### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 123, 27 pages      arXiv:1806.08650      https://doi.org/10.3842/SIGMA.2018.123
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

### On Solutions of the Fuji-Suzuki-Tsuda System

Pavlo Gavrylenko abc, Nikolai Iorgov ad and Oleg Lisovyy e
a) Bogolyubov Institute for Theoretical Physics, 03143 Kyiv, Ukraine
b) Center for Advanced Studies, Skolkovo Institute of Science and Technology, 143026 Moscow, Russia
c) National Research University Higher School of Economics, International Laboratory of Representation Theory and Mathematical Physics, Moscow, Russia
e) Institut Denis-Poisson, Université de Tours, Parc de Grandmont, 37200 Tours, France

Received June 22, 2018, in final form October 30, 2018; Published online November 11, 2018

Abstract
We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlevé VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for $c=N-1$.

Key words: isomonodromic deformations; Painlevé equations; Fredholm determinants.

pdf (617 kb)   tex (57 kb)

References

1. 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.
2. Bershtein M.A., Shchechkin A.I., Bilinear equations on Painlevé $\tau$ functions from CFT, Comm. Math. Phys. 339 (2015), 1021-1061, arXiv:1406.3008.
3. Borodin A., Deift P., Fredholm determinants, Jimbo-Miwa-Ueno $\tau$-functions, and representation theory, Comm. Pure Appl. Math. 55 (2002), 1160-1230, math-ph/0111007.
4. Borodin A., Olshanski G., Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, Ann. of Math. 161 (2005), 1319-1422, math.RT/0109194.
5. Cafasso M., Gavrylenko P., Lisovyy O., Tau functions as Widom constants, Comm. Math. Phys., to appear, arXiv:1712.08546.
6. Chekhov L., Mazzocco M., Isomonodromic deformations and twisted Yangians arising in Teichmüller theory, Adv. Math. 226 (2011), 4731-4775, arXiv:0909.5350.
7. Deift P., Its A., Kapaev A., Zhou X., On the algebro-geometric integration of the Schlesinger equations, Comm. Math. Phys. 203 (1999), 613-633.
8. Dubrovin B., Geometry of $2$D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, hep-th/9407018.
9. Dubrovin B., Mazzocco M., Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000), 55-147, math.AG/9806056.
10. Fateev V.A., Litvinov A.V., Integrable structure, W-symmetry and AGT relation, J. High Energy Phys. 2012 (2012), no. 1, 051, 39 pages, arXiv:1109.4042.
11. Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents: the Riemann-Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, Amer. Math. Soc., Providence, RI, 2006.
12. Fuji K., Suzuki T., Drinfeld-Sokolov hierarchies of type $A$ and fourth order Painlevé systems, Funkcial. Ekvac. 53 (2010), 143-167, arXiv:0904.3434.
13. Gamayun O., Iorgov N., Lisovyy O., Conformal field theory of Painlevé VI, J. High Energy Phys. 2012 (2012), no. 10, 038, 25 pages, arXiv:1207.0787.
14. Gavrylenko P., Iorgov N., Lisovyy O., Higher rank isomonodromic deformations and $W$-algebras, arXiv:1801.09608.
15. Gavrylenko P., Lisovyy O., Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions, Comm. Math. Phys. 363 (2018), 1-58, arXiv:1608.00958.
16. Gavrylenko P., Lisovyy O., Pure ${\rm SU}(2)$ gauge theory partition function and generalized Bessel kernel, in String-Math 2016, Proc. Sympos. Pure Math., Vol. 98, Amer. Math. Soc., Providence, RI, 2018, 181-205, arXiv:1705.01869.
17. Iorgov N., Lisovyy O., Teschner J., Isomonodromic tau-functions from Liouville conformal blocks, Comm. Math. Phys. 336 (2015), 671-694, arXiv:1401.6104.
18. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function, Phys. D 2 (1981), 306-352.
19. Kitaev A.V., Korotkin D.A., On solutions of the Schlesinger equations in terms of $\Theta$-functions, Int. Math. Res. Not. 1998 (1998), 877-905, math-ph/9810007.
20. Korotkin D.A., Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices, Math. Ann. 329 (2004), 335-364, math-ph/0306061.
21. Lisovyy O., Tykhyy Y., Algebraic solutions of the sixth Painlevé equation, J. Geom. Phys. 85 (2014), 124-163, arXiv:0809.4873.
22. Mano T., Tsuda T., Hermite-Padé approximation, isomonodromic deformation and hypergeometric integral, Math. Z. 285 (2017), 397-431, arXiv:1502.06695.
23. Mazzocco M., Picard and Chazy solutions to the Painlevé VI equation, Math. Ann. 321 (2001), 157-195, math.AG/9901054.
24. Mironov A., Morozov A., On AGT relation in the case of ${\rm U}(3)$, Nuclear Phys. B 825 (2010), 1-37, arXiv:0908.2569.
25. Nekrasov N.A., Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), 831-864, hep-th/0206161.
26. Okamoto K., Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{{\rm VI}}$, Ann. Mat. Pura Appl. 146 (1987), 337-381.
27. Suzuki T., A particular solution of a Painlevé system in terms of the hypergeometric function $_{n+1}F_n$, SIGMA 6 (2010), 078, 11 pages, arXiv:1002.2685.
28. Suzuki T., A class of higher order Painlevé systems arising from integrable hierarchies of type $A$, in Algebraic and geometric aspects of integrable systems and random matrices, Contemp. Math., Vol. 593, Amer. Math. Soc., Providence, RI, 2013, 125-141, arXiv:1002.2685.
29. Tsuda T., From KP/UC hierarchies to Painlevé equations, Internat. J. Math. 23 (2012), 1250010, 59 pages, arXiv:1004.1347.
30. Tsuda T., Hypergeometric solution of a certain polynomial Hamiltonian system of isomonodromy type, Q. J. Math. 63 (2012), 489-505, arXiv:1005.4130.
31. Tsuda T., UC hierarchy and monodromy preserving deformation, J. Reine Angew. Math. 690 (2014), 1-34, arXiv:1007.3450.
32. Wyllard N., $A_{N-1}$ conformal Toda field theory correlation functions from conformal ${\mathcal N}=2$ ${\rm SU}(N)$ quiver gauge theories, J. High Energy Phys. 2009 (2009), no. 11, 002, 22 pages, arXiv:0907.2189.
33. Yamada Y., A quantum isomonodromy equation and its application to ${\mathcal N}=2$ ${\rm SU}(N)$ gauge theories, J. Phys. A: Math. Theor. 44 (2011), 055403, 9 pages, arXiv:1011.0292.