Abstract
The Nekrasov-Shatashvili limit of the \( \mathcal{N}=2 \) SU(2) pure gauge (Ω-deformed) super Yang-Mills theory encodes the information about the spectrum of the Mathieu operator. On the other hand, the Mathieu equation emerges entirely within the frame of two-dimensional conformal field theory (2d CFT) as the classical limit of the null vector decoupling equation for some degenerate irregular block. Therefore, it seems to be possible to investigate the spectrum of the Mathieu operator employing the techniques of 2d CFT. To exploit this strategy, a full correspondence between the Mathieu equation and its realization within 2d CFT has to be established. In our previous paper [1], we have found that the expression of the Mathieu eigenvalue given in terms of the classical irregular block exactly coincides with the well known weak coupling expansion of this eigenvalue in the case in which the auxiliary parameter is the noninteger Floquet exponent. In the present work we verify that the formula for the corresponding eigenfunction obtained from the irregular block reproduces the so-called Mathieu exponent from which the noninteger order elliptic cosine and sine functions may be constructed. The derivation of the Mathieu equation within the formalism of 2d CFT is based on conjectures concerning the asymptotic behaviour of irregular blocks in the classical limit. A proof of these hypotheses is sketched. Finally, we speculate on how it could be possible to use the methods of 2d CFT in order to get from the irregular block the eigenvalues of the Mathieu operator in other regions of the coupling constant.
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M. Piatek and A.R. Pietrykowski, Classical irregular block, \( \mathcal{N}=2 \) pure gauge theory and Mathieu equation, JHEP 12 (2014) 032 [arXiv:1407.0305] [INSPIRE].
N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].
J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011) 471 [arXiv:1005.2846] [INSPIRE].
K.K. Kozlowski and J. Teschner, TBA for the Toda chain, arXiv:1006.2906 [INSPIRE].
C. Meneghelli and G. Yang, Mayer-Cluster Expansion of Instanton Partition Functions and Thermodynamic Bethe Ansatz, JHEP 05 (2014) 112 [arXiv:1312.4537] [INSPIRE].
J. Teschner and G.S. Vartanov, Supersymmetric gauge theories, quantization of \( {\mathrm{\mathcal{M}}}_{\mathrm{flat}} \) and conformal field theory, Adv. Theor. Math. Phys. 19 (2015) 1 [arXiv:1302.3778] [INSPIRE].
A. Belavin and V. Belavin, AGT conjecture and Integrable structure of Conformal field theory for c = 1, Nucl. Phys. B 850 (2011) 199 [arXiv:1102.0343] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Integrable structure, W-symmetry and AGT relation, JHEP 01 (2012) 051 [arXiv:1109.4042] [INSPIRE].
V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE].
T.-S. Tai, Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants, JHEP 10 (2010) 107 [arXiv:1008.4332] [INSPIRE].
K. Muneyuki, T.-S. Tai, N. Yonezawa and R. Yoshioka, Baxter’s T − Q equation, SU(N)/SU(2)N − 3 correspondence and Ω-deformed Seiberg-Witten prepotential, JHEP 09 (2011) 125 [arXiv:1107.3756] [INSPIRE].
H. Itoyama and R. Yoshioka, Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models, arXiv:1507.00260 [INSPIRE].
R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].
F. Fucito, J.F. Morales, D.R. Pacifici and R. Poghossian, Gauge theories on Ω-backgrounds from non commutative Seiberg-Witten curves, JHEP 05 (2011) 098 [arXiv:1103.4495] [INSPIRE].
K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].
G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, Phys. Lett. B 691 (2010) 111 [arXiv:0909.4031] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin Systems via beta-deformed Matrix Models, arXiv:1104.4016 [INSPIRE].
M. Piatek, Classical conformal blocks from TBA for the elliptic Calogero-Moser system, JHEP 06 (2011) 050 [arXiv:1102.5403] [INSPIRE].
F. Ferrari and M. Piatek, Liouville theory, N = 2 gauge theories and accessory parameters, JHEP 05 (2012) 025 [arXiv:1202.2149] [INSPIRE].
F. Ferrari and M. Piatek, On a singular Fredholm-type integral equation arising in N = 2 super Yang-Mills theories, Phys. Lett. B 718 (2013) 1142 [arXiv:1202.5135] [INSPIRE].
F. Ferrari and M. Piatek, On a path integral representation of the Nekrasov instanton partition function and its Nekrasov-Shatashvili limit, Can. J. Phys. 92 (2014) 267 [arXiv:1212.6787] [INSPIRE].
M.-C. Tan, M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems, JHEP 07 (2013) 171 [arXiv:1301.1977] [INSPIRE].
J. Teschner, Exact results on \( \mathcal{N}=2 \) supersymmetric gauge theories, arXiv:1412.7145.
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].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in proceedings of XVIth International Congress on Mathematical Physics, Prague, Czech Republic (2009) [arXiv:0908.4052] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [INSPIRE].
N. Nekrasov, A. Rosly and S. Shatashvili, Darboux coordinates, Yang-Yang functional and gauge theory, Nucl. Phys. Proc. Suppl. 216 (2011) 69 [arXiv:1103.3919] [INSPIRE].
N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional \( \mathcal{N}=2 \) quiver gauge theories, arXiv:1211.2240 [INSPIRE].
N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
N. Wyllard, A N − 1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [INSPIRE].
D. Gaiotto, Asymptotically free \( \mathcal{N}=2 \) theories and irregular conformal blocks, J. Phys. Conf. Ser. 462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [INSPIRE].
V. Alba and A. Morozov, Non-conformal limit of AGT relation from the 1-point torus conformal block, JETP Lett. 90 (2009) 708 [arXiv:0911.0363] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Proving the AGT relation for N f = 0, 1, 2 antifundamentals, JHEP 06 (2010) 046 [arXiv:1004.1841] [INSPIRE].
D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].
D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, arXiv:1211.1287 [INSPIRE].
C.-N. Yang and C.P. Yang, Thermodynamics of one-dimensional system of bosons with repulsive delta function interaction, J. Math. Phys. 10 (1969) 1115 [INSPIRE].
G. Başar and G.V. Dunne, Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems, JHEP 02 (2015) 160 [arXiv:1501.05671] [INSPIRE].
A.-K. Kashani-Poor and J. Troost, Pure \( \mathcal{N}=2 \) super Yang-Mills and exact WKB, JHEP 08 (2015) 160 [arXiv:1504.08324] [INSPIRE].
M. Piatek, Classical torus conformal block, \( \mathcal{N}={2}^{*} \) twisted superpotential and the accessory parameter of Lamé equation, JHEP 03 (2014) 124 [arXiv:1309.7672] [INSPIRE].
E.T. Whittaker, On the General Solution of Mathieu’s Equation, Proc. Edin. Math. Soc. 32 (1914) 75.
D. Frenkel, R. Portugal, Algebraic Methods to Compute Mathieu Functions, J. Phys. A 34 (2001) 3541.
R. Sips, Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales, Trans. Am. Math. Soc. 66 (1949) 93.
H. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, World Scientific (2006).
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
G.W. Moore and N. Seiberg, Polynomial Equations for Rational Conformal Field Theories, Phys. Lett. B 212 (1988) 451 [INSPIRE].
G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].
G. Felder, J. Fröhlich and G. Keller, On the Structure of Unitary Conformal Field Theory 1. Existence of Conformal Blocks, Commun. Math. Phys. 124 (1989) 417 [INSPIRE].
G. Felder, J. Fröhlich and G. Keller, On the Structure of Unitary Conformal Field Theory. 2. Representation Theoretic Approach, Commun. Math. Phys. 130 (1990) 1 [INSPIRE].
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].
J. Teschner, A Lecture on the Liouville vertex operators, Int. J. Mod. Phys. A 19S2 (2004) 436 [hep-th/0303150] [INSPIRE].
V.G. Kac, Contravariant form for infinite-dimensional Lie algebras and superalgebras, Lect. Notes Phys. 94 (1979) 441.
B.L. Feigin and D.B. Fuks, Invariant skew symmetric differential operators on the line and verma modules over the Virasoro algebra, Funct. Anal. Appl. 16 (1982) 114 [Funkt. Anal. Ego Prilozh. 16 (1982) 47] [INSPIRE].
B.L. Feigin, D.B. Fuchs, Representations of the Virasoro algebra, in Representations of Lie groups and related topics, A.M. Vershik and D.P. Zhelobenko eds., Gordon and Breach, London U.K. (1990).
C.B. Thorn, Computing the Kac Determinant Using Dual Model Techniques and More About the No-Ghost Theorem, Nucl. Phys. B 248 (1984) 551 [INSPIRE].
V.G. Kac and M. Wakimoto, Unitarizable highest weight representations of the Virasoro, Neveu-Schwarz and Ramond algebras, Lect. Notes Phys. 261 (1986) 345.
V.G. Kac, A.K. Raina, Bombay lectures on highest weight representations of infinite dimensional Lie algebras, Adv. Ser. Math. Phys. 2, World Scientific Publishing (2013).
B. Feigin and D. Fuchs, Representations of the Virasoro algebra, Adv. Stud. Contemp. Math. 7 (1990) 465.
E. Felinska, Z. Jaskolski and M. Kosztolowicz, Whittaker pairs for the Virasoro algebra and the Gaiotto-BMT states, J. Math. Phys. 53 (2012) 033504 [Erratum ibid. 53 (2012) 129902] [arXiv:1112.4453] [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
A.B. Zamolodchikov, Conformal Field Theory And Critical Phenomena In Two-Dimensional Systems, Sov. Sci. Rev. A 10 (1989) 269.
D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
A. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, Sov. Phys. JEPT 63 (1986) 1061.
A. Litvinov, S. Lukyanov, N. Nekrasov and A. Zamolodchikov, Classical Conformal Blocks and Painleve VI, JHEP 07 (2014) 144 [arXiv:1309.4700] [INSPIRE].
N.W. McLachlan, Theory and application of Mathieu functions, Clarendon Press, Oxford U.K. (1947).
S. Yanagida, Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial, J.Algebra 333 (2011) 273.
L. Hadasz, Z. Jaskolski and M. Piatek, Analytic continuation formulae for the BPZ conformal block, Acta Phys. Polon. B 36 (2005) 845 [hep-th/0409258] [INSPIRE].
W. Boenkost and F. Constantinescu, Vertex operators in Hilbert space, J. Math. Phys. 34 (1993) 3607 [INSPIRE].
W. Boenkost, Vertex-Operatoren, Darstellungen der Virasoro-Algebra und konforme Quantenfeldtheorie, dissertation, Frankfurt am Main, Germany (1994), hep-th/9412231.
W. Boenkost, Vertex operators are not closeable, Rev. Math. Phys. 7 (1995) 51 [hep-th/9401004] [INSPIRE].
F. Constantinescu and G. Scharf, Smeared and unsmeared chiral vertex operators, Commun. Math. Phys. 200 (1999) 275 [hep-th/9712174] [INSPIRE].
C. Rim and H. Zhang, Classical Virasoro irregular conformal block, JHEP 07 (2015) 163 [arXiv:1504.07910] [INSPIRE].
C. Rim and H. Zhang, Classical Virasoro irregular conformal block II, JHEP 09 (2015) 097 [arXiv:1506.03561] [INSPIRE].
A. Mironov, A. Morozov and S. Shakirov, Conformal blocks as Dotsenko-Fateev Integral Discriminants, Int. J. Mod. Phys. A 25 (2010) 3173 [arXiv:1001.0563] [INSPIRE].
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Piątek, M., Pietrykowski, A.R. Classical limit of irregular blocks and Mathieu functions. J. High Energ. Phys. 2016, 115 (2016). https://doi.org/10.1007/JHEP01(2016)115
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DOI: https://doi.org/10.1007/JHEP01(2016)115