On Bethe equations of 2d conformal field theory

Abstract

We study the higher spin algebras of two-dimensional conformal field theory from the perspective of quantum integrability. Starting from Maulik-Okounkov instanton R-matrix and applying the procedure of algebraic Bethe ansatz, we obtain infinite commuting families of Hamiltonians of quantum ILW hierarchy parametrized by the shape of the auxiliary torus. We calculate explicitly the first five of these Hamiltonians. Then, we numerically verify that their joint spectrum can be parametrized by solutions of Litvinov's Bethe ansatz equations and we conjecture a general formula for the joint spectrum of all ILW Hamiltonians, based on results of Feigin, Jimbo, Miwa and Mukhin. There are two interesting degeneration limits, the infinitely thick and the infinitely thin auxiliary torus. In one of these limits, the ILW hierarchy degenerates to Yangian or Benjamin-Ono hierarchy and the Bethe equations can be easily solved. The other limit is singular but we can nevertheless extract local Hamiltonians corresponding to quantum KdV or KP hierarchy. Litvinov's Bethe equations in this local limit provide an alternative to Bethe ansatz equations of Bazhanov, Lukyanov and Zamolodchikov, but are more transparent, work at any rank and are manifestly symmetric under triality symmetry of $\mathcal{W}_{1+\infty}$. Finally, we illustrate analytic properties of the solutions of Bethe equations in minimal models, in particular for Lee-Yang CFT. The analytic structure of Bethe roots is very rich as it captures the representation theory of $\mathcal{W}_N$ minimal models.

Figures (8)

• Figure 1: $\mathcal{R}$-matrix as a defect (blue) coupling two chiral CFTs, the quantum space (left) is considered to be on a cylinder while the auxiliary space (right) is torus with shape controlled by parameter $q$. The family of Hamiltonians acting on the cylinder on the left is controlled by the shape of the auxiliary torus on the right. In our construction, the torus comes with a choice of a cycle, so it makes sense to talk about thick or thin torus. By a thin torus we mean a torus glued from a long cylinder equipped with a cycle. The limits of infinitely thick or thin torus correspond to Yangian-Benjamin-Ono family (glued from a long cylidner) or Korteweg–De Vries-Kadomtsev–Petviashvili-Bazhanov-Lukyanov-Zamolodchikov family of Hamiltonians (glued from a short cylinder). For generic shapes $q$ we get Hamiltonians from the family of intermediate long wave equation.
• Figure 2: Illustration of the fusion of elementary $R$-matrices for $(N,\bar{N})=(5,4)$. We should multiply $R_{j\bar{k}}$ starting from the upper left corner and proceeding all the way to the lower right corner. Since the elementary factors that are neither on same row nor on the same column commute, the multiplication column-by-column or row-by-row gives the same answer.
• Figure 3: Periodic plane partition corresponding to $\Delta=0$ primary of Lee-Yang model
• Figure 4: Periodic plane partition corresponding to $\Delta=0$ primary of Ising model
• Figure 5: Periodic plane partition corresponding to $\Delta=-\frac{1}{5}$ primary of Lee-Yang model