Liouville reflection operator, affine Yangian and Bethe ansatz

Abstract

In these notes we study integrable structure of conformal field theory by means of Liouville reflection operator/Maulik-Okounkov $R$-matrix. We discuss the relation between $RLL$ and current realization of the affine Yangian of $\mathfrak{gl}(1)$. We construct the family of commuting transfer matrices related to the Intermediate Long Wave hierarchy and derive Bethe ansatz equations for their spectra discovered by Nekrasov and Okounkov and independently by one of the authors. Our derivation mostly follows the one by Feigin, Jimbo, Miwa and Mukhin, but is adapted to the conformal case.

arXiv comment46 pages

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Cited by in corpus (18)

• Gauge/Bethe correspondence from quiver BPS algebras
• Quantum toroidal algebras and solvable structures in gauge/string theory
• Affine Yangian of $\mathfrak{gl}(2)$ and integrable structures of superconformal field theory
• A Note on Quiver Yangians and $\mathcal{R}$-Matrices
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• Gauge origami and quiver W-algebras
• The $R$-matrix of the quantum toroidal algebra $U_{q,t}({\stackrel{..}{\mathrm{gl}}}_1)$ in the Fock module
• Integrable structure of $\text{BCD}$ conformal field theory and boundary Bethe ansatz for affine Yangian
• BPS States Meet Generalized Cohomology
• More on Affine Dynkin Quiver Yangians
• Quiver Yangians and $\mathcal{W}$-Algebras for Generalized Conifolds
• Shifted quantum groups and matter multiplets in supersymmetric gauge theories
• Generalized Gibbs Ensemble of 2D CFTs with U(1) Charge from the AGT Correspondence
• An Overview of Crystals and Double Quiver Yangians
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• ODE/IQFT correspondence for the generalized affine $\mathfrak{sl}(2)$ Gaudin model
• Bethe/Gauge Correspondence for $A_N$ Spin Chains with Integrable Boundaries
• Symmetric deformed 2D/3D Hurwitz-Kontsevich model and affine Yangian of gl(1)