Abstract
In this paper we explore a new approach to studying three-dimensional \( \mathcal{N}=4 \) super-Yang-Mills on a lattice. Our strategy is to complexify the Donaldson-Witten twist of four-dimensional \( \mathcal{N}=2 \) super-Yang-Mills to make it amenable to a lattice formulation and we find that lattice gauge invariance forces the model to live in at most three dimensions. We analyze the renormalization of the lattice theory and show that uncomplexified three-dimensional \( \mathcal{N}=4 \) super-Yang-Mills can be reached in the continuum limit by supplementing the lattice action with appropriate mass terms.
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References
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] [hep-th/9407087] [INSPIRE].
D.B. Kaplan and M. Ünsal, A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges, JHEP 09 (2005) 042 [hep-lat/0503039] [INSPIRE].
S. Catterall, Lattice formulation of N = 4 super Yang-Mills theory, JHEP 06 (2005) 027 [hep-lat/0503036] [INSPIRE].
P. Becher and H. Joos, The Dirac-Kähler Equation and Fermions on the Lattice, Z. Phys. C 15 (1982) 343 [INSPIRE].
S. Catterall, From Twisted Supersymmetry to Orbifold Lattices, JHEP 01 (2008) 048 [arXiv:0712.2532] [INSPIRE].
S. Catterall, D.B. Kaplan and M. Ünsal, Exact lattice supersymmetry, Phys. Rept. 484 (2009) 71 [arXiv:0903.4881] [INSPIRE].
T. Appelquist and R.D. Pisarski, High-Temperature Yang-Mills Theories and Three-Dimensional Quantum Chromodynamics, Phys. Rev. D 23 (1981) 2305 [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].
B. Assel and S. Cremonesi, The Infrared Physics of Bad Theories, SciPost Phys. 3 (2017) 024 [arXiv:1707.03403] [INSPIRE].
A.G. Cohen, D.B. Kaplan, E. Katz and M. Ünsal, Supersymmetry on a Euclidean space-time lattice. 2. Target theories with eight supercharges, JHEP 12 (2003) 031 [hep-lat/0307012] [INSPIRE].
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
M. Blau and G. Thompson, Aspects of N T ≥ 2 topological gauge theories and D-branes, Nucl. Phys. B 492 (1997) 545 [hep-th/9612143] [INSPIRE].
A. Joseph, Lattice formulation of three-dimensional \( \mathcal{N}=4 \) gauge theory with fundamental matter fields, JHEP 09 (2013) 046 [arXiv:1307.3281] [INSPIRE].
A. D’Adda, I. Kanamori, N. Kawamoto and K. Nagata, Exact Extended Supersymmetry on a Lattice: Twisted N = 4 Super Yang-Mills in Three Dimensions, Nucl. Phys. B 798 (2008) 168 [arXiv:0707.3533] [INSPIRE].
K. Nagata, On the continuum and lattice formulations of N = 4 D = 3 twisted super Yang-Mills, JHEP 01 (2008) 041 [arXiv:0710.5689] [INSPIRE].
F. Bruckmann and M. de Kok, Noncommutativity approach to supersymmetry on the lattice: SUSY quantum mechanics and an inconsistency, Phys. Rev. D 73 (2006) 074511 [hep-lat/0603003] [INSPIRE].
F. Bruckmann, S. Catterall and M. de Kok, A Critique of the Link Approach to Exact Lattice Supersymmetry, Phys. Rev. D 75 (2007) 045016 [hep-lat/0611001] [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in The mathematical beauty of physics: A memorial volume for Claude Itzykson. Proceedings, Conference, Saclay, France, June 5-7, 1996, pp. 333-366, hep-th/9607163 [INSPIRE].
S. Weinberg, The quantum theory of fields. Vol. 3: Supersymmetry, Cambridge University Press, New York, NY, U.S.A. (2013).
F. Sugino, A lattice formulation of superYang-Mills theories with exact supersymmetry, JHEP 01 (2004) 015 [hep-lat/0311021] [INSPIRE].
L. Baulieu, N = 4 Yang-Mills theory as a complexification of the N = 2 theory, Nucl. Phys. Proc. Suppl. 192-193 (2009) 27 [arXiv:0906.1289] [INSPIRE].
E.H. Saidi, Twisted 3D N = 4 Supersymmetric YM on deformed \( {\mathbb{A}}^3 \) Lattice, arXiv:1407.3854 [INSPIRE].
AuroraScience collaboration, M. Cristoforetti, F. Di Renzo and L. Scorzato, New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble, Phys. Rev. D 86 (2012) 074506 [arXiv:1205.3996] [INSPIRE].
T. Appelquist and J. Carazzone, Infrared Singularities and Massive Fields, Phys. Rev. D 11 (1975) 2856 [INSPIRE].
S. Catterall, E. Dzienkowski, J. Giedt, A. Joseph and R. Wells, Perturbative renormalization of lattice N = 4 super Yang-Mills theory, JHEP 04 (2011) 074 [arXiv:1102.1725] [INSPIRE].
S. Catterall and A. Veernala, Spontaneous supersymmetry breaking in two dimensional lattice super QCD, JHEP 10 (2015) 013 [arXiv:1505.00467] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
B. Assel and J. Gomis, Mirror Symmetry And Loop Operators, JHEP 11 (2015) 055 [arXiv:1506.01718] [INSPIRE].
K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
D. Karabali and V.P. Nair, A Gauge invariant Hamiltonian analysis for nonAbelian gauge theories in (2+1)-dimensions, Nucl. Phys. B 464 (1996) 135 [hep-th/9510157] [INSPIRE].
A. Agarwal and V.P. Nair, Supersymmetry and Mass Gap in 2+1 Dimensions: A Gauge Invariant Hamiltonian Analysis, Phys. Rev. D 85 (2012) 085011 [arXiv:1201.6609] [INSPIRE].
S. Sen, S. Sen, J.C. Sexton and D.H. Adams, A geometric discretization scheme applied to the Abelian Chern-Simons theory, Phys. Rev. E 61 (2000) 3174 [hep-th/0001030] [INSPIRE].
B. Assel, C. Bachas, J. Estes and J. Gomis, Holographic Duals of D = 3 N = 4 Superconformal Field Theories, JHEP 08 (2011) 087 [arXiv:1106.4253] [INSPIRE].
P. McFadden and K. Skenderis, The Holographic Universe, J. Phys. Conf. Ser. 222 (2010) 012007 [arXiv:1001.2007] [INSPIRE].
S. Catterall, J. Giedt and A. Joseph, Twisted supersymmetries in lattice \( \mathcal{N}=4 \) super Yang-Mills theory, JHEP 10 (2013) 166 [arXiv:1306.3891] [INSPIRE].
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Giedt, J., Lipstein, A.E. 3d \( \mathcal{N}=4 \) super-Yang-Mills on a lattice. J. High Energ. Phys. 2018, 162 (2018). https://doi.org/10.1007/JHEP03(2018)162
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DOI: https://doi.org/10.1007/JHEP03(2018)162