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James E. Tener

Publications and source records attributed to James E. Tener.

At least 19 recordsLinked to original sources
arXiv2026

A genus zero functorial CFT associated to the vacuum sector of a conformal net

Starting from an arbitrary conformal net, we construct a genus zero functorial conformal field theory whose Hilbert space is the vacuum sector of the net. Specifically, we construct an algebra for the operad of little discs and conformal embeddings with values in the category of Hilbert spaces and bounded linear maps. We work with closed discs, and our conformal embeddings explicitly allow the image of an incoming disc to overlap with the boundary of the outgoing disc. As an application, we show that all conformal nets satisfy the trace class condition: if $L_0$ is the conformal Hamiltonian of the net, then the operators $r^{L_0}$ are trace class whenever $0 \le r < 1$. In particular, the $L_0$-eigenspaces of a conformal net are automatically finite-dimensional.

math.OA↗
arXiv2026

The Bisognano-Wichmann property for non-unitary Wightman conformal field theories

The Bisognano-Wichmann and Haag duality properties for algebraic quantum field theories are often studied using the powerful tools of Tomita-Takesaki modular theory for nets of operator algebras. In this article, we study analogous properties of nets of algebras generated by smeared Wightman fields, for potentially non-unitary theories. In light of recent work constructing Wightman field theories for (non-unitary) Möbius vertex algebras, we obtain a broadly applicable non-unitary version of the Bisognano-Wichmann property. In this setting we do not have access to the traditional tools of Hilbert space functional analysis, like functional calculus. Instead, results analogous to those of Tomita-Takesaki theory are derived `by hand' from the Wightman axioms. As an application, we demonstrate Haag duality for nets of smeared Wightman fields.

math-ph↗
arXiv2025

Every conformal net has an associated unitary VOA

Unitary vertex operator algebras (VOAs) and conformal nets are the two most prominent mathematical axiomatizations of two-dimensional unitary chiral conformal field theories. They are conjectured to be equivalent, but a rigorous comparison has proven challenging. We resolve one direction of the conjecture by showing that every conformal net has an associated unitary VOA. We also show that every representation of a conformal net in which the generator of rotation acts with discrete spectrum and finite-dimensional eigenspaces yields a unitary module of the corresponding VOA. A talk describing our results is available at: https://www.youtube.com/watch?v=f_LhNSeiiaE .

math.OA↗
arXiv2025

Non-unitary Wightman CFTs and non-unitary vertex algebras

We give an equivalence of categories between: (i) Möbius vertex algebras which are equipped with a choice of generating family of quasiprimary vectors, and (ii) (not-necessarily-unitary) Möbius-covariant Wightman conformal field theories on the unit circle. We do not impose any technical restrictions on the theories considered (such as finite-dimensional conformal weight spaces or simplicity), yielding the most general equivalence between these two axiomatizations of two-dimensional chiral conformal field theory. This provides new opportunities to study non-unitary vertex algebras using the lens of algebraic conformal field theory and operator algebras, which we demonstrate by establishing a non-unitary version of the Reeh-Schlieder theorem.

math-ph↗
arXiv2025

Integrating positive energy representations of the Virasoro algebra

We show that every unitary positive energy representation W of the Virasoro algebra exponentiates to a holomorphic *-representation of the semigroup of annuli by bounded operators on the Hilbert space completion of W. We use this to show that every representation of the Virasoro conformal net also carries a representation of the semigroup of annuli of the same kind.

math.FA↗
arXiv2024

The Segal-Neretin semigroup of annuli

The Lie algebra of vector fields on $S^1$ integrates to the Lie group of diffeomorphisms of $S^1$. It is well known since the work of Segal and Neretin that there is no Lie group whose Lie algebra is the complexification of vector fields on $S^1$. A substitute for that non-existent group is provided by the complex semigroup whose elements are annuli: genus zero Riemann surfaces with two boundary circles parametrized by $S^1$. The group $\mathrm{Diff}(S^1)$ sits at the boundary of that semigroup, and can be thought of as annuli which are completely thin, i.e. with empty interior. In this paper, we consider an enlargement of the semigroup of annuli, denoted $\mathrm{Ann}$, where the annuli are allowed to be partially thin: their two boundary circles are allowed to touch each other along an arbitrary closed subset. We prove that every (partially thin) annulus $A\in \mathrm{Ann}$ is the time-ordered exponential of a path with values in the cone of inward pointing complexified vector fields on $S^1$, and use that fact to construct a central extension \[ 0\to \mathbb{C} \times \mathbb{Z} \to \tilde{\mathrm{Ann}} \to \mathrm{Ann} \to 0 \] that integrates the universal (Virasoro) central extension of the Lie algebra of vector fields on $S^1$. In later work, we will prove that every unitary positive energy representations of the Virasoro algebra integrates to a holomorphic representation of $\tilde{\mathrm{Ann}}$ by bounded operators on a Hilbert space.

math.DG↗
arXiv2024

Fusion and positivity in chiral conformal field theory

In this article we show that the conformal nets corresponding to WZW models are rational, resolving a long-standing open problem. Specifically, we show that the Jones-Wassermann subfactors associated with these models have finite index. This result was first conjectured in the early 90s but had previously only been proven in special cases, beginning with Wassermann's landmark results in type A. The proof relies on a new framework for the systematic comparison of tensor products (a.k.a. `fusion') of conformal net representations with the corresponding tensor product of vertex operator algebra modules. This framework is based on the geometric technique of `bounded localized vertex operators,' which realizes algebras of observables via insertion operators localized in partially thin Riemann surfaces. We obtain a general method for showing that Jones-Wassermann subfactors have finite index, and apply it to additional families of important examples beyond WZW models. We also consider applications to a class of positivity phenomena for VOAs, and use this to outline a program for identifying unitary tensor product theories of VOAs and conformal nets even for badly-behaved models.

math-ph↗
arXiv2022

Unitary vertex algebras and Wightman conformal field theories

We prove an equivalence between the following notions: (i) unitary Möbius vertex algebras, and (ii) Wightman conformal field theories on the circle (with finite-dimensional conformal weight spaces) satisfying an additional condition that we call uniformly bounded order. Reading this equivalence in one direction, we obtain new analytic and operator-theoretic information about vertex operators. In the other direction we characterize OPEs of Wightman fields and show they satisfy the axioms of a vertex algebra. As an application we establish new results linking unitary vertex operator algebras with conformal nets.

math-ph↗
arXiv2020

Classification of extremal vertex operator algebras with two simple modules

In recent work, Wang and the third author defined a class of 'extremal' vertex operator algebras (VOAs), consisting of those with at least two simple modules and conformal dimensions as large as possible for the central charge. In this article we show that there are exactly 15 character vectors of extremal VOAs with two simple modules. All but one of the 15 character vectors is realized by a previously known VOA. The last character vector is realized by a new VOA with central charge 33.

math-ph↗
arXiv2019

Representation theory in chiral conformal field theory: from fields to observables

This article develops new techniques for understanding the relationship between the three different mathematical formulations of two-dimensional chiral conformal field theory: conformal nets (axiomatizing local observables), vertex operator algebras (axiomatizing fields), and Segal CFTs. It builds upon previous work which introduced a geometric interpolation procedure for constructing conformal nets from VOAs via Segal CFT, simultaneously relating all three frameworks. In this article, we extend this construction to study the relationship between the representation theory of conformal nets and the representation theory of vertex operator algebras. We define a correspondence between representations in the two contexts, and show how to construct representations of conformal nets from VOAs. We also show that this correspondence is rich enough to relate the respective 'fusion product' theories for conformal nets and VOAs, by constructing local intertwiners (in the sense of conformal nets) from intertwining operators (in the sense of VOAs). We use these techniques to show that all WZW conformal nets can be constructed using our geometric interpolation procedure.

math-ph↗
arXiv2019

Geometric realization of algebraic conformal field theories

We explore new connections between the fields and local observables in two dimensional chiral conformal field theory. We show that in a broad class of examples, the von Neumann algebras of local observables (a conformal net) can be obtained from the fields (a unitary vertex operator algebra) via a continuous geometric interpolation procedure involving Graeme Segal's functorial definition of conformal field theory. In particular, we construct conformal nets from these unitary vertex operator algebras by showing that 'geometrically mollified' versions of the fields yield bounded, local operators on the Hilbert space completion of the vertex algebra. This work is inspired by Henriques' picture of conformal nets arising from degenerate Riemann surfaces.

math-ph↗
arXiv2017

On classification of extremal non-holomorphic conformal field theories

Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category $\mathcal{C}$ and a central charge $c$. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category $\mathcal{C}$, there exists a unitary chiral conformal field theory $V$ so that its modular tensor category $\mathcal{C}_V$ is $\mathcal{C}$. In this paper, we initiate a mathematical program in and around this conjecture. We define a class of extremal vertex operator algebras with minimal conformal dimensions as large as possible for their central charge, and non-trivial representation theory. We show that there are finitely many different characters of extremal vertex operator algebras V possessing at most three different irreducible modules. Moreover, we list all of the possible characters for such vertex operator algebras with $c$ at most 48.

math-ph↗
arXiv2016

Singular values of weighted composition operators and second quantization

We study a semigroup of weighted composition operators on the Hardy space of the disk $H^2(\mathbb{D})$, and more generally on the Hardy space $H^2(U)$ attached to a simply connected domain $U$ with smooth boundary. Motivated by conformal field theory, we establish bounds on the singular values (approximation numbers) of these weighted composition operators. As a byproduct we obtain estimates on the singular values of the restriction operator (embedding operator) $H^2(V) \to H^2(U)$ when $U \subset V$ and the boundary of $U$ touches that of $V$. Moreover, using the connection between the weighted composition operators and restriction operators, we show that these operators exhibit an analog of the Fisher-Micchelli phenomenon for non-compact operators.

math.FA↗
arXiv2016

Construction of the unitary free fermion Segal CFT

In this article, we provide a detailed construction and analysis of the mathematical conformal field theory of the free fermion, defined in the sense of Graeme Segal. We verify directly that the operators assigned to disks with two disks removed correspond to vertex operators, and use this to deduce analytic properties of the vertex operators. One of the main tools used in the construction is the Cauchy transform for Riemann surfaces, for which we establish several properties analogous to those of the classical Cauchy transform in the complex plane.

math-ph↗
arXiv2012

Unitary equivalence to a complex symmetric matrix: low dimensions

A matrix $T \in \M_n(\C)$ is \emph{UECSM} if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize $4 \times 4$ nilpotent matrices which are UECSM and we settle an open problem which has lingered in the $3 \times 3$ case. We conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above

math.FA↗
arXiv2011

Unitary equivalence of a matrix to its transpose

Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisingly, the naive assertion that a matrix is UET if and only if it is unitarily equivalent to a complex symmetric matrix (i.e., $T = T^t$) holds for matrices 7x7 and smaller, but fails for matrices 8x8 and larger.

math.FA↗
arXiv2011

Subfactors of index less than 5, part 4: vines

We eliminate 38 infinite families of possible principal graphs as part of the classification of subfactors up to index 5. A number-theoretic result of Calegari-Morrison-Snyder, generalizing Asaeda-Yasuda, reduces each infinite family to a finite number of cases. We provide algorithms for computing the effective constants that are required for this result, and we obtain 28 possible principal graphs. The Ostrik d-number test and an algebraic integer test reduce this list to 7 graphs in the index range (4,5) which actually occur as principal graphs.

math.OA↗