arXiv subjects

Hiromu Nakano

Publications and source records attributed to Hiromu Nakano.

5 recordsLinked to original sources

arXiv2026

Projective covers of the simple modules for the triplet $W$-algebra $\mathcal{W}_{p_+,p_-}$

We study the structure of the abelian category of modules for the triplet $W$-algebra $\mathcal{W}_{p_+,p_-}$. Using the logarithmic deformation by Fjelstad et al., we construct logarithmic $\mathcal{W}_{p_+,p_-}$-modules that have $L_0$ nilpotent rank three or two. By using the structure of these logarithmic modules and the results on logarithmic Virasoro modules by Kytölä and Ridout, we compute ${\rm Ext}^1$ groups between certain indecomposable modules and simple modules. Based on these ${\rm Ext}^1$ groups we determine the structure of the projective covers of all $\mathcal{W}_{p_+,p_-}$-simple modules.

math.RT↗

arXiv2026

Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras

We prove that the categories of weight modules over the simple $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator algebras at fractional admissible levels and central charges are rigid (and hence the categories of weight modules are braided ribbon categories) and that the decomposition formulae of fusion products of simple projective modules conjectured by Thomas Creutzig, David Ridout and collaborators hold (including when the decomposition involves summands that are indecomposable yet not simple). In addition to solving this old open problem, we develop new techniques for the construction of intertwining operators by means of integrating screening currents over certain cycles, which are expected to be of independent interest, due to their applicability to many other algebras. In the example of $\mathfrak{sl}(2)$ these new techniques allow us to give explicit formulae for a logarithmic intertwining operator from a pair of simple projective modules to the projective cover of the tensor unit, namely, the vertex operator algebra as a module over itself.

math.QA↗

arXiv2026

Derivations on the triplet $W$-algebras with $\mathfrak{sl}_2$-symmetry

We construct derivations on the triplet $W$-algebras $\mathcal{W}_{p_+,p_-}$ by refining the Frobenius homomorphisms of Tsuchiya-Wood and show that the property of the Adamović-Milas derivation for $\mathcal{W}_{2,p}$ extends to our derivations. As an application, we show that the $\mathfrak{sl}_2$-symmetry of $\mathcal{W}_{p_+,p_-}$ arises naturally from our construction. We further show that our method applies to the triplet $W$-superalgebra $\mathcal{SW}(m)$ and that the full automorphism group ${\rm Aut}(\mathcal{SW}(m))$ is $PSL_2(\mathbb{C})\times \mathbb{Z}_2$.

math-ph↗

arXiv2024

Tensor structure on the module category of the triplet superalgebra $\mathcal{SW}(m)$

We discuss the tensor structure on the category of modules of the $N=1$ triplet vertex operator superalgebra $\mathcal{SW}(m)$ introduced by Adamović and Milas. Based on the theory of vertex tensor supercategories, we determine the structure of fusion products between the simple and projective $\mathcal{SW}(m)$-modules and show that the tensor supercategory on $\mathcal{SW}(m)$-mod is rigid. Technically, explicit solutions of a fourth-order Fuchsian differential equation are important to show the rigidity of $\mathcal{SW}(m)$-modules. We construct solutions of this Fuchsian differential equation using the theory of the Dotsenko-Fateev integrals developed by Sussman.

math.QA↗

arXiv2023

Fusion rules for the triplet $W$-algebra $\mathcal{W}_{p_+,p_-}$

We study the structure of fusion rules for the triplet $W$-algebra $\mathcal{W}_{p_+,p_-}$. By using the vertex tensor category theory developed by Huang, Lepowsky and Zhang, we rederive certain non-semisimple fusion rules given by Gaberdiel-Runkel-Wood and Rasmussen. We further show that certain rank two and three indecomposable modules are self-dual.

math.QA↗