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Webs of Quantum Algebra Representations in 5d \({\mathcal {N}}=1\) Super Yang–Mills

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Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1 (LT-XII/QTS-X 2017)

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Abstract

Instanton partition functions of 5d \({\mathcal {N}}=1\) Super Yang–Mills reduced on \(S^1\) are engineered in type IIB string theory from webs of (p, q)-branes. Branes intersections are associated to the (refined) topological vertex, while the web diagram provides gluing rules. These partition functions are covariant under the action of a quantum toroidal algebra, the Ding–Iohara–Miki algebra. In fact, a web of representations can be associated to the brane web diagram, where (p, q)-branes correspond to representations of levels (q, p), and topological vertices to intertwiners. Using this correspondence, the \({\mathcal {T}}\)-operator of a new type of quantum integrable systems can be constructed. Its vacuum expectation value reproduces the Nekrasov instanton partition function, while further insertion of algebra elements provides the qq-characters.

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Notes

  1. 1.

    The AGT-correspondence with Liouville/Toda 2d CFT [4, 34] is another facet of this problem, promoting this duality to a triality but giving only limited insight on the formulation of an \(\epsilon _2\)-deformed integrable system.

  2. 2.

    It is noted that the representation (0, m) is not simply obtained from the coproduct of (0, 1) representations in our conventions where all Young diagrams \(\lambda _l\) appear in a symmetric way.

  3. 3.

    The presence of this extra factor is due to the fact that the Lax matrix comes from a wrongly normalized R-matrix that does not satisfy the two additional relations

    $$\begin{aligned} (\varDelta \otimes 1){\mathcal {R}}={\mathcal {R}}_{13}{\mathcal {R}}_{23},\quad (1\otimes \varDelta ){\mathcal {R}}={\mathcal {R}}_{13}{\mathcal {R}}_{12}. \end{aligned}$$
    (13)
  4. 4.

    The covariance properties of this operator, sometimes called vertical intertwiner, were analyzed in [12, 13], where it has also been used to define the qq-characters.

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Acknowledgements

I would like to thank my collaborators D. Fioravanti, M. Fukuda, K. Harada, Y. Matsuo, H. Zhang, R.-D. Zhu, with whom I had the pleasure to study various aspects of instanton partition functions and quantum algebras.

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Authors and Affiliations

  1. Korea Institute for Advanced Study (KIAS), Quantum Universe Center (QUC), 85 Hoegiro, Dongdaemun-gu, Seoul, South Korea

    Jean-Emile Bourgine

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Correspondence to Jean-Emile Bourgine .

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  1. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Vladimir Dobrev

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Bourgine, JE. (2018). Webs of Quantum Algebra Representations in 5d \({\mathcal {N}}=1\) Super Yang–Mills. In: Dobrev, V. (eds) Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1 . LT-XII/QTS-X 2017. Springer Proceedings in Mathematics & Statistics, vol 263. Springer, Singapore. https://doi.org/10.1007/978-981-13-2715-5_11

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