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Register a new userPrincipal $hat{sl}(3)$ subspaces and quantum Toda Hamiltonian
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We study a class of representations of the Lie algebra of Laurent polynomials with values in the nilpotent subalgebra of sl(3). We derive Weyl-type (bosonic) character formulas for these representations. We establish a connection between the bosonic formulas and the Whittaker vector in the Verma module for the quantum group $U_v sl(3)$. We also obtain a fermionic formula for an eigenfunction of the sl(3) quantum Toda Hamiltonian.
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We use the Whittaker vectors and the Drinfeld Casimir element to show that eigenfunctions of the difference Toda Hamiltonian can be expressed via fermionic formulas. Motivated by the combinatorics of the fermionic formulas we use the representation theory of the quantum groups to prove a number of identities for the coefficients of the eigenfunctions.
For $mathfrak g$ a Kac-Moody algebra of affine type, we show that there is an $text{Aut}, mathcal O$-equivariant identification between $text{Fun},text{Op}_{mathfrak g}(D)$, the algebra of functions on the space of ${mathfrak g}$-opers on the disc, and $Wsubset pi_0$, the intersection of kernels of screenings inside a vacuum Fock module $pi_0$. This kernel $W$ is generated by two states: a conformal vector, and a state $delta_{-1}left|0right>$. We show that the latter endows $pi_0$ with a canonical notion of translation $T^{text{(aff)}}$, and use it to define the densities in $pi_0$ of integrals of motion of classical Conformal Affine Toda field theory. The $text{Aut},mathcal O$-action defines a bundle $Pi$ over $mathbb P^1$ with fibre $pi_0$. We show that the product bundles $Pi otimes Omega^j$, where $Omega^j$ are tensor powers of the canonical bundle, come endowed with a one-parameter family of holomorphic connections, $ abla^{text{(aff)}} - alpha T^{text{(aff)}}$, $alphain mathbb C$. The integrals of motion of Conformal Affine Toda define global sections $[mathbf v_j dt^{j+1} ] in H^1(mathbb P^1, Piotimes Omega^j, abla^{text{(aff)}})$ of the de Rham cohomology of $ abla^{mathrm{(aff)}}$. Any choice of ${mathfrak g}$-Miura oper $chi$ gives a connection $ abla^{mathrm{(aff)}}_chi$ on $Omega^j$. Using coinvariants, we define a map $mathsf F_chi$ from sections of $Pi otimes Omega^j$ to sections of $Omega^j$. We show that $mathsf F_chi abla^{text{(aff)}} = abla^{text{(aff)}}_chi mathsf F_chi$, so that $mathsf F_chi$ descends to a well-defined map of cohomologies. Under this map, the classes $[mathbf v_j dt^{j+1} ]$ are sent to the classes in $H^1(mathbb P^1, Omega^j, abla^{text{(aff)}}_chi)$ defined by the ${mathfrak g}$-oper underlying $chi$.
We provide a ribbon tensor equivalence between the representation category of small quantum SL(2), at parameter q=exp($pi$ i/p), and the representation category of the triplet vertex operator algebra at integral parameter p>1. We provide similar quantum group equivalences for representation categories associated to the Virasoro, and singlet vertex operator algebras at central charge c=1-6(p-1)^2/p. These results resolve a number of fundamental conjectures coming from studies of logarithmic CFTs in type A_1.
In this paper we study an approximation of tensor product of irreducible integrable $hat{mathfrak{sl}_2}$ representations by infinite fusion products. This gives an approximation of the corresponding coset theories. As an application we represent characters of spaces of these theories as limits of certain restricted Kostka polynomials. This leads to the bosonic (which is known) and fermionic (which is new) formulas for the $hat{mathfrak{sl}_2}$ branching functions.
We discuss the quantization of the $widehat{mathfrak{sl}}_2$ coset vertex operator algebra $mathcal{W}D(2,1;alpha)$ using the bosonization technique. We show that after quantization there exist three families of commuting integrals of motion coming from three copies of the quantum toroidal algebra associated to ${mathfrak{gl}}_2$.
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