arXiv subjects

Colton Griffin

Publications and source records attributed to Colton Griffin.

6 recordsLinked to original sources

arXiv2026

Modular Zhu algebra theory and Virasoro vertex algebras

We develop the representation theory of vertex algebras over arbitrary commutative rings as part of a relative theory, suitable for various applications such as the study of modular forms, modular tensor categories, and sheaves of coinvariants and conformal blocks. Towards this end, we construct Zhu algebras (and higher analogues) and mode transition algebras over arbitrary rings via the universal enveloping algebra. We prove that these constructions are compatible with base change and give rationality criteria for M\"obius vertex algebras over arbitrary fields satisfying mild assumptions. We apply this framework to study Virasoro vertex operator algebras over arbitrary fields. Using integral forms and base change, we extend the rationality of the discrete series Virasoro VOAs from $\mathbb{C}$ to arbitrary fields of characteristic zero. In positive characteristic, the Virasoro theory exhibits surprising new phenomena: the so-called restricted Virasoro VOA is $C_2$-cofinite and has a semisimple Zhu algebra, but it is not rational. Moreover, we find that the simple quotient with central charge $c\in \mathbb{F}_p$ is holomorphic for $p=3,5$ and in most cases for $p=7$.

math.QA↗

arXiv2025

Cohomological vertex algebras

Vertex algebras (and their modules) can be described as vector spaces together with a linear operator-valued series in one parameter $z$. With the interpretation of $z$ as a coordinate at a point on a curve, one can construct algebraic structures on the moduli space of curves from $V$-modules. Here we propose a generalization of vertex algebras involving linear operators in parameters $z_1,\ldots,z_n$. One may interpret these as being the components of a set of coordinates on an $n$-dimensional algebraic variety. These are referred to as cohomological vertex algebras (CVAs): the formal punctured 1-disk underlying a vertex algebra is replaced by a ring modeling the cohomology of certain modifications of the formal $n$-disk. We prove several structural theorems for CVAs and give a definition of cohomological vertex operator algebras (CVOAs). Using a reconstruction theorem for CVAs, we provide basic examples such as the $\beta\gamma$-system, the Heisenberg CVA, and the affine Kac-Moody CVAs. We use these constructions to describe BRST reduction, leading to an analog of W-algebras.

math.QA↗

arXiv2023

Use of statistically leinert sets to calculate return probabilities of random walks in F_s1 x F_s2

Hastings first presented bounds on the second largest eigenvalue for matrices in a Hermitian complete positive map in 2007. In this work we extend his work to tighten these bounds. To do this, we introduce the idea of Statistically Leinert Sets to modify the generating functions presented in Woess in 1986 and recompute the radii of convergence in his paper in 1986. We primarily use techniques from combinatorics and calculate norms using the ideas presented by Akemann and Ostrang in their paper in 1976.

math.PR↗

arXiv2023

Approximating projections by quantum operations

Using techniques from semidefinite programming, we study the problem of finding a closest quantum channel to the projection onto a matricial subsystem. We derive two invariants of matricial subsystems which are related to the quantum Lovász theta function of Duan, Severini, and Winter.

quant-ph↗

arXiv2022

Constructing Approximately Diagonal Quantum Gates

We study a method of producing approximately diagonal 1-qubit gates. For each positive integer, the method provides a sequence of gates that are defined iteratively from a fixed diagonal gate and an arbitrary gate. These sequences are conjectured to converge to diagonal gates doubly exponentially fast and are verified for small integers. We systemically study this conjecture and prove several important partial results. Some techniques are developed to pave the way for a final resolution of the conjecture. The sequences provided here have applications in quantum search algorithms, quantum circuit compilation, generation of leakage-free entangled gates in topological quantum computing, etc.

quant-ph↗

arXiv2022

An Index for Inclusions of Operator Systems

Inspired by a well-known characterization of the index of an inclusion of II$_1$ factors due to Pimsner and Popa, we define an index-type invariant for inclusions of operator systems. We compute examples of this invariant, show that it is multiplicative under minimal tensor products, and explain how it generalizes the quantum Lovász theta invariant for a matricial system defined by Duan, Severini, and Winter.

math.OA↗