Title: An operator algebraic construction of 2D extensions of the Heisenberg conformal net
Abstract: Algebraic quantum field theory (AQFT) is a mathematical formalism to rigorously describe physical models in quantum field theory, namely the branch of theoretical physics that studies subatomic particles and their interactions. In such formalism, physical observables are encoded in a family of operator algebras, called a net of operator algebras, which verify a set of physically motivated axioms, in particular locality. A local net is such that operator algebras associated with space-like separated bounded regions commute.
Constructing physically relevant models in the AQFT formalism is usually a difficult task. Thus, one may attempt to construct first models which are richer in symmetries, e.g., conformal field theories in low-dimensional space-time. In two dimensions, one can see a 2D conformal field theory as an extension of the product of a (non-necessarily the same) left and right chiral conformal net, the so-called chiral components. In this talk, we discuss the explicit construction of conformal nets of operator algebras in two dimensions that extend the Heisenberg chiral conformal net, which is a realisation of the canonical commutation relations.
An important tool to investigate nets of operator algebras in the AQFT formalism is given by studying their admissible representations in the sense of Doplicher-Haag-Roberts (DHR) representation theory, used to construct the operator algebras in the 2D conformal net extending the Heisenberg chiral conformal net. To this aim, we introduce charged fields acting as (twisted) shifts between equivalence classes of irreducible DHR representations. Since the irreducible DHR representations are all automorphisms, one can label the equivalence classes by an additive subgroup $Q$ of the real linear space of charges, which are vectors in $\mathbb{R}^N$. In our context, Q encodes the interplay between the DHR representation theory of the left and right chiral components.
When $Q$ is made such that left and right charges coincide (diagonal case), a local 2D conformal net extending the Heisenberg chiral conformal net can be obtained by including the untwisted charge shift operators. In the case when $Q$ is an even lattice (non-diagonal case), we recover locality of the 2D conformal net by introducing a 2-cocycle on $Q$ used to define the twisted charged fields.
This talk is based on a joint project with L. Giorgetti, Y. Tanimoto, arXiv:2301.12310, arXiv:2506.01008.