Quantum Groups and
Algebraic Quantum Field Theory
Titles and abstracts of talks
Monday
Yasuyuki Kawahigashi. Tensor networks, commuting squares and higher relative commutants of subfactors
A bi-unitary connection in subfactor theory of Jones producing a subfactor of finite depth gives a 4-tensor appearing in a recent work on 2-dimensional topological order and anyons. A subfactor arising from a quantum group at a root of unity is such an example. Physicists have a special projection called a projector matrix product operator in this setting. We prove that the range of this projection of length k is naturally identified with the k-th higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between 2-dimensional topological order and subfactor theory.
Ingo Runkel. Quantum groups and chiral CFT
In certain classes of examples, the representation category of a chiral conformal field theory, that is, of a vertex operator algebra, is braided-equivalent to that of a quantum group. I will review some examples and non-examples, as well as recent results and conjectures in this direction. Time permitting, I will remark on applications to three-dimensional topological quantum field theory.
Christian Voigt. Quantum graphs and operator algebras
In this talk I will give an introduction to quantum graphs, a relatively recent concept with connections to graph theory, quantum groups, and quantum information. I will discuss some examples and peculiarities, and then explain how one can associate operator algebras to quantum graphs, in analogy to the construction of Cuntz-Krieger algebras.
Tuesday
Anna Kula. From Hochschild Cohomology to Levy Processes on Compact Quantum Groups, and back
It was shown by Schürmann in 1990ies that the classification of Levy processes on a *-bialgebra A (e.g. on a CQG-algebra) heavily relies on the cohomological properties of A. I will present that, conversely, the studies of Levy Processes on the universal unitary quantum groups U_Q^+, for Q in M_n(C), can help in finding the first and second Hochschild cohomology groups of the CQG-algebras of U_Q^+.
Kenny De Commer. Quantum SL(2,R) and its representations
The theory of quantum deformations of compact Lie groups has been very successful, but a corresponding concrete theory for non-compact semisimple Lie groups is still not fully developed in the analytic setting. In this talk, we will argue that such a theory becomes available, both on the algebraic and analytic level, once one lets go of the idea that the quantization should be a quantum group. Indeed, we show that by combining a variation of the Drinfeld double construction with the work on quantum symmetric spaces due to Letzter, Kolb and others, quantizations of non-compact semisimple Lie groups can be constructed as coideal subalgebras inside the quantization of their complexification. We will explain this construction in detail for the simplest example of SL(2,R). We then show how the representation theory of quantum SL(2,R) relates to the representation theory of classical SL(2,R). This is joint work with Joel Right Dzokou Talla.
Isabelle Baraquin. De Finetti Theorems
In probability theory, de Finetti Theorem states that exchangeable random variables are conditionally independent. In this talk, we will present some similar results in noncommutative probability. After looking at the quantum case, we will study a de Finetti theorem in the dual unitary group.
Sang-Gyun Youn. On the rapid decay property of orthogonal free quantum groups
The rapid decay property (RD) of discrete groups is a fundamental tool in the study of reduced group C*-algebras, and allows one to compare the operator norm of convolution operators with much simpler L2-norms. This property was studied for orthogonal free quantum groups by means of ‘quantum RD’ and ‘twisted quantum RD’, and it was known that the twisted quantum RD holds for all amenable orthogonal free quantum groups. On the other hand, it has turned out that the twisted quantum RD does not hold for any non-amenable orthogonal free quantum groups of non-Kac type, whereas a weakened RD property is always satisfied. Moreover, this weakened RD allows us to get (almost) optimal time for ultracontractivity of heat semigroups. This talk is based on a recent joint work with Michael Brannan and Roland Vergnioux.
Wednesday
Luca Giorgetti. Generalized gauge symmetries and local discrete subfactors
Global gauge group symmetries are not enough to describe all the subnets of a given local conformal net. The main reason is that the DHR braiding is typically not a symmetry in 1 or 1+1 spacetime dimensions. In the talk, I will report on recent works where we study the structure of the “generalized gauge symmetries” of a given inclusion of local conformal nets (not necessarily rational / with finite index). At the level of a single “local” subfactor N < M, the generalized gauge symmetries are the UCP maps on M (e.g. automorphisms) which leave N elementwise fixed. They suffice to describe N as the fixed point subalgebra of M, they have the structure of a compact hypergroup and they give a complete Galois theory for the intermediate algebras N < P < M. I will mention two consequences of our analysis: a rigidity result about the overlap with compact quantum group symmetries and some applications to the subfactor theoretical Fourier transform.
Yoh Tanimoto. Unitary modules and conformal nets associated with the W_3-algebra with c >= 2
The W_3-algebra is a higher spin extension of the Virasoro algebra. One can consider lowest weight modules and invariant sesquilinear forms, and the question of unitarity (positive-definiteness of the invariant sesquilinear form). We show that, for the lowest weights c >= 2, h=w=0, the module is unitary and we construct a conformal net associated with each of such modules.
Gandalf Lechner. Singular half-sided modular inclusion and deformation quantization
A chiral half of a conformal field theory can be described by a system (net) of von Neumann algebras attached to intervals on the circle that transforms covariantly under a representation of the Möbius group. Under favourable circumstances, the whole net can be reconstructed from a single von Neumann algebra and a representation of the 1d translation group. This structure can also be formulated as a half-sided modular inclusion, and an essential requirement is that this inclusion is not singular, i.e. does not have a trivial relative commutant. So far the only known example of a singular half-sided modular inclusion was constructed with the help of free probability (Longo-Tanimoto-Ueda 2019). In this talk I will explain the general setting of half-sided inclusions and conformal field theory on the circle, and then present a technique for producing new singular inclusions by a deformation procedure.
Thursday
Karl-Henning Rehren. Quantum fields with uncountable superselection structure and braid statistics in four dimensions
In the course of analyzing the infrared problems of Quantum Electro Dynamics (QED), one discovers a new type of quantum field that trespass the limitations of standard axiomatic frameworks of QFT. These fields allow a non-perturbative construction based on "infrared limits" of Weyl algebras. Among other things of interest in the context of QED, they satisfy braid group statistics -- which (by the DHR analysis) was believed to be impossible in algebraic QFT.
Kasia Rejzner. A new construction of nets of C*-algebras for interacting QFT: Symmetries and anomalies
In a recent paper, Buchholz and Fredenhagen proposed a formulation of interacting quantum field theory in terms of a net of C*-algebras generated by unitaries that are interpreted as local S-matrices. In an upcoming paper by Brunetti, Duetsch, Fredenhagen and myself, we investigate how symmetries and anomalies can be described in this framework. In particular, we provide a unitary quantum version of Noether's theorem. In my talk I will give a status report on these results.
Vincenzo Morinelli. Covariant homogeneous nets of standard subspaces of Lie groups
In Algebraic Quantum Field Theory (AQFT), a canonical algebraic construction of the fundamental free field models was provided by Brunetti Guido and Longo in 2002. The Brunetti-Guido-Longo (BGL) construction relies on the identification of spacetime regions called wedges and one-parameter groups of Poincaré symmetries called boosts, the Bisognano-Wichmann property and the CPT-theorem. The last two properties make geometrically meaningful the Tomita-Takesaki theory. In this talk we recall this fundamental structure and explain how the one-particle picture can be generalized. The BGL-construction can start just by considering the Poincaré symmetry group and forgetting about the spacetime. Then it is natural to ask what kind of Lie groups can support a one-particle net and in general a QFT. Given a Z2-graded Lie group we define a local poset of abstract wedge regions. We provide a classification of the simple Lie algebras supporting abstract wedges in relation with some special wedge configurations. This allows us to exhibit an analog of the Haag-Kastler axioms for one-particle nets undergoing the action of such general Lie groups without referring to any specific spacetime. This set of axioms supports a first quantization net obtained by generalizing the BGL-construction. The construction is possible for a large family of Lie groups and provides several new models.
Daniela Cadamuro. Relative entropy of coherent states on general CCR algebras
In QFT the total entropy of a state is generically infinite, so one considers the relative entropy between two states with reference to a subalgebra of the observables, such as the von Neumann algebra associated with a double cone or a spacelike wedge. Such entropy can be computed using Tomita-Takesaki modular theory. In this talk, we study the relative entropy for a subalgebra of a generic CCR algebra between a general (possibly mixed) quasifree state and its coherent excitations, and give a formula for this entropy in terms of single-particle modular data. We also investigate changes of the relative entropy along subalgebras arising from an increasing family of symplectic subspaces, and study lower estimates for the second derivative of the relative entropy along this family, which replace the usual notion of convexity of the entropy. Our main input is a regularity condition for the family of subspaces (“differential modular position”) which generalizes the notion of half-sided modular inclusions. Examples include thermal states for the conformal U(1)-current.
Friday
Sergey Neshveyev. Topological boundaries of C*-tensor categories
Motivated by recent interest in developing a Furstenberg type boundary theory in various noncommutative contexts, I will explain how one can associate a Furstenberg-Hamana boundary to every rigid C*-tensor category with simple unit. For a large class of tensor categories this boundary turns out to coincide with a Poisson type boundary arising from an earlier joint work with Makoto Yamashita. For the representation categories of the q-deformations of compact Lie groups this can be viewed as a categorical (or quantum group theoretic) analogue of the classical computation of the Furstenberg boundaries of complex semisimple Lie groups by Furstenberg and Moore. (Joint work with Erik Habbestad and Lucas Hataishi.)
Makoto Yamashita. Categorical quantization of symmetric spaces and reflection equation
Reflection equation is a powerful guiding principle to quantize Poisson homogenous spaces into actions of quantum groups. I will explain a connection between (modified) Knizhnik-Zamolodchikov equations (due to Leibman, Golubeva-Leksin, Enriquez-Etingof) and coideal subalgebra of the quantized universal enveloping algebra (due to Letzter, Kolb, Balagovic-Kolb) through reflection equations. This is an analogue of the famous Kohno-Drinfeld theorem for the type B braid groups, and the formality principle plays a key role in the proof similar to works of Calaque and Brochier. Based on joint works with Kenny De Commer, Sergey Neshveyv, and Lars Tuset.
Atibur Rahaman. Quantum E(2) groups and contraction: the braided analogues
In the first half of the talk we will focus on the construction a family of q deformations of E(2) group for nonzero complex parameters |q| < 1 as locally compact braided quantum groups over the circle group T viewed as a quasi-triangular quantum group with respect to the unitary R-matrix R(m, n) :=(q/q̄)mn for all m, n ∈ Z. For real 0 < |q| < 1, we show that the deformation coincides with Woronowicz’s Eq (2) groups. The quantum SU(2) and E(2) groups are related via a contraction procedure. We will devote the second half of the talk to describe the braided analogue of the contraction procedure between braided SUq (2) and Eq (2) groups in the spirit of Woronowicz’s quantum analogue of the classic Inönü-Wigner group contraction. Also, as an application, we will show that the contraction procedure applied to Uq (2) groups yields the bosonization of braided Eq (2) groups. This is a joint work with Sutanu Roy.
Sutanu Roy. Homogeneous quantum symmetries of finite quantum spaces over the circle group
Suppose D is a noncommutative and finite dimensional C*-algebra carrying a continuous trace preserving action of the circle group. We show that the (compact) quantum symmetries of (the system) D are captured by braided compact quantum groups G over the circle group. The braiding is governed by a certain R-matrix on the group of integers. In particular, if the circle action is trivial then G coincides with Wang's quantum group of automorphisms of D. Furthermore, the bosonisation of G coincides with the quantum symmetry group of the crossed product C*-algebra D by the circle action.