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Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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We study certain representations of quantum toroidal $\mathfrak{gl}_1$ algebra for $q=t$. We construct explicit bosonization of the Fock modules $\mathcal{F}_u^{(n',n)}$ with a nontrivial slope $n'/n$. As a vector space, it is naturally identified with the basic level $1$ representation of affine $\mathfrak{gl}_n$. We also study twisted $W$-algebras of $\mathfrak{sl}_n$ acting on these Fock modules. As an application, we prove the relation on $q$-deformed conformal blocks which was conjectured in the study of $q$-deformation of isomonodromy/CFT correspondence.
Keywords: quantum algebras, toroidal algebras, $W$-algebras, conformal blocks, Nekrasov partition function, Whittaker vector. 
@article{SIGMA_2020_16_a76,
     author = {Mikhail Bershtein and Roman Gonin},
     title = {Twisted {Representations} of {Algebra} of $q${-Difference} {Operators,} {Twisted} $q$-$W$ {Algebras} and {Conformal} {Blocks}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a76/}
}
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Mikhail Bershtein; Roman Gonin. Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a76/

[1] Awata H., Feigin B., Shiraishi J., “Quantum algebraic approach to refined topological vertex”, J. High Energy Phys., 2012:3 (2012), 041, 35 pp., arXiv: 1112.6074 | DOI | MR | Zbl

[2] Awata H., Yamada Y., “Five-dimensional AGT conjecture and the deformed Virasoro algebra”, J. High Energy Phys., 2010:1 (2010), 125, 11 pp., arXiv: 0910.4431 | DOI | MR | Zbl

[3] Bergeron F., Garsia A. M., Haiman M., Tesler G., “Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions”, Methods Appl. Anal., 16 (1999), 363–420, arXiv: alg-geom/9705013 | DOI | MR

[4] Bershtein M., Gavrylenko P., Marshakov A., “Twist-field representations of W-algebras, exact conformal blocks and character identities”, J. High Energy Phys., 2018:8 (2018), 108, 55 pp., arXiv: 1705.00957 | DOI | MR | Zbl

[5] Bershtein M., Gavrylenko P., Marshakov A., “Cluster Toda lattices and Nekrasov functions”, Theoret. and Math. Phys., 198 (2019), 157–188, arXiv: 1804.10145 | DOI | MR | Zbl

[6] Bershtein M., Gonin R., Twisted and non-twisted deformed Virasoro algebra via vertex operators of ${U}_q(\widehat{\mathfrak{sl}}_2)$, arXiv: 2003.12472

[7] Bershtein M., Shchechkin A., “Bilinear equations on Painlevé $\tau$ functions from CFT”, Comm. Math. Phys., 339 (2015), 1021–1061, arXiv: 1406.3008 | DOI | MR | Zbl

[8] Bershtein M., Shchechkin A., “Bäcklund transformation of Painlevé ${\rm III}(D_8)$ $\tau$ function”, J. Phys. A: Math. Theor., 50 (2017), 115205, 31 pp., arXiv: 1608.02568 | DOI | MR | Zbl

[9] Bershtein M., Shchechkin A., “$q$-deformed Painlevé $\tau$ function and $q$-deformed conformal blocks”, J. Phys. A: Math. Theor., 50 (2017), 085202, 22 pp., arXiv: 1608.02566 | DOI | MR | Zbl

[10] Bershtein M., Shchechkin A., “Painlevé equations from Nakajima-Yoshioka blowup relations”, Lett. Math. Phys., 109 (2019), 2359–2402, arXiv: 1811.04050 | DOI | MR | Zbl

[11] Bonelli G., Grassi A., Tanzini A., “Quantum curves and $q$-deformed Painlevé equations”, Lett. Math. Phys., 109 (2019), 1961–2001, arXiv: 1710.11603 | DOI | MR | Zbl

[12] Burban I., Schiffmann O., “On the Hall algebra of an elliptic curve, I”, Duke Math. J., 161 (2012), 1171–1231, arXiv: math.AG/0505148 | DOI | MR | Zbl

[13] Carlsson E., Mellit A., “A proof of the shuffle conjecture”, J. Amer. Math. Soc., 31 (2018), 661–697, arXiv: 1508.06239 | DOI | MR | Zbl

[14] Fairlie D. B., Fletcher P., Zachos C. K., “Trigonometric structure constants for new infinite-dimensional algebras”, Phys. Lett. B, 218 (1989), 203–206 | DOI | MR | Zbl

[15] Feigin B., Feigin E., Jimbo M., Miwa T., Mukhin E., “Quantum continuous $\mathfrak{gl}_\infty$: semiinfinite construction of representations”, Kyoto J. Math., 51 (2011), 337–364, arXiv: 1002.3113 | DOI | MR | Zbl

[16] Feigin B., Feigin E., Jimbo M., Miwa T., Mukhin E., “Quantum continuous $\mathfrak{gl}_\infty$: tensor products of Fock modules and ${\mathcal W}_n$-characters”, Kyoto J. Math., 51 (2011), 365–392, arXiv: 1002.3113 | DOI | MR | Zbl

[17] Feigin B., Frenkel E., “Quantum $\mathcal W$-algebras and elliptic algebras”, Comm. Math. Phys., 178 (1996), 653–678, arXiv: q-alg/9508009 | DOI | MR | Zbl

[18] Feigin B., Hashizume K., Hoshino A., Shiraishi J., Yanagida S., “A commutative algebra on degenerate $\mathbb{CP}^1$ and Macdonald polynomials”, J. Math. Phys., 50 (2009), 095215, 42 pp., arXiv: 0904.2291 | DOI | MR | Zbl

[19] Feigin B., Hoshino A., Shibahara J., Shiraishi J., Yanagida S., “Kernel function and quantum algebras”, Representation Theory and Combinatorics, RIMS Kôkyûroku, 1689, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010, 133–152, arXiv: 1002.2485

[20] Feigin B., Jimbo M., Miwa T., Mukhin E., “Branching rules for quantum toroidal $\mathfrak{gl}_n$”, Adv. Math., 300 (2016), 229–274, arXiv: 1309.2147 | DOI | MR | Zbl

[21] Frenkel I. B., Kac V. G., “Basic representations of affine Lie algebras and dual resonance models”, Invent. Math., 62 (1980), 23–66 | DOI | MR | Zbl

[22] Fujii S., Minabe S., “A combinatorial study on quiver varieties”, SIGMA, 13 (2017), 052, 28 pp., arXiv: math.AG/0510455 | DOI | MR | Zbl

[23] Gamayun O., Iorgov N., Lisovyy O., “Conformal field theory of Painlevé VI”, J. High Energy Phys., 2012:10 (2012), 038, 25 pp., arXiv: 1207.0787 | DOI | MR

[24] Gavrilenko P. G., Marshakov A. V., “Free fermions, $W$-algebras, and isomonodromic deformations”, Theoret. and Math. Phys., 187 (2016), 649–677, arXiv: 1605.04554 | DOI | MR

[25] Gavrylenko P., Iorgov N., Lisovyy O., “Higher-rank isomonodromic deformations and $W$-algebras”, Lett. Math. Phys., 110 (2020), 327–364, arXiv: 1801.09608 | DOI | MR | Zbl

[26] Gavrylenko P., Lisovyy O., “Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions”, Comm. Math. Phys., 363 (2018), 1–58, arXiv: 1608.00958 | DOI | MR | Zbl

[27] Golenishcheva-Kutuzova M., Lebedev D., “Vertex operator representation of some quantum tori Lie algebras”, Comm. Math. Phys., 148 (1992), 403–416 | DOI | MR | Zbl

[28] Gorsky E., Neguţ A., “Refined knot invariants and Hilbert schemes”, J. Math. Pures Appl., 104 (2015), 403–435, arXiv: 1304.3328 | DOI | MR | Zbl

[29] Gorsky E., Neguţ A., “Infinitesimal change of stable basis”, Selecta Math. (N.S.), 23 (2017), 1909–1930, arXiv: 1510.07964 | DOI | MR | Zbl

[30] Iorgov N., Lisovyy O., Teschner J., “Isomonodromic tau-functions from Liouville conformal blocks”, Comm. Math. Phys., 336 (2015), 671–694, arXiv: 1401.6104 | DOI | MR | Zbl

[31] Jimbo M., Nagoya H., Sakai H., “CFT approach to the $q$-Painlevé VI equation”, J. Integrable Syst., 2 (2017), xyx009, 27 pp., arXiv: 1706.01940 | DOI | MR | Zbl

[32] Kac V. G., Kazhdan D. A., Lepowsky J., Wilson R. L., “Realization of the basic representations of the Euclidean Lie algebras”, Adv. Math., 42 (1981), 83–112 | DOI | MR | Zbl

[33] Kac V. G., Radul A., “Quasifinite highest weight modules over the Lie algebra of differential operators on the circle”, Comm. Math. Phys., 157 (1993), 429–457, arXiv: hep-th/9308153 | DOI | MR | Zbl

[34] Kac V. G., Raina A. K., Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, 29, 2nd ed., World Sci. Publ. Co., Inc., Teaneck, NJ, 2013 | DOI | MR | Zbl

[35] Lepowsky J., Wilson R. L., “Construction of the affine Lie algebra $A_{1}^{{}}(1)$”, Comm. Math. Phys., 62 (1978), 43–53 | DOI | MR | Zbl

[36] Macdonald I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995 | MR

[37] Matsuhira Y., Nagoya H., “Combinatorial expressions for the tau functions of $q$-Painlevé V and III equations”, SIGMA, 15 (2019), 074, 17 pp., arXiv: 1811.03285 | DOI | MR | Zbl

[38] Miki K., “A $(q,\gamma)$ analog of the $W_{1+\infty}$ algebra”, J. Math. Phys., 48 (2007), 123520, 35 pp. | DOI | MR | Zbl

[39] Nagoya H., “Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations”, J. Math. Phys., 56 (2015), 123505, 24 pp., arXiv: 1505.02398 | DOI | MR | Zbl

[40] Neguţ A., Quantum algebras and cyclic quiver varieties, Ph.D. Thesis, Columbia University, 2015, arXiv: 1504.06525 | MR

[41] Neguţ A., “Moduli of flags of sheaves and their $K$-theory”, Algebr. Geom., 2 (2015), 19–43, arXiv: 1209.4242 | DOI | MR | Zbl

[42] Neguţ A., W-algebras associated to surfaces, arXiv: 1710.03217 | MR

[43] Neguţ A., “The $q$-AGT-W relations via shuffle algebras”, Comm. Math. Phys., 358 (2018), 101–170, arXiv: 1608.08613 | DOI | MR | Zbl

[44] Shiraishi J., “Free field constructions for the elliptic algebra $\mathcal{A}_{q,p}(\widehat{\rm sl}_2)$ and Baxter's eight-vertex model”, Internat. J. Modern Phys. A, 19, suppl. (2004), 363–380, arXiv: math.QA/0302097 | DOI | MR | Zbl

[45] Shiraishi J., Kubo H., Awata H., Odake S., “A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions”, Lett. Math. Phys., 38 (1996), 33–51, arXiv: q-alg/9507034 | DOI | MR | Zbl

[46] Taki M., On AGT-W conjecture and $q$-deformed ${W}$-algebra, arXiv: 1403.7016

[47] Tsymbaliuk A., “The affine Yangian of $\mathfrak{gl}_1$ revisited”, Adv. Math., 304 (2017), 583–645, arXiv: 1404.5240 | DOI | MR | Zbl

[48] Zamolodchikov Al.B., “Conformal scalar field on the hyperelliptic curve and critical Ashkin–Teller multipoint correlation functions”, Nuclear Phys. B, 285 (1987), 481–503 | DOI | MR