Abstract
We consider the AGT correspondence in the context of the conformal field theory \( {{\mathrm{\mathcal{M}}}^{p, p}}^{\prime}\otimes {\mathrm{\mathcal{M}}}^{\mathcal{H}} \), where \( {{\mathrm{\mathcal{M}}}^{p, p}}^{\prime } \) is the minimal model based on the Virasoro algebra \( {{\mathcal{V}}^{p, p}}^{\prime } \) labeled by two co-prime integers {p, p′}, 1 < p < p′, and \( {\mathrm{\mathcal{M}}}^{\mathcal{H}} \) is the free boson theory based on the Heisenberg algebra \( \mathcal{H} \). Using Nekrasov’s instanton partition functions without modification to compute conformal blocks in \( {{\mathrm{\mathcal{M}}}^{p, p}}^{\prime}\otimes {\mathrm{\mathcal{M}}}^{\mathcal{H}} \) leads to ill-defined or incorrect expressions.
Let \( {\mathrm{\mathcal{B}}}_n^{p, p\prime, \mathcal{H}} \) be a conformal block in \( {{\mathrm{\mathcal{M}}}^{p, p}}^{\prime}\otimes {\mathrm{\mathcal{M}}}^{\mathcal{H}} \), with n consecutive channels χ ι , ι \( = 1, \cdot p \cdot p \cdot p, \) n, and let χ ι carry states from \( {\mathcal{H}}_{r_{\iota},{s}_{\iota}}^{p, p\prime}\otimes \mathrm{\mathcal{F}} \), where \( {\mathcal{H}}_{r_{\iota},{s}_{\iota}}^{p, p\prime } \) is an irreducible highest- weight \( {\mathcal{V}}^{p, p\prime } \) -representation, labeled by two integers {r ι , s ι }, 0 < r ι < p, 0 < s ι < p′, and \( \mathrm{\mathcal{F}} \) is the Fock space of \( \mathcal{H} \).
We show that restricting the states that flow in χ ι , ι = 1, · · · , n, to states labeled by partition pairs \( \left\{{Y}_1^{\iota},{Y}_2^{\iota}\right\} \) that satisfy \( {Y}_{2,\sigma}^{{}_{\iota, \mathrm{T}}}-{Y}_{1,\sigma +{r}_{\iota}-1}^{\iota, \mathrm{T}}\ge 1-{s}_{\iota} \), and \( {Y}_{1,\sigma}^{\iota, \mathrm{T}}-{Y}_{2,\sigma + p-{r}_{\iota}-1}^{\iota, \mathrm{T}}\ge 1- p^{\prime }+{s}_{\iota} \),where \( {Y}_{i,\sigma}^{\iota, \mathrm{T}} \) is the σ-column of \( {Y}_i^{\iota},\kern0.5em i\in \left\{1,2\right\} \), we obtain a well-defined expression that we identify with \( {\mathrm{\mathcal{B}}}_n^{p, p\prime, \mathcal{H}} \). We check the correctness of this expression for 1. Any 1-point \( {\mathrm{\mathcal{B}}}_1^{p, p\prime, \mathcal{H}} \) on the torus, when the operator insertion is the identity, and 2. The 6-point \( {\mathrm{\mathcal{B}}}_3^{3,4,\mathcal{H}} \) on the sphere that involves six Ising magnetic operators.
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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].
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].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
A. Mironov, A. Morozov and S. Shakirov, A direct proof of AGT conjecture at β = 1, JHEP 02 (2011) 067 [arXiv:1012.3137] [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].
R. Santachiara, private communication.
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].
R. Santachiara and A. Tanzini, Moore-Read Fractional Quantum Hall wavefunctions and SU(2) quiver gauge theories, Phys. Rev. D 82 (2010) 126006 [arXiv:1002.5017] [INSPIRE].
W.H. Burge, Restricted partition pairs, J. Comb. Theory. Ser. A 63 (1993) 210.
O. Foda, K.S.M. Lee and T.A. Welsh, A Burge tree of Virasoro type polynomial identities, Int. J. Mod. Phys. A 13 (1998) 4967 [q-alg/9710025] [INSPIRE].
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous \( \mathfrak{g}{\mathfrak{l}}_{\infty } \) : Tensor products of Fock modules and \( {\mathcal{W}}_n \) -characters, Kyoto J. Math 51 (2011) 365. [arXiv:1002.3113].
S. Kanno, Y. Matsuo and H. Zhang, Extended Conformal Symmetry and Recursion Formulae for Nekrasov Partition Function, JHEP 08 (2013) 028 [arXiv:1306.1523] [INSPIRE].
B. Nienhuis, Coulomb gas representations of phase transitions in two dimensions, Phase Trans. Critical Phenom. 11 (1987) 1.
Vl.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B 240 (1984) 312.
E. Ardonne and G. Sierra, Chiral correlators of the Ising conformal field theory, J. Phys. A 43 (2010) 505402 [arXiv:1008.2863] [INSPIRE].
I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Clarendon Press Oxford (1995).
M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Am. Math Soc. 14 (2001) 941 [math/0010246].
B. Feigin, A. Hoshino, J. Shibahara, J. Shiraishi and S. Yanagida, Kernel function and quantum algebras, RIMS kôkyûroku 1689 (2010) 133 [arXiv:1002.2485].
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].
K.B. Alkalaev and V.A. Belavin, Conformal blocks of \( {\mathcal{W}}_n \) Minimal Models and AGT correspondence, arXiv:1404.7094 [INSPIRE].
B. Estienne, V. Pasquier, R. Santachiara and D. Serban, Conformal blocks in Virasoro and W theories: Duality and the Calogero-Sutherland model, Nucl. Phys. B 860 (2012) 377 [arXiv:1110.1101] [INSPIRE].
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Bershtein, M., Foda, O. AGT, Burge pairs and minimal models. J. High Energ. Phys. 2014, 177 (2014). https://doi.org/10.1007/JHEP06(2014)177
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DOI: https://doi.org/10.1007/JHEP06(2014)177