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