Skip to main content
SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Journal of High Energy Physics
  3. Article

Crystal bases and three-dimensional 𝒩 = 4 Coulomb branches

  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 11 March 2022
  • volume 2022, Article number: 73 (2022)
Download PDF

You have full access to this open access article

Journal of High Energy Physics Aims and scope Submit manuscript
Crystal bases and three-dimensional 𝒩 = 4 Coulomb branches
Download PDF
  • Leonardo Santilli  ORCID: orcid.org/0000-0002-2497-48681 &
  • Miguel Tierz2,1 

  • 132 Accesses

  • 2 Citations

  • 1 Altmetric

  • Explore all metrics

  • Cite this article

A preprint version of the article is available at arXiv.

Abstract

We establish and develop a correspondence between certain crystal bases (Kashiwara crystals) and the Coulomb branch of three-dimensional 𝒩 = 4 gauge theories. The result holds for simply-laced, non-simply laced and affine quivers. Two equivalent derivations are given in the non-simply laced case, either by application of the axiomatic rules or by folding a simply-laced quiver. We also study the effect of turning on real masses and the ensuing simplification of the crystal. We present a multitude of explicit examples of the equivalence. Finally, we put forward a correspondence between infinite crystals and Hilbert spaces of theories with isolated vacua.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. M. R. Douglas and G. W. Moore, D-branes, quivers, and ALE instantons, hep-th/9603167 [INSPIRE].

  2. H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994) 365 [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  3. V. G. Kac, Infinite-dimensional Lie algebras, 3 ed., Cambridge university press, (1990), [DOI].

  4. N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in Conference on the Mathematical Beauty of Physics (In Memory of C. Itzykson), (1996), pp. 333–366 [hep-th/9607163] [INSPIRE].

  5. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. N. J. Hitchin, A. Karlhede, U. Lindström and M. Roček, HyperKähler Metrics and Supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  7. M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb Branch of 3d 𝒩 = 4 Theories, Commun. Math. Phys. 354 (2017) 671 [arXiv:1503.04817] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. 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].

    Article  ADS  Google Scholar 

  9. 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].

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Braverman, M. Finkelberg and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional 𝒩 = 4 gauge theories, II, Adv. Theor. Math. Phys. 22 (2018) 1071 [arXiv:1601.03586] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Braverman, M. Finkelberg and H. Nakajima, Coulomb branches of 3d 𝒩 = 4 quiver gauge theories and slices in the affine Grassmannian, Adv. Theor. Math. Phys. 23 (2019) 75 [arXiv:1604.03625] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Finkelberg, Doule affine Grassmannians and Coulomb branches of 3d N = 4 quiver gauge theories, in International Congress of Mathematicians, World Scientific, Singapore (2018), pp. 1279–1298 [arXiv:1712.03039] [INSPIRE].

  13. A. Braverman, M. Finkelberg and H. Nakajima, Line bundles over Coulomb branches, arXiv:1805.11826 [INSPIRE].

  14. J. Kamnitzer, P. Tingley, B. Webster, A. Weekes and O. Yacobi, On category O for affine Grassmannian slices and categorified tensor products, Proc. Lond. Math. Soc. 119 (2019) 1179 [arXiv:1806.07519].

  15. D. Muthiah and A. Weekes, Symplectic leaves for generalized affine Grassmannian slices, arXiv:1902.09771 [INSPIRE].

  16. A. Weekes, Generators for Coulomb branches of quiver gauge theories, arXiv:1903.07734 [INSPIRE].

  17. H. Nakajima and A. Weekes, Coulomb branches of quiver gauge theories with symmetrizers, arXiv:1907.06552 [INSPIRE].

  18. A. Dancer, A. Hanany and F. Kirwan, Symplectic duality and implosions, arXiv:2004.09620 [INSPIRE].

  19. A. Weekes, Quiver gauge theories and symplectic singularities, Adv. Math. 396 (2022) 108185 [arXiv:2005.01702] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Zhou, Note on some properties of generalized affine Grassmannian slices, arXiv:2011.04109 [INSPIRE].

  21. M. Dedushenko, Y. Fan, S. S. Pufu and R. Yacoby, Coulomb Branch Operators and Mirror Symmetry in Three Dimensions, JHEP 04 (2018) 037 [arXiv:1712.09384] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. M. Dedushenko, Y. Fan, S. S. Pufu and R. Yacoby, Coulomb Branch Quantization and Abelianized Monopole Bubbling, JHEP 10 (2019) 179 [arXiv:1812.08788] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. C. Beem, W. Peelaers and L. Rastelli, Deformation quantization and superconformal symmetry in three dimensions, Commun. Math. Phys. 354 (2017) 345 [arXiv:1601.05378] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  25. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling and Z. Zhong, Branes, Quivers, and the Affine Grassmannian, arXiv:2102.06190 [INSPIRE].

  27. G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Am. Math. Soc. 3 (1990) 447.

  28. G. Lusztig, Canonical bases arising from quantized enveloping algebras. II, Prog. Theor. Phys. Suppl. 102 (1991) 175.

  29. G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Am. Math. Soc. 4 (1991) 365.

  30. M. Kashiwara, Crystalizing the Q Analog of Universal Enveloping Algebras, Commun. Math. Phys. 133 (1990) 249 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  31. M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 62 (1990) 465.

  32. M. Kashiwara, On crystal bases, in Canadian Math. Conf. Proc., vol. 16, (Providence, RI), p. 155-197, AMS, (1995).

  33. G. Grojnowski and I. Lusztig, A comparison of bases of quantized enveloping algebras, Contemp. Math. 153 (1993) 11.

  34. A. Braverman and D. Gaitsgory, Crystals via the affine Grassmannian, Duke Math. J. 107 (2001) 561 [math/9909077].

  35. D. Bump and A. Schilling, Crystal Bases. World Scientific, Singapore (2017), [DOI].

  36. J. R. Stembridge, A local characterization of simply-laced crystals, Trans. Am. Math. Soc. 355 (2003) 4807.

  37. D. Bump, A. Schilling and B. Salisbury, Lie Methods and Related Combinatorics in Sage, in Sage Thematic Tutorials, SAGE, (2015).

  38. M. Bullimore, T. Dimofte, D. Gaiotto, J. Hilburn and H.-C. Kim, Vortices and Vermas, Adv. Theor. Math. Phys. 22 (2018) 803 [arXiv:1609.04406] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Bump, Lie Groups, vol. 225 of Graduate texts in mathematics, Springer-Verlag, New York, U.S.A., (2013), [DOI].

  40. V. Ginzburg, Lectures on Nakajima’s Quiver Varieties, [arXiv:0905.0686].

  41. D. Tong, Three-dimensional gauge theories and ADE monopoles, Phys. Lett. B 448 (1999) 33 [hep-th/9803148] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. H. Kraft and C. Procesi, On the geometry of conjugacy classes in classical groups, Comment. Math. Helv. 57 (1982) 539.

  43. P. Z. Kobak and A. Swann, Classical nilpotent orbits as hyperkähler quotients, Int. J. Math. 07 (1996) 193.

  44. Y. Namikawa, A characterization of nilpotent orbit closures among symplectic singularities, Math. Ann. 370 (2018) 811 [arXiv:1603.06105].

  45. A. Hanany and R. Kalveks, Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits, JHEP 06 (2016) 130 [arXiv:1601.04020] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. S. Cabrera and A. Hanany, Branes and the Kraft-Procesi Transition, JHEP 11 (2016) 175 [arXiv:1609.07798] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  47. S. Cabrera, A. Hanany and Z. Zhong, Nilpotent orbits and the Coulomb branch of Tσ(G) theories: special orthogonal vs orthogonal gauge group factors, JHEP 11 (2017) 079 [arXiv:1707.06941] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. A. Hanany and M. Sperling, Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional 𝒩 = 4 theories, JHEP 08 (2018) 189 [arXiv:1806.01890] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. A. Hanany and R. Kalveks, Quiver Theories and Hilbert Series of Classical Slodowy Intersections, Nucl. Phys. B 952 (2020) 114939 [arXiv:1909.12793] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  50. A. Bourget et al., The Higgs mechanism — Hasse diagrams for symplectic singularities, JHEP 01 (2020) 157 [arXiv:1908.04245] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  51. J. F. Grimminger and A. Hanany, Hasse diagrams for 3d 𝒩 = 4 quiver gauge theories — Inversion and the full moduli space, JHEP 09 (2020) 159 [arXiv:2004.01675] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. A. Beauville, Symplectic singularities, Invent. Math. 139 (2000) 541 [math/9903070].

  53. D. Kaledin, Symplectic singularities from the Poisson point of view, J. Reine Angew. Math. 2006 (2006) 135 [math/0310186].

  54. K. A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  55. M. Aganagic, Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves, arXiv:2004.14518 [INSPIRE].

  56. M. Aganagic, Knot Categorification from Mirror Symmetry, Part II: Lagrangians, arXiv:2105.06039 [INSPIRE].

  57. J. Hong and S.-J. Kang, Introduction to Quantum Groups and Crystal Bases, vol. 42 of Graduate Studies in Mathematics, AMS, Providence, RI, U.S.A. (2002), [DOI].

  58. M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997) 9 [q-alg/9606009].

  59. I. Mirković and M. Vybornov, On quiver varieties and affine Grassmannians of type A, Compt. Rend. Math. 336 (2003) 207 [math/0206084].

  60. A. Dranowski, Comparing two perfect bases, Ph.D. Thesis, University of Toronto, Canada (2020).

  61. A. Berenstein and D. Kazhdan, Geometric and Unipotent Crystals, in Visions in Mathematics. GAFA, Special Volume, Part I, N. Alon, J. Bourgain, A. Connes, M. Gromov and V. Milman, eds., Modern Birkhauser Classics, pp. 188–236. Birkhauser, Basel, Germany (2010). [math/9912105]. [DOI].

  62. V. Krylov, Integrable Crystals and Restriction to Levi Subgroups Via Generalized Slices in the Affine Grassmannian, Funct. Anal. Appl. 52 (2018) 113 [arXiv:1709.00391].

  63. S. Cabrera and A. Hanany, Quiver Subtractions, JHEP 09 (2018) 008 [arXiv:1803.11205] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. M. Kashiwara, Similarity of Crystal Bases, in Lie algebras and their representations (Seoul 1995), vol. 16 of Contemp. Math., pp. 177–186. AMS, Providence, RI, U.S.A. (1996).

  65. A. Dey, A. Hanany, P. Koroteev and N. Mekareeya, On Three-Dimensional Quiver Gauge Theories of Type B, JHEP 09 (2017) 067 [arXiv:1612.00810] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. A. Bourget, A. Hanany and D. Miketa, Quiver origami: discrete gauging and folding, JHEP 01 (2021) 086 [arXiv:2005.05273] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  67. S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992) 499.

  68. M. Shimozono, Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties, J. Algebr. Comb. 15 (2002) 151 [math/9804039].

  69. M. Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993) 839.

  70. P. Littelmann, Crystal Graphs and Young Tableaux, J. Algebra 175 (1995) 65.

  71. D. Gaiotto and T. Okazaki, Sphere correlation functions and Verma modules, JHEP 02 (2020) 133 [arXiv:1911.11126] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  72. M. Bullimore, S. Crew and D. Zhang, Boundaries, Vermas, and Factorisation, JHEP 04 (2021) 263 [arXiv:2010.09741] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  73. L. Santilli and M. Tierz, Exact results and Schur expansions in quiver Chern-Simons-matter theories, JHEP 10 (2020) 022 [arXiv:2008.00465] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. A. Kapustin, B. Willett and I. Yaakov, Nonperturbative Tests of Three-Dimensional Dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  75. T. Nishioka, Y. Tachikawa and M. Yamazaki, 3d Partition Function as Overlap of Wavefunctions, JHEP 08 (2011) 003 [arXiv:1105.4390] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  76. S. Benvenuti and S. Pasquetti, 3D-partition functions on the sphere: exact evaluation and mirror symmetry, JHEP 05 (2012) 099 [arXiv:1105.2551] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  77. J. G. Russo and M. Tierz, Quantum phase transition in many-flavor supersymmetric QED3, Phys. Rev. D 95 (2017) 031901 [arXiv:1610.08527] [INSPIRE].

    Article  ADS  Google Scholar 

  78. L. Santilli and M. Tierz, SQED3 and SQCD3: Phase transitions and integrability, Phys. Rev. D 100 (2019) 061702 [arXiv:1906.09917] [INSPIRE].

    Article  ADS  Google Scholar 

  79. A. Hanany and N. Mekareeya, Complete Intersection Moduli Spaces in N = 4 Gauge Theories in Three Dimensions, JHEP 01 (2012) 079 [arXiv:1110.6203] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  80. S. Benvenuti, A. Hanany and N. Mekareeya, The Hilbert Series of the One Instanton Moduli Space, JHEP 06 (2010) 100 [arXiv:1005.3026] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. Y. Chen and N. Mekareeya, The Hilbert series of U/SU SQCD and Toeplitz Determinants, Nucl. Phys. B 850 (2011) 553 [arXiv:1104.2045] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  82. L. Santilli and M. Tierz, Exact equivalences and phase discrepancies between random matrix ensembles, J. Stat. Mech. 2008 (2020) 083107 [arXiv:2003.10475] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  83. J. E. Anderson, A polytope calculus for semisimple groups, Duke Math. J. 116 (2003) 567 [math/0110225].

  84. J. Kamnitzer, Mirkovic-Vilonen cycles and polytopes, Annals Math. 171 (2010) 245 [math/0501365].

  85. I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Annals Math. 166 (2007) 95 [math/0401222].

  86. B. Webster, Weighted Khovanov-Lauda-Rouquier algebras, Doc Math. 24 (2019) 209 [arXiv:1209.2463].

  87. P. Tingley and B. Webster, Mirković-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras, Compositio Math. 152 (2016) 1648 [arXiv:1210.6921].

  88. P. Littelmann, Paths and root operators in representation theory, Annals Math 142 (1995) 499.

  89. P. Biane, P. Bougerol and N. O’Connell, Littelmann paths and Brownian paths, Duke Math. J. 130 (2005) 127 [math/0403171].

  90. S. de Haro and M. Tierz, Brownian motion, Chern-Simons theory, and 2-D Yang-Mills, Phys. Lett. B 601 (2004) 201 [hep-th/0406093] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  91. M. van Beest, A. Bourget, J. Eckhard and S. Schäfer-Nameki, (Symplectic) Leaves and (5d Higgs) Branches in the Poly(go)nesian Tropical Rain Forest, JHEP 11 (2020) 124 [arXiv:2008.05577] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  92. G. Ferlito, A. Hanany, N. Mekareeya and G. Zafrir, 3d Coulomb branch and 5d Higgs branch at infinite coupling, JHEP 07 (2018) 061 [arXiv:1712.06604] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  93. S. Cabrera, A. Hanany and F. Yagi, Tropical Geometry and Five Dimensional Higgs Branches at Infinite Coupling, JHEP 01 (2019) 068 [arXiv:1810.01379] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  94. C. Closset, S. Schäfer-Nameki and Y.-N. Wang, Coulomb and Higgs Branches from Canonical Singularities: Part 0, JHEP 02 (2021) 003 [arXiv:2007.15600] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Grupo de Física Matemática, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Edifício C6, 1749-016, Campo Grande, Lisboa, Portugal

    Leonardo Santilli & Miguel Tierz

  2. Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias, 3, 28040, Madrid, Spain

    Miguel Tierz

Authors
  1. Leonardo Santilli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miguel Tierz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leonardo Santilli.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2111.05206

Rights and permissions

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.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Santilli, L., Tierz, M. Crystal bases and three-dimensional 𝒩 = 4 Coulomb branches. J. High Energ. Phys. 2022, 73 (2022). https://doi.org/10.1007/JHEP03(2022)073

Download citation

  • Received: 12 January 2022

  • Accepted: 23 February 2022

  • Published: 11 March 2022

  • DOI: https://doi.org/10.1007/JHEP03(2022)073

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Supersymmetric Gauge Theory
  • Field Theories in Lower Dimensions
  • Differential and Algebraic Geometry

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

Not affiliated

Springer Nature

© 2023 Springer Nature