Coulomb branches of quiver gauge theories with symmetrizers. (English) Zbl 1508.14011

Nakajima, Hiraku; Weekes, Alex

J. Eur. Math. Soc. (JEMS) 25, No. 1, 203-230 (2023).

Let \(I\) be a finite set. A symmetrizable Cartan matrix \((c_{ij})_{i,j\in I}\) satisfies the following: \(c_{ii} = 2\) for all \(i\in I\), \(c_{ij}\in\mathbb{Z}_{\le 0}\) for all \(i\neq j\), and there is \((d_i)\in \mathbb{Z}_{>0}^I\) such that \(d_i c_{ij} = d_j c_{ji}\) for all \(i,j\). When \(d_i = 1\) for any \(i\in I\), a mathematical definition of the Coulomb branch of a \(3\)-dimensional \(\mathcal N=4\) quiver gauge theory associated with two \(I\)-graded vector spaces \(V = \bigoplus V_i\) and \(W = \bigoplus W_i\) has been given in [A. Braverman et al., Adv. Theor. Math. Phys. 22, No. 5, 1071–1147 (2018; Zbl 1479.81043)] and [H. Nakajima, Adv. Theor. Math. Phys. 20, No. 3, 595–669 (2016; Zbl 1433.81121)], and has been studied in [A. Braverman et al., Adv. Theor. Math. Phys. 23, No. 1, 75–166 (2019; Zbl 1479.81044)].
In this paper, the authors generalize the mathematical definition of Coulomb branches of \(3\)-dimensional \(\mathcal{N}= 4\) supersymmetric (SUSY) quiver gauge theories to the cases with symmetrizers. Here, they realize a symmetrizable Cartan matrix by a folding of a graph. This folding gives a finite group action on the Coulomb branch of the quiver gauge theory of the unfolded graph. Then one may define the Coulomb branch of the symmetrizable theory as the corresponding fixed point subscheme. The authors obtain generalized affine Grassmannian slices of type BCFG as examples of the construction, and their deformation quantizations via truncated shifted Yangians. They also study modules over these quantizations and relate them to the lower triangular part of the quantized enveloping algebra of type \(ADE\).

Reviewer: Mee Seong Im (Annapolis)

MSC:

14D21	Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
16G20	Representations of quivers and partially ordered sets
17B37	Quantum groups (quantized enveloping algebras) and related deformations
81T13	Yang-Mills and other gauge theories in quantum field theory

Keywords:

affine Grassmannian; Coulomb branches; zastava spaces; shifted Yangian; quivers with symmetrizers; quiver gauge theories

Citations:

Zbl 1479.81043; Zbl 1433.81121; Zbl 1479.81044

References:

[1]	Braverman, A., Dobrovolska, G., Finkelberg, M.: Gaiotto-Witten superpotential and Whit-taker D-modules on monopoles. Adv. Math. 300, 451-472 (2016) Zbl 1383.14012 MR 3534838 · Zbl 1383.14012
[2]	Braverman, A., Finkelberg, M.: Semi-infinite Schubert varieties and quantum K-theory of flag manifolds. J. Amer. Math. Soc. 27, 1147-1168 (2014) Zbl 1367.17011 MR 3230820 · Zbl 1367.17011
[3]	Braverman, A., Finkelberg, M., Nakajima, H.: Towards a mathematical definition of Coulomb branches of 3-dimensional N D 4 gauge theories, II. Adv. Theor. Math. Phys. 22, 1071-1147 (2018) MR 3952347 · Zbl 1479.81043
[4]	Braverman, A., Finkelberg, M., Nakajima, H.: Coulomb branches of 3d N D 4 quiver gauge theories and slices in the affine Grassmannian. Adv. Theor. Math. Phys. 23, 75-166 (2019) MR 4020310 · Zbl 1479.81044
[5]	Cremonesi, S., Ferlito, G., Hanany, A., Mekareeya, N.: Coulomb branch and the moduli space of instantons. J. High Energy Phys. 2014, no. 12, art. 103, 35 pp.
[6]	Cremonesi, S., Hanany, A., Zaffaroni, A.: Monopole operators and Hilbert series of Coulomb branches of 3d N D 4 gauge theories. J. High Energy Phys. 2014, no. 1, art. 5, 33 pp.
[7]	De Campos Affonso, H.: Bow varieties for the symplectic group. Master thesis, Kyoto Univ. (2018)
[8]	Dlab, V., Ringel, C. M.: The preprojective algebra of a modulated graph. In: Representation Theory, II (Ottawa, Ont., 1979), Lecture Notes in Math. 832, Springer, Berlin, 216-231 (1980) Zbl 0489.16024 MR 607155 · Zbl 0489.16024
[9]	Finkelberg, M., Kamnitzer, J., Pham, K., Rybnikov, L., Weekes, A.: Comultiplication for shif-ted Yangians and quantum open Toda lattice. Adv. Math. 327, 349-389 (2018) Zbl 1384.81045 MR 3761996 · Zbl 1384.81045
[10]	Finkelberg, M., Kuznetsov, A., Markarian, N., Mirković, I.: A note on a symplectic structure on the space of G-monopoles. Comm. Math. Phys. 201, 411-421 (1999) Zbl 0981.53083 MR 1682218 · Zbl 0981.53083
[11]	Finkelberg, M., Mirković, I.: Semi-infinite flags. I. Case of global curve P 1 . In: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. (2) 194, Amer. Math. Soc., Providence, RI, 81-112 (1999) Zbl 1076.14512 MR 1729360 · Zbl 1076.14512
[12]	Finkelberg, M., Tsymbaliuk, A.: Multiplicative slices, relativistic Toda and shifted quantum affine algebras. In: Representations and Nilpotent Orbits of Lie Algebraic Systems, Progr. Math. 330, Birkhäuser/Springer, Cham, 133-304 (2019) Zbl 1436.17021 MR 3971731 · Zbl 1436.17021
[13]	Frenkel, E., Reshetikhin, N.: The q-characters of representations of quantum affine algebras and deformations of W -algebras. In: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemp. Math. 248, Amer. Math. Soc., Providence, RI, 163-205 (1999) Zbl 0973.17015 MR 1745260 · Zbl 0973.17015
[14]	Fulton, W., Harris, J.: Representation Theory. Grad. Texts in Math. 129, Springer, New York (1991) Zbl 0744.22001 MR 1153249 · Zbl 0744.22001
[15]	Geiß, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable Cartan matrices III: Convolution algebras. Represent. Theory 20, 375-413 (2016) Zbl 1362.16018 MR 3555157 · Zbl 1362.16018
[16]	Geiss, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable Cartan matrices I: Foundations. Invent. Math. 209, 61-158 (2017) Zbl 1395.16006 MR 3660306 · Zbl 1395.16006
[17]	Geiß, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable Cartan matrices II: Change of symmetrizers. Int. Math. Res. Notices 2018, 2866-2898 Zbl 1408.16012 MR 3801499 · Zbl 1408.16012
[18]	Geiss, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable Cartan matrices IV: Crystal graphs and semicanonical functions. Selecta Math. (N.S.) 24, 3283-3348 (2018) Zbl 1434.16007 MR 3848021 · Zbl 1434.16007
[19]	Geiß, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable Cartan matrices V: Caldero-Chapoton formulas. Proc. London Math. Soc. (3) 117, 125-148 (2018) Zbl 06928214 MR 3830892 · Zbl 1468.13051
[20]	Geiß, C., Leclerc, B., Schröer, J.: Rigid modules and Schur roots. Math. Z. 295, 1245-1277 (2020) Zbl 07238516 MR 4125687 · Zbl 1475.16020
[21]	Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a class of representations of the Yangian and moduli space of monopoles. Comm. Math. Phys. 260, 511-525 (2005) Zbl 1142.53069 MR 2182434 · Zbl 1142.53069
[22]	Hernandez, D., Leclerc, B.: A cluster algebra approach to q-characters of Kirillov-Reshetikhin modules. J. Eur. Math. Soc. 18, 1113-1159 (2016) Zbl 1405.17028 MR 3500832 · Zbl 1405.17028
[23]	Kac, V. G.: Infinite Dimensional Lie Algebras. 3rd ed., Cambridge Univ. Press, Cambridge (1990) Zbl 0716.17022 MR 1104219 · Zbl 0716.17022
[24]	Kamnitzer, J., Webster, B., Weekes, A., Yacobi, O.: Yangians and quantizations of slices in the affine Grassmannian. Algebra Number Theory 8, 857-893 (2014) Zbl 1325.14068 MR 3248988 · Zbl 1325.14068
[25]	Kashiwara, M., Kim, M., Oh, S.-j.: Monoidal categories of modules over quantum affine algebras of type A and B. Proc. London Math. Soc. (3) 118, 43-77 (2019) Zbl 07015002 MR 3898986 · Zbl 1472.17054
[26]	Kim, J.-A., Shin, D.-U.: Nakajima monomials, strip bundles, and the crystals B.1/ over U q .B 1 /, U q .C 1 /, and U q .D 1 /. Comm. Algebra 40, 697-718 (2012) Zbl 1267.17014 MR 2889489 · Zbl 1267.17014
[27]	Kimura, T., Pestun, V.: Fractional quiver W-algebras. Lett. Math. Phys. 108, 2425-2451 (2018) Zbl 1402.81245 MR 3861382 · Zbl 1402.81245
[28]	Knight, H.: Spectra of tensor products of finite-dimensional representations of Yangians. J. Algebra 174, 187-196 (1995) Zbl 0868.17009 MR 1332866 · Zbl 0868.17009
[29]	Krylov, V. V.: Integrable crystals and a restriction on a Levi subgroup via generalized slices in the affine Grassmannian. Funktsional. Anal. i Prilozhen. 52, no. 2, 40-65 (2018) (in Russian) Zbl 1423.14086 MR 3799411
[30]	Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc. 4, 365-421 (1991) Zbl 0738.17011 MR 1088333 · Zbl 0738.17011
[31]	Nakajima, H.: Towards geometric Satake correspondence for Kac-Moody algebras -Cherkis bow varieties and affine Lie algebras of type A. arXiv:1810.04293 (2018)
[32]	Nakajima, H.: Quiver varieties and t-analogs of q-characters of quantum affine algebras. Ann. of Math. (2) 160, 1057-1097 (2004) Zbl 1140.17015 MR 2144973 · Zbl 1140.17015
[33]	Nakajima, H.: Towards a mathematical definition of Coulomb branches of 3-dimensional N D 4 gauge theories, I. Adv. Theor. Math. Phys. 20, 595-669 (2016) Zbl 1433.81121 MR 3565863 · Zbl 1433.81121
[34]	Nakajima, H.: Modules of quantized Coulomb branches. In preparation (2019)
[35]	Nandakumar, V., Tingley, P.: Quiver varieties and crystals in symmetrizable type via modu-lated graphs. Math. Res. Lett. 25, 159-180 (2018) Zbl 1457.17014 MR 3818618 · Zbl 1457.17014
[36]	Okounkov, A.: Lectures on K-theoretic computations in enumerative geometry. In: Geometry of Moduli Spaces and Representation Theory, IAS/Park City Math. Ser. 24, Amer. Math. Soc., Providence, RI, 251-380 (2017) Zbl 1402.19001 MR 3752463 · Zbl 1402.19001
[37]	Varagnolo, M., Vasserot, E.: Double affine Hecke algebras and affine flag manifolds, I. In: Affine Flag Manifolds and Principal Bundles, Trends Math., Birkhäuser/Springer, Basel, 233-289 (2010) Zbl 1242.20007 MR 3013034 · Zbl 1242.20007
[38]	Webster, B.: Koszul duality between Higgs and Coulomb categories O. arXiv:1611.06541 (2016)
[39]	Webster, B.: Gelfand-Tsetlin modules in the Coulomb context. arXiv:1904.05415 (2019)
[40]	Weekes, A.: Generators for Coulomb branches of quiver gauge theories. arXiv:1903.07734 (2019)