Monoidal Categories Enriched in Braided Monoidal Categories

We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal{V}$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld centre of some monoidal category $\mathcal{T}$. Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather satisfy a braided interchange relation. Strikingly, our classification is slightly more general than what one might have anticipated in terms of strong monoidal functors $\mathcal{V}\to Z(\mathcal{T})$. We would like to understand this further; in a future article, we show that the functor is strong if and only if the enriched category is ‘complete’ in a certain sense. Nevertheless it remains to understand what non-complete enriched categories may look like. One should think of our construction as a generalization of de-equivariantization, which takes a strong monoidal functor ${\mathsf {Rep}}(G) \to Z(\mathcal{T})$ for some finite group $G$ and a monoidal category $\mathcal{T}$, and produces a new monoidal category $\mathcal{T} _{{/\hspace{-2px}/}G}$. In our setting, given any braided oplax monoidal functor $\mathcal{V} \to Z(\mathcal{T})$, for any braided $\mathcal{V}$, we produce $\mathcal{T} _{{/\hspace{-2px}/}\mathcal{V}}$: this is not usually an ‘honest’ monoidal category, but is instead $\mathcal{V}$-enriched. If $\mathcal{V}$ has a braided lax monoidal functor to ${\mathsf {Vec}}$, we can use this to reduce the enrichment to ${\mathsf {Vec}}$, and this recovers de-equivariantization as a special case. This is the published version of arXiv:1701.00567.

A class of braided monoidal categories via quasitriangular Hopf π-crossed coproduct algebras

Let π be a group and (H = {Hα}α∈π, μ, η) a Hopf π-algebra. First, we introduce the concept of quasitriangular Hopf π-algebra, and then prove that the left H-π-module category [Formula: see text], where (H, R) is a quasitriangular Hopf π-algebra, is a braided monoidal category. Second, we give the construction of Hopf π-crossed coproduct algebra [Formula: see text]. At last, the necessary and sufficient conditions for [Formula: see text] to be a quasitriangular Hopf π-algebra are derived, and in this case, [Formula: see text] is a braided monoidal category.

Instantons and vortices on noncommutative toric varieties

We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.