Abstract
In this paper we investigate the relation between conformal blocks of Liouville CFT and the topological string partition functions of the rank one trinion theory T2. The partition functions exhibit jumps when passing from one chamber in the parameter space to another. Such jumps can be attributed to a change of the integration contour in the free field representation of Liouville conformal blocks. We compare the partition functions of the T2 theories representing trifundamental half hypermultiplets in N = 2, d = 4 field theories to the partition functions associated to bifundamental hypermultiplets. We find that both are related to the same Liouville conformal blocks up to inessential factors. In order to establish this picture we combine and compare results obtained using topological vertex techniques, matrix models and topological recursion. We furthermore check that the partition functions obtained by gluing two T2 vertices can be represented in terms of a four point Liouville conformal block. Our results indicate that the T2 vertex offers a useful starting point for developing an analog of the instanton calculus for SUSY gauge theories with trifundamental hypermultiplets.
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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.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].
N. Nekrasov, BPS/CFT correspondence V: BPZ and KZ equations from qq-characters, arXiv:1711.11582 [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
Y. Tachikawa, A brief review of the 2d/ 4d correspondences, J. Phys. A 50 (2017) 443012 [arXiv:1608.02964] [INSPIRE].
V.A. Fateev and A.V. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [arXiv:0912.0504] [INSPIRE].
A. Mironov and A. Morozov, Proving AGT relations in the large-c limit, Phys. Lett. B 682 (2009) 118 [arXiv:0909.3531] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Proving the AGT relation for Nf = 0, 1, 2 antifundamentals, JHEP 06 (2010) 046 [arXiv:1004.1841] [INSPIRE].
A. Mironov, A. Morozov and S. Shakirov, Towards a proof of AGT conjecture by methods of matrix models, Int. J. Mod. Phys. A 27 (2012) 1230001 [arXiv:1011.5629] [INSPIRE].
A. Mironov, A. Morozov and S. Shakirov, A direct proof of AGT conjecture at beta = 1, JHEP 02 (2011) 067 [arXiv:1012.3137] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Integrable structure, W-symmetry and AGT relation, JHEP 01 (2012) 051 [arXiv:1109.4042] [INSPIRE].
H. Kanno, K. Maruyoshi, S. Shiba and M. Taki, W3 irregular states and isolated N = 2 superconformal field theories, JHEP 03 (2013) 147 [arXiv:1301.0721] [INSPIRE].
S. Mironov, A. Morozov and Y. Zenkevich, Generalized Jack polynomials and the AGT relations for the SU(3) group, JETP Lett. 99 (2014) 109 [arXiv:1312.5732] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
L. Hollands, C.A. Keller and J. Song, Towards a 4d/2d correspondence for Sicilian quivers, JHEP 10 (2011) 100 [arXiv:1107.0973] [INSPIRE].
S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE].
S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4-D N = 2 gauge theories: 1., Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110] [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
H. Awata and H. Kanno, Refined BPS state counting from Nekrasov’s formula and Macdonald functions, Int. J. Mod. Phys. A 24 (2009) 2253 [arXiv:0805.0191] [INSPIRE].
R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings, and N = 2 Gauge Systems, arXiv:0909.2453 [INSPIRE].
M.C.N. Cheng, R. Dijkgraaf and C. Vafa, Non-Perturbative Topological Strings And Conformal Blocks, JHEP 09 (2011) 022 [arXiv:1010.4573] [INSPIRE].
R. Schiappa and N. Wyllard, An A(r) threesome: Matrix models, 2d CFTs and 4d N = 2 gauge theories, J. Math. Phys. 51 (2010) 082304 [arXiv:0911.5337] [INSPIRE].
K. Maruyoshi, β-Deformed Matrix Models and 2d/ 4d Correspondence, in New Dualities of Supersymmetric Gauge Theories, J. Teschner, ed., pp. 121–157 (2016), DOI [arXiv:1412.7124] [INSPIRE].
N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [INSPIRE].
A. Gorsky, S. Gukov and A. Mironov, SUSY field theories, integrable systems and their stringy/brane origin. 2., Nucl. Phys. B 518 (1998) 689 [hep-th/9710239] [INSPIRE].
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
L. Bao, V. Mitev, E. Pomoni, M. Taki and F. Yagi, Non-Lagrangian Theories from Brane Junctions, JHEP 01 (2014) 175 [arXiv:1310.3841] [INSPIRE].
H. Hayashi, H.-C. Kim and T. Nishinaka, Topological strings and 5d TN partition functions, JHEP 06 (2014) 014 [arXiv:1310.3854] [INSPIRE].
V. Mitev and E. Pomoni, Toda 3-Point Functions From Topological Strings, JHEP 06 (2015) 049 [arXiv:1409.6313] [INSPIRE].
M. Isachenkov, V. Mitev and E. Pomoni, Toda 3-Point Functions From Topological Strings II, JHEP 08 (2016) 066 [arXiv:1412.3395] [INSPIRE].
D. Gaiotto and J. Maldacena, The gravity duals of N = 2 superconformal field theories, JHEP 10 (2012) 189 [arXiv:0904.4466] [INSPIRE].
A. Iqbal and C. Vafa, BPS Degeneracies and Superconformal Index in Diverse Dimensions, Phys. Rev. D 90 (2014) 105031 [arXiv:1210.3605] [INSPIRE].
H. Hayashi and G. Zoccarato, Exact partition functions of Higgsed 5d TN theories, JHEP 01 (2015) 093 [arXiv:1409.0571] [INSPIRE].
Y. Tachikawa, A review of the TN theory and its cousins, PTEP 2015 (2015) 11B102 [arXiv:1504.01481] [INSPIRE].
M. Aganagic, N. Haouzi, C. Kozcaz and S. Shakirov, Gauge/Liouville Triality, arXiv:1309.1687 [INSPIRE].
M. Aganagic, N. Haouzi and S. Shakirov, An-Triality, arXiv:1403.3657 [INSPIRE].
C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, The vertex on a strip, Adv. Theor. Math. Phys. 10 (2006) 317 [hep-th/0410174] [INSPIRE].
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
O. Aharony, A. Hanany and B. Kol, Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, Instanton counting and Chern-Simons theory, Adv. Theor. Math. Phys. 7 (2003) 457 [hep-th/0212279] [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, SU(N) geometries and topological string amplitudes, Adv. Theor. Math. Phys. 10 (2006) 1 [hep-th/0306032] [INSPIRE].
M. Taki, Refined Topological Vertex and Instanton Counting, JHEP 03 (2008) 048 [arXiv:0710.1776] [INSPIRE].
H. Awata and H. Kanno, Changing the preferred direction of the refined topological vertex, J. Geom. Phys. 64 (2013) 91 [arXiv:0903.5383] [INSPIRE].
Y. Zenkevich, Refined toric branes, surface operators and factorization of generalized Macdonald polynomials, JHEP 09 (2017) 070 [arXiv:1612.09570] [INSPIRE].
T. Kimura, H. Mori and Y. Sugimoto, Refined geometric transition and qq-characters, JHEP 01 (2018) 025 [arXiv:1705.03467] [INSPIRE].
P. Sulkowski, Crystal model for the closed topological vertex geometry, JHEP 12 (2006) 030 [hep-th/0606055] [INSPIRE].
I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, (1998).
Y. Konishi and S. Minabe, Flop invariance of the topological vertex, Int. J. Math. 19 (2008) 27 [math/0601352] [INSPIRE].
V. Mitev, E. Pomoni, M. Taki and F. Yagi, Fiber-Base Duality and Global Symmetry Enhancement, JHEP 04 (2015) 052 [arXiv:1411.2450] [INSPIRE].
I. Coman, E. Pomoni and J. Teschner, From quantum curves to topological string partition functions, arXiv:1811.01978 [INSPIRE].
B. Feigin and E. Frenkel, Quantum W algebras and elliptic algebras, Commun. Math. Phys. 178 (1996) 653 [q-alg/9508009] [INSPIRE].
E. Frenkel and N. Reshetikhin, Deformations of W-algebras associated to simple Lie algebras, q-alg/9708006.
A.B. Zamolodchikov, Three-point function in the minimal Liouville gravity, Theor. Math. Phys. 142 (2005) 183 [hep-th/0505063] [INSPIRE].
S. Ribault and R. Santachiara, Liouville theory with a central charge less than one, JHEP 08 (2015) 109 [arXiv:1503.02067] [INSPIRE].
T. Eguchi and H. Kanno, Topological strings and Nekrasov’s formulas, JHEP 12 (2003) 006 [hep-th/0310235] [INSPIRE].
R. Dijkgraaf and C. Vafa, On geometry and matrix models, Nucl. Phys. B 644 (2002) 21 [hep-th/0207106] [INSPIRE].
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Num. Theor. Phys. 1 (2007) 347 [math-ph/0702045] [INSPIRE].
B. Eynard, A short overview of the “Topological recursion”, arXiv:1412.3286 [INSPIRE].
K. Iwaki, T. Koike and Y. Takei, Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin’s Topological Recursion — Part I: For the Weber Equation, arXiv:1805.10945.
K. Iwaki, T. Koike and Y. Takei, Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin’s Topological Recursion — Part II: For the Confluent Family of Hypergeometric Equations, arXiv:1810.02946.
V.S. Adamchik, Contributions to the Theory of the Barnes Function, math/0308086.
M. Fukuda, Y. Ohkubo and J. Shiraishi, Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction, arXiv:1903.05905 [INSPIRE].
A. Neguţ, The q-AGT-W relations via shuffle algebras, Commun. Math. Phys. 358 (2018) 101 [arXiv:1608.08613] [INSPIRE].
I. Coman, M. Gabella and J. Teschner, Line operators in theories of class \( \mathcal{S} \), quantized moduli space of flat connections, and Toda field theory, JHEP 10 (2015) 143 [arXiv:1505.05898] [INSPIRE].
I. Coman, E. Pomoni and J. Teschner, Toda conformal blocks, quantum groups, and flat connections, Commun. Math. Phys. 375 (2019) 1117 [arXiv:1712.10225] [INSPIRE].
M. Taki, Seiberg Duality, 5d SCFTs and Nekrasov Partition Functions, arXiv:1401.7200 [INSPIRE].
F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP 01 (2010) 088 [arXiv:0909.1327] [INSPIRE].
E. Gava, K.S. Narain, M. Muteeb and V.I. Giraldo-Rivera, N = 2 gauge theories on the hemisphere H S4 , Nucl. Phys. B 920 (2017) 256 [arXiv:1611.04804] [INSPIRE].
M. Dedushenko, Gluing II: Boundary Localization and Gluing Formulas, arXiv:1807.04278 [INSPIRE].
S. Pasquetti, Factorisation of N = 2 Theories on the Squashed 3-Sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, JHEP 12 (2014) 177 [arXiv:1211.1986] [INSPIRE].
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].
I. Coman-Lohi, On generalisations of the AGT correspondence for non-Lagrangian theories of class S, Ph.D. Thesis, Hamburg University, Hamburg, Germany (2018), DOI [INSPIRE].
A.N. Kirillov, Dilogarithm identities, Prog. Theor. Phys. Suppl. 118 (1995) 61 [hep-th/9408113] [INSPIRE].
T.H. Koornwinder, Jacobi functions as limit cases of q-ultraspherical polynomials, J. Math. Anal. Appl. 148 (1990) 44.
S.O. Warnaar, A Selberg integral for the Lie algebra An , arXiv:0708.1193.
S.O. Warnaar, The sl3 Selberg integral, Adv. Math. 224 (2010) 499 [arXiv:0901.4176].
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Coman, I., Pomoni, E. & Teschner, J. Trinion conformal blocks from topological strings. J. High Energ. Phys. 2020, 78 (2020). https://doi.org/10.1007/JHEP09(2020)078
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DOI: https://doi.org/10.1007/JHEP09(2020)078