Branes and the Kraft-Procesi transition: classical case

  • Santiago CabreraEmail author
  • Amihay Hanany
Open Access
Regular Article - Theoretical Physics


Moduli spaces of a large set of 3d \( \mathcal{N}=4 \) effective gauge theories are known to be closures of nilpotent orbits. This set of theories has recently acquired a special status, due to Namikawa’s theorem. As a consequence of this theorem, closures of nilpotent orbits are the simplest non-trivial moduli spaces that can be found in three dimensional theories with eight supercharges. In the early 80’s mathematicians Hanspeter Kraft and Claudio Procesi characterized an inclusion relation between nilpotent orbit closures of the same classical Lie algebra. We recently [1] showed a physical realization of their work in terms of the motion of D3-branes on the Type IIB superstring embedding of the effective gauge theories. This analysis is restricted to A-type Lie algebras. The present note expands our previous discussion to the remaining classical cases: orthogonal and symplectic algebras. In order to do so we introduce O3-planes in the superstring description. We also find a brane realization for the mathematical map between two partitions of the same integer number known as collapse. Another result is that basic Kraft-Procesi transitions turn out to be described by the moduli space of orthosymplectic quivers with varying boundary conditions.


Brane Dynamics in Gauge Theories Global Symmetries Field Theories in Lower Dimensions Supersymmetric Gauge Theory 


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]
    S. Cabrera and A. Hanany, Branes and the Kraft-Procesi transition, JHEP 11 (2016) 175 [arXiv:1609.07798] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    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].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    H. Kraft and C. Procesi, Minimal singularities in GL n, Invent. Math. 62 (1980) 503.ADSCrossRefGoogle Scholar
  4. [4]
    H. Kraft and C. Procesi, On the geometry of conjugacy classes in classical groups, Comm. Math. Helv. 57 (1982) 539.MathSciNetCrossRefGoogle Scholar
  5. [5]
    A. Giveon and D. Kutasov, Brane dynamics and gauge theory, Rev. Mod. Phys. 71 (1999) 983 [hep-th/9802067] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    B. Feng and A. Hanany, Mirror symmetry by O3 planes, JHEP 11 (2000) 033 [hep-th/0004092] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes Congress. Intern. Math. 2 (1970) 279.MathSciNetGoogle Scholar
  8. [8]
    P. Slodowy, Simple singularities and simple algebraic groups, Lect. Notes Math. 815 (1980) 1.MathSciNetCrossRefGoogle Scholar
  9. [9]
    F. Carta and H. Hayashi, Hilbert series and mixed branches of T [SU(N)] theory, JHEP 02 (2017) 037 [arXiv:1609.08034] [INSPIRE].
  10. [10]
    B. Assel and S. Cremonesi, The infrared physics of bad theories, SciPost Phys. 3 (2017) 024 [arXiv:1707.03403] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    R. Steinberg, Classes of elements of semisimple algebraic groups, in Proceedings of the International Congress of Mathematicians, (1966), pg. 277.Google Scholar
  12. [12]
    D. Luna, Slices étales (in French), Mém. Soc. Math. France 33 (1973) 81.CrossRefGoogle Scholar
  13. [13]
    D.H. Collingwood and W.M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold, (1993).Google Scholar
  14. [14]
    N. Spaltenstein, Classes unipotentes et sous-groupes de Borel (in French), Lect. Notes Math. 946 (1982) 1.Google Scholar
  15. [15]
    R. Carter, Finite groups of Lie type: conjugacy classes and complex characters, John Wiley and Sons, U.S.A., (1985).zbMATHGoogle Scholar
  16. [16]
    W.M. McGovern, The adjoint representation and the adjoint action, in Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, Springer-Verlag, Berlin Heidelberg Germany, (2002).zbMATHGoogle Scholar
  17. [17]
    B. Fu, D. Juteau, P. Levy and E. Sommers, Generic singularities of nilpotent orbit closures, arXiv:1502.05770.
  18. [18]
    W. Nahm, A simple formalism for the BPS monopole, Phys. Lett. B 90 (1980) 413 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    C. Bachas, J. Hoppe and B. Pioline, Nahm equations, N = 1∗ domain walls and D strings in AdS 5 × S 5, JHEP 07 (2001) 041 [hep-th/0007067] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    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].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    S. Gukov and E. Witten, Rigid surface operators, Adv. Theor. Math. Phys. 14 (2010) 87 [arXiv:0804.1561] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  22. [22]
    S.-S. Kim, J. Lindman Hornlund, J. Palmkvist and A. Virmani, Extremal solutions of the S 3 model and nilpotent orbits of G 2(2), JHEP 08 (2010) 072 [arXiv:1004.5242] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    M.B. Green, S.D. Miller and P. Vanhove, Small representations, string instantons and Fourier modes of Eisenstein series, J. Number Theor. 146 (2015) 187 [arXiv:1111.2983] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  24. [24]
    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 [arXiv:1203.2930] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    A. Bourget and J. Troost, Counting the massive vacua of N = 1∗ super Yang-Mills theory, JHEP 08 (2015) 106 [arXiv:1506.03222] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    J.J. Heckman, T. Rudelius and A. Tomasiello, 6D RG flows and nilpotent hierarchies, JHEP 07 (2016) 082 [arXiv:1601.04078] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    Y. Namikawa, A characterization of nilpotent orbit closures among symplectic singularities, arXiv:1603.06105.
  28. [28]
    H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994) 365.MathSciNetCrossRefGoogle Scholar
  29. [29]
    K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    P.B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Diff. Geom. 32 (1990) 473 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  31. [31]
    H. Kraft and C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math. 53 (1979) 227.ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    E. Witten, Baryons and branes in anti-de Sitter space, JHEP 07 (1998) 006 [hep-th/9805112] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  33. [33]
    N.J. Evans, C.V. Johnson and A.D. Shapere, Orientifolds, branes and duality of 4D gauge theories, Nucl. Phys. B 505 (1997) 251 [hep-th/9703210] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    A. Hanany and A. Zaffaroni, Issues on orientifolds: on the brane construction of gauge theories with SO(2N) global symmetry, JHEP 07 (1999) 009 [hep-th/9903242] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    A. Hanany and B. Kol, On orientifolds, discrete torsion, branes and M-theory, JHEP 06 (2000) 013 [hep-th/0003025] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    D. Barbasch and D.A. Vogan, Unipotent representations of complex semisimple groups, Ann. Math. 121 (1985) 41.MathSciNetCrossRefGoogle Scholar
  37. [37]
    P. Achar, An order-reversing duality map for conjugacy classes in Lusztigs canonical quotient, math.RT/0203082.
  38. [38]
    I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford mathematical monographs, Clarendon Press, Oxford U.K., (1995).Google Scholar
  39. [39]
    P. Goddard, J. Nuyts and D.I. Olive, Gauge theories and magnetic charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, T pσ(G) theories and their Hilbert series, JHEP 01 (2015) 150 [arXiv:1410.1548] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    G. Ferlito and A. Hanany, A tale of two cones: the Higgs branch of Sp(n) theories with 2n flavours, arXiv:1609.06724 [INSPIRE].
  42. [42]
    S. Cabrera, A. Hanany and Z. Zhong, Nilpotent orbits and the Coulomb branch of T σ (G) theories: special orthogonal vs orthogonal gauge group factors, JHEP 11 (2017) 079 [arXiv:1707.06941] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, I, Adv. Theor. Math. Phys. 20 (2016) 595 [arXiv:1503.03676] [INSPIRE].
  45. [45]
    A. Braverman, M. Finkelberg and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II, arXiv:1601.03586 [INSPIRE].
  46. [46]
    S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    A. Hanany and R. Kalveks, Quiver theories for moduli spaces of classical group nilpotent orbits, JHEP 06 (2016) 130 [arXiv:1601.04020] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    A. Bourget and A. Pini, Non-connected gauge groups and the plethystic program, JHEP 10 (2017) 033 [arXiv:1706.03781] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: the plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    A. Hanany and M. Sperling, Coulomb branches for rank 2 gauge groups in 3d N = 4 gauge theories, JHEP 08 (2016) 016 [arXiv:1605.00010] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    A. Hanany and M. Sperling, Algebraic properties of the monopole formula, JHEP 02 (2017) 023 [arXiv:1611.07030] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Theoretical Physics, The Blackett LaboratoryImperial College LondonLondonU.K.

Personalised recommendations