Abstract
We show how to combine higgsed topological vertices introduced in [7] with conventional refined topological vertices. We demonstrate that the extended formalism describes very general interacting D5-NS5-D3 brane systems. In particular, we introduce new types of intertwining operators of Ding-Iohara-Miki algebra between different types of Fock representations corresponding to the crossings of NS5 and D5 branes. As a byproduct we obtain an algebraic description of the Hanany-Witten brane creation effect, give an efficient recipe to compute the brane factors in 3d \( \mathcal{N} \) = 2 and \( \mathcal{N} \) = 4 quiver gauge theories and demonstrate how 3d S-duality appears in our setup.
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References
H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994) 365 [INSPIRE].
H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Annals Math. 145 (1997) 379.
H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998) 515.
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].
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].
A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [INSPIRE].
Y. Zenkevich, Higgsed network calculus, JHEP 08 (2021) 149 [arXiv:1812.11961] [INSPIRE].
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 W1+∞ algebra, J. Math. Phys. 48 (2007) 123520.
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].
H. Awata, B. Feigin and J. Shiraishi, Quantum Algebraic Approach to Refined Topological Vertex, JHEP 03 (2012) 041 [arXiv:1112.6074] [INSPIRE].
Y. Zenkevich, \( \mathfrak{gl} \)N Higgsed networks, arXiv:1912.13372 [INSPIRE].
T. Kimura and V. Pestun, Quiver W-algebras, Lett. Math. Phys. 108 (2018) 1351 [arXiv:1512.08533] [INSPIRE].
T. Kimura and V. Pestun, Quiver elliptic W-algebras, Lett. Math. Phys. 108 (2018) 1383 [arXiv:1608.04651] [INSPIRE].
T. Kimura and V. Pestun, Fractional quiver W-algebras, Lett. Math. Phys. 108 (2018) 2425 [arXiv:1705.04410] [INSPIRE].
D. Gaiotto and M. Rapčák, Vertex Algebras at the Corner, JHEP 01 (2019) 160 [arXiv:1703.00982] [INSPIRE].
T. Procházka and M. Rapčák, Webs of W-algebras, JHEP 11 (2018) 109 [arXiv:1711.06888] [INSPIRE].
T. Procházka and M. Rapčák, \( \mathcal{W} \)-algebra modules, free fields, and Gukov-Witten defects, JHEP 05 (2019) 159 [arXiv:1808.08837] [INSPIRE].
M. Rapčák, Y. Soibelman, Y. Yang and G. Zhao, Cohomological Hall algebras, vertex algebras and instantons, Commun. Math. Phys. 376 (2019) 1803 [arXiv:1810.10402] [INSPIRE].
M. Rapčák, On extensions of \( \mathfrak{gl} \)(\( \hat{m\mid n} \)) Kac-Moody algebras and Calabi-Yau singularities, JHEP 01 (2020) 042 [arXiv:1910.00031] [INSPIRE].
M. Rapčák, Y. Soibelman, Y. Yang and G. Zhao, Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds, arXiv:2007.13365 [INSPIRE].
D. Gaiotto and M. Rapčák, Miura operators, degenerate fields and the M2-M5 intersection, arXiv:2012.04118 [INSPIRE].
M. Bershtein, B. Feigin and G. Merzon, Plane partitions with a “pit”: generating functions and representation theory, Sel. Math. New Ser. 24 (2018) 21 [arXiv:1512.08779].
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].
J.-E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang and R.-D. Zhu, Coherent states in quantum \( \mathcal{W} \)1+∞ 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 \( \mathcal{N} \) = 1 instanton partition functions and qq-characters, JHEP 11 (2017) 034 [arXiv:1703.10759] [INSPIRE].
J.-E. Bourgine, Webs of Quantum Algebra Representations in 5d \( \mathcal{N} \) = 1 Super Yang-Mills, Springer Proc. Math. Stat. 263 (2017) 209 [INSPIRE].
J.-E. Bourgine, M. Fukuda, Y. Matsuo and R.-D. Zhu, Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver, JHEP 12 (2017) 015 [arXiv:1709.01954] [INSPIRE].
J.E. Bourgine and K. Zhang, A note on the algebraic engineering of 4D \( \mathcal{N} \) = 2 super Yang-Mills theories, Phys. Lett. B 789 (2019) 610 [arXiv:1809.08861] [INSPIRE].
J.-E. Bourgine, Fiber-base duality from the algebraic perspective, JHEP 03 (2019) 003 [arXiv:1810.00301] [INSPIRE].
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous \( \mathfrak{gl} \)∞: Semi-infinite construction of representations, Kyoto J. Math. 51 (2011) 337 [arXiv:1002.3100].
H. Awata et al., Toric Calabi-Yau threefolds as quantum integrable systems. \( \mathcal{R} \)-matrix and \( \mathcal{RTT} \) 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].
H. Awata, H. Kanno, A. Mironov, A. Morozov, K. Suetake and Y. Zenkevich, The MacMahon R-matrix, JHEP 04 (2019) 097 [arXiv:1810.07676] [INSPIRE].
A. Neguţ, The R-matrix of the quantum toroidal algebra, arXiv:2005.14182 [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Finite Type Modules and Bethe Ansatz for Quantum Toroidal \( \mathfrak{gl} \)1, Commun. Math. Phys. 356 (2017) 285 [arXiv:1603.02765] [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum toroidal \( \mathfrak{gl} \)1 and Bethe ansatz, J. Phys. A 48 (2015) 244001 [arXiv:1502.07194] [INSPIRE].
S.M. Khoroshkin and V.N. Tolstoy, Universal R-matrix for quantized (super)algebras, Commun. Math. Phys. 141 (1991) 599.
V.N. Tolstoy and S.M. Khoroshkin, The universal R-matrix for quantum untwisted affine Lie algebras, Funct. Anal. Appl. 26 (1992) 69.
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. Nedelin, S. Pasquetti and Y. Zenkevich, T[SU(N)] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences, JHEP 02 (2019) 176 [arXiv:1712.08140] [INSPIRE].
F. Aprile, S. Pasquetti and Y. Zenkevich, Flipping the head of T[SU(N)]: mirror symmetry, spectral duality and monopoles, JHEP 04 (2019) 138 [arXiv:1812.08142] [INSPIRE].
M. Fukuda, Y. Ohkubo and J. Shiraishi, Non-Stationary Ruijsenaars Functions for κ = t−1/N and Intertwining Operators of Ding-Iohara-Miki Algebra, SIGMA 16 (2020) 116 [arXiv:2002.00243] [INSPIRE].
B. Assel, Hanany-Witten effect and SL(2, ℤ) dualities in matrix models, JHEP 10 (2014) 117 [arXiv:1406.5194] [INSPIRE].
K. Costello, Supersymmetric gauge theory and the Yangian, arXiv:1303.2632 [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge Theory and Integrability, I, ICCM Not. 06 (2018) 46 [arXiv:1709.09993] [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge Theory and Integrability, II, ICCM Not. 06 (2018) 120 [arXiv:1802.01579] [INSPIRE].
K. Costello and M. Yamazaki, Gauge Theory And Integrability, III, arXiv:1908.02289 [INSPIRE].
V.V. Bazhanov and S.M. Sergeev, Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups, J. Phys. A 39 (2006) 3295 [hep-th/0509181] [INSPIRE].
P. Gavrylenko, M. Semenyakin and Y. Zenkevich, Solution of tetrahedron equation and cluster algebras, JHEP 05 (2021) 103 [arXiv:2010.15871] [INSPIRE].
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Zenkevich, Y. Mixed network calculus. J. High Energ. Phys. 2021, 27 (2021). https://doi.org/10.1007/JHEP12(2021)027
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DOI: https://doi.org/10.1007/JHEP12(2021)027