×

Pure \(\mathrm{SU}(2)\) gauge theory partition function and generalized Bessel kernel. (English) Zbl 1448.34163

Kashani-Poor, Amir-Kian (ed.) et al., String-Math 2016. Proceedings of the conference, Collège de France, Paris, France, June 27 – July 2, 2016. Providence, RI: American Mathematical Society (AMS); Boston, MA: International Press. Proc. Symp. Pure Math. 98, 181-205 (2018).
Summary: We show that the dual partition function of the pure \(\mathcal{N}=2 \mathrm{SU}(2)\) gauge theory in the self-dual \(\Omega\)-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painlevé III equation of type \(D_8\) (radial sine-Gordon equation). In particular, the principal minor expansion of the Fredholm determinant yields Nekrasov combinatorial sums over pairs of Young diagrams.
For the entire collection see [Zbl 1394.14001].

MSC:

34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
33E17 Painlevé-type functions
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34M56 Isomonodromic deformations for ordinary differential equations in the complex domain
81T45 Topological field theories in quantum mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alday, Luis F., Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys., 167-197 (2010) · Zbl 1185.81111
[2] Apikyan, S. A., Conformal blocks, related to conformally invariant Ramond states of a free scalar field, Soviet Phys. JETP. Zh. \`Eksper. Teoret. Fiz., 34-45 (1987)
[3] Bershtein, M. A., Bilinear equations on Painlev\'e \(\tau\) functions from CFT, Comm. Math. Phys., 1021-1061 (2015) · Zbl 1332.34141
[4] Bershtein, M. A., B\"acklund transformation of Painlev\'e \({\rm III}(D_8) \tau\) function, J. Phys. A, 115205, 31 pp. (2017) · Zbl 1396.33038
[5] Bonelli, Giulio, Seiberg-Witten theory as a Fermi gas, Lett. Math. Phys., 1-30 (2017) · Zbl 1390.70067
[6] G. Bonelli, A. Grassi, and A. Tanzini, New results in \(\mathcal N=2\) theories from non-perturbative string, Ann. Henri Poincar\'e 19 (2018), No. 3, 743-774. DOI: 10.1007/s00023-017-0643-5. · Zbl 1386.81137
[7] Bonelli, Giulio, On Painlev\'e/gauge theory correspondence, Lett. Math. Phys., 2359-2413 (2017) · Zbl 1380.34130
[8] Bonelli, Giulio, Wild quiver gauge theories, J. High Energy Phys., 031, front matter+30 pp. (2012) · Zbl 1309.81144
[9] L. Chekhov and M. Mazzocco, Colliding holes in Riemann surfaces and quantum cluster algebras, Nonlinearity 31 (2018), No. 54. DOI:10.1088/1361-6544/aa9729. · Zbl 1390.13066
[10] L. Chekhov, M. Mazzocco, and V. Rubtsov, Painlev\'e monodromy manifolds, decorated character varieties and cluster algebras, Int. Math. Res. Not. 2017 (2017), 7639-7691. DOI:10.1093/imrn/rnw219. · Zbl 1405.30044
[11] Fokas, Athanassios S., Painlev\'e transcendents, Mathematical Surveys and Monographs, xii+553 pp. (2006), American Mathematical Society, Providence, RI · Zbl 1111.34001
[12] D. Gaiotto, Asymptotically free \(\mathcal N=2\) theories and irregular conformal blocks, arXiv:0908.0307 [hep-th].
[13] Gaiotto, D., Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, J. High Energy Phys., 050, front matter + 76 pp. (2012) · Zbl 1397.81305
[14] Gamayun, O., Conformal field theory of Painlev\'e VI, J. High Energy Phys., 038, front matter + 24 pp. (2012) · Zbl 1397.81307
[15] Gamayun, O., How instanton combinatorics solves Painlev\'e VI, V and IIIs, J. Phys. A, 335203, 29 pp. (2013) · Zbl 1282.34096
[16] P. Gavrylenko, O. Lisovyy, Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions, (2016); arXiv:1608.00958 [math-ph]. · Zbl 1414.34072
[17] Gavrilenko, P. G., Free fermions, \(W\)-algebras, and isomonodromic deformations, Theoret. and Math. Phys.. Teoret. Mat. Fiz., 232-262 (2016) · Zbl 1346.81118
[18] Grassi, Alba, Topological strings from quantum mechanics, Ann. Henri Poincar\'e, 3177-3235 (2016) · Zbl 1365.81094
[19] Iorgov, N., Isomonodromic tau-functions from Liouville conformal blocks, Comm. Math. Phys., 671-694 (2015) · Zbl 1311.30029
[20] Its, Alexander, Connection problem for the sine-Gordon/Painlev\'e III tau function and irregular conformal blocks, Int. Math. Res. Not. IMRN, 8903-8924 (2015) · Zbl 1329.34140
[21] Its, Alexander R., The isomonodromic deformation method in the theory of Painlev\'e equations, Lecture Notes in Mathematics, iv+313 pp. (1986), Springer-Verlag, Berlin · Zbl 0592.34001
[22] Its, Alexander, Connection problem for the tau-function of the sine-Gordon reduction of Painlev\'e-III equation via the Riemann-Hilbert approach, Int. Math. Res. Not. IMRN, 6856-6883 (2016) · Zbl 1404.34104
[23] Jimbo, Michio, Monodromy problem and the boundary condition for some Painlev\'e equations, Publ. Res. Inst. Math. Sci., 1137-1161 (1982) · Zbl 0535.34042
[24] Jimbo, Michio, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and \(\tau \)-function, Phys. D, 306-352 (1981) · Zbl 1194.34167
[25] Knizhnik, V. G., Multiloop amplitudes in the theory of quantum strings and complex geometry, Soviet Phys. Uspekhi. Uspekhi Fiz. Nauk, 401-453 (1989)
[26] McCoy, Barry M., Painlev\'e functions of the third kind, J. Mathematical Phys., 1058-1092 (1977) · Zbl 0353.33008
[27] Moore, Gregory, Geometry of the string equations, Comm. Math. Phys., 261-304 (1990) · Zbl 0727.35134
[28] Nagoya, Hajime, Irregular conformal blocks, with an application to the fifth and fourth Painlev\'e equations, J. Math. Phys., 123505, 24 pp. (2015) · Zbl 1328.81195
[29] H. Nagoya, Conformal blocks and Painlev\'e functions, arXiv:1611.08971 [math-ph]. · Zbl 1328.81195
[30] Nekrasov, Nikita A., Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys., 831-864 (2003) · Zbl 1056.81068
[31] Nekrasov, Nikita A., The unity of mathematics. Seiberg-Witten theory and random partitions, Progr. Math., 525-596 (2006), Birkh\"auser Boston, Boston, MA · Zbl 1233.14029
[32] Niles, David Gregory, The Riemann-Hilbert-Birkhoff inverse monodromy problem and connection formulae for the third Painleve transcendents, 89 pp. (2009), ProQuest LLC, Ann Arbor, MI
[33] Novoksh\"enov, V. Yu., The asymptotic behavior of the general real solution of the third Painlev\'e equation, Dokl. Akad. Nauk SSSR, 1161-1165 (1985) · Zbl 0674.34048
[34] Sato, Mikio, Holonomic quantum fields. I, Publ. Res. Inst. Math. Sci., 223-267 (1978) · Zbl 0383.35066
[35] Seiberg, N., Electric-magnetic duality, monopole condensation, and confinement in \(N=2\) supersymmetric Yang-Mills theory, Nuclear Phys. B, 19-52 (1994) · Zbl 0996.81510
[36] Teschner, J., Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, Adv. Theor. Math. Phys., 471-564 (2011) · Zbl 1442.81059
[37] Tracy, Craig A., Asymptotics of a \(\tau \)-function arising in the two-dimensional Ising model, Comm. Math. Phys., 297-311 (1991) · Zbl 0734.60106
[38] Zamolodchikov, Al. B., Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation functions, Nuclear Phys. B, 481-503 (1987)
[39] Zamolodchikov, Al. B., Painlev\'e III and \(2\) D polymers, Nuclear Phys. B, 427-456 (1994) · Zbl 1020.81889
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.