Abstract
In this article we use 5-brane junctions to study the 5D T N SCFTs corresponding to the 5D \( \mathcal{N} \) = 1 uplift of the 4D \( \mathcal{N} \) = 2 strongly coupled gauge theories, which are obtained by compactifying N M5 branes on a sphere with three full punctures. Even though these theories have no Lagrangian description, by using the 5-brane junctions proposed by Benini, Benvenuti and Tachikawa, we are able to derive their Seiberg-Witten curves and Nekrasov partition functions. We cross-check our results with the 5D super-conformal index proposed by Kim, Kim and Lee. Through the AGTW correspondence, we discuss the relations between 5D superconformal indices and n-point functions of the q-deformed W N Toda theories.
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References
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
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. 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].
F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP 01 (2010) 088 [arXiv:0909.1327] [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].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2., Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
O.J. Ganor, A test of the chiral E 8 current algebra on a 6 − D noncritical string, Nucl. Phys. B 479 (1996) 197 [hep-th/9607020] [INSPIRE].
A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, hep-th/9607139 [INSPIRE].
O.J. Ganor, Toroidal compactification of heterotic 6 − D noncritical strings down to four-dimensions, Nucl. Phys. B 488 (1997) 223 [hep-th/9608109] [INSPIRE].
O.J. Ganor, D.R. Morrison and N. Seiberg, Branes, Calabi-Yau spaces and toroidal compactification of the N = 1 six-dimensional E 8 theory, Nucl. Phys. B 487 (1997) 93 [hep-th/9610251] [INSPIRE].
J. Minahan, D. Nemeschansky, C. Vafa and N. Warner, E strings and N = 4 topological Yang-Mills theories, Nucl. Phys. B 527 (1998) 581 [hep-th/9802168] [INSPIRE].
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [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].
B. Kol, 5 − D field theories and M-theory, JHEP 11 (1999) 026 [hep-th/9705031] [INSPIRE].
A. Brandhuber, N. Itzhaki, J. Sonnenschein, S. Theisen and S. Yankielowicz, On the M-theory approach to (compactified) 5 − D field theories, Phys. Lett. B 415 (1997) 127 [hep-th/9709010] [INSPIRE].
L. Bao, E. Pomoni, M. Taki and F. Yagi, M5-Branes, Toric Diagrams and Gauge Theory Duality, JHEP 04 (2012) 105 [arXiv:1112.5228] [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].
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].
O. DeWolfe, A. Hanany, A. Iqbal and E. Katz, Five-branes, seven-branes and five-dimensional E(n) field theories, JHEP 03 (1999) 006 [hep-th/9902179] [INSPIRE].
M.R. Gaberdiel and B. Zwiebach, Exceptional groups from open strings, Nucl. Phys. B 518 (1998) 151 [hep-th/9709013] [INSPIRE].
M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, Del Pezzo surfaces and type-I-prime theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].
J. Minahan, D. Nemeschansky and N. Warner, Investigating the BPS spectrum of noncritical E(n) strings, Nucl. Phys. B 508 (1997) 64 [hep-th/9705237] [INSPIRE].
T. Eguchi and K. Sakai, Seiberg-Witten curve for the E string theory, JHEP 05 (2002) 058 [hep-th/0203025] [INSPIRE].
T. Eguchi and K. Sakai, Seiberg-Witten curve for E string theory revisited, Adv. Theor. Math. Phys. 7 (2004) 419 [hep-th/0211213] [INSPIRE].
J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E 6 global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].
J.A. Minahan and D. Nemeschansky, Superconformal fixed points with E(n) global symmetry, Nucl. Phys. B 489 (1997) 24 [hep-th/9610076] [INSPIRE].
J. Minahan, D. Nemeschansky and N. Warner, Partition functions for BPS states of the noncritical E 8 string, Adv. Theor. Math. Phys. 1 (1998) 167 [hep-th/9707149] [INSPIRE].
D.-E. Diaconescu and B. Florea, The ruled vertex and nontoric del Pezzo surfaces, JHEP 12 (2006) 028 [hep-th/0507240] [INSPIRE].
D.-E. Diaconescu and B. Florea, The ruled vertex and D-E degenerations, JHEP 09 (2005) 044 [hep-th/0507058] [INSPIRE].
Y. Konishi and S. Minabe, Local Gromov-Witten invariants of cubic surfaces via NEF toric degeneration, math/0607187 [INSPIRE].
K. Mohri, Exceptional string: instanton expansions and Seiberg-Witten curve, Rev. Math. Phys. 14 (2002) 913 [hep-th/0110121] [INSPIRE].
K. Sakai, Topological string amplitudes for the local half K3 surface, arXiv:1111.3967 [INSPIRE].
K. Sakai, Seiberg-Witten prepotential for E-string theory and global symmetries, JHEP 09 (2012) 077 [arXiv:1207.5739] [INSPIRE].
M.-X. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M- and [p, q]-strings, JHEP 11 (2013) 112 [arXiv:1308.0619] [INSPIRE].
M.R. Gaberdiel, T. Hauer and B. Zwiebach, Open string-string junction transitions, Nucl. Phys. B 525 (1998) 117 [hep-th/9801205] [INSPIRE].
O. DeWolfe and B. Zwiebach, String junctions for arbitrary Lie algebra representations, Nucl. Phys. B 541 (1999) 509 [hep-th/9804210] [INSPIRE].
A. Iqbal, Selfintersection number of BPS junctions in backgrounds of three-branes and seven-branes, JHEP 10 (1999) 032 [hep-th/9807117] [INSPIRE].
O. DeWolfe, T. Hauer, A. Iqbal and B. Zwiebach, Constraints on the BPS spectrum of N =2, D = 4 theories with A-D-E flavor symmetry, Nucl. Phys. B 534 (1998) 261 [hep-th/9805220] [INSPIRE].
O. DeWolfe, T. Hauer, A. Iqbal and B. Zwiebach, Uncovering the symmetries on [p,q] seven-branes: beyond the Kodaira classification, Adv. Theor. Math. Phys. 3 (1999) 1785 [hep-th/9812028] [INSPIRE].
O. DeWolfe, T. Hauer, A. Iqbal and B. Zwiebach, Uncovering infinite symmetries on [p, q] 7-branes: kac-Moody algebras and beyond, Adv. Theor. Math. Phys. 3 (1999) 1835 [hep-th/9812209] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
M. Taki, Refined Topological Vertex and Instanton Counting, JHEP 03 (2008) 048 [arXiv:0710.1776] [INSPIRE].
A. Iqbal and C. Vafa, BPS Degeneracies and Superconformal Index in Diverse Dimensions, arXiv:1210.3605 [INSPIRE].
H.-C. Kim, S.-S. Kim and K. Lee, 5-dim Superconformal Index with Enhanced En Global Symmetry, JHEP 10 (2012) 142 [arXiv:1206.6781] [INSPIRE].
A. Gadde, E. Pomoni and L. Rastelli, The Veneziano Limit of N = 2 Superconformal QCD: towards the String Dual of N = 2 SU(N c) SYM with N f = 2N c, arXiv:0912.4918 [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].
O. Bergman, D. Rodriguez-Gomez and G. Zafrir, Discrete theta and the 5d superconformal index, arXiv:1310.2150 [INSPIRE].
H. Awata and Y. Yamada, Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [INSPIRE].
H. Awata and Y. Yamada, Five-dimensional AGT Relation and the Deformed beta-ensemble, Prog. Theor. Phys. 124 (2010) 227 [arXiv:1004.5122] [INSPIRE].
A. Mironov, A. Morozov, S. Shakirov and A. Smirnov, Proving AGT conjecture as HS duality: extension to five dimensions, Nucl. Phys. B 855 (2012) 128 [arXiv:1105.0948] [INSPIRE].
H. Itoyama, T. Oota and R. Yoshioka, 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space, Nucl. Phys. B 877 (2013) 506 [arXiv:1308.2068] [INSPIRE].
F. Nieri, S. Pasquetti and F. Passerini, 3d & 5d gauge theory partition functions as q-deformed CFT correlators, arXiv:1303.2626 [INSPIRE].
H. Hayashi, H.-C. Kim and T. Nishinaka, Topological strings and 5d T N partition functions, arXiv:1310.3854 [INSPIRE].
N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems, Nucl. Phys. B 531 (1998) 323 [hep-th/9609219] [INSPIRE].
L. Bao, E. Pomoni, M. Taki and F. Yagi, M5-branes, toric diagrams and gauge theory duality, Int. J. Mod. Phys. Conf. Ser. 21 (2013) 136.
A. Iqbal, C. Kozcaz and K. Shabbir, Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory, Nucl. Phys. B 838 (2010) 422 [arXiv:0803.2260] [INSPIRE].
A. Iqbal and C. Kozcaz, Refined Topological Strings and Toric Calabi-Yau Threefolds, arXiv:1210.3016 [INSPIRE].
K. Hori et al., Mirror symmetry, Clay mathematics monographs, Vol. 1, Oxford University Press, (2003).
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].
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).
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d Topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d Superconformal Index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [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].
L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, Phys. Rev. Lett. 105 (2010) 141601 [arXiv:0909.4776] [INSPIRE].
M.-C. Tan, An M-Theoretic Derivation of a 5d and 6d AGT Correspondence and Relativistic and Elliptized Integrable Systems, JHEP 12 (2013) 031 [arXiv:1309.4775] [INSPIRE].
G. Bonelli, A. Tanzini and J. Zhao, Vertices, Vortices and Interacting Surface Operators, JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].
G. Bonelli, A. Tanzini and J. Zhao, The Liouville side of the Vortex, JHEP 09 (2011) 096 [arXiv:1107.2787] [INSPIRE].
P. Bowcock and G. Watts, Null vectors, three point and four point functions in conformal field theory, Theor. Math. Phys. 98 (1994) 350 [hep-th/9309146] [INSPIRE].
V. Fateev and A. Litvinov, Correlation functions in conformal Toda field theory. I., JHEP 11 (2007) 002 [arXiv:0709.3806] [INSPIRE].
V. Fateev and A. Litvinov, Correlation functions in conformal Toda field theory II, JHEP 01 (2009) 033 [arXiv:0810.3020] [INSPIRE].
C.A. Keller, N. Mekareeya, J. Song and Y. Tachikawa, The ABCDEFG of Instantons and W-algebras, JHEP 03 (2012) 045 [arXiv:1111.5624] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. 1., Invent. Math. 162 (2005) 313 [math/0306198] [INSPIRE].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].
D. Rodríguez-Gómez and G. Zafrir, On the 5d instanton index as a Hilbert series, Nucl. Phys. B 878 (2014) 1 [arXiv:1305.5684] [INSPIRE].
D. Gaiotto and S.S. Razamat, Exceptional Indices, JHEP 05 (2012) 145 [arXiv:1203.5517] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
C. Vafa, Supersymmetric Partition Functions and a String Theory in 4 Dimensions, arXiv:1209.2425 [INSPIRE].
H. Awata and H. Kanno, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [INSPIRE].
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Bao, L., Mitev, V., Pomoni, E. et al. Non-Lagrangian theories from brane junctions. J. High Energ. Phys. 2014, 175 (2014). https://doi.org/10.1007/JHEP01(2014)175
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DOI: https://doi.org/10.1007/JHEP01(2014)175