Self-conjugate QCD

  • Mohamed M. AnberEmail author
Open Access
Regular Article - Theoretical Physics


We carry out a systematic study of SU(6) Yang-Mills theory endowed with fermions in the adjoint and 3-index antisymmetric mixed-representation. The fermion bilinear in the 3-index antisymmetric representation vanishes identically, which leads to interesting new phenomena. We first study the theory on a small circle, i.e., on \( {\mathrm{\mathbb{R}}}^3\times {\mathbbm{S}}_L^1 \), employing symmetry-twisted boundary conditions and semi-classical techniques. We find that the ground state is 3-fold degenerate, which can be explained as a consequence of a 1-form/0-form mixed ’t Hooft anomaly. In addition, the theory may admit massless bosonic and fermionic degrees of freedom, depending on the number of flavors, and confines the electric probes in the infrared. Empowered by ’t Hooft anomaly matching conditions along with the 2-loop β-function, we further examine the possible infrared symmetry realizations on ℝ4 for various number of adjoint and 3-index antisymmetric fermions. The infrared theory is either a conformal field theory, which is expected for a large number of flavors, or it is confining with or without chiral symmetry breaking. In a few cases, we are able to give enough evidence for adiabatic continuity between the small- and large-circle limits.


Anomalies in Field and String Theories Field Theories in Lower Dimensions Nonpert urbative Effects Confinement 


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]
    G. ’t Hooft et al. eds., Recent developments in gauge theories, in Proceedings, Nato Advanced Study Institute, Cargese, France, 26 August–8 September 1979 [NATO Sci. Ser.B 59 (1980) 1] [INSPIRE].
  2. [2]
    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP02 (2015) 172 [arXiv:1412.5148] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal and temperature, JHEP05 (2017) 091 [arXiv:1703 .00501] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Y. Tanizaki, Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking, JHEP08 (2018) 171 [arXiv:1807.07666] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    F. Benini, C. Córdova and P.-S. Hsin, On 2-group global symmetries and their anomalies, JHEP03 (2019) 118 [arXiv:1803.09336] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    C. Choi, D. Delmastro, J. Gomis and Z. Komargodski, Dynamics of QCD 3with rank-two quarks and duality, arXiv:1810.07720 [INSPIRE].
  7. [7]
    Z. Komargodski, T. Sulejmanpasic and M. Ünsal, Walls, anomalies and deconfinement in quantum antiferromagnets, Phys. Rev.B 97 (2018) 054418 [arXiv:1706.05731] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    H. Shimizu and K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition, Phys. Rev.D 97 (2018) 105011 [arXiv:1706 .06104] [INSPIRE].ADSGoogle Scholar
  9. [9]
    Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, Comments on Abelian Higgs models and persistent order, SciPost Phys.6 (2019) 003 [arXiv: 1705 .04786] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    Y. Kikuchi and Y. Tanizaki, Global inconsistency, ’t Hooft anomaly and level crossing in quantum mechanics, PTEP2017 (2017) 113B05 [arXiv:1708 .01962] [INSPIRE].Google Scholar
  11. [11]
    K. Aitken, A. Cherman and M. Ünsal, Dihedral symmetry in SU(N) Yang-Mills theory, arXiv: 1804.05845 [INSPIRE].
  12. [12]
    Y. Tanizaki and T. Sulejmanpasic, Anomaly and global inconsistency matching: θ-angles, SU(3)/U(1)2nonlinear σ-model, SU(3) chains and its generalizations, Phys. Rev.B 98 (2018) 115126 [arXiv:1805.11423] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    T. Sulejmanpasic and Y. Tanizaki, C-P-T anomaly matching in bosonic quantum field theory and spin chains, Phys. Rev.B 97 (2018) 144201 [arXiv:1802 .02153] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    Y. Tanizaki, Y. Kikuchi, T. Misumi and N. Sakai, Anomaly matching for the phase diagram of massless Z N-QCD, Phys. Rev.D 97 (2018) 054012 [arXiv:1711.10487] [INSPIRE].ADSGoogle Scholar
  15. [15]
    M.M. Anber and E. Poppitz, Domain walls in high-T SU(N) super Yang-Mills theory and QCD(adj), JHEP05 (2019) 151 [arXiv:1811.10642] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    M.M. Anber and E. Poppitz, Anomaly matching, (axial) Schwinger models and high- T super Yang-Mills domain walls, JHEP09 (2018) 076 [arXiv:1807 .00093] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    A. Karasik and Z. Komargodski, The hi-fundamental gauge theory in 3 + 1 dimensions: the vacuum structure and a cascade, JHEP05 (2019) 144 [arXiv:1904 .09551] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    C. Córdova, D.S. Freed, H.T. Lam and N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications I, arXiv:1905.09315 [INSPIRE].
  19. [19]
    C. Córdova, D.S. Freed, H.T. Lam and N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications II, arXiv:1905 .13361 [INSPIRE].
  20. [20]
    T. Misumi, Y. Tanizaki and M. Ünsal, Fractional θ angle, ’t Hooft anomaly and quantum instantons in charge-q multi-flavor Schwinger model, JHEP07 (2019) 018 [arXiv: 1905 .05781] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    H. Nishimura and Y. Tanizaki, High-temperature domain walls of QCD with imaginary chemical potentials, JHEP06 (2019) 040 [arXiv:1903. 04014] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    G.V. Dunne, Y. Tanizaki and M. Ünsal, Quantum distillation of Hilbert spaces, semi-classics and anomaly matching, JHEP08 (2018) 068 [arXiv: 1803 .02430] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    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
  24. [24]
    A. Cherman, T. Schäfer and M. Ünsal, Chiral lagrangian from duality and monopole operators in compactified QCD, Phys. Rev. Lett.117 (2016) 081601 [arXiv: 1604 . 06108] [INSPIRE].
  25. [25]
    M.M. Anber, E. Poppitz and M. Ünsal, 2d affine XY-spin modelj4d gauge theory duality and deconfinement, JHEP04 (2012) 040 [arXiv: 1112.6389] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  26. [26]
    E. Poppitz, T. Schäfer and M. Ünsal, Continuity, deconfinement and (super) Yang-Mills theory, JHEP10 (2012) 115 [arXiv: 1205. 0290] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    M.M. Anber, S. Collier and E. Poppitz, The SU(3)/Z 3QCD(adj) deconfinement transition via the gauge theory/‘affine’ XY-model duality, JHEP01 (2013) 126 [arXiv:1211. 2824] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  28. [28]
    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
  29. [29]
    M.M. Anber, S. Collier, E. Poppitz, S. Strimas-Mackey and B. Teeple, Deconfinement in N = 1 super Yang-Mills theory on R 3 × S 1via dual-Coulomb gas and “affine” XY-model, JHEP11 (2013) 142 [arXiv:1310 .3522] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    M.M. Anber, E. Poppitz and B. Teeple, Deconfinement and continuity between thermal and (super) Yang-Mills theory for all gauge groups, JHEP09 (2014) 040 [arXiv:1406 . 1199] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    G.V. Dunne and M. Ünsal, New nonperturbative methods in quantum field theory: from large-N orbifold equivalence to bions and resurgence, Ann. Rev. Nucl. Part. Sci.66 (2016) 245 [arXiv: 1601.03414] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    M.M. Anber and T. Sulejmanpasic, The renormalon diagram in gauge theories on R 3 × S 1, JHEP01 (2015) 139 [arXiv:1410 . 0121] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    M.M. Anber and B.J. Kolligs, Entanglement entropy, dualities and deconfinement in gauge theories, JHEP08 (2018) 175 [arXiv:1804 . 01956] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    K. Aitken, A. Cherman, E. Poppitz and L.G. Yaffe, QCD on a small circle, Phys. Rev.D 96 (2017) 096022 [arXiv:1707 . 08971] [INSPIRE].ADSGoogle Scholar
  35. [35]
    Y. Tanizaki, T. Misumi and N. Sakai, Circle compactification and ’t Hooft anomaly, JHEP12 (2017) 056 [arXiv:1710 .08923] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    M. Hongo, T. Misumi and Y. Tanizaki, Phase structure of the twisted SU(3) /U(1)2flag σ-model on R × S 1, JHEP02 (2019) 070 [arXiv:1812 . 02259] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    M.M. Anber and E. Poppitz, Two-flavor adjoint QCD, Phys. Rev.D 98 (2018) 034026 [arXiv: 1805 .12290] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    M.M. Anber and L. Vincent-Genod, Classification of compactified su(N c) gauge theories with fermions in all representations, JHEP12 (2017) 028 [arXiv: 1704. 08277] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    D.D. Dietrich and F. Sannino, Conformal window of SU(N) gauge theories with fermions in higher dimensional representations, Phys. Rev.D 75 (2007) 085018 [hep-ph/0611341] [INSPIRE].ADSMathSciNetGoogle Scholar
  40. [40]
    E. Poppitz and Y. Shang, Chiral lattice gauge theories via mirror-fermion decoupling: a mission (im)possible?, Int. J. Mod. Phys.A 25 (2010) 2761 [arXiv: 1003 .5896] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    S. Bolognesi and K. Konishi, Dynamics and symmetries in chiral SU(N) gauge theories, arXiv: 1906 . 01485 [INSPIRE].
  42. [42]
    T.A. Ryttov and R. Shrock, Ultraviolet to infrared evolution and nonperturbative behavior of SU(N) ⨂ SU(N − 4) ⨂ U(1) chiral gauge theories, Phys. Rev.D 100 (2019) 055009 [arXiv: 1906 . 04255] [INSPIRE].ADSGoogle Scholar
  43. [43]
    S. Yamaguchi, ’t Hooft anomaly matching condition and chiral symmetry breaking without bilinear condensate, JHEP01 (2019) 014 [arXiv: 1811.09390] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    E. Poppitz and M. Ünsal, Conformality or confinement: (IR)relevance of topological excitations, JHEP09 (2009) 050 [arXiv:0906. 5156] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    E. Poppitz and M. Ünsal, Conformality or confinement (II): one-flavor CFTs and mixed-representation QCD, JHEP12 (2009) 011 [arXiv:0910.1245] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    J.C. Myers and M.C. Ogilvie, Phase diagrams of SU(N) gauge theories with fermions in various representations, JHEP07 (2009) 095 [arXiv:0903. 4638] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    D.J. Gross and W. Taylor, Twists and Wilson loops in the string theory of two-dimensional QCD, Nucl. Phys.B 403 (1993) 395 [hep-th/9303046] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    W.E. Caswell, Asymptotic behavior of non-Abelian gauge theories to two loop order, Phys. Rev. Lett.33 (1974) 244 [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    T. Appelquist, K.D. Lane and U. Mahanta, On the ladder approximation for spontaneous chiral symmetry breaking, Phys. Rev. Lett.61 (1988) 1553 [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    C. Córdova and T.T. Dumitrescu, Candidate phases for SU(2) adjoint QCD 4with two flavors from N = 2 supersymmetric Yang-Mills theory, arXiv: 1806 . 09592 [INSPIRE].
  51. [51]
    Z. Bi and T. Senthil, Adventure in topological phase transitions in 3 + 1-D: non-Abelian deconfined quantum criticalities and a possible duality, Phys. Rev.X 9 (2019) 021034 [arXiv : 1808 . 07465] [INSPIRE].CrossRefGoogle Scholar
  52. [52]
    Z. Wan and J. Wang, Adjoint QCD 4, deconfined critical phenomena, symmetry-enriched topological quantum field theory and higher symmetry-extension, Phys. Rev.D 99 (2019) 065013 [arXiv:1812 . 11955] [INSPIRE].ADSMathSciNetGoogle Scholar
  53. [53]
    E. Poppitz and T.A. Ryttov, A possible phase for adjoint QCD, arXiv: 1904.11640 [INSPIRE].
  54. [54]
    E. Poppitz and M. Ünsal, Chiral gauge dynamics and dynamical supersymmetry breaking, JHEP07 (2009) 060 [arXiv:0905 .0634] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    P.C. Argyres and M. Ünsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion and renormalon effects, JHEP08 (2012) 063 [arXiv: 1206 .1890] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    M.F. Atiyah and I.M. Singer, The index of elliptic operators: I, Ann. Math. 87 (1968) 484.MathSciNetzbMATHCrossRefGoogle Scholar
  57. [57]
    T.M.W. Nye and M.A. Singer, An L 2index theorem for Dirac operators on S 1 × R 3, submitted to J. Funct. Anal. (2000) [math.DG/0009144] [INSPIRE].
  58. [58]
    E. Poppitz and M. Ünsal, Index theorem for topological excitations on R 3 × S 1and Chern- Simons theory, JHEP03 (2009) 027 [arXiv:0812. 2085] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    T.C. Kraan and P. van Baal, Periodic instantons with nontrivial holonomy, Nucl. Phys.B 533 (1998) 627 [hep-th/9805168] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  60. [60]
    M.M. Anber and E. Poppitz, Microscopic structure of magnetic bions, JHEP06 (2011) 136 [arXiv: 1105 .0940] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  61. [61]
    M.M. Anber and E. Poppitz, On the global structure of deformed Yang-Mills theory and QCD(adj) on R 3 × S 1, JHEP10 (2015) 051 [arXiv:1508 .00910] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  62. [62]
    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
  63. [63]
    G. ’t Hooft, A property of electric and magnetic flux in non-Abelian gauge theories, Nucl. Phys.B 153 (1979) 141 [INSPIRE].
  64. [64]
    T.D. Cohen, Center symmetry and area laws, Phys. Rev.D 90 (2014) 047703 [arXiv: 1407 .4128] [INSPIRE].ADSGoogle Scholar
  65. [65]
    G. Bergner, P. Giudice, G. Münster, I. Montvay and S. Piemonte, The light bound states of supersymmetric SU(2) Yang-Mills theory, JHEP03 (2016) 080 [arXiv:1512 .07014] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  66. [66]
    T. Banks and A. Zaks, On the phase structure of vector-like gauge theories with massless fermions, Nucl. Phys.B 196 (1982) 189 [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    L.-F. Li, Group theory of the spontaneously broken gauge symmetries, Phys. Rev.D 9 (1974) 1723 [INSPIRE].ADSMathSciNetGoogle Scholar
  68. [68]
    C. Csáki and H. Murayama, Discrete anomaly matching, Nucl. Phys.B 515 (1998) 114 [hep-th/9710105] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  69. [69]
    J.L. Cardy, Is there a c theorem in four-dimensions?, Phys. Lett.B 215 (1988) 749 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  70. [70]
    Z. Wan and J. Wang, Higher anomalies, higher symmetries and cobordisms I: classification of higher-symmetry-protected topological states and their boundary fermionic jbosonic anomalies via a generalized cobordism theory, arXiv:1812.11967 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsLewis & Clark CollegePortlandU.S.A.

Personalised recommendations