Abstract
We introduce an R-matrix acting on the tensor product of MacMahon representations of Ding-Iohara-Miki (DIM) algebra \( {U}_{q,t}\left({\widehat{\widehat{\mathfrak{gl}}}}_1\right) \). This R-matrix acts on pairs of 3d Young diagrams and retains the nice symmetry of the DIM algebra under the permutation of three deformation parameters q, t−1 and \( \frac{t}{q} \). We construct the intertwining operator for a tensor product of the horizontal Fock representation and the vertical MacMahon representation and show that the intertwiners are permuted using the MacMahon R-matrix.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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) 123520.
V. Ginzburg, M. Kapranov and E. Vasserot, Langlands reciprocity for algebraic surfaces, Math. Res. Lett. 2 (1995) 147 [q-alg/9502013].
M. Varagnolo and E. Vasserot, Schur duality in the toroidal setting, Commun. Math. Phys. 182 (1996) 469 [q-alg/9506026].
R.V. Moody, S.E. Rao and T. Yokonuma, Toroidal Lie algebras and vertex representations, Geom. Dedicata 35 (1990) 283.
G. Lockhart and C. Vafa, Superconformal Partition Functions and Non-perturbative Topological Strings, JHEP 10 (2018) 051 [arXiv:1210.5909] [INSPIRE].
A. Iqbal and K. Shabbir, Elliptic CY3folds and Non-Perturbative Modular Transformation, Eur. Phys. J. C 76 (2016) 148 [arXiv:1510.03332] [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].
F. Nieri, Y. Pan and M. Zabzine, q-Virasoro modular triple, Commun. Math. Phys. 366 (2019) 397 [arXiv:1710.07170] [INSPIRE].
M. Bershtein, B. Feigin and G. Merzon, Plane partitions with a “pit”: Generating functions and representation theory, Selecta Math. 24 (2018) 21 [arXiv:1512.08779].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum toroidal \( \mathfrak{g}{\mathfrak{l}}_1 \) algebra: plane partitions, Kyoto J. Math. 52 (2012) 621 [arXiv:1110.5310].
H. Awata et al., Toric Calabi-Yau threefolds as quantum integrable systems. \( \mathrm{\mathcal{R}} \) -matrix and \( \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} \) relations, JHEP 10 (2016) 047 [arXiv:1608.05351] [INSPIRE].
H. Awata et al., Anomaly in \( \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} \) relation for DIM algebra and network matrix models, Nucl. Phys. B 918 (2017) 358 [arXiv:1611.07304] [INSPIRE].
A. Smirnov, On the Instanton R-matrix, Commun. Math. Phys. 345 (2016) 703 [arXiv:1302.0799] [INSPIRE].
H. Awata, B. Feigin and J. Shiraishi, Quantum Algebraic Approach to Refined Topological Vertex, JHEP 03 (2012) 041 [arXiv:1112.6074] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The Refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
S. Gukov, A. Iqbal, C. Kozcaz and C. Vafa, Link Homologies and the Refined Topological Vertex, Commun. Math. Phys. 298 (2010) 757 [arXiv:0705.1368] [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].
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 H. Kanno, Changing the preferred direction of the refined topological vertex, J. Geom. Phys. 64 (2013) 91 [arXiv:0903.5383] [INSPIRE].
M. Taki, Refined Topological Vertex and Instanton Counting, JHEP 03 (2008) 048 [arXiv:0710.1776] [INSPIRE].
M. Taki, Flop Invariance of Refined Topological Vertex and Link Homologies, arXiv:0805.0336 [INSPIRE].
H. Awata and H. Kanno, Macdonald operators and homological invariants of the colored Hopf link, J. Phys. A 44 (2011) 375201 [arXiv:0910.0083] [INSPIRE].
I.G. Macdonald, Symmetric functions and Hall polynomials, second edition, Oxford Mathematical Monographs, Oxford University Press, Oxford U.K. (1995).
Y. Zenkevich, 3d field theory, plane partitions and triple Macdonald polynomials, arXiv:1712.10300 [INSPIRE].
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, M. Jimbo, T. Miwa and E. Mukhin, Finite Type Modules and Bethe Ansatz for Quantum Toroidal \( \mathfrak{g}{\mathfrak{l}}_1 \), Commun. Math. Phys. 356 (2017) 285 [arXiv:1603.02765] [INSPIRE].
B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi and S. Yanagida, A commutative algebra on degenerate ℂℙ1 and Macdonald polynomials, J. Math. Phys. 50 (2009) 095215 [arXiv:0904.2291].
A. Tsymbaliuk, The affine Yangian of \( \mathfrak{g}{\mathfrak{l}}_1 \) revisited, Adv. Math. 304 (2017) 583 [arXiv:1404.5240] [INSPIRE].
E. Mukhin, V. Tarasov and A. Varchenko, Bispectral and (\( \mathfrak{g}{\mathfrak{l}}_N,\mathfrak{g}{\mathfrak{l}}_M \)) Dualities, math.QA/0510364.
E. Mukhin, V. Tarasov and A. Varchenko, Bispectral and (\( \mathfrak{g}{\mathfrak{l}}_N,\mathfrak{g}{\mathfrak{l}}_M \)) dualities, discrete versus differential, Adv. Math. 218 (2008) 216 [math.QA/0605172].
A. Mironov, A. Morozov, Y. Zenkevich and A. Zotov, Spectral Duality in Integrable Systems from AGT Conjecture, JETP Lett. 97 (2013) 45 [arXiv:1204.0913] [INSPIRE].
A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral Duality Between Heisenberg Chain and Gaudin Model, Lett. Math. Phys. 103 (2013) 299 [arXiv:1206.6349] [INSPIRE].
A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral dualities in XXZ spin chains and five dimensional gauge theories, JHEP 12 (2013) 034 [arXiv:1307.1502] [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].
M. Aganagic, N. Haouzi, C. Kozcaz and S. Shakirov, Gauge/Liouville Triality, arXiv:1309.1687 [INSPIRE].
M. Aganagic, N. Haouzi and S. Shakirov, A n -Triality, arXiv:1403.3657 [INSPIRE].
M. Aganagic and N. Haouzi, ADE Little String Theory on a Riemann Surface (and Triality), arXiv:1506.04183 [INSPIRE].
T. Procházka, \( \mathcal{W} \) -symmetry, topological vertex and affine Yangian, JHEP 10 (2016) 077 [arXiv:1512.07178] [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].
H. Awata et al., Explicit examples of DIM constraints for network matrix models, JHEP 07 (2016) 103 [arXiv:1604.08366] [INSPIRE].
T. Kimura and V. Pestun, Quiver W-algebras, Lett. Math. Phys. 108 (2018) 1351 [arXiv:1512.08533] [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, Prog. Theor. Exp. Phys. 2016 (2016) 123B05 [arXiv:1606.08020] [INSPIRE].
A. Nedelin and M. Zabzine, q-Virasoro constraints in matrix models, JHEP 03 (2017) 098 [arXiv:1511.03471] [INSPIRE].
H. Awata et al., Generalized Knizhnik-Zamolodchikov equation for Ding-Iohara-Miki algebra, Phys. Rev. D 96 (2017) 026021 [arXiv:1703.06084] [INSPIRE].
H. Awata, H. Kanno, A. Mironov, A. Morozov, K. Suetake and Y. Zenkevich, (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces, JHEP 03 (2018) 192 [arXiv:1712.08016] [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: 1810.07676
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Awata, H., Kanno, H., Mironov, A. et al. The MacMahon R-matrix. J. High Energ. Phys. 2019, 97 (2019). https://doi.org/10.1007/JHEP04(2019)097
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2019)097