Abstract
In this paper, we study 5d \( \mathcal{N} \) = 1 Sp(N) gauge theory with Nf (≤ 2N + 3) flavors based on 5-brane web diagram with O5-plane. On the one hand, we discuss Seiberg-Witten curve based on the dual graph of the 5-brane web with O5-plane. On the other hand, we compute the Nekrasov partition function based on the topological vertex formalism with O5-plane. Rewriting it in terms of profile functions, we obtain the saddle point equation for the profile function after taking thermodynamic limit. By introducing the resolvent, we derive the Seiberg-Witten curve and its boundary conditions as well as its relation to the prepotential in terms of the cycle integrals. They coincide with those directly obtained from the dual graph of the 5-brane web with O5-plane. This agreement gives further evidence for mirror symmetry which relates Nekrasov partition function with Seiberg-Witten curve in the case with orientifold plane and shed light on the non-toric Calabi-Yau 3-folds including D-type singularities.
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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. 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].
N. A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. 1., Invent. Math. 162 (2005) 313 [math/0306198] [INSPIRE].
A. Braverman and P. Etingof, Instanton counting via affine Lie algebras II: From Whittaker vectors to the Seiberg-Witten prepotential, math/0409441 [INSPIRE].
E. Gasparim and C.-C.M. Liu, The Nekrasov Conjecture for Toric Surfaces, Commun. Math. Phys. 293 (2010) 661 [arXiv:0808.0884] [INSPIRE].
N. C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [INSPIRE].
A. Karch, D. Lüst and D.J. Smith, Equivalence of geometric engineering and Hanany-Witten via fractional branes, Nucl. Phys. B 533 (1998) 348 [hep-th/9803232] [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].
H. Hayashi, H.-C. Kim and T. Nishinaka, Topological strings and 5d TN partition functions, JHEP 06 (2014) 014 [arXiv:1310.3854] [INSPIRE].
H. Hayashi and G. Zoccarato, Topological vertex for Higgsed 5d TN theories, JHEP 09 (2015) 023 [arXiv:1505.00260] [INSPIRE].
A. Hanany and A. Zaffaroni, Issues on orientifolds: On the brane construction of gauge theories with SO(2N) global symmetry, JHEP 07 (1999) 009 [hep-th/9903242] [INSPIRE].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [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. Brandhuber, J. Sonnenschein, S. Theisen and S. Yankielowicz, M theory and Seiberg-Witten curves: Orthogonal and symplectic groups, Nucl. Phys. B 504 (1997) 175 [hep-th/9705232] [INSPIRE].
K. Landsteiner, E. Lopez and D. A. Lowe, N = 2 supersymmetric gauge theories, branes and orientifolds, Nucl. Phys. B 507 (1997) 197 [hep-th/9705199] [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].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Discrete theta angle from an O5-plane, JHEP 11 (2017) 041 [arXiv:1707.07181] [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].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. II. K-theoretic partition function, math/0505553 [INSPIRE].
T. Eguchi and H. Kanno, Topological strings and Nekrasov’s formulas, JHEP 12 (2003) 006 [hep-th/0310235] [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].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The Topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
J. Li, C.-C.M. Liu, K. Liu and J. Zhou, A Mathematical theory of the topological vertex, Geom. Topol. 13 (2009) 527 [math/0408426] [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].
S.-S. Kim and F. Yagi, Topological vertex formalism with O5-plane, Phys. Rev. D 97 (2018) 026011 [arXiv:1709.01928] [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].
A. Fayyazuddin and M. Spalinski, The Seiberg-Witten differential from M-theory, Nucl. Phys. B 508 (1997) 219 [hep-th/9706087] [INSPIRE].
M. Henningson and P. Yi, Four-dimensional BPS spectra via M-theory, Phys. Rev. D 57 (1998) 1291 [hep-th/9707251] [INSPIRE].
A. Mikhailov, BPS states and minimal surfaces, Nucl. Phys. B 533 (1998) 243 [hep-th/9708068] [INSPIRE].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [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].
H. Hayashi and R.-D. Zhu, More on topological vertex formalism for 5-brane webs with O5-plane, JHEP 04 (2021) 292 [arXiv:2012.13303] [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. Dedushenko and E. Witten, Some Details On The Gopakumar-Vafa and Ooguri-Vafa Formulas, Adv. Theor. Math. Phys. 20 (2016) 1 [arXiv:1411.7108] [INSPIRE].
S. Codesido, A. Grassi and M. Mariño, Spectral Theory and Mirror Curves of Higher Genus, Annales Henri Poincaré 18 (2017) 559 [arXiv:1507.02096] [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].
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].
Y. Konishi and S. Minabe, Flop invariance of the topological vertex, Int. J. Math. 19 (2008) 27 [math/0601352] [INSPIRE].
M. Taki, Flop Invariance of Refined Topological Vertex and Link Homologies, arXiv:0805.0336 [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, The Vertex on a strip, Adv. Theor. Math. Phys. 10 (2006) 317 [hep-th/0410174] [INSPIRE].
Y. Konishi and S. Minabe, Local Gromov-Witten invariants of cubic surfaces via NEF toric degeneration, math/0607187 [INSPIRE].
O. Bergman, D. Rodríguez-Gómez and G. Zafrir, Discrete θ and the 5d superconformal index, JHEP 01 (2014) 079 [arXiv:1310.2150] [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].
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].
C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [Addendum ibid. 04 (2016) 094] [arXiv:1406.6793] [INSPIRE].
S.-S. Kim, M. Taki and F. Yagi, Tao Probing the End of the World, PTEP 2015 (2015) 083B02 [arXiv:1504.03672] [INSPIRE].
T. J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].
S. Shadchin, Saddle point equations in Seiberg-Witten theory, JHEP 10 (2004) 033 [hep-th/0408066] [INSPIRE].
A. Klemm and P. Sulkowski, Seiberg-Witten theory and matrix models, Nucl. Phys. B 819 (2009) 400 [arXiv:0810.4944] [INSPIRE].
N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
T. Ishii and K. Sakai, Thermodynamic limit of the Nekrasov-type formula for E-string theory, JHEP 02 (2014) 087 [arXiv:1312.1050] [INSPIRE].
B. Haghighat and W. Yan, M-strings in thermodynamic limit: Seiberg-Witten geometry, arXiv:1607.07873 [INSPIRE].
X. Zhang, Seiberg-Witten geometry of four-dimensional N = 2 SO-USp quiver gauge theories, Phys. Rev. D 100 (2019) 125015 [arXiv:1910.10104] [INSPIRE].
N. A. Nekrasov and S. L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in 16th International Congress on Mathematical Physics, Prague Czechia (2009), pg. 265 [arXiv:0908.4052] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B 405 (1993) 279 [hep-th/9302103] [INSPIRE].
L. Gottsche, H. Nakajima and K. Yoshioka, K-theoretic Donaldson invariants via instanton counting, Pure Appl. Math. Quart. 5 (2009) 1029 [math/0611945] [INSPIRE].
H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blowup, III: Blow-up formula from wall-crossing, Kyoto J. Math. 51 (2011) 263 [arXiv:0911.1773] [INSPIRE].
L. Gottsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki’s formula and instanton counting, Publ. Res. Inst. Math. Sci. Kyoto 47 (2011) 307 [arXiv:1001.5024] [INSPIRE].
C. A. Keller and J. Song, Counting Exceptional Instantons, JHEP 07 (2012) 085 [arXiv:1205.4722] [INSPIRE].
J. Gu, M.-x. Huang, A.-K. Kashani-Poor and A. Klemm, Refined BPS invariants of 6d SCFTs from anomalies and modularity, JHEP 05 (2017) 130 [arXiv:1701.00764] [INSPIRE].
M.-x. Huang, K. Sun and X. Wang, Blowup Equations for Refined Topological Strings, JHEP 10 (2018) 196 [arXiv:1711.09884] [INSPIRE].
J. Gu, B. Haghighat, K. Sun and X. Wang, Blowup Equations for 6d SCFTs. I, JHEP 03 (2019) 002 [arXiv:1811.02577] [INSPIRE].
J. Kim, S.-S. Kim, K.-H. Lee, K. Lee and J. Song, Instantons from Blow-up, JHEP 11 (2019) 092 [Erratum ibid. 06 (2020) 124] [arXiv:1908.11276] [INSPIRE].
J. Gu, A. Klemm, K. Sun and X. Wang, Elliptic blowup equations for 6d SCFTs. Part II. Exceptional cases, JHEP 12 (2019) 039 [arXiv:1905.00864] [INSPIRE].
J. Gu, B. Haghighat, A. Klemm, K. Sun and X. Wang, Elliptic blowup equations for 6d SCFTs. Part III. E-strings, M-strings and chains, JHEP 07 (2020) 135 [arXiv:1911.11724] [INSPIRE].
J. Gu, B. Haghighat, A. Klemm, K. Sun and X. Wang, Elliptic Blowup Equations for 6d SCFTs. IV: Matters, arXiv:2006.03030 [INSPIRE].
H.-C. Kim, M. Kim, S.-S. Kim and K.-H. Lee, Bootstrapping BPS spectra of 5d/6d field theories, JHEP 04 (2021) 161 [arXiv:2101.00023] [INSPIRE].
N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems, Nucl. Phys. B 531 (1998) 323 [hep-th/9609219] [INSPIRE].
A. E. Lawrence and N. Nekrasov, Instanton sums and five-dimensional gauge theories, Nucl. Phys. B 513 (1998) 239 [hep-th/9706025] [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].
V. Mitev, E. Pomoni, M. Taki and F. Yagi, Fiber-Base Duality and Global Symmetry Enhancement, JHEP 04 (2015) 052 [arXiv:1411.2450] [INSPIRE].
S. Cheng and S.-S. Kim, Refined topological vertex for 5d Sp(N) gauge theories with antisymmetric matter, arXiv:1809.00629 [INSPIRE].
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Li, X., Yagi, F. Thermodynamic limit of Nekrasov partition function for 5-brane web with O5-plane. J. High Energ. Phys. 2021, 4 (2021). https://doi.org/10.1007/JHEP06(2021)004
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DOI: https://doi.org/10.1007/JHEP06(2021)004