Letters in Mathematical Physics

, Volume 108, Issue 4, pp 1137–1146 | Cite as

Cluster-enriched Yang–Baxter equation from SUSY gauge theories

  • Masahito Yamazaki


We propose a new generalization of the Yang–Baxter equation, where the R-matrix depends on cluster y-variables in addition to the spectral parameters. We point out that we can construct solutions to this new equation from the recently found correspondence between Yang–Baxter equations and supersymmetric gauge theories. The \(S^2\) partition function of a certain 2d \({\mathcal {N}}=(2,2)\) quiver gauge theory gives an R-matrix, whereas its FI parameters can be identified with the cluster y-variables.


Integrable model Cluster algebras Quiver gauge theory Supersymmetry 

Mathematics Subject Classification

16T25 13F60 81Q60 



The author would like to thank Wenbin Yan for collaboration in [12], which leads the author to the current project. He would also like to thank the organizers and the audience of the workshop “Integrability in Gauge and String Theory 2015” (Imperial college), where the results of this paper were announced during the author’s talk. This research is supported in part by the WPI Research Center Initiative (MEXT, Japan), by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, by JSPS KAKENHI (15K17634) and by JSPS-NRF Joint Research Project.


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Kavli IPMU (WPI)University of TokyoKashiwaJapan
  2. 2.Center for the Fundamental Laws of NatureHarvard UniversityCambridgeUSA

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