194,473 research outputs found
An Explicit Framework for Interaction Nets
Interaction nets are a graphical formalism inspired by Linear Logic
proof-nets often used for studying higher order rewriting e.g. \Beta-reduction.
Traditional presentations of interaction nets are based on graph theory and
rely on elementary properties of graph theory. We give here a more explicit
presentation based on notions borrowed from Girard's Geometry of Interaction:
interaction nets are presented as partial permutations and a composition of
nets, the gluing, is derived from the execution formula. We then define
contexts and reduction as the context closure of rules. We prove strong
confluence of the reduction within our framework and show how interaction nets
can be viewed as the quotient of some generalized proof-nets
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.Comment: 79 pages. v2: minor revisions and update
A Congruence for Petri Nets
We introduce a way of viewing Petri nets as open systems. This is done by considering a bicategory of cospans over a category of p/t nets and embeddings. We derive a labelled transition system (LTS) semantics for such nets using GIPOs and characterise the resulting congruence. Technically, our results are similar to the recent work by Milner on applying the theory of bigraphs to Petri Nets. The two main differences are that we treat p/t nets instead of c/e nets and we deal directly with a category of nets instead of encoding them into bigraphs
On the representation theory of Virasoro Nets
We discuss various aspects of the representation theory of the local nets of
von Neumann algebras on the circle associated with positive energy
representations of the Virasoro algebra (Virasoro nets). In particular we
classify the local extensions of the Virasoro net for which the
restriction of the vacuum representation to the Virasoro subnet is a direct sum
of irreducible subrepresentations with finite statistical dimension (local
extensions of compact type). Moreover we prove that if the central charge
is in a certain subset of , including , and , the irreducible representation with lowest weight of the
corresponding Virasoro net has infinite statistical dimension. As a consequence
we show that if the central charge is in the above set and satisfies then the corresponding Virasoro net has no proper local extensions of
compact type.Comment: 34 page
Scaling limit for subsystems and Doplicher-Roberts reconstruction
Given an inclusion of (graded) local nets, we analyse the
structure of the corresponding inclusion of scaling limit nets , giving conditions, fulfilled in free field theory, under which the
unicity of the scaling limit of implies that of the scaling limit of .
As a byproduct, we compute explicitly the (unique) scaling limit of the
fixpoint nets of scalar free field theories. In the particular case of an
inclusion of local nets with the same canonical field net , we
find sufficient conditions which entail the equality of the canonical field
nets of and .Comment: 31 page
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