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.
M. R. Douglas and G. W. Moore, D-branes, quivers, and ALE instantons, hep-th/9603167 [INSPIRE].
H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994) 365 [INSPIRE].
V. G. Kac, Infinite-dimensional Lie algebras, 3 ed., Cambridge university press, (1990), [DOI].
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].
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].
N. J. Hitchin, A. Karlhede, U. Lindström and M. Roček, HyperKähler Metrics and Supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].
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].
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].
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].
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].
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].
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].
A. Braverman, M. Finkelberg and H. Nakajima, Line bundles over Coulomb branches, arXiv:1805.11826 [INSPIRE].
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].
D. Muthiah and A. Weekes, Symplectic leaves for generalized affine Grassmannian slices, arXiv:1902.09771 [INSPIRE].
A. Weekes, Generators for Coulomb branches of quiver gauge theories, arXiv:1903.07734 [INSPIRE].
H. Nakajima and A. Weekes, Coulomb branches of quiver gauge theories with symmetrizers, arXiv:1907.06552 [INSPIRE].
A. Dancer, A. Hanany and F. Kirwan, Symplectic duality and implosions, arXiv:2004.09620 [INSPIRE].
A. Weekes, Quiver gauge theories and symplectic singularities, Adv. Math. 396 (2022) 108185 [arXiv:2005.01702] [INSPIRE].
Y. Zhou, Note on some properties of generalized affine Grassmannian slices, arXiv:2011.04109 [INSPIRE].
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].
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].
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].
A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
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].
A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling and Z. Zhong, Branes, Quivers, and the Affine Grassmannian, arXiv:2102.06190 [INSPIRE].
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Am. Math. Soc. 3 (1990) 447.
G. Lusztig, Canonical bases arising from quantized enveloping algebras. II, Prog. Theor. Phys. Suppl. 102 (1991) 175.
G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Am. Math. Soc. 4 (1991) 365.
M. Kashiwara, Crystalizing the Q Analog of Universal Enveloping Algebras, Commun. Math. Phys. 133 (1990) 249 [INSPIRE].
M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 62 (1990) 465.
M. Kashiwara, On crystal bases, in Canadian Math. Conf. Proc., vol. 16, (Providence, RI), p. 155-197, AMS, (1995).
G. Grojnowski and I. Lusztig, A comparison of bases of quantized enveloping algebras, Contemp. Math. 153 (1993) 11.
A. Braverman and D. Gaitsgory, Crystals via the affine Grassmannian, Duke Math. J. 107 (2001) 561 [math/9909077].
D. Bump and A. Schilling, Crystal Bases. World Scientific, Singapore (2017), [DOI].
J. R. Stembridge, A local characterization of simply-laced crystals, Trans. Am. Math. Soc. 355 (2003) 4807.
D. Bump, A. Schilling and B. Salisbury, Lie Methods and Related Combinatorics in Sage, in Sage Thematic Tutorials, SAGE, (2015).
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].
D. Bump, Lie Groups, vol. 225 of Graduate texts in mathematics, Springer-Verlag, New York, U.S.A., (2013), [DOI].
V. Ginzburg, Lectures on Nakajima’s Quiver Varieties, [arXiv:0905.0686].
D. Tong, Three-dimensional gauge theories and ADE monopoles, Phys. Lett. B 448 (1999) 33 [hep-th/9803148] [INSPIRE].
H. Kraft and C. Procesi, On the geometry of conjugacy classes in classical groups, Comment. Math. Helv. 57 (1982) 539.
P. Z. Kobak and A. Swann, Classical nilpotent orbits as hyperkähler quotients, Int. J. Math. 07 (1996) 193.
Y. Namikawa, A characterization of nilpotent orbit closures among symplectic singularities, Math. Ann. 370 (2018) 811 [arXiv:1603.06105].
A. Hanany and R. Kalveks, Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits, JHEP 06 (2016) 130 [arXiv:1601.04020] [INSPIRE].
S. Cabrera and A. Hanany, Branes and the Kraft-Procesi Transition, JHEP 11 (2016) 175 [arXiv:1609.07798] [INSPIRE].
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].
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].
A. Hanany and R. Kalveks, Quiver Theories and Hilbert Series of Classical Slodowy Intersections, Nucl. Phys. B 952 (2020) 114939 [arXiv:1909.12793] [INSPIRE].
A. Bourget et al., The Higgs mechanism — Hasse diagrams for symplectic singularities, JHEP 01 (2020) 157 [arXiv:1908.04245] [INSPIRE].
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].
A. Beauville, Symplectic singularities, Invent. Math. 139 (2000) 541 [math/9903070].
D. Kaledin, Symplectic singularities from the Poisson point of view, J. Reine Angew. Math. 2006 (2006) 135 [math/0310186].
K. A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
M. Aganagic, Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves, arXiv:2004.14518 [INSPIRE].
M. Aganagic, Knot Categorification from Mirror Symmetry, Part II: Lagrangians, arXiv:2105.06039 [INSPIRE].
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].
M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997) 9 [q-alg/9606009].
I. Mirković and M. Vybornov, On quiver varieties and affine Grassmannians of type A, Compt. Rend. Math. 336 (2003) 207 [math/0206084].
A. Dranowski, Comparing two perfect bases, Ph.D. Thesis, University of Toronto, Canada (2020).
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].
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].
S. Cabrera and A. Hanany, Quiver Subtractions, JHEP 09 (2018) 008 [arXiv:1803.11205] [INSPIRE].
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).
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].
A. Bourget, A. Hanany and D. Miketa, Quiver origami: discrete gauging and folding, JHEP 01 (2021) 086 [arXiv:2005.05273] [INSPIRE].
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.
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].
M. Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993) 839.
P. Littelmann, Crystal Graphs and Young Tableaux, J. Algebra 175 (1995) 65.
D. Gaiotto and T. Okazaki, Sphere correlation functions and Verma modules, JHEP 02 (2020) 133 [arXiv:1911.11126] [INSPIRE].
M. Bullimore, S. Crew and D. Zhang, Boundaries, Vermas, and Factorisation, JHEP 04 (2021) 263 [arXiv:2010.09741] [INSPIRE].
L. Santilli and M. Tierz, Exact results and Schur expansions in quiver Chern-Simons-matter theories, JHEP 10 (2020) 022 [arXiv:2008.00465] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Nonperturbative Tests of Three-Dimensional Dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].
T. Nishioka, Y. Tachikawa and M. Yamazaki, 3d Partition Function as Overlap of Wavefunctions, JHEP 08 (2011) 003 [arXiv:1105.4390] [INSPIRE].
S. Benvenuti and S. Pasquetti, 3D-partition functions on the sphere: exact evaluation and mirror symmetry, JHEP 05 (2012) 099 [arXiv:1105.2551] [INSPIRE].
J. G. Russo and M. Tierz, Quantum phase transition in many-flavor supersymmetric QED3, Phys. Rev. D 95 (2017) 031901 [arXiv:1610.08527] [INSPIRE].
L. Santilli and M. Tierz, SQED3 and SQCD3: Phase transitions and integrability, Phys. Rev. D 100 (2019) 061702 [arXiv:1906.09917] [INSPIRE].
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].
S. Benvenuti, A. Hanany and N. Mekareeya, The Hilbert Series of the One Instanton Moduli Space, JHEP 06 (2010) 100 [arXiv:1005.3026] [INSPIRE].
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].
L. Santilli and M. Tierz, Exact equivalences and phase discrepancies between random matrix ensembles, J. Stat. Mech. 2008 (2020) 083107 [arXiv:2003.10475] [INSPIRE].
J. E. Anderson, A polytope calculus for semisimple groups, Duke Math. J. 116 (2003) 567 [math/0110225].
J. Kamnitzer, Mirkovic-Vilonen cycles and polytopes, Annals Math. 171 (2010) 245 [math/0501365].
I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Annals Math. 166 (2007) 95 [math/0401222].
B. Webster, Weighted Khovanov-Lauda-Rouquier algebras, Doc Math. 24 (2019) 209 [arXiv:1209.2463].
P. Tingley and B. Webster, Mirković-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras, Compositio Math. 152 (2016) 1648 [arXiv:1210.6921].
P. Littelmann, Paths and root operators in representation theory, Annals Math 142 (1995) 499.
P. Biane, P. Bougerol and N. O’Connell, Littelmann paths and Brownian paths, Duke Math. J. 130 (2005) 127 [math/0403171].
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].
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].
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].
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].
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].
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.
About this article
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
- Supersymmetric Gauge Theory
- Field Theories in Lower Dimensions
- Differential and Algebraic Geometry