Abstract
We show the refinement of the prescription for the geometric transition in the refined topological string theory and, as its application, discuss a possibility to describe qq-characters from the string theory point of view. Though the suggested way to operate the refined geometric transition has passed through several checks, it is additionally found in this paper that the presence of the preferred direction brings a nontrivial effect. We provide the modified formula involving this point. We then apply our prescription of the refined geometric transition to proposing the stringy description of doubly quantized Seiberg-Witten curves called qq-characters in certain cases.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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].
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].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The Topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [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].
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].
R. Gopakumar and C. Vafa, M theory and topological strings. 1., hep-th/9809187 [INSPIRE].
R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].
M. Kameyama and S. Nawata, Refined large-N duality for knots, arXiv:1703.05408 [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [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].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in \( \mathcal{N}=2 \) gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, JHEP 03 (2016) 181 [arXiv:1512.05388] [INSPIRE].
N. Nekrasov, BPS/CFT correspondence II: Instantons at crossroads, moduli and compactness theorem, Adv. Theor. Math. Phys. 21 (2017) 503 [arXiv:1608.07272] [INSPIRE].
N. Nekrasov, BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters, Commun. Math. Phys. (2017) 1 [arXiv:1701.00189] [INSPIRE].
E. Frenkel and N. Reshetikhin, The q-characters of representations of quantum affine algebras and deformations of \( \mathcal{W} \) -algebras, in Contemporary Mathematics. Vol. 248: Recent Developments in Quantum Affine Algebras and Related Topics, AMS Press, Providence U.S.A. (1999), pg. 163 [math/9810055].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
T. Kimura and V. Pestun, Quiver W-algebras, arXiv:1512.08533 [INSPIRE].
T. Kimura and V. Pestun, Quiver elliptic W-algebras, arXiv:1608.04651 [INSPIRE].
T. Kimura, Double quantization of Seiberg-Witten geometry and W-algebras, arXiv:1612.07590 [INSPIRE].
J.-E. Bourgine, Y. Matsuo and H. Zhang, Holomorphic field realization of SH c and quantum geometry of quiver gauge theories, JHEP 04 (2016) 167 [arXiv:1512.02492] [INSPIRE].
J.-E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang and R.-D. Zhu, Coherent states in quantum \( {\mathcal{W}}_{1+\infty } \) algebra and qq-character for 5d Super Yang-Mills, PTEP 2016 (2016) 123B05 [arXiv:1606.08020] [INSPIRE].
J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo and R.-D. Zhu, (p, q)-webs of DIM representations, 5d N = 1 instanton partition functions and qq-characters, JHEP 11 (2017) 034 [arXiv:1703.10759] [INSPIRE].
A. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, Phys. Lett. B 762 (2016) 196 [arXiv:1603.05467] [INSPIRE].
H. Awata et al., Explicit examples of DIM constraints for network matrix models, JHEP 07 (2016) 103 [arXiv:1604.08366] [INSPIRE].
H.-C. Kim, Line defects and 5d instanton partition functions, JHEP 03 (2016) 199 [arXiv:1601.06841] [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].
M. Taki, Surface Operator, Bubbling Calabi-Yau and AGT Relation, JHEP 07 (2011) 047 [arXiv:1007.2524] [INSPIRE].
H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String, Adv. Theor. Math. Phys. 16 (2012) 725 [arXiv:1008.0574] [INSPIRE].
G. Bonelli, A. Tanzini and J. Zhao, Vertices, Vortices and Interacting Surface Operators, JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].
H.-Y. Chen and A. Sinkovics, On Integrable Structure and Geometric Transition in Supersymmetric Gauge Theories, JHEP 05 (2013) 158 [arXiv:1303.4237] [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].
C. Vafa, Brane/anti-brane systems and U(N |M) supergroup, hep-th/0101218 [INSPIRE].
B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-Strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [INSPIRE].
B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D 89 (2014) 046003 [arXiv:1310.1185] [INSPIRE].
D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions, JHEP 10 (2016) 012 [arXiv:1412.2781] [INSPIRE].
Y. Zenkevich, Refined toric branes, surface operators and factorization of generalized Macdonald polynomials, JHEP 09 (2017) 070 [arXiv:1612.09570] [INSPIRE].
H. Mori and Y. Sugimoto, Surface Operators from M-strings, Phys. Rev. D 95 (2017) 026001 [arXiv:1608.02849] [INSPIRE].
N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Am. Math. Soc. 14 (2001) 145 [math/9912158].
G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Am. Math. Soc. 4 (1991) 365.
V. Ginzburg and É. Vasserot, Langlands reciprocity for affine quantum groups of type A n , Int. Math. Res. Not. (1993) 67.
R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].
F. Fucito, J.F. Morales, D.R. Pacifici and R. Poghossian, Gauge theories on Ω-backgrounds from non commutative Seiberg-Witten curves, JHEP 05 (2011) 098 [arXiv:1103.4495] [INSPIRE].
F. Fucito, J.F. Morales and D. Ricci Pacifici, Deformed Seiberg-Witten Curves for ADE Quivers, JHEP 01 (2013) 091 [arXiv:1210.3580] [INSPIRE].
V. Mikhaylov and E. Witten, Branes and supergroups, Commun. Math. Phys. 340 (2015) 699 [arXiv:1410.1175] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, \( \mathcal{N}=6 \) superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].
H. Hayashi and K. Ohmori, 5d/6d DE instantons from trivalent gluing of web diagrams, JHEP 06 (2017) 078 [arXiv:1702.07263] [INSPIRE].
T. Kimura, Linking loops in ABJM and refined theory, JHEP 07 (2015) 030 [arXiv:1503.01462] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Elliptic Hypergeometry of Supersymmetric Dualities, Commun. Math. Phys. 304 (2011) 797 [arXiv:0910.5944] [INSPIRE].
F. Nieri, An elliptic Virasoro symmetry in 6d, Lett. Math. Phys. 107 (2017) 2147 [arXiv:1511.00574] [INSPIRE].
A. Iqbal, C. Kozcaz and S.-T. Yau, Elliptic Virasoro Conformal Blocks, arXiv:1511.00458 [INSPIRE].
I. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Clarendon Press, Oxford U.K. (1998).
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].
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: 1705.03467
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
Kimura, T., Mori, H. & Sugimoto, Y. Refined geometric transition and qq-characters. J. High Energ. Phys. 2018, 25 (2018). https://doi.org/10.1007/JHEP01(2018)025
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2018)025