219 research outputs found
Realization of rigid C-tensor categories via Tomita bimodules
Starting from a (small) rigid C-tensor category with simple
unit, we construct von Neumann algebras associated to each of its objects.
These algebras are factors and can be either semifinite (of type II or
II, depending on whether the spectrum of the category is finite or
infinite) or they can be of type III, . The choice
of type is tuned by the choice of Tomita structure (defined in the paper) on
certain bimodules we use in the construction. Moreover, if the spectrum is
infinite we realize the whole tensor category directly as endomorphisms of
these algebras, with finite Jones index, by exhibiting a fully faithful unitary
tensor functor where is a
factor (of type II or III).
The construction relies on methods from free probability (full Fock space,
amalgamated free products), it does not depend on amenability assumptions, and
it can be applied to categories with uncountable spectrum (hence it provides an
alternative answer to a conjecture of Yamagami \cite{Y3}). Even in the case of
uncountably generated categories, we can refine the previous equivalence to
obtain realizations on -finite factors as endomorphisms (in the type
III case) and as bimodules (in the type II case).
In the case of trivial Tomita structure, we recover the same algebra obtained
in \cite{PopaS} and \cite{AMD}, namely the (countably generated) free group
factor if the given category has denumerable spectrum, while we
get the free group factor with uncountably many generators if the spectrum is
infinite and non-denumerable.Comment: 39 page
A planar algebraic description of conditional expectations
Let be a unital inclusion of arbitrary von
Neumann algebras. We give a 2-{}-categorical/planar algebraic description
of normal faithful conditional expectations
with finite index and their
duals by means of the
solutions of the conjugate equations for the inclusion morphism
and its conjugate morphism
. In particular, the theory of
index for conditional expectations admits a 2-{}-categorical formulation
in full generality. Moreover, we show that a pair
as above can be described by a Q-system,
and vice versa. These results are due to Longo in the subfactor/simple tensor
unit case [Lon90, Thm.\ 5.2], [Lon94, Thm.\ 5.1].Comment: 20 page
Quantum Operations in Algebraic QFT
Conformal Quantum Field Theories (CFT) in 1 or 1+1 spacetime dimensions
(respectively called chiral and full CFTs) admit several "axiomatic"
(mathematically rigorous and model-independent) formulations. In this note, we
deal with the von Neumann algebraic formulation due to Haag and Kastler, mainly
restricted to the chiral CFT setting. Irrespectively of the chosen formulation,
one can ask the questions: given a theory , how many and which are
the possible extensions or subtheories
? How to construct and classify them, and
study their properties? Extensions are typically described in the language of
algebra objects in the braided tensor category of representations of
, while subtheories require different ideas.
In this paper, we review recent structural results on the study of
subtheories in the von Neumann algebraic formulation (conformal subnets) of a
given chiral CFT (conformal net). Furthermore, building on [BDG23], we provide
a "quantum Galois theory" for conformal nets analogous to the one for Vertex
Operator Algebras (VOA). We also outline the case of 3+1 dimensional Algebraic
Quantum Field Theories (AQFT). The aforementioned results make use of families
of (extreme) vacuum state preserving unital completely positive maps acting on
the net of von Neumann algebras, hereafter called quantum operations. These are
natural generalizations of the ordinary vacuum preserving gauge automorphisms,
hence they play the role of "generalized global gauge symmetries". Quantum
operations suffice to describe all possible conformal subnets of a given
conformal net with the same central charge.Comment: Invited contribution to the conference proceedings of Functional
Analysis, Approximation Theory and Numerical Analysis, FAATNA20>22, Matera,
Italy, 202
Braided Actions of DHR Categories and Reconstruction of Chiral Conformal Field Theories
Quantum Field Theory (QFT) is our modern understanding of particles and matter at small scales, where quantum behaviour replaces macroscopical phenomena, which are much closer to intuition, and dynamics is rather driven by the more fundamental interactions between fields. Using quantum fields one can describe particle production, annihilation and scattering processes and they can all together be cast into the Standard Model of particle physics. The latter gives a recipe to predict cross sections of high-energy collisions which fit remarkably well with experimental data. On the other hand our mathematical understanding of the framework, and how to replace diverging series, ad hoc renormalized or truncated to get finite numbers, is still a deep open question.
Since the early days it was clear that quantum fields, even when they arise from the classical picture of Lagrangian functionals and actions, are more singular objects than those employed in classical physics. Their values in points of spacetime, i.e., their point-like dependence as operator-valued functions, is easily seen to clash with their realizability as operators on an Hilbert space on one hand, on the other hand it is neither dictated by physics. The structure of spacetime itself, at very small scales, is by now out of our experimental reach.
In order to overcome the previous difficulties the notion of field can be relaxed to that of an (unbounded) operator-valued distribution (Wightman axiomatization), elevating the smearing with test functions to an essential feature of a local quantum theory. This generalization introduces more difficult mathematical objects (distributions, compared to functions) but which can be rigorously (without ambiguities) treated, and which are suitable enough to obtain a complete scattering theory, once a Wightman QFT is assigned.
In the same spirit, but with different mathematics, QFTs can be dealt with using techniques from the theory of operator algebras. The first main characteristics of the algebraic approach (AQFT) is that one describes local measurements or observable fields and regards them as the primary objects of interest to study matter, particles and fundamental interactions, relegating the non-observable quantities to theoretical tools. Secondly, one treats them by means of bounded operators on Hilbert space (e.g., by considering bounded functions of the fields), advantageous at least for the analysis of the framework. More in details, physically relevant quantities such as observables (and states) of a QFT are described in terms of abstract operator algebras associated to open bounded regions of spacetime (âlocal algebrasâ). By abstract we mean independent of any specific Hilbert space realization, and then we regard the choice of different representations of the local algebras as the choice of different states (mathematically speaking via the GNS construction). In particular these objects encode both quantum behaviour, in their intrinsic non-commutativity, and Einsteinâs causality principle, in the triviality of commutation relations between local algebras sitting at space-like distances. This second approach is what this thesis is devoted to.
The relation between these two formalisms is not completely understood, from distributions to local algebras one has to take care of spectral commuta- tion relations on suitable domains, vice versa one should control the scaling limits of the local algebras in order to exploit the distributional âpoint-likeâ generators. In both cases, and (theoretically) in any other mathematically sound description of QFT, consequences become proofs, and different features of models or more general model-independent principles (particle content, covariance, local commutation relations) can be separated and analysed. Moreover, beyond the needs of rigorous description of models, the âaxiomaticâ approaches to QFT have the advantage of being more independent from classical analogies, like field equations and Lagrangians. Fields themselves are not an essential input to model local measurements obeying the constraints of Einsteinâs special relativity and quantum theory.
AQFT can be thought of as being divided into two lines, the first aims to the construction of models (either in low or high dimensions, both starting from physical counterparts or using the theory of operator algebras), the second is devoted to the analysis of the assumptions and of the possibly new mathematical structures arising from them. The work presented in this thesis has been developed and expresses its contribution in the second line of research. Our aim is to introduce new invariants for local quantum field theories, more specifically to complete a well established construction (the DHR construction) which associates a certain category of representations (collection of superselection sectors together with their fusion rules, exchange symmetry, statistics) to any local quantum field theory, once the latter is
formulated as a local net of algebras
Minimal index and dimension for inclusions of von Neumann algebras with finite-dimensional centers
The notion of index for inclusions of von Neumann algebras goes back to a
seminal work of Jones on subfactors of type . In the absence of a
trace, one can still define the index of a conditional expectation associated
to a subfactor and look for expectations that minimize the index. This value is
called the minimal index of the subfactor. We report on our analysis, contained
in [GL19], of the minimal index for inclusions of arbitrary von Neumann
algebras (not necessarily finite, nor factorial) with finite-dimensional
centers. Our results generalize some aspects of the Jones index for
multi-matrix inclusions (finite direct sums of matrix algebras), e.g., the
minimal index always equals the squared norm of a matrix, that we call
\emph{matrix dimension}, as it is the case for multi-matrices with respect to
the Bratteli inclusion matrix. We also mention how the theory of minimal index
can be formulated in the purely algebraic context of rigid 2--categories.Comment: Invited contribution to the Proceedings of the 27th International
Conference in Operator Theory (OT27), Timi\c{s}oara, 201
Realization of rigid C*-bicategories as bimodules over type II von Neumann algebras
We prove that every rigid C*-bicategory with finite-dimensional centers
(finitely decomposable horizontal units) can be realized as Connes' bimodules
over finite direct sums of II factors. In particular, we realize every
multitensor C*-category as bimodules over a finite direct sum of II
factors.Comment: 20 page
Single-Layer versus Multilayer Preplanned Lightpath Restoration
Special Issue on âOptical Networksâ October 200
Wightman Fields for Two-Dimensional Conformal Field Theories with Pointed Representation Category
Two-dimensional full conformal field theories have been studied in various mathematical frameworks, from algebraic, operator-algebraic to categorical. In this work, we focus our attention on theories with chiral components having pointed braided tensor representation subcategories, namely having automorphisms whose equivalence classes necessarily form an abelian group. For such theories, we exhibit the explicit Hilbert space structure and construct primary fields as Wightman fields for the two-dimensional full theory. Given a finite collection of chiral components with automorphism categories with trivial total braiding, we also construct a local extension of their tensor product as a chiral component. We clarify the relations with the Longo-Rehren construction, and illustrate these results with concrete examples including the U(1)current
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