Cotangent spaces and separating re-embeddings

Given an affine algebra [Formula: see text], where [Formula: see text] is a polynomial ring over a field [Formula: see text] and [Formula: see text] is an ideal in [Formula: see text], we study re-embeddings of the affine scheme [Formula: see text], i.e. presentations [Formula: see text] such that [Formula: see text] is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials [Formula: see text] in the ideal [Formula: see text] which are coherently separating in the sense that they are of the form [Formula: see text] with an indeterminate [Formula: see text] which divides neither a term in the support of [Formula: see text] nor in the support of [Formula: see text] for [Formula: see text]. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of [Formula: see text]. The dimension of the cotangent space of [Formula: see text] at a [Formula: see text]-linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of [Formula: see text] and found an optimal re-embedding.

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.

Exploring new boundary conditions for $$\mathcal {N}=(1,1)$$ extended higher-spin $$AdS_3$$ supergravity

In this paper, we present a candidate for $$\mathcal {N}=(1,1)$$ N = ( 1 , 1 ) extended higher-spin $$AdS_3$$ A d S 3 supergravity with the most general boundary conditions discussed by Grumiller and Riegler recently. We show that the asymptotic symmetry algebra consists of two copies of the $$\mathfrak {osp}(3|2)_k$$ osp ( 3 | 2 ) k affine algebra in the presence of the most general boundary conditions. Furthermore, we impose some certain restrictions on gauge fields on the most general boundary conditions and that leads us to the supersymmetric extension of the Brown–Henneaux boundary conditions. We eventually see that the asymptotic symmetry algebra reduces to two copies of the $$\mathcal {SW}(\frac{3}{2},2)$$ SW ( 3 2 , 2 ) algebra for $$\mathcal {N}=(1,1)$$ N = ( 1 , 1 ) extended higher-spin supergravity.

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.

Feigin–Frenkel Image of Witten–Kontsevich Points

The Witten–Kontsevich KdV tau function of topological gravity has a generalization to an arbitrary Drinfeld–Sokolov hierarchy associated to a simple complex Lie algebra. Using the Feigin–Frenkel isomorphism we describe the affine opers describing such generalized Witten–Kontsevich functions in terms of Segal–Sugawara operators associated to the Langlands dual Lie algebra. In the case where the Lie algebra is simply laced there is a second role these Segal–Sugawara operators play: their action, in the basic representation of the affine algebra associated to the Lie algebra, singles out the Witten–Kontsevich tau function within the phase space. We show that these two Langlands dual roles of Segal–Sugawara operators correspond to a duality between the first and last operator for a complete set of Segal–Sugawara operators.