Geodesic
Non-Stationary Ruijsenaars Functions for $\kappa=t^{-1/N}$ and Intertwining Operators of Ding–Iohara–Miki Algebra
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We construct the non-stationary Ruijsenaars functions (affine analogue of the Macdonald functions) in the special case $\kappa=t^{-1/N}$, using the intertwining operators of the Ding–Iohara–Miki algebra (DIM algebra) associated with $N$-fold Fock tensor spaces. By the $S$-duality of the intertwiners, another expression is obtained for the non-stationary Ruijsenaars functions with $\kappa=t^{-1/N}$, which can be regarded as a natural elliptic lift of the asymptotic Macdonald functions to the multivariate elliptic hypergeometric series. We also investigate some properties of the vertex operator of the DIM algebra appearing in the present algebraic framework; an integral operator which commutes with the elliptic Ruijsenaars operator, and the degeneration of the vertex operators to the Virasoro primary fields in the conformal limit $q \rightarrow 1$.
Keywords: Macdonald function, Rujisenaars function, Ding–Iohara–Miki algebra. 
@article{SIGMA_2020_16_a115,
     author = {Masayuki Fukuda and Yusuke Ohkubo and Jun'ichi Shiraishi},
     title = {Non-Stationary {Ruijsenaars} {Functions} for $\kappa=t^{-1/N}$ and {Intertwining} {Operators} of {Ding{\textendash}Iohara{\textendash}Miki} {Algebra}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a115/}
}
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Masayuki Fukuda; Yusuke Ohkubo; Jun'ichi Shiraishi. Non-Stationary Ruijsenaars Functions for $\kappa=t^{-1/N}$ and Intertwining Operators of Ding–Iohara–Miki Algebra. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a115/

[1] Alba V. A., Fateev V. A., Litvinov A. V., Tarnopolskiy G. M., “On combinatorial expansion of the conformal blocks arising from AGT conjecture”, Lett. Math. Phys., 98 (2011), 33–64, arXiv: 1012.1312 | DOI | MR | Zbl

[2] Alday L. F., Gaiotto D., Tachikawa Y., “Liouville correlation functions from four-dimensional gauge theories”, Lett. Math. Phys., 91 (2010), 167–197, arXiv: 0906.3219 | DOI | MR | Zbl

[3] Atai F., Langmann E., “Series solutions of the non-stationary Heun equation”, SIGMA, 14 (2018), 011, 32 pp., arXiv: 1609.02525 | DOI | MR | Zbl

[4] Atai F., Langmann E., “Exact solutions by integrals of the non-stationary elliptic Calogero–Sutherland equation”, J. Integrable Syst., 5 (2020), xyaa001, 26 pp., arXiv: 1908.00529 | DOI | MR | Zbl

[5] Awata H., Feigin B., Hoshino A., Kanai M., Shiraishi J., Yanagida S., “Notes on Ding–Iohara algebra and AGT conjecture”, RIMS Kōkyūroku, 1765 (2011), 12–32, arXiv: 1106.4088

[6] 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

[7] Awata H., Fujino H., Ohkubo Y., “Crystallization of deformed Virasoro algebra, Ding–Iohara–Miki algebra, and 5D AGT correspondence”, J. Math. Phys., 58 (2017), 071704, 25 pp., arXiv: 1512.08016 | DOI | MR | Zbl

[8] Awata H., Kanno H., “Refined BPS state counting from Nekrasov's formula and Macdonald functions”, Internat. J. Modern Phys. A, 24 (2009), 2253–2306, arXiv: 0805.0191 | DOI | MR | Zbl

[9] Awata H., Kanno H., “Changing the preferred direction of the refined topological vertex”, J. Geom. Phys., 64 (2013), 91–110, arXiv: 0903.5383 | DOI | MR | Zbl

[10] Awata H., Matsuo Y., Odake S., Shiraishi J., “Collective field theory, Calogero–Sutherland model and generalized matrix models”, Phys. Lett. B, 347 (1995), 49–55, arXiv: hep-th/9411053 | DOI | MR | Zbl

[11] Braverman A., Finkelberg M., Shiraishi J., “Macdonald polynomials, Laumon spaces and perverse coherent sheaves”, Perspectives in Representation Theory, Contemp. Math., 610, Amer. Math. Soc., Providence, RI, 2014, 23–41, arXiv: 1206.3131 | DOI | MR | Zbl

[12] Carlsson E., Nekrasov N., Okounkov A., “Five dimensional gauge theories and vertex operators”, Mosc. Math. J., 14 (2014), 39–61, arXiv: 1308.2465 | DOI | MR | Zbl

[13] Ding J., Iohara K., “Generalization of Drinfeld quantum affine algebras”, Lett. Math. Phys., 41 (1997), 181–193, arXiv: q-alg/9608002 | DOI | MR | Zbl

[14] 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: 1002.3100 | DOI | MR | Zbl

[15] 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

[16] Feigin B., Hoshino A., Shibahara J., Shiraishi J., Yanagida S., “Kernel function and quantum algebra”, RIMS Kōkyūroku, 1689 (2010), 133–152, arXiv: 1002.2485

[17] Feigin B., Kojima T., Shiraishi J., Watanabe H., The integrals of motion for the deformed Virasoro algebra, arXiv: 0705.0427

[18] Feigin B., Kojima T., Shiraishi J., Watanabe H., The integrals of motion for the deformed $W$-algebra $W_{q,t}(\hat{sl}_N)$, arXiv: 0705.0627

[19] Feigin B. L., Tsymbaliuk A. I., “Equivariant $K$-theory of Hilbert schemes via shuffle algebra”, Kyoto J. Math., 51 (2011), 831–854, arXiv: 0904.1679 | DOI | MR | Zbl

[20] Felder G., Varchenko A., “Hypergeometric theta functions and elliptic Macdonald polynomials”, Int. Math. Res. Not., 2004 (2004), 1037–1055, arXiv: math.QA/0309452 | DOI | MR | Zbl

[21] Fukuda M., Ohkubo Y., Shiraishi J., “Generalized Macdonald functions on Fock tensor spaces and duality formula for changing preferred direction”, Comm. Math. Phys., 380 (2020), 1–70, arXiv: 1903.05905 | DOI | MR | Zbl

[22] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[23] Hausel T., Rodriguez-Villegas F., “Mixed Hodge polynomials of character varieties”, Invent. Math., 174 (2008), 555–624, arXiv: math.AG/0612668 | DOI | MR | Zbl

[24] Iqbal A., Kozçaz C., Vafa C., “The refined topological vertex”, J. High Energy Phys., 2009:10 (2009), 069, 58 pp., arXiv: hep-th/0701156 | DOI | MR

[25] Kojima T., Shiraishi J., “The integrals of motion for the deformed $W$-algebra $W_{q,t}(\widehat{gl_N})$. II Proof of the commutation relations”, Comm. Math. Phys., 283 (2008), 795–851, arXiv: 0709.2305 | DOI | MR | Zbl

[26] Langmann E., Noumi M., Shiraishi J., “Basic properties of non-stationary Ruijsenaars functions”, SIGMA, 16 (2020), 105, 26 pp., arXiv: 2006.07171 | DOI | MR

[27] Macdonald I. G., Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, 2nd ed., The Clarendon Press, Oxford University Press, New York, 2015 | MR | Zbl

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

[29] Morozov A., Smirnov A., “Towards the proof of AGT relations with the help of the generalized Jack polynomials”, Lett. Math. Phys., 104 (2014), 585–612, arXiv: 1307.2576 | DOI | MR | Zbl

[30] Nedelin A., Pasquetti S., Zenkevich Y., “$T[{\rm SU}(N)]$ duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences”, J. High Energy Phys., 2019:2 (2019), 176, 57 pp., arXiv: 1712.08140 | DOI | MR | Zbl

[31] Noumi M., Shiraishi J., A direct approach to the bispectral problem for the Ruijsenaars–Macdonald $q$-difference operators, arXiv: 1206.5364

[32] Ohkubo Y., “Generalized Jack and Macdonald polynomials arising from AGT conjecture”, J. Phys. Conf. Ser., 804 (2017), 012036, 7 pp., arXiv: 1404.5401 | DOI

[33] Ohkubo Y., “Kac determinant and singular vector of the level $N$ representation of Ding–Iohara–Miki algebra”, Lett. Math. Phys., 109 (2019), 33–60, arXiv: 1706.02243 | DOI | MR | Zbl

[34] Olshanetsky M. A., Perelomov A. M., “Quantum integrable systems related to Lie algebras”, Phys. Rep., 94 (1983), 313–404 | DOI | MR

[35] Rains E. M., Warnaar S. O., “A Nekrasov–Okounkov formula for Macdonald polynomials”, J. Algebraic Combin., 48 (2018), 1–30, arXiv: 1606.04613 | DOI | MR | Zbl

[36] Ruijsenaars S. N. M., “Complete integrability of relativistic Calogero–Moser systems and elliptic function identities”, Comm. Math. Phys., 110 (1987), 191–213 | DOI | MR | Zbl

[37] Ruijsenaars S. N. M., “Zero-eigenvalue eigenfunctions for differences of elliptic relativistic Calogero–Moser Hamiltonians”, Theoret. and Math. Phys., 146 (2006), 25–33 | DOI | MR | Zbl

[38] Ruijsenaars S. N. M., “Hilbert–Schmidt operators vs. integrable systems of elliptic Calogero–Moser type. I The eigenfunction identities”, Comm. Math. Phys., 286 (2009), 629–657 | DOI | MR | Zbl

[39] Ruijsenaars S. N. M., “Hilbert–Schmidt operators vs. integrable systems of elliptic Calogero–Moser type. II The $A_{N-1}$ case: first steps”, Comm. Math. Phys., 286 (2009), 659–680 | DOI | MR | Zbl

[40] Shiraishi J., A commutative family of integral transformations and basic hypergeometric Series. I. Eigenfunctions, arXiv: math.QA/0501251

[41] Shiraishi J., A commutative family of integral transformations and basic hypergeometric series. II. Eigenfunctions and quasi-eigenfunctions, arXiv: math.QA/0502228

[42] Shiraishi J., “A conjecture about raising operators for Macdonald polynomials”, Lett. Math. Phys., 73 (2005), 71–81, arXiv: math.QA/0503727 | DOI | MR | Zbl

[43] Shiraishi J., “A family of integral transformations and basic hypergeometric series”, Comm. Math. Phys., 263 (2006), 439–460 | DOI | MR | Zbl

[44] Shiraishi J., “Affine screening operators, affine Laumon spaces and conjectures concerning non-stationary Ruijsenaars functions”, J. Integrable Syst., 4 (2019), xyz010, 30 pp., arXiv: 1903.07495 | DOI | MR

[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] Zenkevich Y., Higgsed network calculus, arXiv: 1812.11961