Abstract
Instanton partition functions of \( \mathcal{N}=1 \) 5d Super Yang-Mills reduced on S1 can be engineered in type IIB string theory from the (p, q)-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting on the representation spaces of levels (1, n) ⊗ (0, m) → (1, n + m), thereby generalizing to higher rank m the original construction. It allows us to use a folded version of the usual (p, q)-web diagram, bringing great simplifications to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal representations. We further develop a method to build qq-characters of linear quivers based on the horizontal action of DIM elements. While fundamental qq-characters can be built using the coproduct, higher ones require the introduction of a (quantum) Weyl reflection acting on tensor products of DIM generators.
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References
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].
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].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [INSPIRE].
N. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
A. Morozov and Y. Zenkevich, Decomposing Nekrasov Decomposition, JHEP 02 (2016) 098 [arXiv:1510.01896] [INSPIRE].
A. Mironov, A. Morozov and Y. Zenkevich, Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings, JHEP 05 (2016) 121 [arXiv:1603.00304] [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].
O. Schiffmann and E. Vasserot, Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A 2, Publ. Math. IHÉS 118 (2013) 213 [arXiv:1202.2756].
J.-t. Ding and K. Iohara, Generalization and deformation of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997) 181 [INSPIRE].
K. Miki, A (q, γ) analog of the W 1+∞ algebra, J. Math. Phys. 48 (2007) 3520.
B. Feigin and A. Tsymbaliuk, Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra, arXiv:0904.1679.
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous \( \mathfrak{g}{\mathfrak{l}}_{\infty } \) : Semi-infinite construction of representations, Kyoto J. Math. 51 (2011) 337 [arXiv:1002.3100].
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous gl ∞ : Tensor products of Fock modules and W n characters, arXiv:1002.3113 [INSPIRE].
M. Fukuda, S. Nakamura, Y. Matsuo and R.-D. Zhu, SH c realization of minimal model CFT: triality, poset and Burge condition, JHEP 11 (2015) 168 [arXiv:1509.01000] [INSPIRE].
D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, arXiv:1211.1287 [INSPIRE].
A. Smirnov, On the Instanton R-matrix, Commun. Math. Phys. 345 (2016) 703 [arXiv:1302.0799] [INSPIRE].
H. Awata et al., Toric Calabi-Yau threefolds as quantum integrable systems. ℛ-matrix and \( \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} \) relations, JHEP 10 (2016) 047 [arXiv:1608.05351] [INSPIRE].
H. Awata et al., Anomaly in RTT relation for DIM algebra and network matrix models, Nucl. Phys. B 918 (2017) 358 [arXiv:1611.07304] [INSPIRE].
N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [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, arXiv:1701.00189 [INSPIRE].
N. Nekrasov and N.S. Prabhakar, Spiked Instantons from Intersecting D-branes, Nucl. Phys. B 914 (2017) 257 [arXiv:1611.03478] [INSPIRE].
H.-C. Kim, Line defects and 5d instanton partition functions, JHEP 03 (2016) 199 [arXiv:1601.06841] [INSPIRE].
H. Knight, Spectra of Tensor Products of Finite Dimensional Representations of Yangians,” J. Algebra 174 (1995) 187.
E. Frenkel and N. Reshetikhin, The q-characters of representations of quantum affine algebras and deformations of W-algebras, math/9810055.
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].
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. Awata, B. Feigin and J. Shiraishi, Quantum Algebraic Approach to Refined Topological Vertex, JHEP 03 (2012) 041 [arXiv:1112.6074] [INSPIRE].
B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi and S. Yanagida, A commutative algebra on degenerate CP 1 and Macdonald polynomials, J. Math. Phys. 50 (2009) 095215 [arXiv:0904.2291].
M.A. Rieffel, C * -algebras associated with irrational rotations, Pacific J. Math. 93 (1981) 415.
M.A. Rieffel, Projective modules over higher-dimensional noncommutative tori, Can. J. Math. 40 (1988) 257.
A. Grassi, Y. Hatsuda and M. Mariño, Topological Strings from Quantum Mechanics, Annales Henri Poincaré 17 (2016) 3177 [arXiv:1410.3382] [INSPIRE].
S. Kanno, Y. Matsuo and H. Zhang, Extended Conformal Symmetry and Recursion Formulae for Nekrasov Partition Function, JHEP 08 (2013) 028 [arXiv:1306.1523] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Triality in Minimal Model Holography, JHEP 07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
D. Altschuler, M. Bauer and H. Saleur, Level rank duality in nonunitary coset theories, J. Phys. A 23 (1990) L789 [INSPIRE].
A. Kuniba, T. Nakanishi and J. Suzuki, Ferromagnetizations and antiferromagnetizations in RSOS models, Nucl. Phys. B 356 (1991) 750 [INSPIRE].
I. . Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Clarendon Press, Oxford U.K. (1995).
J.-E. Bourgine and D. Fioravanti, Non-linear integral equation and quantum integrability in the Nekrasov-Shatashvili limit, to appear.
A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, Prog. Math. 244 (2006) 597 [hep-th/0309208] [INSPIRE].
J.-E. Bourgine and D. Fioravanti, Omega-deformed Seiberg-Witten relations, to appear.
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 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].
M. Taki, On AGT-W Conjecture and q-Deformed W-Algebra, arXiv:1403.7016 [INSPIRE].
H. Awata, B. Feigin, A. Hoshino, M. Kanai, J. Shiraishi and S. Yanagida, Notes on Ding-Iohara algebra and AGT conjecture, arXiv:1106.4088 [INSPIRE].
B.L. Feigin and A.I. Tsymbaliuk, Equivariant K-theory of Hilbert schemes via shuffle algebra, Kyoto J. Math. 51 (2011) 831 [arXiv:0904.1679].
D. Gaiotto, Asymptotically free \( \mathcal{N}=2 \) theories and irregular conformal blocks, J. Phys. Conf. Ser. 462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [INSPIRE].
H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [INSPIRE].
H. Kanno and M. Taki, Generalized Whittaker states for instanton counting with fundamental hypermultiplets, JHEP 05 (2012) 052 [arXiv:1203.1427] [INSPIRE].
E. Carlsson and A. Okounkov, Exts and Vertex Operators, arXiv:0801.2565.
J. Shiraishi, H. Kubo, H. Awata and S. Odake, A Quantum deformation of the Virasoro algebra and the Macdonald symmetric functions, Lett. Math. Phys. 38 (1996) 33 [q-alg/9507034] [INSPIRE].
A. Kapustin, D(n) quivers from branes, JHEP 12 (1998) 015 [hep-th/9806238] [INSPIRE].
H. Hayashi and K. Ohmori, 5d/6d DE instantons from trivalent gluing of web diagrams, JHEP 06 (2017) 078 [arXiv:1702.07263] [INSPIRE].
O. Foda and J.-F. Wu, A Macdonald refined topological vertex, J. Phys. A 50 (2017) 294003 [arXiv:1701.08541] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar, W. Li and C. Peng, Higher Spins and Yangian Symmetries, JHEP 04 (2017) 152 [arXiv:1702.05100] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York U.S.A. (1997).
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Bourgine, JE., Fukuda, M., Harada, K. et al. (p, q)-webs of DIM representations, 5d \( \mathcal{N}=1 \) instanton partition functions and qq-characters. J. High Energ. Phys. 2017, 34 (2017). https://doi.org/10.1007/JHEP11(2017)034
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DOI: https://doi.org/10.1007/JHEP11(2017)034