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Register a new userGalois Correspondence and Fourier Analysis on Local Discrete Subfactors
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Discrete subfactors include a particular class of infinite index subfactors and all finite index ones. A discrete subfactor is called local when it is braided and it fulfills a commutativity condition motivated by the study of inclusion of Quantum Field Theories in the algebraic Haag-Kastler setting. In [BDG21], we proved that every irreducible local discrete subfactor arises as the fixed point subfactor under the action of a canonical compact hypergroup. In this work, we prove a Galois correspondence between intermediate von Neumann algebras and closed subhypergroups, and we study the subfactor theoretical Fourier transform in this context. Along the way, we extend the main results concerning $alpha$-induction and $sigma$-restriction for braided subfactors previously known in the finite index case.
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Conformal inclusions of chiral conformal field theories, or more generally inclusions of quantum field theories, are described in the von Neumann algebraic setting by nets of subfactors, possibly with infinite Jones index if one takes non-rational theories into account. With this situation in mind, we study in a purely subfactor theoretical context a certain class of braided discrete subfactors with an additional commutativity constraint, that we call locality, and which corresponds to the commutation relations between field operators at space-like distance in quantum field theory. Examples of subfactors of this type come from taking a minimal action of a compact group on a factor and considering the fixed point subalgebra. We show that to every irreducible local discrete subfactor $mathcal{N}subsetmathcal{M}$ of type ${I!I!I}$ there is an associated canonical compact hypergroup (an invariant for the subfactor) which acts on $mathcal{M}$ by unital completely positive (ucp) maps and which gives $mathcal{N}$ as fixed points. To show this, we establish a duality pairing between the set of all $mathcal{N}$-bimodular ucp maps on $mathcal{M}$ and a certain commutative unital $C^*$-algebra, whose spectrum we identify with the compact hypergroup. If the subfactor has depth 2, the compact hypergroup turns out to be a compact group. This rules out the occurrence of compact emph{quantum} groups acting as global gauge symmetries in local conformal field theory.
A theorem is derived which (i) provides a new class of subfactors which may be interpreted as generalized asymptotic subfactors, and which (ii) ensures the existence of two-dimensional local quantum field theories associated with certain modular invariant matrices.
For a finite-index $mathrm{II}_1$ subfactor $N subset M$, we prove the existence of a universal Hopf $ast$-algebra (or, a discrete quantum group in the analytic language) acting on $M$ in a trace-preserving fashion and fixing $N$ pointwise. We call this Hopf $ast$-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse.
We construct analogs of the embedding of orthogonal and symplectic groups into unitary groups in the context of fusion categories. At least some of the resulting module categories also appear in boundary conformal field theory. We determine when these categories are unitarizable, and explicitly calculate the index and principal graph of the resulting subfactors.
Growing out of the initial connections between subfactors and knot theory that gave rise to the Jones polynomial, Jones axiomatization of the standard invariant of an extremal finite index $II_1$ subfactor as a spherical $C^*$-planar algebra, presented in arXiv:math.QA/9909027, is the most elegant and powerful description available. We make the natural extension of this axiomatization to the case of finite index subfactors of arbitrary type. We also provide the first steps toward a limited planar structure in the infinite index case. The central role of rotations, which provide the main non-trivial part of the planar structure, is a recurring theme throughout this work. In the finite index case the axioms of a $C^*$-planar algebra need to be weakened to disallow rotation of internal discs, giving rise to the notion of a rigid $C^*$-planar algebra. We show that the standard invariant of any finite index subfactor has a rigid $C^*$-planar algebra structure. We then show that rotations can be re-introduced with associated correction terms entirely controlled by the Radon-Nikodym derivative of the two canonical states on the first relative commutant, $N cap M$. By deforming a rigid $C^*$-planar algebra to obtain a spherical $C^*$-planar algebra and lifting the inverse construction to the subfactor level we show that any rigid $C^*$-planar algebra arises as the standard invariant of a finite index $II_1$ subfactor equipped with a conditional expectation, which in general is not trace preserving. Jones results thus extend completely to the general finite index case. We conclude by applying our machinery to the $II_1$ case, shedding new light on the rotations studied by Huang [11] and touching briefly on the work of Popa [29]. (continued in article)
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