We study the abelian category $W(p)$-mod of modules over the triplet $W$ algebra $W(p)$. We construct the projective covers $\mathcal{P}_s^{\pm}$ of all the simple objects $\mathcal{X}_s^{\pm}$, $1 \leq s \leq p$, in the category $W(p)$-mod. By using the structure of these projective modules, we show that $W(p)$-mod is a category which is equivalent to the abelian category of the finite-dimensional modules for the restricted quantum group $\bar{U}_q (sl_2)$ at $q = e^{\frac{\pi i}{p}}$. This Kazdan–Lusztig type correspondence was conjectured by Feigin et al. [FGST1], [FGST2].

We study the (Ramond twisted) representations of the affine $W$-algebra $\mathcal{W}^k (\bar{\mathfrak{g}}, f)$ in the case that $f$ admits a good even grading. We establish the vanishing and the almost irreducibility of the corresponding BRST cohomology. This confirms some of the recent conjectures of Kac and Wakimoto [KW08]. In type $A$, our results give the characters of all irreducible ordinary (Ramond twisted) representations of $\mathcal{W}^k (\mathfrak{sl}_n, f)$ for all nilpotent elements $f$ and all non-critical $k$, and prove the existence of modular invariant representations conjectured in [KW08].

The discrete quantum Sine-Gordon model at roots of unity remarkably combines a classical integrable system with an integrable quantum spin system, whose parameters obey classical equations of motion. We show that the fundamental $R$-matrix of the model (which satisfies a difference property Yang–Baxter equation) naturally splits into a product of a singular "classical" part and a finite dimensional quantum part. The classical part of the $R$-matrix itself satisfies the quantum Yang–Baxter equation, and therefore can be factored out producing, however, a certain "twist" of the quantum part. We show that the resulting equation exactly coincides with the star-triangle relation of the $N$-state chiral Potts model. The associated spin model on the whole lattice is, in fact, more general than the chiral Potts and reduces to the latter only for the simplest (constant) classical background. In a general case the model is inhomogeneous: its Boltzmann weights are determined by non-trivial background solutions of the equations of motion of the classical discrete sine-Gordon model.

When we describe string propagation on non-compact or singular Calabi–Yau manifolds by CFT, continuous as well as discrete representations appear in the theory. These representations mix in an intricate way under the modular transformations. In this article, we propose a method of combining discrete and continuous representations so that the resulting combinations have a simpler modular behavior and can be used as conformal blocks of the theory. We compute elliptic genera of ALE spaces and obtain results which agree with those suggested from the decompactification of K3 surface. Consistency of our approach is assured by some remarkable identity of theta functions.

We include in the appendix some new materials on the representation theory of $\mathcal{N} = 4$ superconformal algebra.

We give expressions for the characters of $(1, p)$ logarithmic conformal field models in the Gordon-type form. The formulas are obtained in terms of "quasiparticles" that are Virasoro $\Phi_{2,1}$ primary fields and generalize the symplectic fermions.

We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac–Moody algebras. Our experience with the Gaudin models associated to finite-dimensional simple Lie algebras suggests that the common eigenvalues of the mutually commuting quantum Hamiltonians in a model associated to an affine algebra $\widehat{\mathfrak{g}}$ should be encoded by affine opers associated to the Langlands dual affine algebra ${}^L\widehat{\mathfrak{g}}$. This leads us to some concrete predictions for the spectra of the quantum Hamiltonians of the soliton systems. In particular, for the KdV system the corresponding affine opers may be expressed as Schrödinger operators with spectral parameter, and our predictions in this case match those recently made by Bazhanov, Lukyanov and Zamolodchikov. This suggests that this and other recently found examples of the correspondence between quantum integrals of motion and differential operators may be viewed as special cases of the Langlands duality.

Based on the works by Kajiwara, Noumi and Yamada, we propose a canonically quantized version of the rational Weyl group representation which originally arose as symmetries or the Bäcklund transformations in Painlevé equations. We thereby propose a quantization of discrete Painlevé VI equation as a discrete Hamiltonian flow commuting with the action of $W (D_4^{(1)})$.

We shall construct the quantized $q$-analogues of the birational Weyl group actions arising from nilpotent Poisson algebras, which are conceptual generalizations, proposed by Noumi and Yamada, of the Bäcklund transformations for Painlevé equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra for any symmetrizable generalized Cartan matrix. Then non-integral powers of the image of the Chevalley generators generate the quantized $q$-analogue of the birational Weyl group action. Using the same method, we shall reconstruct the quantized Bäcklund transformations of $q$-Painlevé equations constructed by Hasegawa. We shall also prove that any subquotient integral domain of a quantized universal enveloping algebra of finite or affine type is an Ore domain.

The $q$-character is a strong tool to study finite-dimensional representations of quantum affine algebras. However, the explicit formula of the $q$-character of a given representation has not been known so far. Frenkel and Mukhin proposed the iterative algorithm which generates the $q$-character of a given irreducible representation starting from its highest weight monomial. The algorithm is known to work for various classes of representations. In this note, however, we give an example in which the algorithm fails to generate the $q$-character.

We review the notion of differential Fay identities and demonstrate, through case studies, its new role in integrable hierarchies of the KP type. These identities are known to be a convenient tool for deriving dispersionless Hirota equations. We show that differential (or, in the case of the Toda hierarchy, difference) Fay identities play a more fundamental role. Namely, they are nothing but a generating functional expression of the full set of auxiliary linear equations, hence substantially equivalent to the integrable hierarchies themselves. These results are illustrated for the KP, Toda, BKP and DKP hierarchies. As a byproduct, we point out some new features of the DKP hierarchy and its dispersionless limit.