On the global structure of deformed Yang-Mills theory and QCD(adj) on \( {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}^1 \)

Open Access
Regular Article - Theoretical Physics


Spatial compactification on \( {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}_L^1 \) at small \( {\mathbb{S}}^1 \)-size L often leads to a calculable vacuum structure, where various “topological molecules” are responsible for confinement and the realization of the center and discrete chiral symmetries. Within this semiclassically calculable framework, we study how distinct theories with the same \( \mathrm{S}\mathrm{U}\left({N}_c\right)/{\mathrm{\mathbb{Z}}}_k \) gauge group (labeled by “discrete θ-angles”) arise upon gauging of appropriate \( {\mathrm{\mathbb{Z}}}_k \) subgroups of the one-form global center symmetry of an SU(N c ) gauge theory. We determine the possible \( {\mathrm{\mathbb{Z}}}_k \) actions on the local electric and magnetic effective degrees of freedom, find the ground states, and use domain walls and confining strings to give a physical picture of the vacuum structure of the different \( \mathrm{S}\mathrm{U}\left({N}_c\right)/{\mathrm{\mathbb{Z}}}_k \) theories. Some of our results reproduce ones from earlier supersymmetric studies, but most are new and do not invoke supersymmetry. We also study a further finite-temperature compactification to \( {\mathrm{\mathbb{R}}}^2\times {\mathbb{S}}_{\beta}^1\times {\mathbb{S}}_L^1 \). We argue that, in deformed Yang-Mills theory, the effective theory near the deconfinement temperature β c L exhibits an emergent Kramers-Wannier duality and that it exchanges high- and low-temperature theories with different global structure, sharing features with both the Ising model and S-duality in \( \mathcal{N}=4 \) supersymmetric Yang-Mills theory.


Confinement Solitons Monopoles and Instantons Nonperturbative Effects Discrete and Finite Symmetries 


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.


  1. [1]
    P. Goddard, J. Nuyts and D.I. Olive, Gauge theories and magnetic charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  2. [2]
    A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    P. de Forcrand and O. Jahn, Comparison of SO(3) and SU(2) lattice gauge theory, Nucl. Phys. B 651 (2003) 125 [hep-lat/0211004] [INSPIRE].CrossRefADSGoogle Scholar
  4. [4]
    O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  5. [5]
    G. ’t Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B 153 (1979) 141 [INSPIRE].
  6. [6]
    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  7. [7]
    E. Witten, Supersymmetric index in four-dimensional gauge theories, Adv. Theor. Math. Phys. 5 (2002) 841 [hep-th/0006010] [INSPIRE].Google Scholar
  8. [8]
    A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].CrossRefADSGoogle Scholar
  9. [9]
    Y. Tachikawa, Magnetic discrete gauge field in the confining vacua and the supersymmetric index, JHEP 03 (2015) 035 [arXiv:1412.2830] [INSPIRE].CrossRefADSGoogle Scholar
  10. [10]
    A. Amariti, C. Klare, D. Orlando and S. Reffert, The M-theory origin of global properties of gauge theories, arXiv:1507.04743 [INSPIRE].
  11. [11]
    M. Shifman and M. Ünsal, QCD-like Theories on R 3 × S 1 : a smooth journey from small to large r(S 1) with double-trace deformations, Phys. Rev. D 78 (2008) 065004 [arXiv:0802.1232] [INSPIRE].ADSGoogle Scholar
  12. [12]
    M. Ünsal and L.G. Yaffe, Center-stabilized Yang-Mills theory: confinement and large-N volume independence, Phys. Rev. D 78 (2008) 065035 [arXiv:0803.0344] [INSPIRE].ADSGoogle Scholar
  13. [13]
    M. Ünsal, Magnetic bion condensation: a new mechanism of confinement and mass gap in four dimensions, Phys. Rev. D 80 (2009) 065001 [arXiv:0709.3269] [INSPIRE].ADSGoogle Scholar
  14. [14]
    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].MathSciNetCrossRefADSGoogle Scholar
  15. [15]
    M.M. Anber, E. Poppitz and T. Sulejmanpasic, Strings from domain walls in supersymmetric Yang-Mills theory and adjoint QCD, Phys. Rev. D 92 (2015) 021701 [arXiv:1501.06773] [INSPIRE].ADSGoogle Scholar
  16. [16]
    D. Simic and M. Ünsal, Deconfinement in Yang-Mills theory through toroidal compactification with deformation, Phys. Rev. D 85 (2012) 105027 [arXiv:1010.5515] [INSPIRE].ADSGoogle Scholar
  17. [17]
    M.M. Anber, E. Poppitz and M. Ünsal, 2D affine XY-spin model/4D gauge theory duality and deconfinement, JHEP 04 (2012) 040 [arXiv:1112.6389] [INSPIRE].CrossRefADSGoogle Scholar
  18. [18]
    J. Liao and E. Shuryak, Strongly coupled plasma with electric and magnetic charges, Phys. Rev. C 75 (2007) 054907 [hep-ph/0611131] [INSPIRE].ADSGoogle Scholar
  19. [19]
    E. Poppitz and M. Ünsal, Conformality or confinement: (IR)relevance of topological excitations, JHEP 09 (2009) 050 [arXiv:0906.5156] [INSPIRE].CrossRefADSGoogle Scholar
  20. [20]
    E. Poppitz and M. Ünsal, Conformality or confinement (II): one-flavor CFTs and mixed-representation QCD, JHEP 12 (2009) 011 [arXiv:0910.1245] [INSPIRE].CrossRefADSGoogle Scholar
  21. [21]
    M.M. Anber and E. Poppitz, Microscopic structure of magnetic bions, JHEP 06 (2011) 136 [arXiv:1105.0940] [INSPIRE].CrossRefADSGoogle Scholar
  22. [22]
    E. Poppitz and M. Ünsal, Seiberg-Witten andPolyakov-likemagnetic bion confinements are continuously connected, JHEP 07 (2011) 082 [arXiv:1105.3969] [INSPIRE].CrossRefADSGoogle Scholar
  23. [23]
    E. Poppitz, T. Schäfer and M. Ünsal, Continuity, deconfinement and (super) Yang-Mills theory, JHEP 10 (2012) 115 [arXiv:1205.0290] [INSPIRE].CrossRefADSGoogle Scholar
  24. [24]
    M. Ünsal, Theta dependence, sign problems and topological interference, Phys. Rev. D 86 (2012) 105012 [arXiv:1201.6426] [INSPIRE].ADSGoogle Scholar
  25. [25]
    J. Greensite, An introduction to the confinement problem, Lect. Notes Phys. 821 (2011) 1 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  26. [26]
    M. Dierigl and A. Pritzel, Topological model for domain walls in (super-)Yang-Mills theories, Phys. Rev. D 90 (2014) 105008 [arXiv:1405.4291] [INSPIRE].ADSGoogle Scholar
  27. [27]
    T. Azeyanagi, M. Hanada, M. Ünsal and R. Yacoby, Large-N reduction in QCD-like theories with massive adjoint fermions, Phys. Rev. D 82 (2010) 125013 [arXiv:1006.0717] [INSPIRE].ADSGoogle Scholar
  28. [28]
    T. Misumi and T. Kanazawa, Adjoint QCD on \( {\mathrm{\mathbb{R}}}^3\times {S}^1 \) with twisted fermionic boundary conditions, JHEP 06 (2014) 181 [arXiv:1405.3113] [INSPIRE].CrossRefADSGoogle Scholar
  29. [29]
    G. Bergner and S. Piemonte, Compactified \( \mathcal{N}=1 \) supersymmetric Yang-Mills theory on the lattice: continuity and the disappearance of the deconfinement transition, JHEP 12 (2014) 133 [arXiv:1410.3668] [INSPIRE].CrossRefADSGoogle Scholar
  30. [30]
    J.C. Myers and M.C. Ogilvie, New phases of SU(3) and SU(4) at finite temperature, Phys. Rev. D 77 (2008) 125030 [arXiv:0707.1869] [INSPIRE].ADSGoogle Scholar
  31. [31]
    D.J. Gross, R.D. Pisarski and L.G. Yaffe, QCD and instantons at finite temperature, Rev. Mod. Phys. 53 (1981) 43 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  32. [32]
    P.C. Argyres and M. Ünsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion and renormalon effects, JHEP 08 (2012) 063 [arXiv:1206.1890] [INSPIRE].CrossRefADSGoogle Scholar
  33. [33]
    M.M. Anber, E. Poppitz and B. Teeple, Deconfinement and continuity between thermal and (super) Yang-Mills theory for all gauge groups, JHEP 09 (2014) 040 [arXiv:1406.1199] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  34. [34]
    E. Poppitz and M. Ünsal, Index theorem for topological excitations on R 3 × S 1 and Chern-Simons theory, JHEP 03 (2009) 027 [arXiv:0812.2085] [INSPIRE].CrossRefADSGoogle Scholar
  35. [35]
    M.M. Anber and T. Sulejmanpasic, The renormalon diagram in gauge theories on R 3 × S 1, JHEP 01 (2015) 139 [arXiv:1410.0121] [INSPIRE].CrossRefADSGoogle Scholar
  36. [36]
    E. Thomas and A.R. Zhitnitsky, Topological susceptibility and contact term in QCD. A toy model, Phys. Rev. D 85 (2012) 044039 [arXiv:1109.2608] [INSPIRE].ADSGoogle Scholar
  37. [37]
    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].MATHMathSciNetCrossRefADSGoogle Scholar
  38. [38]
    E. Poppitz, T. Schäfer and M. Ünsal, Universal mechanism of (semi-classical) deconfinement and theta-dependence for all simple groups, JHEP 03 (2013) 087 [arXiv:1212.1238] [INSPIRE].CrossRefADSGoogle Scholar
  39. [39]
    L. Álvarez-Gaumé and M. Mariño, More on softly broken N = 2 QCD, Int. J. Mod. Phys. A 12 (1997) 975 [hep-th/9606191] [INSPIRE].CrossRefADSGoogle Scholar
  40. [40]
    N.J. Evans, S.D.H. Hsu and M. Schwetz, Phase transitions in softly broken N = 2 SQCD at nonzero theta angle, Nucl. Phys. B 484 (1997) 124 [hep-th/9608135] [INSPIRE].CrossRefADSGoogle Scholar
  41. [41]
    A.M. Polyakov, Quark confinement and topology of gauge groups, Nucl. Phys. B 120 (1977) 429 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  42. [42]
    M.M. Anber, S. Collier and E. Poppitz, The SU(3)/Z(3) QCD(adj) deconfinement transition via the gauge theory/affineXY-model duality, JHEP 01 (2013) 126 [arXiv:1211.2824] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  43. [43]
    G.V. Dunne, I.I. Kogan, A. Kovner and B. Tekin, Deconfining phase transition in (2 + 1)-dimensions: The Georgi-Glashow model, JHEP 01 (2001) 032 [hep-th/0010201] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  44. [44]
    M.M. Anber, The abelian confinement mechanism revisited: new aspects of the Georgi-Glashow model, Annals Phys. 341 (2014) 21 [arXiv:1308.0027] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  45. [45]
    M.M. Anber, S. Collier, E. Poppitz, S. Strimas-Mackey and B. Teeple, Deconfinement in \( \mathcal{N}=1 \) super Yang-Mills theory on R 3 × S 1 via dual-Coulomb gas andaffineXY-model, JHEP 11 (2013) 142 [arXiv:1310.3522] [INSPIRE].CrossRefADSGoogle Scholar
  46. [46]
    P. Lecheminant, A.O. Gogolin and A.A. Nersesyan, Criticality in selfdual sine-Gordon models, Nucl. Phys. B 639 (2002) 502 [cond-mat/0203294] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  47. [47]
    Y.V. Kovchegov and D.T. Son, Critical temperature of the deconfining phase transition in (2 + 1)-d Georgi-Glashow model, JHEP 01 (2003) 050 [hep-th/0212230] [INSPIRE].CrossRefADSGoogle Scholar
  48. [48]
    P. Lecheminant, Nature of the deconfining phase transition in the 2+1-dimensional SU(N) Georgi-Glashow model, Phys. Lett. B 648 (2007) 323 [hep-th/0610046] [INSPIRE].CrossRefADSGoogle Scholar
  49. [49]
    A. Kapustin, Wilson-t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].MathSciNetADSGoogle Scholar
  50. [50]
    L.P. Kadanoff, Lattice Coulomb gas representations of two-dimensional problems, J. Phys. A 11 (1978) 1399 [INSPIRE].MathSciNetADSGoogle Scholar
  51. [51]
    C. Korthals-Altes, A. Kovner and M.A. Stephanov, Spatialt Hooft loop, hot QCD and Z(N) domain walls, Phys. Lett. B 469 (1999) 205 [hep-ph/9909516] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  52. [52]
    G. ’t Hooft, On the phase transition towards permanent quark confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
  53. [53]
    E. Witten, Lectures on QFT, in Quantum fields and strings: A course for mathematicians, P. Deligne et al. eds., American Mathematical Society, U.S.A. (2000).Google Scholar
  54. [54]
    H. Reinhardt, Ont Hoofts loop operator, Phys. Lett. B 557 (2003) 317 [hep-th/0212264] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  55. [55]
    J. Gomis, T. Okuda and D. Trancanelli, Quantumt Hooft operators and S-duality in N = 4 super Yang-Mills, Adv. Theor. Math. Phys. 13 (2009) 1941 [arXiv:0904.4486] [INSPIRE].MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Institute de Théorie des Phénomenès Physiques, École Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Department of PhysicsUniversity of TorontoTorontoCanada

Personalised recommendations