Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

Gabriele Rembado

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.

Détail

Keywords: quantum integrable systems, quiver varieties, quantum Hamiltonian reduction.
Mots-clés : isomonodromic deformations, deformation quantisation

Bibliographie

[1] Axelrod S., Della Pietra S., Witten E., “Geometric quantization of Chern–Simons gauge theory”, J. Differential Geom., 33 (1991), 787–902 | DOI | MR | Zbl

[2] Baumann P., The $q$-Weyl group of a $q$-Schur algebra, hal-00143359

[3] Boalch P. P., “Symplectic manifolds and isomonodromic deformations”, Adv. Math., 163 (2001), 137–205, arXiv: 2002.00052 | DOI | MR | Zbl

[4] Boalch P. P., “Riemann–Hilbert for tame complex parahoric connections”, Transform. Groups, 16 (2011), 27–50, arXiv: 1003.3177 | DOI | MR | Zbl

[5] Boalch P. P., “Simply-laced isomonodromy systems”, Publ. Math. Inst. Hautes Études Sci., 116 (2012), 1–68, arXiv: 1107.0874 | DOI | MR | Zbl

[6] Boalch P. P., “Geometry and braiding of Stokes data; fission and wild character varieties”, Ann. of Math., 179 (2014), 301–365, arXiv: 1111.6228 | DOI | MR | Zbl

[7] Boalch P. P., Yamakawa D., Twisted wild character varieties, arXiv: 1512.08091

[8] Collingwood D. H., McGovern W. M., Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993 | MR | Zbl

[9] Donin J., Mudrov A., “Explicit equivariant quantization on coadjoint orbits of ${\rm GL}(n,{\mathbb C})$”, Lett. Math. Phys., 62 (2002), 17–32, arXiv: math.QA/0206049 | DOI | MR | Zbl

[10] Donin J., Mudrov A., “Quantum coadjoint orbits in ${\rm gl}(n)$”, Czechoslovak J. Phys., 52 (2002), 1207–1212 | DOI | MR | Zbl

[11] Etingof P., “Calogero–Moser systems and representation theory”, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2007 | DOI | MR

[12] Etingof P., Schiffmann O., Lectures on quantum groups, Lectures in Mathematical Physics, International Press, Boston, MA, 1998 | MR | Zbl

[13] Felder G., Markov Y., Tarasov V., Varchenko A., “Differential equations compatible with KZ equations”, Math. Phys. Anal. Geom., 3 (2000), 139–177, arXiv: math.QA/0001184 | DOI | MR | Zbl

[14] Harnad J., “Dual isomonodromic deformations and moment maps to loop algebras”, Comm. Math. Phys., 166 (1994), 337–365, arXiv: hep-th/9301076 | DOI | MR | Zbl

[15] Hitchin N. J., “Flat connections and geometric quantization”, Comm. Math. Phys., 131 (1990), 347–380 | DOI | MR | Zbl

[16] Jimbo M., Miwa T., Môri Y., Sato M., “Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent”, Phys. D, 1 (1980), 80–158 | DOI | MR | Zbl

[17] Kirillov Jr. A.A., Quiver representations and quiver varieties, Graduate Studies in Mathematics, 174, Amer. Math. Soc., Providence, RI, 2016 | DOI | MR | Zbl

[18] Knizhnik V. G., Zamolodchikov A. B., “Current algebra and Wess–Zumino model in two dimensions”, Nuclear Phys. B, 247 (1984), 83–103 | DOI | MR | Zbl

[19] Le Bruyn L., Procesi C., “Semisimple representations of quivers”, Trans. Amer. Math. Soc., 317 (1990), 585–598 | DOI | MR | Zbl

[20] Losev I., Deformations of symplectic singularities and orbit method for semisimple Lie algebras, arXiv: 1605.00592

[21] Millson J. J., Toledano Laredo V., “Casimir operators and monodromy representations of generalised braid groups”, Transform. Groups, 10 (2005), 217–254, arXiv: math.QA/0305062 | DOI | MR | Zbl

[22] Nagoya H., Yamada Y., “Symmetries of quantum Lax equations for the Painlevé equations”, Ann. Henri Poincaré, 15 (2014), 313–344, arXiv: 1206.5963 | DOI | MR | Zbl

[23] Rembado G., “Simply-laced quantum connections generalising KZ”, Comm. Math. Phys., 368 (2019), 1–54, arXiv: 1704.08616 | DOI | MR | Zbl

[24] Schedler T., “A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver”, Int. Math. Res. Not., 2005 (2005), 725–760, arXiv: math.QA/0406200 | DOI | MR | Zbl

[25] Schedler T., “Deformations of algebras in noncommutative geometry”, Noncommutative algebraic geometry, Math. Sci. Res. Inst. Publ., 64, Cambridge University Press, New York, 2016, 71–165, arXiv: 1212.0914 | MR | Zbl

[26] Schlesinger L., “Über die Lösungen gewisser linearer Differentialgleichungen als Funktionen der singulären Punkte”, J. Reine Angew. Math., 129 (1905), 287–294 | DOI | MR

[27] Toledano Laredo V., “A Kohno–Drinfeld theorem for quantum Weyl groups”, Duke Math. J., 112 (2002), 421–451, arXiv: math.QA/0009181 | DOI | MR | Zbl

[28] Witten E., “Quantization of Chern–Simons gauge theory with complex gauge group”, Comm. Math. Phys., 137 (1991), 29–66 | DOI | MR | Zbl