Abstract
We consider the computation of the topological string partition function for 5-brane web diagrams with an O7−-plane. Since upon quantum resolution of the orientifold plane these diagrams become non-toric web diagrams without the orientifold we are able to apply the topological vertex to obtain the Nekrasov partition function of the corresponding 5d theory. We apply this procedure to the case of 5d SU(N) theories with one hypermultiplet in the antisymmetric representation and to the case of 5d pure USp(2N) theories. For these cases we discuss the dictionary between parameters and moduli of the 5d gauge theory and lengths of 5-branes in the web diagram and moreover we perform comparison of the results obtained via application of the topological vertex and the one obtained via localisation techniques, finding in all instances we consider perfect agreement.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [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].
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].
O. Bergman, D. Rodríguez-Gómez and G. Zafrir, 5-brane webs, symmetry enhancement and duality in 5d supersymmetric gauge theory, JHEP 03 (2014) 112 [arXiv:1311.4199] [INSPIRE].
G. Zafrir, Duality and enhancement of symmetry in 5d gauge theories, JHEP 12 (2014) 116 [arXiv:1408.4040] [INSPIRE].
O. Bergman and G. Zafrir, Lifting 4d dualities to 5d, JHEP 04 (2015) 141 [arXiv:1410.2806] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, A new 5d description of 6d D-type minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [INSPIRE].
O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, JHEP 12 (2015) 163 [arXiv:1507.03860] [INSPIRE].
G. Zafrir, Brane webs, 5d gauge theories and 6d \( \mathcal{N}=\left(1,0\right) \) SCFT’s, JHEP 12 (2015) 157 [arXiv:1509.02016] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 6d SCFTs, 5d dualities and Tao web diagrams, arXiv:1509.03300 [INSPIRE].
K. Ohmori and H. Shimizu, S 1 /T 2 compactifications of 6d \( \mathcal{N}=\left(1,0\right) \) theories and brane webs, JHEP 03 (2016) 024 [arXiv:1509.03195] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, More on 5d descriptions of 6d SCFTs, JHEP 10 (2016) 126 [arXiv:1512.08239] [INSPIRE].
G. Zafrir, Brane webs and O5-planes, JHEP 03 (2016) 109 [arXiv:1512.08114] [INSPIRE].
G. Zafrir, Brane webs in the presence of an O(5) − -plane and 4d class S theories of type D, JHEP 07 (2016) 035 [arXiv:1602.00130] [INSPIRE].
Y. Tachikawa, Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories, PTEP 2015 (2015) 043B06 [arXiv:1501.01031] [INSPIRE].
G. Zafrir, Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO and exceptional gauge theories, JHEP 07 (2015) 087 [arXiv:1503.08136] [INSPIRE].
K. Yonekura, Instanton operators and symmetry enhancement in 5d supersymmetric quiver gauge theories, JHEP 07 (2015) 167 [arXiv:1505.04743] [INSPIRE].
D. Gaiotto and H.-C. Kim, Duality walls and defects in 5d \( \mathcal{N}=1 \) theories, JHEP 01 (2017) 019 [arXiv:1506.03871] [INSPIRE].
B. Kol, 5D field theories and M-theory, JHEP 11 (1999) 026 [hep-th/9705031] [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].
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.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [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].
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].
H. Hayashi, H.-C. Kim and T. Nishinaka, Topological strings and 5d T N partition functions, JHEP 06 (2014) 014 [arXiv:1310.3854] [INSPIRE].
H. Hayashi and G. Zoccarato, Exact partition functions of Higgsed 5d T N theories, JHEP 01 (2015) 093 [arXiv:1409.0571] [INSPIRE].
S.-S. Kim, M. Taki and F. Yagi, Tao probing the end of the world, PTEP 2015 (2015) 083B02 [arXiv:1504.03672] [INSPIRE].
H. Hayashi and G. Zoccarato, Topological vertex for Higgsed 5d T N theories, JHEP 09 (2015) 023 [arXiv:1505.00260] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Equivalence of several descriptions for 6d SCFT, JHEP 01 (2017) 093 [arXiv:1607.07786] [INSPIRE].
A. Sen, F theory and orientifolds, Nucl. Phys. B 475 (1996) 562 [hep-th/9605150] [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].
V. Mitev, E. Pomoni, M. Taki and F. Yagi, Fiber-base duality and global symmetry enhancement, JHEP 04 (2015) 052 [arXiv:1411.2450] [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].
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].
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].
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].
O. Bergman, D. Rodríguez-Gómez and G. Zafrir, Discrete θ and the 5d superconformal index, JHEP 01 (2014) 079 [arXiv:1310.2150] [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. Nakajima and K. Yoshioka, Instanton counting on blowup. 1, Invent. Math. 162 (2005) 313 [math/0306198] [INSPIRE].
S. Shadchin, On certain aspects of string theory/gauge theory correspondence, hep-th/0502180 [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, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
S. Benvenuti, A. Hanany and N. Mekareeya, The Hilbert series of the one instanton moduli space, JHEP 06 (2010) 100 [arXiv:1005.3026] [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].
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].
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].
C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [arXiv:1406.6793] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1609.07381
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hayashi, H., Zoccarato, G. Partition functions of web diagrams with an O7−-plane. J. High Energ. Phys. 2017, 112 (2017). https://doi.org/10.1007/JHEP03(2017)112
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2017)112