Abstract
We continue our study of the \( \mathcal{N}={1}^{\ast } \) supersymmetric gauge theory on \( {\mathbb{R}}^{2,1}\times {S}^1 \) and its relation to elliptic integrable systems. Upon compactification on a circle, we show that the semi-classical analysis of the massless and massive vacua depends on the classification of nilpotent orbits, as well as on the conjugacy classes of the component group of their centralizer. We demonstrate that semi-classically massless vacua can be lifted by Wilson lines in unbroken discrete gauge groups. The pseudo-Levi subalgebras that play a classifying role in the nilpotent orbit theory are also key in defining generalized Inozemtsev limits of (twisted) elliptic integrable systems. We illustrate our analysis in the \( \mathcal{N}={1}^{\ast } \) theories with gauge algebras su(3), su(4), so(5) and for the exceptional gauge algebra G 2. We map out modular duality diagrams of the massive and massless vacua. Moreover, we provide an analytic description of the branches of massless vacua in the case of the su(3) and the so(5) theory. The description of these branches in terms of the complexified Wilson lines on the circle invokes the Eichler-Zagier technique for inverting the elliptic Weierstrass function. After fine-tuning the coupling to elliptic points of order three, we identify the Argyres-Douglas singularities of the su(3) \( \mathcal{N}={1}^{\ast } \) theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
E. Witten, Toroidal compactification without vector structure, JHEP 02 (1998) 006 [hep-th/9712028] [INSPIRE].
A. Keurentjes, Nontrivial flat connections on the 3 torus I: G 2 and the orthogonal groups, JHEP 05 (1999) 001 [hep-th/9901154] [INSPIRE].
V.G. Kac and A.V. Smilga, Vacuum structure in supersymmetric Yang-Mills theories with any gauge group, in The many faces of the superworld, M.A. Shifman ed., (1999), pg. 185 [hep-th/9902029] [INSPIRE].
E. Witten, Supersymmetric index in four-dimensional gauge theories, Adv. Theor. Math. Phys. 5 (2002) 841 [hep-th/0006010] [INSPIRE].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
S.G. Naculich, H.J. Schnitzer and N. Wyllard, Vacuum states of N = 1∗ mass deformations of N = 4 and N = 2 conformal gauge theories and their brane interpretations, Nucl. Phys. B 609 (2001) 283 [hep-th/0103047] [INSPIRE].
N. Wyllard, A note on S-duality for the N = 1∗ Sp(2n) and SO(2n + 1) super-Yang-Mills theories, JHEP 06 (2007) 077 [hep-th/0703246] [INSPIRE].
A. Bourget and J. Troost, Counting the massive vacua of N = 1∗ super Yang-Mills theory, JHEP 08 (2015) 106 [arXiv:1506.03222] [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in The mathematical beauty of physics, Saclay France (1996), pg. 333 [hep-th/9607163] [INSPIRE].
N. Dorey, An elliptic superpotential for softly broken N = 4 supersymmetric Yang-Mills theory, JHEP 07 (1999) 021 [hep-th/9906011] [INSPIRE].
E. D’Hoker and D.H. Phong, Calogero-Moser Lax pairs with spectral parameter for general Lie algebras, Nucl. Phys. B 530 (1998) 537 [hep-th/9804124] [INSPIRE].
E. D’Hoker and D.H. Phong, Calogero-Moser and Toda systems for twisted and untwisted affine Lie algebras, Nucl. Phys. B 530 (1998) 611 [hep-th/9804125] [INSPIRE].
S.P. Kumar and J. Troost, Geometric construction of elliptic integrable systems and N = 1∗ superpotentials, JHEP 01 (2002) 020 [hep-th/0112109] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities for orthogonal groups, JHEP 08 (2013) 099 [arXiv:1307.0511] [INSPIRE].
A. Bourget and J. Troost, Duality and modularity in elliptic integrable systems and vacua of N =1∗ gauge theories, JHEP 04 (2015) 128 [arXiv:1501.05074] [INSPIRE].
P. Bala and R. Carter, Classes of unipotent elements in simple algebraic groups I, Math. Proc. Camb. Phil. Soc. 79 (1976) 401.
P. Bala and R. Carter, Classes of unipotent elements in simple algebraic groups II, Math. Proc. Camb. Phil. Soc. 80 (1976) 1.
E. Sommers, A generalization of the Bala-Carter theorem for nilpotent orbits, Int. Math. Res. Notices 11 (1998) 539.
V. Inozemtsev, The finite Toda lattices, Commun. Math. Phys. 121 (1989) 629.
D. Collingwood and W. McGovern, Nilpotent orbits in semisimple Lie algebra, CRC Press, U.S.A. (1993).
R. Carter, Finite groups of Lie type, John Wiley and Sons, U.S.A. (1985).
R. Lawther and D. Testerman, Centres of centralizers of unipotent elements in simple algebraic groups, American Mathematical Society, U.S.A. (2011).
M. Liebeck and G. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, American Mathematical Society, U.S.A. (2012).
A. Onishchik and E. Vinberg, Lie groups and algebraic groups, Springer, Germany (1990).
E. Corrigan and R. Sasaki, Quantum versus classical integrability in Calogero-Moser systems, J. Phys. A 35 (2002) 7017 [hep-th/0204039] [INSPIRE].
S. Odake and R. Sasaki, Polynomials associated with equilibrium positions in Calogero-Moser systems, J. Phys. A 35 (2002) 8283 [hep-th/0206172] [INSPIRE].
S.P. Khastgir, R. Sasaki and K. Takasaki, Calogero-Moser models. 4. Limits to Toda theory, Prog. Theor. Phys. 102 (1999) 749 [hep-th/9907102] [INSPIRE].
V. Kac, Infinite dimensional Lie algebras, Springer, Germany (1983).
O. Aharony, N. Dorey and S.P. Kumar, New modular invariance in the N = 1∗ theory, operator mixings and supergravity singularities, JHEP 06 (2000) 026 [hep-th/0006008] [INSPIRE].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
S. Terashima and S.-K. Yang, Confining phase of N = 1 supersymmetric gauge theories and N =2 massless solitons, Phys. Lett. B 391 (1997) 107 [hep-th/9607151] [INSPIRE].
P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
K.-M. Lee, Instantons and magnetic monopoles on R 3 × S 1 with arbitrary simple gauge groups, Phys. Lett. B 426 (1998) 323 [hep-th/9802012] [INSPIRE].
A. Hanany and J. Troost, Orientifold planes, affine algebras and magnetic monopoles, JHEP 08 (2001) 021 [hep-th/0107153] [INSPIRE].
S. Kim, K.-M. Lee, H.-U. Yee and P. Yi, The N = 1∗ theories on R 1+2 × S 1 with twisted boundary conditions, JHEP 08 (2004) 040 [hep-th/0403076] [INSPIRE].
N.M. Davies, T.J. Hollowood and V.V. Khoze, Monopoles, affine algebras and the gluino condensate, J. Math. Phys. 44 (2003) 3640 [hep-th/0006011] [INSPIRE].
M. Eichler and D. Zagier, On the zeros of the Weierstrass P-function, Math. Ann. 258 (1982) 399.
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: 1511.03116
Unité Mixte du CNRS et de l’Ecole Normale Supérieure associée à l’université Pierre et Marie Curie 6, UMR 8549.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bourget, A., Troost, J. On the \( \mathcal{N}={1}^{\ast } \) gauge theory on a circle and elliptic integrable systems. J. High Energ. Phys. 2016, 97 (2016). https://doi.org/10.1007/JHEP01(2016)097
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)097