Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik–Zamolodchikov Equation

Hidetoshi Awata ; Koji Hasegawa ; Hiroaki Kanno ; Ryo Ohkawa ; Shamil Shakirov ; Jun'ichi Shiraishi ; Yasuhiko Yamada

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

We show that Shakirov's non-stationary difference equation, when it is truncated, implies the quantum Knizhnik–Zamolodchikov ($q$-KZ) equation for $U_{\mathsf v}\bigl(A_1^{(1)}\bigr)$ with generic spins. Namely, we can tune mass parameters so that the Hamiltonian acts on the space of finite Laurent polynomials. Then the representation matrix of the Hamiltonian agrees with the $R$-matrix, or the quantum $6j$ symbols. On the other hand, we prove that the $K$ theoretic Nekrasov partition function from the affine Laumon space is identified with the well-studied Jackson integral solution to the $q$-KZ equation. Combining these results, we establish that the affine Laumon partition function gives a solution to Shakirov's equation, which was a conjecture in our previous paper. We also work out the base-fiber duality and four-dimensional limit in relation with the $q$-KZ equation.

Détail

Keywords: quantum affine algebra, non-stationary difference equation, quantum Knizhnik–Zamolodchikov equation.
Mots-clés : affine Laumon space

Bibliographie

[1] Aganagic M., Frenkel E., Okounkov A., “Quantum $q$-Langlands correspondence”, Trans. Moscow Math. Soc., 79 (2018), 1–83, arXiv: 1701.03146 | DOI | MR | Zbl

[2] Aganagic M., Shakirov Sh., “Knot homology and refined Chern–Simons index”, Comm. Math. Phys., 333 (2015), 187–228, arXiv: 1105.5117 | DOI | MR | Zbl

[3] Agarwal A.K., Andrews G.E., Bressoud D.M., “The Bailey lattice”, J. Indian Math. Soc. (N.S.), 51 (1987), 57–73 | MR | Zbl

[4] Alday L.F., Gaiotto D., Gukov S., Tachikawa Y., Verlinde H., Loop and surface operators in ${\mathcal N}=2$ gauge theory and Liouville modular geometry, J. High Energy Phys., 2010:1 (2010), 113, 50 pp., arXiv: 0909.0945 | DOI | MR | Zbl

[5] 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

[6] Alday L.F., Tachikawa Y., “Affine ${\rm SL}(2)$ conformal blocks from 4d gauge theories”, Lett. Math. Phys., 94 (2010), 87–114, arXiv: 1005.4469 | DOI | MR | Zbl

[7] Andrews G.E., “Connection coefficient problems and partitions”, Relations Between Combinatorics and Other Parts of Mathematics (Ohio State Univ., Columbus, Ohio, 1978), Proc. Sympos. Pure Math., 34, American Mathematical Society, Providence, RI, 1979, 1–24 | DOI | MR

[8] Aomoto K., Kato Y., “Gauss decomposition of connection matrices for symmetric $A$-type Jackson integrals”, Selecta Math. (N.S.), 1 (1995), 623–666 | DOI | MR | Zbl

[9] Awata H., Fuji H., Kanno H., Manabe M., Yamada Y., “Localization with a surface operator, irregular conformal blocks and open topological string”, Adv. Theor. Math. Phys., 16 (2012), 725–804, arXiv: 1008.0574 | DOI | MR | Zbl

[10] Awata H., Hasegawa K., Kanno H., Ohkawa R., Shakirov S., Shiraishi J., Yamada Y., “Non-stationary difference equation and affine Laumon space: quantization of discrete Painlevé equation”, SIGMA, 19 (2023), 089, 47 pp., arXiv: 2211.16772 | DOI | MR | Zbl

[11] 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

[12] Awata H., Kanno H., Mironov A., Morozov A., “On a complete solution of the quantum Dell system”, J. High Energy Phys., 2020:4 (2020), 212, 30 pp., arXiv: 1912.12897 | DOI | MR | Zbl

[13] 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

[14] Bosnjak G., Mangazeev V.V., “Construction of $R$-matrices for symmetric tensor representations related to $U_q(\widehat{sl_n})$”, J. Phys. A, 49 (2016), 495204, 19 pp., arXiv: 1607.07968 | DOI | MR | Zbl

[15] Braverman A., “Instanton counting via affine Lie algebras. I Equivariant $J$-functions of (affine) flag manifolds and Whittaker vectors”, Algebraic Structures and Moduli Spaces, CRM Proc. Lecture Notes, 38, American Mathematical Society, Providence, RI, 2004, 113–132, arXiv: math.AG/0401409 | DOI | MR | Zbl

[16] Braverman A., Etingof P., “Instanton counting via affine Lie algebras. II. From Whittaker vectors to the Seiberg–Witten prepotential”, Studies in Lie Theory, Progr. Math., 243, Birkhäuser Boston, Boston, MA, 2006, 61–78, arXiv: math.AG/0409441 | DOI | MR | Zbl

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

[18] Bressoud D.M., “A matrix inverse”, Proc. Amer. Math. Soc., 88 (1983), 446–448 | DOI | MR | Zbl

[19] Bullimore M., Kim H.-C., Koroteev P., “Defects and quantum Seiberg–Witten geometry”, J. High Energy Phys., 2015, no. 5, 095, 78 pp., arXiv: 1412.6081 | DOI | MR

[20] Cotti G., Varchenko A., “Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-theorem”, Integrability, Quantization, and Geometry, v. I, Proc. Sympos. Pure Math., 103, Integrable Systems, American Mathematical Society, Providence, RI, 2021, 101–170, arXiv: 1909.06582 | DOI | MR | Zbl

[21] Di Francesco P., Kedem R., “Macdonald duality and the proof of the quantum Q-system conjecture”, Selecta Math. (N.S.), 30 (2024), 23, 100 pp., arXiv: 2112.09798 | DOI | MR

[22] Di Francesco P., Kedem R., Duality and Macdonald difference operators, arXiv: 2303.04276

[23] Etingof P.I., Frenkel I.B., Kirillov Jr. A.A., Lectures on representation theory and Knizhnik–Zamolodchikov equations, Math. Surveys Monogr., 58, American Mathematical Society, Providence, RI, 1998 | DOI | MR | Zbl

[24] Fateev V.A., Zamolodchikov A.B., “Operator algebra and correlation functions in the two-dimensional ${\rm SU}(2) \times {\rm SU}(2)$ chiral Wess–Zumino model”, Sov. J. Nuclear Phys., 43 (1986), 657–664

[25] Feigin B., Finkelberg M., Negut A., Rybnikov L., “Yangians and cohomology rings of Laumon spaces”, Selecta Math. (N.S.), 17 (2011), 573–607, arXiv: 0812.4656 | DOI | MR | Zbl

[26] Finkelberg M., Rybnikov L., Quantization of Drinfeld Zastava in type A, arXiv: 1009.0676 | MR

[27] Frenkel I.B., Reshetikhin N. Yu., “Quantum affine algebras and holonomic difference equations”, Comm. Math. Phys., 146 (1992), 1–60 | DOI | MR | Zbl

[28] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia Math. Appl., 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[29] Ito M., “$q$-difference systems for the Jackson integral of symmetric Selberg type”, SIGMA, 16 (2020), 113, 31 pp., arXiv: 1910.08393 | DOI | MR | Zbl

[30] Ito M., “Gauss decomposition and $q$-difference equations for Jackson integrals of symmetric Selberg type”, Ryukyu Math. J., 36 (2023), 1–47, arXiv: 2309.17181 | MR | Zbl

[31] Ito M., Forrester P.J., “A bilateral extension of the $q$-Selberg integral”, Trans. Amer. Math. Soc., 369 (2017), 2843–2878, arXiv: 1309.0001 | DOI | MR | Zbl

[32] Ito M., Noumi M., “Connection formula for the Jackson integral of type $A_n$ and elliptic Lagrange interpolation”, SIGMA, 14 (2018), 077, 42 pp., arXiv: 1801.07041 | DOI | MR | Zbl

[33] Kanno H., Tachikawa Y., “Instanton counting with a surface operator and the chain-saw quiver”, J. High Energy Phys., 2011:6 (2011), 119, 24 pp., arXiv: 1105.0357 | DOI | MR | Zbl

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

[35] Laumon G., “Un analogue global du cône nilpotent”, Duke Math. J., 57 (1988), 647–671 | DOI | MR | Zbl

[36] Laumon G., “Faisceaux automorphes liés aux séries d'Eisenstein”, Automorphic Forms, Shimura Varieties, and $L$-functions (Ann Arbor, MI, 1988), v. I, Perspect. Math., 10, Academic Press, Boston, MA, 1990, 227–281 | MR

[37] Macdonald I.G., Symmetric functions and Hall polynomials, Oxford Math. Monogr., 2nd ed., Oxford University Press, New York, 1995 | MR | Zbl

[38] Mangazeev V.V., “On the Yang–Baxter equation for the six-vertex model”, Nuclear Phys. B, 882 (2014), 70–96, arXiv: 1401.6494 | DOI | MR | Zbl

[39] Matsuo A., “Jackson integrals of Jordan–Pochhammer type and quantum Knizhnik–Zamolodchikov equations”, Comm. Math. Phys., 151 (1993), 263–273 | DOI | MR | Zbl

[40] Matsuo A., “Quantum algebra structure of certain Jackson integrals”, Comm. Math. Phys., 157 (1993), 479–498 | DOI | MR | Zbl

[41] Mimachi K., “Holonomic $q$-difference system of the first order associated with a Jackson integral of Selberg type”, Duke Math. J., 73 (1994), 453–468 | DOI | MR | Zbl

[42] Nagoya H., “Hypergeometric solutions to Schrödinger equations for the quantum Painlevé equations”, J. Math. Phys., 52 (2011), 083509, 16 pp., arXiv: 1109.1645 | DOI | MR | Zbl

[43] Neguţ A., Affine Laumon spaces and integrable systems, arXiv: 1112.1756

[44] Neguţ A., “Affine Laumon spaces and a conjecture of Kuznetsov”, Ann. Sci. Éc. Norm. Supér., 55 (2022), 739–789, arXiv: 1811.01011 | DOI | MR | Zbl

[45] Nekrasov N., “BPS/CFT correspondence IV: sigma models and defects in gauge theory”, Lett. Math. Phys., 109 (2019), 579–622, arXiv: 1711.11011 | DOI | MR | Zbl

[46] Nekrasov N., BPS/CFT correspondence V: BPZ and KZ equations from $qq$-characters, arXiv: 1711.11582

[47] Nekrasov N., Tsymbaliuk A., “Surface defects in gauge theory and KZ equation”, Lett. Math. Phys., 112 (2022), 28, 53 pp., arXiv: 2103.12611 | DOI | MR | Zbl

[48] Reshetikhin N., “Jackson-type integrals, Bethe vectors, and solutions to a difference analog of the Knizhnik–Zamolodchikov system”, Lett. Math. Phys., 26 (1992), 153–165 | DOI | MR | Zbl

[49] Reshetikhin N., “The Knizhnik–Zamolodchikov system as a deformation of the isomonodromy problem”, Lett. Math. Phys., 26 (1992), 167–177 | DOI | MR | Zbl

[50] Rosengren H., “An elementary approach to $6j$-symbols (classical, quantum, rational, trigonometric, and elliptic)”, Ramanujan J., 13 (2007), 131–166, arXiv: math.CA/0312310 | DOI | MR | Zbl

[51] Shakirov Sh., “Non-stationary difference equation for $q$-Virasoro conformal blocks”, Lett. Math. Phys. (to appear) , arXiv: 2111.07939

[52] 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 | Zbl

[53] 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

[54] Tarasov V., Varchenko A., “Landau–Ginzburg mirror, quantum differential equations and $q$KZ difference equations for a partial flag variety”, J. Geom. Phys., 184 (2023), 104711, 58 pp., arXiv: 2203.03039 | DOI | MR | Zbl

[55] Varchenko A., “Quantized Knizhnik–Zamolodchikov equations, quantum Yang–Baxter equation, and difference equations for $q$-hypergeometric functions”, Comm. Math. Phys., 162 (1994), 499–528 | DOI | MR | Zbl