Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.16359

Universal quantum dimensions: $γ$-dependent factors

M. Y. Avetisyan·R. L. Mkrtchyan
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop an approach for computing the universal, in Vogel's sense, quantum dimensions. In a previous work a method for calculating the $γ$-independent part of these dimensions for universal multiplets with nonzero associated members for classical algebras was proposed. In the present paper, this approach is extended to the $γ$-dependent factors. In particular, we derive these contributions for the $E$ universal multiplet, which appears in the universal decomposition of the fourth power of the adjoint representation, thereby obtaining its universal quantum dimension for the first time. The analysis requires knowledge of the quantum dimensions of the classical and $E_8$ members of the multiplet; for the remaining exceptional algebras, the corresponding quantum dimensions are recovered automatically, providing an additional consistency check of the approach. We also partially extend a previously conjectured relation between the $sl$ and $so, sp$ members - formulated in terms of vertical and horizontal sums of Young diagrams - to the exceptional case. Finally, we present an explicit algorithm implementing the proposed method, which enables the derivation of universal formulae for higher-dimensional representations. The universal quantum dimensions provide, in some cases, a unified description of Chern-Simons observables, including knot invariants and partition functions, in a form valid simultaneously for all gauge algebras.

Explore related subjects

math.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

M. Y. Avetisyan, R. L. Mkrtchyan. 2026-07-17. Universal quantum dimensions: $γ$-dependent factors. https://doi.org/10.1016/j.physletb.2026.140466

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Nodal degeneration of chiral algebras II: Local structure and chiral Zhu algebras

Given a universal factorization algebra $\mathcal{A}$, we constructed in our previous paper a derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, together with chiral modules $\\mathfrak{Z}_{\mathcal{A}}^+$ and $\mathfrak{Z}_{\mathcal{A}}^-$ associated to a puncture, and a chiral bimodule $\mathfrak{Z}_{\mathcal{A}}$ associated to a node. Furthermore, these constructions assemble to a factorization $\mathcal{A}$-module over any family of nodal punctured curves. In this paper, we show that in the case where $\mathcal{A}$ is constructed from a quasi-conformal vertex algebra $V$, the zeroth homology algebra $H^0\mathfrak{Z}_{\mathcal{A}}$ is naturally isomorphic to Zhu's associative algebra $A(V)$, and we identify $H^0\mathfrak{Z}_{\mathcal{A}}$ with the bimodule underlying the mode-transition algebra of Damiolini-Gibney-Krashen. We also give an explicit description of the smoothing module $H^0\tilde{\mathfrak{Z}}_{\mathcal{C}}$ which describes the deformation of $H^0\mathfrak{Z}_{\mathcal{A}}$ which we attach to a smoothing family of a nodal curve. We therefore get a geometric interpretation of the Zhu algebra and the mode-transition algebra, as the integration of a factorization algebra over a certain compactification of configuration spaces of punctured nodal curves.

math.QA↗
arXiv2026

Braided Hopf algebroids and Lie algebroids

We construct braided Hopf algebroids in a braided monoidal category over a field $k$ and study its properties using graphical representation. Then we study Ehresmann-Schauenburg Hopf algebroids assocaited to braided Hopf Galois extensions. We introduce braided Lie-Rinehart algebras which generalize the definition in \cite{ALP24, ALP23, ALP25} and study its universal enveloping algebra under symmetric condition. In particular, we introduce Lie-Rinehart algebras (braided Lie-algebroids) associated with braided Hopf algebroids in the category of the module of a triangular Hopf algebra. We then focus on the case for braided Ehresmann-Schauenburg Hopf algebroids and study their right invariant vector fields, which turn out to be isomorphic to the $H$-equivariant vector fields on the quantum principal bundle (Hopf Galois extension) as Lie-Rinehart algebras. Finally, we introduce braided jet Hopf algebroids associate to braided Hopf algebroids in the case that the source and target subalgebra belong to the braided center. Moreover, we show there is a dual pairing between the $k$-order universal enveloping algebra of the braided right invariant vector fields and the $k$-th jet space of a braided Hopf algebroid and further more a skew pairing between the universal enveloping algebra of its braided right invariant vector fields and its jet Hopf algebroid under a finiteness condition.

math.QA↗
arXiv2026

Affine quantum Schur--Weyl duality

Let $\mathpzc K$ be an arbitrary commutative ring containing an invertible element $\varepsilon$. Let ${\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}$ be the extended affine Hecke algebra of type $A$ with Hecke parameter $\varepsilon$, let $Ω_{\mathpzc K}^{\otimes r}$ be the affine tensor space, and let ${\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}$ be the corresponding affine quantum Schur algebra. We first prove that the natural right action of ${\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}$ on $Ω_{\mathpzc K}^{\otimes r}$ is always faithful. Assume further that $\mathpzc K$ is a field of characteristic $0$ and that $\varepsilon$ is not a root of unity. We prove that, for any $n\geq 2$, the natural algebra homomorphism $ξ_r:{\mathcal H}_{\!\vartriangle\!}(r)_{\mathpzc K}\rightarrow\operatorname{End}_{{\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}}(Ω_{\mathpzc K}^{\otimes r})^{\mathrm{op}}$ is an isomorphism. This proves Conjecture~3.8.8 of \cite{DDF}. As an application, we prove the conjecture formulated in \cite[5.2.4]{DDF} concerning the center of the affine quantum Schur algebra. We also prove that ${\mathcal S}_{\!\vartriangle\!}(n,r)_{\mathpzc K}$ is left and right Noetherian whenever $\mathpzc K$ is a Noetherian commutative ring, which verify a conjecture in \cite[Rem. 1.7]{DY}.

math.QA↗