Multiplicative Slices, Relativistic Toda and Shifted Quantum Affine Algebras

  • Michael Finkelberg
  • Alexander Tsymbaliuk
Part of the Progress in Mathematics book series (PM, volume 330)


We introduce the shifted quantum affine algebras. They map homomorphically into the quantized K-theoretic Coulomb branches of \(3d\ {\mathcal N}=4\) SUSY quiver gauge theories. In type A, they are endowed with a coproduct, and they act on the equivariant K-theory of parabolic Laumon spaces. In type A1, they are closely related to the type A open relativistic quantum Toda system.

Mathematics Subject Classification

17B37 81R10 81T13 



We are deeply grateful to M. Bershtein, R. Bezrukavnikov, A. Braverman, A. Brochier, S. Cautis, S. Cherkis, T. Dimofte, P. Etingof, B. Feigin, S. Gautam, M. Gekhtman, V. Ginzburg, D. Hernandez, D. Jordan, J. Kamnitzer, S. Khoroshkin, A. Marshakov, A. Molev, H. Nakajima, V. Pestun, L. Rybnikov, A. Shapiro, A. Weekes, M. Yakimov, whose generous help and advice was crucial in the process of our work. We also thank the anonymous referee.

M.F. was partially funded within the framework of the HSE University Basic Research Program and the Russian Academic Excellence Project ‘5-100’.

A.T. gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University and is extremely grateful to IHES for invitation and wonderful working conditions in the summer 2017, where most of the research for this paper was performed. A.T. also thanks Yale University and Max Planck Institute for Mathematics in Bonn for the hospitality and support in the summer 2018, where the final version of this paper was completed. A.T. was partially supported by the NSF Grants DMS–1502497, DMS–1821185.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Michael Finkelberg
    • 1
    • 2
    • 3
  • Alexander Tsymbaliuk
    • 4
  1. 1.National Research University Higher School of EconomicsDepartment of MathematicsMoscowRussian Federation
  2. 2.Skolkovo Institute of Science and TechnologyMoscowRussian Federation
  3. 3.Institute for Information Transmission ProblemsMoscowRussian Federation
  4. 4.Yale UniversityDepartment of MathematicsNew HavenUSA

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