A quantum cluster algebra approach to representations of simply laced quantum affine algebras

We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q, t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q, t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.

Quantum geometry and θ-angle in five-dimensional super Yang-Mills

Five-dimensional Sp(N) supersymmetric Yang-Mills admits a ℤ2 version of a theta angle θ. In this note, we derive a double quantization of the Seiberg-Witten geometry of $$ \mathcal{N} $$ N = 1 Sp(1) gauge theory at θ = π, on the manifold S1× ℝ4. Crucially, ℝ4 is placed on the Ω-background, which provides the two parameters to quantize the geometry. Physically, we are counting instantons in the presence of a 1/2-BPS fundamental Wilson loop, both of which are wrapping S1. Mathematically, this amounts to proving the regularity of a qq-character for the spin-1/2 representation of the quantum affine algebra $$ {U}_q\left(\hat{A_1}\right) $$ U q A 1 ̂ , with a certain twist due to the θ-angle. We motivate these results from two distinct string theory pictures. First, in a (p, q)-web setup in type IIB, where the loop is characterized by a D3 brane. Second, in a type I′ string setup, where the loop is characterized by a D4 brane subject to an orientifold projection. We comment on the generalizations to the higher rank case Sp(N) when N > 1, and the SU(N) theory at Chern-Simons level κ when N > 2.

Set-theoretical solutions to the reflection equation associated to the quantum affine algebra of type $\boldsymbol{A^{(1)}_{n-1}}$

A trick to obtain a solution to the set-theoretical reflection equation from a known one to the Yang–Baxter equation is applied to crystals and geometric crystals associated to the quantum affine algebra of type $A^{(1)}_{n-1}$.

An Algebra Associated with a Flag in a Subspace Lattice over a Finite Field and the Quantum Affine Algebra

In this paper, we introduce an algebra $\mathcal{H}$ from a subspace lattice with respect to a fixed flag which contains its incidence algebra as a proper subalgebra. We then establish a relation between the algebra $\mathcal{H}$ and the quantum affine algebra $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$, where $q$ denotes the cardinality of the base field. It is an extension of the well-known relation between the incidence algebra of a subspace lattice and the quantum algebra $U_{q^{1/2}}(\mathfrak{sl}_2)$. We show that there exists an algebra homomorphism from $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$ to $\mathcal{H}$ and that any irreducible module for $\mathcal{H}$ is irreducible as an $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$-module.

Scalar products and norm of Bethe vectors for integrable models based on $U_q(\widehat{\mathfrak{gl}}_{n})$

We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra \uqgl{m}. We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, whose factors are characterized by two highest coefficients. We provide different recursions for these highest coefficients.In addition, we show that when the Bethe vectors are on-shell, their norm takes the form of a Gaudin determinant.