arXiv subjects

Andrew Riesen

Publications and source records attributed to Andrew Riesen.

6 recordsLinked to original sources

arXiv2026

Associated Varieties of Finite-Group Orbifolds

Let $V$ be a simple, strongly finitely generated vertex algebra with a connected nonnegative integer grading, and let $G$ be a finite group of grading-preserving automorphisms. In this paper, we prove that $X_{V^G}\cong X_V/G$ as reduced Poisson varieties. More precisely, the canonical surjection $R_{V^G}\longrightarrow (R_V)^G $ has nilpotent kernel. The proof uses a uniform derivative bound in a relative $C_2$-quotient, which also implies that $V^G$ is strongly finitely generated. Consequently, quasi-lisseness and $C_2$-cofiniteness descend to finite-group orbifolds, and hence, by existing results, so does strong rationality.

math.QA↗

arXiv2026

Vertex operator algebra realizations of untwisted Haagerup--Izumi centers

For every odd integer $N\geq3$, we construct an untwisted Haagerup--Izumi fusion category $\mathcal H_N$ on the negative spherical branch and a simple, self-contragredient, regular, $C_2$-cofinite vertex operator algebra $E_N$ of central charge $-16N$, together with an equivalence of modular fusion categories: \[ \operatorname{Rep}(E_N)\simeq Z(\mathcal H_N). \] The VOA $E_N$ is rational, has exactly $N^2+3$ irreducible modules, and is lower bounded and integer graded, with least conformal degree $-(N-1)/2$. The proof begins with a CFT-type cyclic Virasoro extension of central charge $-5N$, whose regularity we establish. Two categorical condensations recover $\mathcal H_N$, and a finite extension by shifted lattice modules removes the auxiliary pointed factor. We identify the complete ribbon modular data of $Z(\mathcal H_N)$ with $\mathrm{MD}_{-}(\mathbb Z/N;\mathbb Z/(N^2+4),b\mapsto b^2/(N^2+4))$ in the notation of Gannon--Schopieray--Yadav, for every odd $N$.

math.QA↗

arXiv2026

The Haagerup VOA at $c=8$

We announce an explicit construction of a unitary, strongly rational vertex operator algebra $U$ of central charge $8$ inside $V_{E_6\oplus A_2}$, with a ribbon equivalence $\mathrm{Rep}(U)\simeq Z(\mathcal H_3)$ realizing the Haagerup double. The vertex algebra is generated by six affine currents and three specified weight-two fields; its weight-one Lie algebra is $\mathfrak{sl}_2\oplus\mathfrak{sl}_2$, at levels $1$ and $39$. We outline the argument, which combines exact lattice calculations, an all-weight strong-generation induction and $C_2$-cofiniteness, semisimple descent for a full corner of Zhu's algebra, modular Galois symmetry, and holomorphic $E_8$ extensions.

math.QA↗

arXiv2026

Interpolating Feigin-Frenkel Duality at the Critical Level to Matrices of Complex Size

In this paper, we extend Feigin-Frenkel duality at the critical level to complex rank by identifying two seemingly unrelated constructions in complex rank. On the affine side, we interpolate Molev's construction of higher Segal-Sugawara vectors and thereby describe the centers of universal affine vertex algebras at the critical level in Deligne's interpolating categories. On the $\mathcal{W}$-side, we construct the classical $\mathcal{W}$-algebras associated with Feigin's Lie algebras of complex rank $\mathfrak{gl}_λ$ and $\mathfrak{po}_λ$ as Poisson vertex algebras, realizing their Drinfeld-Sokolov reduction via an interpolated Adler-Gelfand-Dickey bracket. Upon specialization to positive integer rank in types A, B, and C, this recovers the usual Feigin-Frenkel duality at the critical level. As applications, we obtain a uniform construction of several families of higher Segal-Sugawara vectors for Lie superalgebras and recover a complex-rank analogue of the universal Bethe algebra.

math.QA↗

arXiv2024

Orbifolds of Pointed Vertex Operator Algebras I

By a pointed vertex operator algebra (VOA) we mean one whose modules are all simple currents (i.e. invertible), e.g. lattice VOAs. This paper systematically explores the interplay between their orbifolds and tensor category theory. We begin by supplying an elementary proof of the Dijkgraaf-Witten conjecture, which predicts the representation theory of holomorphic VOA orbifolds. We then apply that argument more generally to the situation where the automorphism subgroup fixes all VOA modules, and relate the result to recent work of Mason-Ng and Naidu. Here our results are complete. We then turn to the other extreme, where the automorphisms act fixed-point freely on the modules, and realize any possible nilpotent group as lattice VOA automorphisms. This affords a considerable generalization of the Tambara-Yamagami categories. We conclude by considering some hybrid actions. In this way we use tensor category theory to organize and generalize systematically several isolated examples and special cases scattered in the literature. Conversely, we show how VOA orbifolds can be used to construct broad classes of braided crossed fusion categories and modular tensor categories.

math.QA↗

arXiv2023

Rational Schur Superalgebras

We develop and study the generalization of rational Schur algebras to the super setting. Similar to the classical case, this provides a new method for studying rational supermodules of the general linear supergroup $GL(m|n)$. Furthermore, we establish a Schur-Weyl duality result for rational Schur superalgebras and conclude that under certain conditions these objects will be semisimple.

math.RT↗