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 Rmatrix and applying the procedure of algebraic Bethe ansatz, we obtain infinite commuting families of Hamiltonians of quantum ILW hierarchy labeled by shape of the auxiliary torus. We calculate explicitly the first five of these Hamiltonians. Then, we numerically verify that their joint spectrum can be parametriezed 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 hiearchy 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 W 1+∞ . 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 reflects the whole representation theory of W N minimal models.

Gauge origami and quiver W-algebras

“…The commutativity in (10.58) implies the existence of commuting Hamiltonians. Constructing the explicit Hamiltonians of the corresponding integrable models for the most general gauge origami system on the lines of [14,73,178,181] are left for future work. Moreover, it would be also interesting to relate the triple bilinear identity discussed in [182] to the triality of the q-characters.…”

“…We also note that the similar Bethe ansatz equations in Conj. 10.6 were proposed in [183] (see also [54,181,217] for related works) by studying the R-matrices of the quiver Yangians following previous works [218][219][220][221][222][223]. It was shown in [183] that under suitable assumptions, there is no universal R-matrix for some representations, and thus there are obstructions in the Bethe/Gauge correspondence.…”

BPS states meet generalized cohomology

“…The basis transformation is given by a sum over instantons carrying atoms from one crystal to the other, and the first order term in this expansion -one-instanton amplitude Ahas a particular form of a classical R-matrix containing the action of raising and lowering operators in the would-be BPS algebra (for a recent discussion of quiver Yangian R-matrix factorization properties see e.g. [107][108][109][110][111] and references therein). In other words we would be able to divide the instanton amplitude in characteristic pieces corresponding to matrix elements of BPS algebra operators and restore the BPS algebra operators themselves.…”