Abstract
We evaluate the Nekrasov partition function of 5d gauge theories engineered by webs of 5-branes, using the refined topological vertex on the dual Calabi-Yau three-folds. The theories include certain non-Lagrangian theories such as the T N theory. The refined topological vertex computation generically contains contributions from decoupled M2-branes which are not charged under the 5d gauge symmetry engineered. We argue that, after eliminating them, the refined topological string partition function agrees with the 5d Nekrasov partition function. We explicitly check this for the T 3 theory as well as Sp(1) gauge theories with N f = 2, 3, 4 flavors. In particular, our method leads to a new expression of the Sp(1) Nekrasov partition functions without any contour integrals. We also develop prescriptions to calculate the partition functions of theories obtained by Higgsing the T N theory. We compute the partition function of the E 7 theory via this prescription, and find the E 7 global symmetry enhancement. We finally discuss a potential application of the refined topological vertex to non-toric web diagrams.
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References
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
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].
A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen, Simple singularities and N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 344 (1995) 169 [hep-th/9411048] [INSPIRE].
P.C. Argyres and A.E. Faraggi, The vacuum structure and spectrum of N = 2 supersymmetric SU(N) gauge theory, Phys. Rev. Lett. 74 (1995) 3931 [hep-th/9411057] [INSPIRE].
A. Hanany and Y. Oz, On the quantum moduli space of vacua of N = 2 supersymmetric SU(N c ) gauge theories, Nucl. Phys. B 452 (1995) 283 [hep-th/9505075] [INSPIRE].
P.C. Argyres, M.R. Plesser and A.D. Shapere, The Coulomb phase of N = 2 supersymmetric QCD, Phys. Rev. Lett. 75 (1995) 1699 [hep-th/9505100] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].
H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series 18, American Mathematical Society, Providence U.S.A. (1999).
N. Nekrasov and A.S. Schwarz, Instantons on noncommutative R 4 and (2, 0) superconformal six-dimensional theory, Commun. Math. Phys. 198 (1998) 689 [hep-th/9802068] [INSPIRE].
M. Mariño and N. Wyllard, A note on instanton counting for N = 2 gauge theories with classical gauge groups, JHEP 05 (2004) 021 [hep-th/0404125] [INSPIRE].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].
H.-C. Kim, S.-S. Kim and K. Lee, 5-dim superconformal index with enhanced E n global symmetry, JHEP 10 (2012) 142 [arXiv:1206.6781] [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].
D. Bashkirov, A comment on the enhancement of global symmetries in superconformal SU(2) gauge theories in 5D, arXiv:1211.4886 [INSPIRE].
A. Iqbal, All genus topological string amplitudes and five-brane webs as Feynman diagrams, hep-th/0207114 [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [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].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
A. Klemm, W. Lerche, P. Mayr, C. Vafa and N.P. Warner, Selfdual strings and N = 2 supersymmetric field theory, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034] [INSPIRE].
D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [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].
S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4D N = 2 gauge theories: 1, Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110] [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, Instanton counting and Chern-Simons theory, Adv. Theor. Math. Phys. 7 (2004) 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].
T. Eguchi and H. Kanno, Topological strings and Nekrasov’s formulas, JHEP 12 (2003) 006 [hep-th/0310235] [INSPIRE].
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].
M. Taki, Refined topological vertex and instanton counting, JHEP 03 (2008) 048 [arXiv:0710.1776] [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].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
D.-E. Diaconescu, B. Florea and N. Saulina, A vertex formalism for local ruled surfaces, Commun. Math. Phys. 265 (2006) 201 [hep-th/0505192] [INSPIRE].
D.-E. Diaconescu and B. Florea, The ruled vertex and nontoric del Pezzo surfaces, JHEP 12 (2006) 028 [hep-th/0507240] [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].
M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
S. Kachru, A. Klemm, W. Lerche, P. Mayr and C. Vafa, Nonperturbative results on the point particle limit of N = 2 heterotic string compactifications, Nucl. Phys. B 459 (1996) 537 [hep-th/9508155] [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].
N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [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].
I. Antoniadis, E. Gava, K.S. Narain and T.R. Taylor, Topological amplitudes in string theory, Nucl. Phys. B 413 (1994) 162 [hep-th/9307158] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 1, hep-th/9809187 [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 2, hep-th/9812127 [INSPIRE].
M. Aganagic, M. Mariño and C. Vafa, All loop topological string amplitudes from Chern-Simons theory, Commun. Math. Phys. 247 (2004) 467 [hep-th/0206164] [INSPIRE].
L. Hollands, C.A. Keller and J. Song, From SO/Sp instantons to W-algebra blocks, JHEP 03 (2011) 053 [arXiv:1012.4468] [INSPIRE].
I. Antoniadis, S. Hohenegger, K.S. Narain and T.R. Taylor, Deformed topological partition function and Nekrasov backgrounds, Nucl. Phys. B 838 (2010) 253 [arXiv:1003.2832] [INSPIRE].
Y. Nakayama and H. Ooguri, Comments on worldsheet description of the Omega background, Nucl. Phys. B 856 (2012) 342 [arXiv:1106.5503] [INSPIRE].
I. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Worldsheet realization of the refined topological string, Nucl. Phys. B 875 (2013) 101 [arXiv:1302.6993] [INSPIRE].
I. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Non-perturbative Nekrasov partition function from string theory, Nucl. Phys. B 880 (2014) 87 [arXiv:1309.6688] [INSPIRE].
M. Aganagic and S. Shakirov, Refined Chern-Simons theory and topological string, arXiv:1210.2733 [INSPIRE].
A. Iqbal and C. Kozcaz, Refined topological strings and toric Calabi-Yau threefolds, arXiv:1210.3016 [INSPIRE].
O. Bergman, D. Rodríguez-Gómez and G. Zafrir, Discrete θ and the 5d superconformal index, JHEP 01 (2014) 079 [arXiv:1310.2150] [INSPIRE].
A. Iqbal and C. Vafa, BPS degeneracies and superconformal index in diverse dimensions, arXiv:1210.3605 [INSPIRE].
Y. Tachikawa, Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting, JHEP 02 (2004) 050 [hep-th/0401184] [INSPIRE].
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].
D. Karp, C.-C.M. Liu and M. Mariño, The local Gromov-Witten invariants of configurations of rational curves, math.AG/0506488 [INSPIRE].
P. Sulkowski, Crystal model for the closed topological vertex geometry, JHEP 12 (2006) 030 [hep-th/0606055] [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].
S.H. Katz, A. Klemm and C. Vafa, M theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999) 1445 [hep-th/9910181] [INSPIRE].
C. Vafa, Supersymmetric partition functions and a string theory in 4 dimensions, arXiv:1209.2425 [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].
H. Awata and H. Kanno, Changing the preferred direction of the refined topological vertex, J. Geom. Phys. 64 (2013) 91 [arXiv:0903.5383] [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
M. Taki, Surface operator, bubbling Calabi-Yau and AGT relation, JHEP 07 (2011) 047 [arXiv:1007.2524] [INSPIRE].
M. Aganagic and S. Shakirov, Knot homology from refined Chern-Simons theory, arXiv:1105.5117 [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].
D. Gaiotto and S.S. Razamat, Exceptional indices, JHEP 05 (2012) 145 [arXiv:1203.5517] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, arXiv:1207.3577 [INSPIRE].
O. Bergman and A. Fayyazuddin, String junction transitions in the moduli space of N = 2 SYM, Nucl. Phys. B 535 (1998) 139 [hep-th/9806011] [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].
N. Mekareeya and D. Rodriguez-Gomez, 5d gauge theories on orbifolds and 4d ‘t Hooft line indices, JHEP 11 (2013) 157 [arXiv:1309.1213] [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].
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 β-ensemble, Prog. Theor. Phys. 124 (2010) 227 [arXiv:1004.5122] [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].
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].
F. Nieri, S. Pasquetti and F. Passerini, 3d & 5d gauge theory partition functions as q-deformed CFT correlators, arXiv:1303.2626 [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].
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].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, The vertex on a strip, Adv. Theor. Math. Phys. 10 (2006) 317 [hep-th/0410174] [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].
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Hayashi, H., Kim, HC. & Nishinaka, T. Topological strings and 5d T N partition functions. J. High Energ. Phys. 2014, 14 (2014). https://doi.org/10.1007/JHEP06(2014)014
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DOI: https://doi.org/10.1007/JHEP06(2014)014