One of the most important callings in present-day mathematics is to provide language to describe and understand the concept of quantum fields.
In functional analysis, this has been tightly coupled with the development of the theory of operator algebras, which originally arose from mathematical foundation of quantum mechanics. Eventually, it led to the notion of quantum symmetry, such as subfactors and algebraic quantum field theories, while also motivating noncommutative geometry.
In more geometric and algebraic fields, this led to the deformation quantization of Poisson manifolds and related structures, which are motivated by the transfer from Hamiltonian mechanics to quantum mechanics. Far beyond this original motivation, it became a guiding principle for development of deformation theory of algebraic structures, combined with theory of operads from topology and the formality principle.
The interaction of viewpoints and paradigms from these fields culminated in a number of fruitful new ideas in the last 40 years, such as the development of quantum groups and tensor categorical theory of quantum fields. To accelerate communication among experts and raise researchers of the next generation in these fields, we organize a summer school at The International Center of Interdisciplinary Science Education at Quy Nhon, with distinguished international researchers as speakers.
Title: Triangles, area and associativity
Abstract: Following a suggestion made, in the nineties, by Alan Weinstein in the context of symmetric spaces, I spent quite a while thinking about deriving associativity, say of a star product, from purely geometric considerations. In this talk, I will present some of my thoughts about that, and, if time permits, show how this leads to locally compact quantum groups.
Title: Topological phases of gapped fermionic ground states
Abstract: In these talks, I will give an introduction to the topological phases/indices associated to low-energy (non-relativistic) fermionic systems. In a more mathematical language, we study ground states of the CAR algebra that satisfy a spectral gap condition and possible group symmetry. After giving some general motivation, I will focus on two special cases: one-dimensional (many-body) lattice fermions and quasifree ground states via the Hartree-Fock-Bogoliubov mean field theory. For the former, techniques from von Neumann algebra and algebraic quantum field theory turn out to be useful. For the latter, we find connections to index theory and can make new extensions via coarse geometry.
Title: Diagram Algebras and their Applications
Abstract: A “diagram algebra” refers to certain classes of algebras which have rich connections to representation theory, statistical mechanics, low dimensional topology, combinatorics, quantization and more. Notable examples include the group algebra of the symmetric group, the Brauer algebra, the partition algebras, the Temperley-Lieb algebra. This talk will be an overview of algebraic structures of this type with illustrations of their origins and applications to other areas. Special emphasis will be on the Temperley-Lieb algebra which serves as a model for the general theory. In particular, a recent orthogonal construction of its representations which has striking combinatorial properties will be presented.
Title: Haploid algebras, Q-systems, and the Schellekens list
Abstract: Q-systems / algebra objects in the tensor category of representations / unitary modules of a given chiral CFT provide an effective way of studying its extensions (those with finite Jones index). More generally, one can talk of Q-systems (special C*-Frobenius algebras), Frobenius algebras, or just algebras in an abstract unitary tensor category. We provide two unitarizability results for such algebras and their isomorphism classes. Namely, we show that a haploid (sometimes also called connected or irreducible) algebra in a unitary tensor category is equivalent to a Q-system if and only if it is rigid (i.e. Frobenius). The haploid condition cannot be dropped from the statement. We also show that isomorphic algebras in the previously mentioned correspondence give rise to unitarily isomorphic Q-systems. Lastly, we present some of our motivating applications to the theory of unitary VOA extensions, and a tensor categorical proof that all the holomorphic VOAs with central charge c=24 corresponding to entries 1-70 in the list of Schellekens are unitary VOAs. Joint work with S. Carpi, T. Gaudio, R. Hillier, https://arxiv.org/abs/2211.12790
Title: Operator Algebras, Tensor Categories and Quantum Field Theory
Abstract: I present various roles of tensor categories in the operator algebraic approach to quantum field theory. Emphasis is put on applications of subfactor theory to 2-dimentional conformal field theory and their relations to condensed matter physics.
Title: An index theorem of lattice Wilson-Dirac operator via higher index theory
Abstract: The lattice Wilson-Dirac operator is a discrete approximation of the twisted Dirac operator on the torus, which is known to remember the index of the continuum Dirac operator. It has been considered in lattice gauge theory as a description of the chiral anomaly. In this talk we introduce a new proof of this theorem based on higher index theory, particularly the monodromy correspondence of almost flat bundles and group quasi-representations.
Title: Modular operator for null plane algebras in free fields
Abstract: We consider the algebras generated by observables in quantum field theory localized in regions in the null plane, in particular null cut regions. For a scalar free field theory, we will show that the one-particle structure can be decomposed into a continuous direct integral of lightlike fibres and the modular operator decomposes accordingly. For this model we also compute the relative entropy of null cut algebras with respect to the vacuum and some coherent states and discuss Quantum Null Energy Conditions.
Title: Noncommutative geometry and semiclassical analysis.
Abstract: Semiclassical analysis and noncommutative geometry are distinct fields within the wider area of quantum theory. Bridges between them have been emerging recently. This lays down on operator ideal techniques that are used in both fields. In this talk we shall present semiclassical Weyl’s laws for Schrödinger operators on noncommutative manifolds (i.e., spectral triples). This shows that well known semiclassical Weyl’s laws in the commutative setting ultimately holds in a purely noncommutative setting. This extends and simplifies previous work of McDonald-Sukochev-Zanin. In particular, this allows us to get semiclassical Weyl’s laws on noncommutative tori of any dimension n ≥ 2, which were only accessible in dimension n ≥ 3 by the MSZ approach. There are numerous other examples as well. The approach relies on spectral asymptotics for some weak Schatten class operators. As a further application of these asymptotics we obtain far reaching extensions of Connes’ integration formulas for noncommutative manifolds. (For Riemannian closed manifolds, Connes’ integration shows that Connes’ NC integral recaptures the Riemannian measure.)
Title: The reasonable effectiveness of mathematical deformation theory in physics.
Abstract: New fundamental physical theories can, so far a posteriori, be seen as emerging from existing ones via some kind of deformation. That is the basis for Flato’s “deformation philosophy", of which the main paradigms are the physics revolutions from the beginning of the twentieth century, quantum mechanics (via deformation quantization) and special relativity. On the basis of these facts we explain how symmetries of hadrons (strongly interacting elementary particles) could “emerge" by deforming in some sense (including quantization) the Anti de Sitter symmetry (AdS), itself a deformation of the Poincaré group of special relativity. The ultimate goal is to base on fundamental principles the dynamics of strong interactions, which originated over half a century ago from empirically guessed “internal" symmetries. We start with a rapid presentation of the physical (hadrons) and mathematical (deformation theory) contexts, including a possible explanation of photons as composites of AdS singletons and of leptons as similar composites. Then we present a “model generating" framework in which AdS would be deformed and quantized (possibly at root of unity and/or in manner not yet mathematically developed with noncommutative “parameters"). That would give (using deformations) a space-time origin to the “internal" symmetries of elementary particles, on which their dynamics were based, and either question, or give a conceptually solid base to, the Standard Model, in line with Einstein's quotation: “The important thing is not to stop questioning. Curiosity has its own reason for existing."
Title: Operator algebraic renormalization: Quantum scaling limits, conformal symmetries, and quantum simulation
Abstract: Quantum field theory (QFT) describes quantum systems with infinitely, even continuously, many degrees of freedom, modeling subtle and complex phenomena in statistical and high-energy physics. Improving the understanding of QFT remains a major challenge — in particular, non-perturbative methods are required for insights into the structure of strongly interacting QFT, enabling, for example, the ab-initio computation of hadron masses.
The renormalization group (RG) in combination with lattice approximations is such a method, effectively truncating systems to finitely many degrees of freedom relevant at the observational scale. The most general RG is formulated in (quantum) statistical mechanics and connects to QFT through scaling limits of critical systems. This general RG is a well-developed and important tool in imaginary-time QFT. It is far less developed in real-time QFT, which is entering a new era of exploration due to tremendous progress in quantum computing.
In this lecture, I present an RG formulation, coined operator-algebraic renormalization, aiming to close this gap by providing access to real-time QFTs via quantum scaling limits. Specifically, I will discuss the recovery of conformal symmetries and applications to quantum simulation of bosonic and fermionic QFTs within this framework.
Title: Tensor categories of Lie type
Abstract: Representation theory of simple Lie groups has interesting deformations in the framework of tensor categories. What kind of categories can have the fusion rules of type A theory (corresponding to finite-dimensional representations of special linear groups) is well understood by the work of Kazhdan and Wenzl (1993). We look at the remaining classical series, namely the fusion rules of type BCD theories (corresponding to orthogonal and symplectic groups), and give a similar classification. Based on ongoing joint work with P. Grossman and S. Neshveyev.
Daniel Sternheimer, Yasuyuki Kawahigashi, Giuseppe Dito, Makoto Yamashita