Abstract
Twisted compactification of the 6d \( \mathcal{N} \) = (2, 0) theories on a punctured Riemann surface give a large class of 4d \( \mathcal{N} \) = 1 and \( \mathcal{N} \) = 2 gauge theories, called class \( \mathcal{S} \). We argue that nonperturbative dynamics of class \( \mathcal{S} \) theories are described by 5d maximal Super-Yang-Mills (SYM) twisted on the Riemann surface. In a sense, twisted 5d SYM might be regarded as a “Lagrangian” for class \( \mathcal{S} \) theories on \( {{\mathbb{R}}^{1,2 }}\times {S^1} \) . First, we show that twisted 5d SYM gives generalized Hitchin’s equations which was proposed recently. Then, we discuss how to identify chiral operators with quantities in twisted 5d SYM. Mesons, or holomorphic moment maps, are identified with operators at punctures which are realized as 3d superconformal theories T ρ [G] coupled to twisted 5d SYM. “Baryons” are identified qualitatively through a study of 4d \( \mathcal{N} \) = 2 Higgs branches. We also derive a simple formula for dynamical superpotential vev which is relevant for BPS domain wall tensions. With these tools, we examine many examples of 4d \( \mathcal{N} \) = 1 theories with several phases such as confining, Higgs, and Coulomb phases, and show perfect agreements between field theories and twisted 5d SYM. Spectral curve is an essential tool to solve generalized Hitchin’s equations, and our results clarify the physical information encoded in the curve.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
K. Hori, H. Ooguri and Y. Oz, Strong coupling dynamics of four-dimensional N = 1 gauge theories from M-theory five-brane, Adv. Theor. Math. Phys. 1 (1998) 1 [hep-th/9706082] [INSPIRE].
E. Witten, Branes and the dynamics of QCD, Nucl. Phys. B 507 (1997) 658 [hep-th/9706109] [INSPIRE].
J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [ar**v:0904.2715] [INSPIRE].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [ar**v:0711.0054] [INSPIRE].
J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E 6 global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, ar**v:0907.3987 [INSPIRE].
A. Kapustin, Solution of N = 2 gauge theories via compactification to three-dimensions, Nucl. Phys. B 534 (1998) 531 [hep-th/9804069] [INSPIRE].
N.J. Hitchin, The selfduality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59 [INSPIRE].
N.J. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) 91 [INSPIRE].
A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
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) [ar**v:0804.1561] [INSPIRE].
D. **e, M5 brane and four dimensional N = 1 theories I, ar**v:1307.5877 [INSPIRE].
G. Bonelli, S. Giacomelli, K. Maruyoshi and A. Tanzini, N = 1 Geometries via M-theory, JHEP 10 (2013) 227 [ar**v:1307.7703] [INSPIRE].
M.R. Douglas, On D = 5 super Yang-Mills theory and (2,0) theory, JHEP 02 (2011) 011 [ar**v:1012.2880] [INSPIRE].
N. Lambert, C. Papageorgakis and M. Schmidt-Sommerfeld, M5-Branes, D4-branes and Quantum 5D super-Yang-Mills, JHEP 01 (2011) 083 [ar**v:1012.2882] [INSPIRE].
S. Bolognesi and K. Lee, Instanton Partons in 5-dim SU(N) Gauge Theory, Phys. Rev. D 84 (2011) 106001 [ar**v:1106.3664] [INSPIRE].
Y. Tachikawa, On S-duality of 5d super Yang-Mills on S 1, JHEP 11 (2011) 123 [ar**v:1110.0531] [INSPIRE].
H.-C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, On instantons as Kaluza-Klein modes of M5-branes, JHEP 12 (2011) 031 [ar**v:1110.2175] [INSPIRE].
Z. Bern et al., D = 5 maximally supersymmetric Yang-Mills theory diverges at six loops, Phys. Rev. D 87 (2013) 025018 [ar**v:1210.7709] [INSPIRE].
N. Lambert, C. Papageorgakis and M. Schmidt-Sommerfeld, Deconstructing (2,0) Proposals, Phys. Rev. D 88 (2013) 026007 [ar**v:1212.3337] [INSPIRE].
D. Bak and A. Gustavsson, One dyonic instanton in 5d maximal SYM theory, JHEP 07 (2013) 021 [ar**v:1305.3637] [INSPIRE].
J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [ar**v:1202.1956] [INSPIRE].
K. Hosomichi, R.-K. Seong and S. Terashima, Supersymmetric Gauge Theories on the Five-Sphere, Nucl. Phys. B 865 (2012) 376 [ar**v:1203.0371] [INSPIRE].
T. Kawano and N. Matsumiya, 5D SYM on 3D Sphere and 2D YM, Phys. Lett. B 716 (2012) 450 [ar**v:1206.5966] [INSPIRE].
J. Källén, J. Qiu and M. Zabzine, The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere, JHEP 08 (2012) 157 [ar**v:1206.6008] [INSPIRE].
H.-C. Kim and S. Kim, M5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [ar**v:1206.6339] [INSPIRE].
S. Terashima, On Supersymmetric Gauge Theories on S 4 × S 1, ar**v:1207.2163 [INSPIRE].
J. Kallen, J. Minahan, A. Nedelin and M. Zabzine, N3 -behavior from 5D Yang-Mills theory, JHEP 10 (2012) 184 [ar**v:1207.3763] [INSPIRE].
Y. Imamura, Supersymmetric theories on squashed five-sphere, PTEP 2013 (2013) 013B04 [ar**v:1209.0561] [INSPIRE].
H.-C. Kim and K. Lee, Supersymmetric M5 Brane Theories on R x CP2, JHEP 07 (2013) 072 [ar**v:1210.0853] [INSPIRE].
Y. Fukuda, T. Kawano and N. Matsumiya, 5D SYM and 2D q-Deformed YM, Nucl. Phys. B 869 (2013) 493 [ar**v:1210.2855] [INSPIRE].
Y. Imamura, Perturbative partition function for squashed S 5, ar**v:1210.6308 [INSPIRE].
H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M5-branes, ar**v:1211.0144 [INSPIRE].
J.A. Minahan, A. Nedelin and M. Zabzine, 5D super Yang-Mills theory and the correspondence to AdS 7 /CFT 6, J. Phys. A 46 (2013) 355401 [ar**v:1304.1016] [INSPIRE].
J. Yagi, 3d TQFT from 6d SCFT, JHEP 08 (2013) 017 [ar**v:1305.0291] [INSPIRE].
S. Lee and M. Yamazaki, 3d Chern-Simons Theory from M5-branes, JHEP 12 (2013) 035 [ar**v:1305.2429] [INSPIRE].
C. Cordova and D.L. Jafferis, Five-Dimensional Maximally Supersymmetric Yang-Mills in Supergravity Backgrounds, ar**v:1305.2886 [INSPIRE].
C. Cordova and D.L. Jafferis, Complex Chern-Simons from M5-branes on the Squashed Three-Sphere, ar**v:1305.2891 [INSPIRE].
H.-C. Kim, S. Kim, S.-S. Kim and K. Lee, The general M5-brane superconformal index, ar**v:1307.7660 [INSPIRE].
D. Gaiotto, G.W. Moore and Y. Tachikawa, On 6d N = (2,0) theory compactified on a Riemann surface with finite area, Prog. Theor. Exp. Phys. 2013 (2013) 013B03 [ar**v:1110.2657] [INSPIRE].
F. Benini, Y. Tachikawa and D. **e, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [ar**v:1007.0992] [INSPIRE].
D. **e and K. Yonekura, Generalized Hitchin system, Spectral curve and \( \mathcal{N} \) = 1 dynamics, JHEP 01 (2014) 001 [ar**v:1310.0467] [INSPIRE].
K. Maruyoshi, M. Taki, S. Terashima and F. Yagi, New Seiberg Dualities from N = 2 Dualities, JHEP 09 (2009) 086 [ar**v:0907.2625] [INSPIRE].
F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP 01 (2010) 088 [ar**v:0909.1327] [INSPIRE].
Y. Tachikawa and K. Yonekura, N = 1 curves for trifundamentals, JHEP 07 (2011) 025 [ar**v:1105.3215] [INSPIRE].
I. Bah and B. Wecht, New N = 1 Superconformal Field Theories In Four Dimensions, JHEP 07 (2013) 107 [ar**v:1111.3402] [INSPIRE].
I. Bah, C. Beem, N. Bobev and B. Wecht, AdS/CFT Dual Pairs from M5-Branes on Riemann Surfaces, Phys. Rev. D 85 (2012) 121901 [ar**v:1112.5487] [INSPIRE].
I. Bah, C. Beem, N. Bobev and B. Wecht, Four-Dimensional SCFTs from M5-Branes, JHEP 06 (2012) 005 [ar**v:1203.0303] [INSPIRE].
C. Beem and A. Gadde, The superconformal index of N = 1 class S fixed points, ar**v:1212.1467 [INSPIRE].
A. Gadde, K. Maruyoshi, Y. Tachikawa and W. Yan, New N = 1 Dualities, JHEP 06 (2013) 056 [ar**v:1303.0836] [INSPIRE].
I. Bah, Quarter-BPS AdS5 solutions in M-theory with a T 2 bundle over a Riemann surface, JHEP 08 (2013) 137 [ar**v:1304.4954] [INSPIRE].
K. Maruyoshi, Y. Tachikawa, W. Yan and K. Yonekura, N = 1 dynamics with T N theory, JHEP 10 (2013) 010 [ar**v:1305.5250] [INSPIRE].
I. Bah and N. Bobev, Linear Quivers and N = 1 SCFTs from M5-branes, ar**v:1307.7104 [INSPIRE].
K. Maruyoshi, Y. Tachikawa, W. Yan and K. Yonekura, Dynamical Supersymmetry Breaking with T N Theory, ar**v:1308.0064 [INSPIRE].
G. Dvali and M.A. Shifman, Domain walls in strongly coupled theories, Phys. Lett. B 396 (1997) 64 [Erratum ibid. B 407 (1997) 452] [hep-th/9612128] [INSPIRE].
D. Gaiotto and J. Maldacena, The gravity duals of N = 2 superconformal field theories, JHEP 10 (2012) 189 [ar**v:0904.4466] [INSPIRE].
O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6d N = (2,0) theories, Int. J. Mod. Phys. A 28 (2013) 1340006 [ar**v:1203.2930] [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 [ar**v:0807.3720] [INSPIRE].
D. Nanopoulos and D. **e, Hitchin Equation, Irregular Singularity and N = 2 Asymptotical Free Theories, ar**v:1005.1350 [INSPIRE].
S. Cecotti and C. Vafa, Classification of complete N = 2 supersymmetric theories in 4 dimensions, Surveys in differential geometry, vol 18 (2013) [ar**v:1103.5832] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP 02 (2012) 031 [ar**v:1112.1691] [INSPIRE].
D. **e, General Argyres-Douglas Theory, JHEP 01 (2013) 100 [ar**v:1204.2270] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Spectral networks, Annales Henri Poincaré 14 (2013) 1643 [ar**v:1204.4824] [INSPIRE].
H. Kanno, K. Maruyoshi, S. Shiba and M. Taki, W 3 irregular states and isolated N = 2 superconformal field theories, JHEP 03 (2013) 147 [ar**v:1301.0721] [INSPIRE].
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) 485-486] [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
K.A. Intriligator and N. Seiberg, Phases of N = 1 supersymmetric gauge theories in four-dimensions, Nucl. Phys. B 431 (1994) 551 [hep-th/9408155] [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, (1992).
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
A. Klemm, W. Lerche, P. Mayr, C. Vafa and N.P. Warner, Selfdual strings and N = 2 supersymmetric field theory, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034] [INSPIRE].
E. Markman, Spectral curves and integrable systems, Compos. Math. 93 (1994) 255.
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [ar**v:0906.0359] [INSPIRE].
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789 [ar**v:0804.2902] [INSPIRE].
N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, HyperKähler Metrics and Supersymmetry, Commun. Math. Phys. 108 (1987) 535 [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].
D. Tong, TASI lectures on solitons: Instantons, monopoles, vortices and kinks, hep-th/0509216 [INSPIRE].
P.C. Argyres, M.R. Plesser and N. Seiberg, The moduli space of vacua of N = 2 SUSY QCD and duality in N = 1 SUSY QCD, Nucl. Phys. B 471 (1996) 159 [hep-th/9603042] [INSPIRE].
N. Arkani-Hamed and H. Murayama, Renormalization group invariance of exact results in supersymmetric gauge theories, Phys. Rev. D 57 (1998) 6638 [hep-th/9705189] [INSPIRE].
D. Finnell and P. Pouliot, Instanton calculations versus exact results in four-dimensional SUSY gauge theories, Nucl. Phys. B 453 (1995) 225 [hep-th/9503115] [INSPIRE].
K.A. Intriligator and N. Seiberg, Lectures on supersymmetric gauge theories and electric-magnetic duality, Nucl. Phys. Proc. Suppl. 45BC (1996) 1 [hep-th/9509066] [INSPIRE].
Y. Tachikawa, N = 2 S-duality via Outer-automorphism Twists, J. Phys. A 44 (2011) 182001 [ar**v:1009.0339] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [ar**v:1305.3924] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities for orthogonal groups, JHEP 08 (2013) 099 [ar**v:1307.0511] [INSPIRE].
N. Seiberg, Naturalness versus supersymmetric nonrenormalization theorems, Phys. Lett. B 318 (1993) 469 [hep-ph/9309335] [INSPIRE].
M. Flato and C. Fronsdal, Representations of Conformal Supersymmetry, Lett. Math. Phys. 8 (1984) 159 [INSPIRE].
V. Dobrev and V. Petkova, All Positive Energy Unitary Irreducible Representations of Extended Conformal Supersymmetry, Phys. Lett. B 162 (1985) 127 [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [INSPIRE].
D. Anselmi, D. Freedman, M.T. Grisaru and A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B 526 (1998) 543 [hep-th/9708042] [INSPIRE].
A. Hanany and N. Mekareeya, Tri-vertices and SU(2)’s, JHEP 02 (2011) 069 [ar**v:1012.2119] [INSPIRE].
D. Gaiotto, A. Neitzke and Y. Tachikawa, Argyres-Seiberg duality and the Higgs branch, Commun. Math. Phys. 294 (2010) 389 [ar**v:0810.4541] [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].
A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen, Simple singularities and N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 344 (1995) 169 [hep-th/9411048] [INSPIRE].
P.C. Argyres and A.E. Faraggi, The vacuum structure and spectrum of N = 2 supersymmetric SU(N) gauge theory, Phys. Rev. Lett. 74 (1995) 3931 [hep-th/9411057] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1310.7943
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Yonekura, K. Supersymmetric gauge theory, (2,0) theory and twisted 5d Super-Yang-Mills. J. High Energ. Phys. 2014, 142 (2014). https://doi.org/10.1007/JHEP01(2014)142
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2014)142