Abstract
We further explore the correspondence between \( \mathcal{N} \) = 2 supersymmetric SU(2) gauge theory with four flavors on ϵ-deformed backgrounds and conformal field theory, with an emphasis on the ϵ-expansion of the partition function natural from a topological string theory point of view. Solving an appropriate null vector decoupling equation in the semi-classical limit allows us to express the instanton partition function as a series in quasi-modular forms of the group Γ(2), with the expected symmetry W(D 4) ⋊ S 3. In the presence of an elementary surface operator, this symmetry is enhanced to an action of \( W\left( {D_4^{(1) }} \right)\rtimes {S_4} \) on the instanton partition function, as we demonstrate via the link between the null vector decoupling equation and the quantum Painlevé VI equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
V. Fateev and A. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [arXiv:0912.0504] [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles, J. Geom. Phys. 61 (2011) 1203 [arXiv:1011.4491] [INSPIRE].
A.-K. Kashani-Poor and J. Troost, The toroidal block and the genus expansion, JHEP 03 (2013) 133 [arXiv:1212.0722] [INSPIRE].
A. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, Sov. Phys. JETP 63 (1986) 1061.
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
G. Giribet, On AGT description of N = 2 SCFT with N(f) = 4, JHEP 01 (2010) 097 [arXiv:0912.1930] [INSPIRE].
A. Zamolodchikov and V. Fateev, Operator Algebra and Correlation Functions in the Two-Dimensional Wess-Zumino SU(2) × SU(2) Chiral Model, Sov. J. Nucl. Phys. 43 (1986) 657 [INSPIRE].
V. Fateev and A. Litvinov, Multipoint correlation functions in Liouville field theory and minimal Liouville gravity, Theor. Math. Phys. 154 (2008) 454 [arXiv:0707.1664] [INSPIRE].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
A. Levin and Olshanetsky, Painlevé-Calogero correspondence, arXiv:alg-geom/9706012.
H. Nagoya, A quantization of the sixth Painlevé equation, Adv. Stud. Pure Math. 55 (2009) 291.
A. Zabrodin and A. Zotov, Quantum Painleve-Calogero Correspondence, J. Math. Phys. 53 (2012) 073507 [arXiv:1107.5672] [INSPIRE].
P. Painlevé, Sur les équations difféérentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Math. 25 (1902) 185.
R. Fuchs, Uber linear homogene Differentialgleichungen zweiter Ordnung mit im endlich gelegene wesentlich singularen Stellen, Math. Ann. 63 (1907) 301.
B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes, CR Ac. Sci. Paris 142 (1906) 266.
H. Nagoya, Realizations of affine Weyl group symmetries on the quantum Painleve equations by fractional calculus, Lett. Math. Phys. 102 (2012) 297 [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
L. Schlesinger, Uber eine Klasse von Differentialsystemen beliebliger Ordnumg mit festen kritischer Punkten, J. fUr Math. 141 (1912) 96.
R. Garnier, Sur des équations différentielles du troisième ordre dont l’intégrale est uniform et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale a ses point critiques fixés, Ann. Sci. de l’ENS 29 (1912) 1.
R. Garnier, Sur une classe de systèmes differentiels abéliens deduits de la théorie des équations linéaires, Rend. Circ. Mat. Palermo 43 (1918-19) 155.
M. Jimbo and T. Miwa, Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II, Physica D 2 (1981) 407.
K. Okamoto, Studies on the Painlevé equations. I. Sixth Painlevé equation, Ann. Mat. Pura Appl. (4) 146 (1987) 337.
H. Nagoya and Y. Yamada, Symmetries of quantum Lax equations for the Painlevé equations, arXiv:1206.5963 [INSPIRE].
P. Boalch, Six results on Painlevé VI, Société Mathématique de France, Séminaires et congrès 14 (2006) 1.
A. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of the conformal block, Theor. Math. Phys. 73 (1987) 1088.
Y. Manin, Sixth Painlevé Equation, Universal Elliptic Curve, and Mirror of \( {{\mathbb{P}}^2} \), AMS Transl. (2) 186 (1998) 131.
M. Olshanetsky and A. Perelomov, Classical integrable finite dimensional systems related to Lie algebras, Phys. Rept. 71 (1981) 313 [INSPIRE].
V. Inozemtsev, Lax Representation with Spectral Parameter on a Torus for Particle Systems, Lett. Math. Phys. 17 (1989) 11.
M. Gaudin, Diagonalisation d’une classe d’Hamiltoniens de spin, J. Physique 37 (1976) 1087.
A. Zotov, Elliptic linear problem for Calogero-Inozemtsev model and Painleve VI equation, Lett. Math. Phys. 67 (2004) 153 [hep-th/0310260] [INSPIRE].
K. Takasaki, Painleve-Calogero correspondence revisited, J. Math. Phys. 42 (2001) 1443 [math/0004118] [INSPIRE].
V.A. Fateev, A. Litvinov, A. Neveu and E. Onofri, Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks, J. Phys. A 42 (2009) 304011 [arXiv:0902.1331] [INSPIRE].
M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, in The Moduli Space of Curves, Dijkgraaf, Faber, vanderGeer eds., Birkhäuser, (1995).
M.-x. Huang, A.-K. Kashani-Poor and A. Klemm, The Ω deformed B-model for rigid \( \mathcal{N} \) = 2 theories, Annales Henri Poincaré 14 (2013) 425 [arXiv:1109.5728] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Conformal field theory and 2-D critical phenomena. 3. Conformal bootstrap and degenerate representations of conformal algebra, ITEP-90-31 (1990).
A. Zamolodchikov, Conformal symmetry in two dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419.
M. Billó, M. Frau, L. Gallot, A. Lerda and I. Pesando, Deformed N = 2 theories, generalized recursion relations and S-duality, JHEP 04 (2013) 039 [arXiv:1302.0686] [INSPIRE].
M. Billó, M. Frau, L. Gallot and A. Lerda, The exact 8d chiral ring from 4d recursion relations, JHEP 11 (2011) 077 [arXiv:1107.3691] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1305.7408
Unité Mixte du CNRS et de l’Ecole Normale Supérieure associée à l’Université Pierre et Marie Curie 6, UMR 8549. (Amir-Kian Kashani-Poor and Jan Troost)
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kashani-Poor, AK., Troost, J. Transformations of Spherical Blocks. J. High Energ. Phys. 2013, 9 (2013). https://doi.org/10.1007/JHEP10(2013)009
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2013)009