrable models, Nucl. Phys. B 450 (1995) 663 [hep-th/9506064] [INSPIRE].
A. Kapustin and Y. Li, D branes in Landau-Ginzburg models and algebraic geometry, JHEP 12 (2003) 005 [hep-th/0210296] [INSPIRE].
I. Brunner, M. Herbst, W. Lerche and B. Scheuner, Landau-Ginzburg realization of open string TFT, JHEP 11 (2006) 043 [hep-th/0305133] [INSPIRE].
K. Hori and J. Walcher, D-branes from matrix factorizations, Comptes Rendus Physique 5 (2004) 1061 [hep-th/0409204] [INSPIRE].
S.J. Gates, 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].
S. Ferrara, Supersymmetric gauge theories in two-dimensions, Lett. Nuovo Cim. 13 (1975) 629 [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten index and wall crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
O. Bergman and E. Avraham, Branes and 2d N = (2, 2) gauge theories with orthogonal and symplectic groups, JHEP 08 (2018) 023 [arXiv:1804.00694] [INSPIRE].
J.P. Gauntlett and N. Kim, M five-branes wrapped on supersymmetric cycles. 2, Phys. Rev. D 65 (2002) 086003 [hep-th/0109039] [INSPIRE].
J. Sanchez Loureda and D.J. Smith, Central charges of wrapped M5-brane backgrounds, JHEP 07 (2006) 043 [hep-th/0604144] [INSPIRE].
J.P. Gauntlett, N. Kim, S. Pakis and D. Waldram, M theory solutions with AdS factors, Class. Quant. Grav. 19 (2002) 3927 [hep-th/0202184] [INSPIRE].
M. Aganagic, K. Hori, A. Karch and D. Tong, Mirror symmetry in (2 + 1)-dimensions and (1 + 1)-dimensions, JHEP 07 (2001) 022 [hep-th/0105075] [INSPIRE].
H.-Y. Chen, H.-Y. Chen and J.-K. Ho, Connecting mirror symmetry in 3d and 2d via localization, Int. J. Mod. Phys. A 29 (2014) 1530004 [arXiv:1312.2361] [INSPIRE].
O. Aharony, S.S. Razamat and B. Willett, From 3d duality to 2d duality, JHEP 11 (2017) 090 [arXiv:1710.00926] [INSPIRE].
A. Karch, D. Tong and C. Turner, Mirror symmetry and bosonization in 2d and 3d, JHEP 07 (2018) 059 [arXiv:1805.00941] [INSPIRE].
H. Jockers and P. Mayr, A 3d gauge theory/quantum k-theory correspondence, Adv. Theor. Math. Phys. 24 (2020) 327 [arXiv:1808.02040] [INSPIRE].
J.P. Gauntlett, N. Kim and D. Waldram, M five-branes wrapped on supersymmetric cycles, Phys. Rev. D 63 (2001) 126001 [hep-th/0012195] [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton, NJ, U.S.A. (1992).
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [AMS/IP Stud. Adv. Math. 1 (1996) 143] [hep-th/9301042] [INSPIRE].
N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59.
C.T. Simpson, Non-Abelian Hodge theory, in Proceedings of the International Congress of Mathematicians, Math. Soc. Japan, Tokyo, Japan (1991), pg. 747.
S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
S. Gukov and E. Witten, Rigid surface operators, Adv. Theor. Math. Phys. 14 (2010) 87 [arXiv:0804.1561] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2009.02304
Rights and permissions
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.
About this article
Cite this article
Okazaki, T., Smith, D.J. Singular BPS boundary conditions in \( \mathcal{N} \) = (2, 2) supersymmetric gauge theories. J. High Energ. Phys. 2021, 43 (2021). https://doi.org/10.1007/JHEP03(2021)043
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2021)043