arXiv subjects

Brett Hungar

Publications and source records attributed to Brett Hungar.

6 recordsLinked to original sources

arXiv2026

Orthonormal bases for higher Hilbert spaces

In our previous article [arxiv:2410.05120], we introduced the notion of a finite dimensional 3-Hilbert space, categorifying Baez's 2-Hilbert spaces. In this article, by further categorifying Baez's higher linear algebra, we provide useful tools for working with 3-Hilbert spaces, including, generalized scalar multiplication, orthonormal bases, and unitary adjoints for operators. We use these tools to endow the $\mathrm{C}^*$-3-category of 3-Hilbert spaces with a self-enrichment. We prove a Unitary Yoneda Lemma/Riesz Representation Theorem for 3-Hilbert spaces: the Yoneda embedding is an isometric equivalence. Finally, we define a unitary version of the Deligne product on 3-Hilbert spaces and prove that it satisfies an isometric version of the folding trick.

math.QA↗

arXiv2025

Anchored Planar Algebras and 3-Categorical Graphical Calculus

Anchored planar algebras, a generalized notion of Vaughan Jones' planar algebras, have recently seen use in higher category theory, functional analysis, and TQFT applications. These algebras are equipped with a natural 3-dimensional graphical calculus. We compare this graphical calculus with the 3-dimensional graphical calculus associated to tricategories, and we show that anchored planar algebras can be thought of as living in a particular tricategory. This allows for more general techniques to be applied to anchored planar algebras and expands the types of diagrams that anchored planar algebras can interpret.

math.CT↗

arXiv2024

Manifestly unitary higher Hilbert spaces

Higher idempotent completion gives a formal inductive construction of the $n$-category of finite dimensional $n$-vector spaces starting with the complex numbers. We propose a manifestly unitary construction of low dimensional higher Hilbert spaces, formally constructing the $\mathrm{C}^*$-3-category of 3-Hilbert spaces from Baez's 2-Hilbert spaces, which itself forms a 3-Hilbert space. We prove that the forgetful functor from 3-Hilbert spaces to 3-vector spaces is fully faithful.

math.QA↗

arXiv2024

Dagger $n$-categories

Category theory provides a unified language for organizing composable operations in many disciplines. In disciplines where unitarity is fundamental -- such as functional analysis, quantum field theory, and quantum logic -- this language must also capture adjoints, leading to the notion of dagger categories. Higher category theory, which extends this framework to encode operations between operations, has recently become indispensable in both theoretical physics and pure mathematics. Finding a higher categorical analogue of a dagger category is therefore key to the foundations of quantum field theory. In this work, we present a coherent definition of \emph{dagger $(\infty,n)$-category} in terms of equivariance data trivialized on parts of the category. Our main example is the bordism $(\infty,n)$-category $\mathbf{Bord}_{n}^X$. This allows us to define (fully-local) \emph{reflection-positive topological quantum field theories} to be higher dagger functors out of $\mathbf{Bord}_{n}^X$.

math.CT↗

arXiv2023

Boundary algebras of the Kitaev Quantum Double model

The recent article [arXiv:2307.12552] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev's Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev's Quantum Double model for a finite group $G$. We identify the boundary nets of algebras with fusion categorical nets associated to $(\mathsf{Hilb}(G),\mathbb{C}[G])$ or $(\mathsf{Rep}(G),\mathbb{C}^G)$ depending on whether the boundary cut is rough or smooth respectively. This allows us to make connections to work of Ogata on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial $G$-symmetry protected topological phase ($G$-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3+1)D Quantum Double model associated to an abelian group.

cond-mat.str-el↗

arXiv2021

Spectra of three-peg Hanoi towers graphs

We consider the relationship between the Laplacians on two sequences of planar graphs, one from the theory of self-similar groups and one from analysis on fractals. By establishing a spectral decimation map between these sequences we give an elementary calculation of the spectrum of the former, which was first computed by Grigorchuk and \v{S}uni\'{c}. Our method also gives a full description of the eigenfunctions.

math.CO↗