Speaker: Edoardo D'Angelo (Università di Milano)

Title:  A locally covariant renormalization group in Lorentzian spacetimes

Abstract: Renormalization group flows, based on functional Polchinski or Wetterich equations, are powerful tools that give access to non-perturbative aspects of strongly coupled QFTs and gravity.  I will provide an overview of a new approach, developed to construct a rigorous renormalization group (RG) flow on Lorentzian manifolds. This approach, based on a local and covariant regularization of the Wetterich equation, highlights its state dependence. I give the main ideas of a proof of local existence of solutions for the RG equation, when a suitable Local Potential Approximation is considered. The proof is based on an application of the renown Nash-Moser theorem. I will also present recent applications of the locally covariant RG equation to the non-perturbative renormalizability of quantum gravity.

Speaker: Ian Charlesworth (Cardiff University, School of Mathematics)

Title: Graph products, ε-independence, and atoms

Abstract: In both classical and free probability theory, the central limit distribution can be modeled on a symmetric or free Fock space. The $q$-deformed Gaussians are the corresponding variables on a $q$-deformed Fock space (being the free semicirculars when $q = 0$ and the classical Gaussians when $q=1$), which raises the question of whether they arise from a central limit-type theorem. To find such a situation, Młotkowski introduced $\varepsilon$-independence as an interpolation between free and classical independence, where distributions (or von Neumann algebras) are assigned to the vertices of a graph with adjacency matrix $\varepsilon$, and are placed in a larger algebra in such a way that they are independent when they correspond to adjacent vertices and free otherwise. The corresponding product operation on von Neumann algebras corresponds to the idea of a graph product of groups, studied by Green. In this talk I will be interested in the following question: when do type I summands appear in the graph product of von Neumann algebras? The answer is pleasantly combinatorial, and can be described based on a family of polynomials built using the cliques in the graph (first arising in work of Cartier--Foata in 1969), and the behaviour of type I summands in the input algebras. This is joint work with David Jekel.

Speaker: Vedran Sohinger (University of Warwick)

Title: Gibbs measures of 1D quintic nonlinear Schrödinger equations as limits of many-body quantum Gibbs states

Abstract: Gibbs measures of nonlinear Schrödinger equations (NLS) are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We study the problem of the derivation of Gibbs measures as mean-field limits of Gibbs states in many-body quantum mechanics.

In earlier joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein, we studied this problem for variants of the cubic NLS with defocusing (positive) interactions. The latter models physically correspond to pair interactions of bosons. In these works, the problem was studied in dimensions d=1,2,3.

In this talk, I will explain how one can obtain an analogous result for the 1D quintic NLS, which corresponds to three-body interactions of bosons. In this setting, we consider focusing interactions, due to which we need to add a truncation in the mass and rescaled particle number. Our methods allow us to obtain a microscopic derivation of the time-dependent correlation functions for the 1D quintic NLS. This is joint work with Andrew Rout.

Speaker: Roberto Conti (Sapienza University of Rome)

Title: Automorphisms of the Cuntz algebras: small subgroups of outer reduced Weyl groups

Abstract: Weyl groups for the Cuntz algebras were introduced implicitly by Cuntz at the end of the seventies. However, they remained largely ignored probably because of computational difficulties until about 30 years later, when the breakthrough work by Conti and Szymanski allowed to determine explicitly a huge number of their elements by a sophisticated condition with a clear combinatorial flavour. In recent times, Brenti, Conti and Nenashev pushed the boundaries of the involved combinatorial structures, obtaining for instance the first enumerative results (at the moment, only for cycles).

In this talk I will report on recent work joint with F. Brenti and G. Nenashev (in preparation) where, building on the combinatorial machinery developed over the last few years, we construct certain subgroups of outer automorphisms of O_n. In particular, we are able to describe in detail the 46 distinct finite groups of outer automorphisms of O_4 lying in the outer reduced Weyl group and maximal at level 2, which were first determined by Szymanski and collaborators by clever computer-assisted methods. The notion of bicompatible subgroup of the permutations of a square grid will play a role in the discussion.

Speaker: Christopher Raymond (University of Hamburg)

Title: Inverse hamiltonian reduction in VOA representation theory

Abstract: Recent interest in vertex operator algebra theory has focused on examples whose representation theory describes conformal field theories with logarithmic divergences in their correlation functions (logCFTs). These VOAs admit non-semisimple representations that play a key role in the CFT, and additionally often feature infinitely many simple representations. A large class of examples of such VOAs are W-algebras associated to affine VOAs at fractional admissible level. However, representations for these VOAs are notoriously difficult to construct in a general way. One approach to solving this problem is known as inverse quantum hamiltonian reduction (IQHR). The aim of this talk will be to introduce the ideas behind IQHR in some accessible examples, and then discuss generalisations.

Wojciech Dybalski (University in Poznań)

Title Exact Schwinger functions for a class of bounded interactions in d≥ 2.

Speaker: Ian Koot  (FAU Erlangen-Nürnberg)

Title: Relative Positions in Half-Sided Modular Inclusions

Abstract: Tomita-Takesaki modular theory has become a powerful tool in the analysis of quantum field theories. Although generally the modular objects are difficult to calculate explicitly, in the setting of Half-sided Modular Inlcusions we have more control over them. The representation theory of a single Half-sided Modular Inclusion is closely related to the canonical commutation relations and is therefore well understood, but it is not so clear what is possible when multiple different half-sided modular inclusions arise within the same standard subspace/von Neumann algebra. After introducing Half-sided Modular Inclusions and their relation to so-called Standard Pairs, I will discuss a recent result which relates inclusions of standard subspaces, both included as half-sided modular inclusions in a surrounding standard subspace, to inclusions of associated complex subspaces. This allows one to relate back to the representation theory to construct concrete examples of non-trivial phenomena, which we also discuss.

Speaker: Antti Knowles  (Université de Genève)

Title: the Euclidean field theories and interacting Bose gases

Colloquium CMTP

Abstract: Euclidean field theories have been extensively studied in the mathematical literature since the sixties, motivated by high-energy physics and statistical mechanics. Formally, such a theory is given by a Gibbs measure associated with a Euclidean action functional over a space of distributions. In this talk I explain how some such theories arise as high-density limits of interacting Bose gases at positive temperature. This provides a rigorous derivation of them starting from a realistic microscopic model of statistical mechanics. I focus on field theories with a quartic, local or nonlocal, interaction in dimensions ≤ 3. Owing to the singularity of the Gaussian free field in dimensions higher than one, the interaction is ill-defined and has to be renormalized by infinite mass and energy counterterms. The proof is based on a functional integral representation of the interacting Bose gas. Based on joint work with Cristina Caraci, Jurg Fröhlich, Alessio Ranallo, Benjamin Schlein,  Vedran Sohinger, and Pedro Torres Giesteira.

Speaker:  Simone Murro (University of Genova)

Title: A pathway to noncommutative Gelfand duality

Abstract:  The duality between algebraic structures and geometric spaces is of paramount importance in mathematics and physics, because provides a dictionary to describe manifolds and variaties in a purely algebraic fashion. In his seminal paper, Gelfand showed that a topological space can be functorially reconstructed from its Banach algebra of continuous functions. Conversely, the Gelfand spectrum of the algebra of continuous functions is homeomorphic to the underlying topological space.

The goal of this talk is to constrcut a sufficiently robust notion of spectrum for general rings that allows one to implement a noncommutative analog of Gelfand duality. Our notion of spectrum, although formally reminiscent of the Grothendieck spectrum, is new. Remarkably, an appropriately refined relative version of our spectrum agrees with the Grothendieck spectrum for finitely generated commutative algebras over the complex numbers, among others. 

This is a joint project with Federico Bambozzi and Matteo Capoferri.

Speaker: Fabio Cipriani (Politecnico di Milano)

Title: Existence/uniqueness of ground state and spectral gap of Hamiltonians by logarithmic Sobolev inequalities

Abstract: we discuss the emergence of logarithmic Sobolev inequalities from energy/entropy inequalities and then derive from them the existence and uniqueness of the ground state of Hamiltonians as well the spectral gap. The method is an infinitesimal extension of the one introduced by Len Gross in case the ground state is a probability or a trace and is based on the monotonicity of the relative entropy.

Speaker: Ginestra Bianconi (Queen Mary University of London)

Title: Gravity from entropy

Abstract: Gravity is derived from an entropic action coupling matter fields with geometry. The fundamental idea is to relate the metric of Lorentzian spacetime to a quantum operator, playing the role of an renormalizable effective density matrix and to describe the matter fields topologically, according to a Dirac-Kähler formalism, as the direct sum of a 0-form, a 1-form and a 2-form. While the geometry of spacetime is defined by its metric, the matter fields can be used to define an alternative metric, the metric induced by the matter fields, which geometrically describes the interplay between spacetime and matter. The proposed entropic action is the quantum relative entropy between the metric of spacetime and the metric induced by the matter fields. The modified Einstein equations obtained from this action reduce to the Einstein equations with zero cosmological constant in the regime of low coupling. By introducing the G-field, which acts as a set of Lagrangian multipliers, the proposed entropic action reduces to a dressed Einstein-Hilbert action with an emergent small and positive cosmological constant only dependent on the G-field. The obtained equations of modified gravity remain second order in the metric and in the G-field. A canonical quantization of this field theory could bring new insights into quantum gravity while further research might clarify the role that the G-field could have for dark matter.

Speaker: Roberto Volpato  (University of Padova - INFN sezione di Padova)

Title: Topological defects in vertex operators algebras

Abstract:  Topological defects in quantum field theory have received considerable attention in the last few years as generalizations of the concept of symmetry. In the context of two dimensional conformal field theory, the properties of topological defects have been studied since the 90s, in particular in a series of works by Froehlich, Fuchs, Runkel and Schweigert. In this talk, I will discuss some applications of these ideas from physics to the theory of vertex operator (super-)algebras. In particular, I will describe some recent results about topological defects in the Frenkel-Lepowsky-Meurman Monstrous module, as well as in the Conway module, i.e. the holomorphic vertex operator superalgebra at central charge 12 with no weight 1/2 states. Finally, I will speculate about possible generalizations of the Moonshine conjectures.
This is partially based on ongoing joint work with Roberta Angius, Stefano Giaccari,and Sarah Harrison.

Speaker: Filippo Bracci (University of Rome Tor Vergata)

Title: The Beurling theorem for finite index shifts and the invariant subspace problem

Abstract: The famous Beurling theorem provides a concrete characterization of closed invariant subspaces for the shift on the Hardy space H^2 in the unit disc, stating that every such space is of the form fH^2 , where f is an inner function. This result can also be interpreted in an operator sense by saying that every closed subspace invariant for the shift is the image of H^2  via an isometry. From this perspective, Beurling’s theorem has been extended by Lax, Halmos, and Rovnyak to shifts of any index, proving that a closed subspace is invariant for a shift if and only if it is the image of the space via a quasi-isometry that commutes with the shift (the so-called Beurling-Lax theorem).

In this talk, I will present a generalization of the “concrete” form of Beurling’s theorem for the shift on the direct finite sum of  H^2. I will show that every closed invariant subspace is given, up to multiplication by an inner function, by the intersection of what we call “determinantal spaces”—which, roughly speaking, are the preimages of shift-invariant subspaces of H^2  via a linear operator commuting with the shift and that are constructed through a determinant of certain matrices with entries given by holomorphic bounded functions. The concreteness of such a structure theorem allows us to prove by rather simple algebraic manipulation, as in the classical Beurling theorem, that the only non-trivial maximal closed shift-invariant subspaces are of codimension one. Using the universality of the (backward) shift in the class of operators with defect less than or equal to the index of the shift, this gives a proof of the following result: every bounded linear operator from a Hilbert space into itself whose defect is finite has a non-trivial closed invariant subspace.

The talk is based on a joint work with Eva Gallardo-Gutierrez

Speaker: Jacopo Bassi (IM PAN, Warsaw)

Title: Applications of measurable dynamics to analytic group theory

Abstract: Biexactness and the (AO)-property can be considered as analytic counterparts of hyperbolicity for discrete groups. Motivated by the problem of determining whether they are equivalent, I will discuss an approach to the study of regularity properties of boundary actions/representations based on measurable dynamics. This approach will be used to study SL(3,Z) and to answer a question posed by C. Anantharaman-Delaroche.

Some bibliography: https://arxiv.org/abs/2111.13885,  https://arxiv.org/abs/2305.16277, https://arxiv.org/abs/2410.01447

Speaker: Fabio Cipriani (Politecnico di Milano)

Title: Energies of vector bundles

Abstract: We introduce energies E(V) of vector bundles V on Riemannian manifolds and more generally on Dirichlet spaces, commutative or not. We then derive a relationship between triviality of V and smallness of E(V).

Speaker: Maria Stella Adamo (FAU Erlangen-Nürnberg)

Title: Osterwalder-Schrader axioms for unitary full VOAs

Abstract: The celebrated Osterwalder-Schrader (OS) reconstruction results provide conditions verified by Euclidean n-point correlation functions to produce a Wightman quantum field theory. This talk aims to show a conformal version of the OS axioms, including the linear growth condition, for n-point correlation functions defined for a reasonable class of unitary full Vertex Operator Algebras (VOAs). Such VOAs can be seen as extensions of commuting chiral and anti-chiral VOAs, introduced to describe compact 2D conformal field theories.

Speaker: Silvia Pappalardi (University of Cologne)

Title: Free probability approaches to quantum many-body dynamics

Abstract: Understanding how to characterize quantum chaotic dynamics is a longstanding question. The universality of chaotic many-body dynamics has long been identified by random matrix theory, which led to the well-established framework of the Eigenstate Thermalization Hypothesis. In this talk, I will discuss recent developments that identify Free Probability --  a generalization of probability theory to non-commuting objects — as a unifying mathematical framework to describe correlations of chaotic many-body systems. I will show how the full version of the Eigenstate Thermalization Hypothesis,  which encompasses all the correlations, can be rationalized and simplified using the language of Free Probability. This approach uncovers unexpected connections between quantum chaos and concepts in quantum information theory, such as unitary designs. 

Speaker: Kang Li (FAU Erlangen-Nürnberg)

Title: Non-commutative Dimension Theories

Abstract: A C*-algebra is often considered as non-commutative space, which is justified by the natural duality between the category of unital, commutative C*-algebras and the category of compact, Hausdorff spaces. Via this natural duality, we transfer Lebesgue covering dimension on compact, Hausdorff spaces to nuclear dimension on unital, commutative C*-algebras. The notion of nuclear dimension for C*-algebras was first introduced by Winter and Zacharias, and it has come to play a central role in the structure and classification for simple nuclear C*-algebras. Indeed, after several decades of work, one of the major achievements in C*-algebra theory was completed: the classification via the Elliott invariant for non-elementary simple separable unital C*-algebras with finite nuclear dimension that satisfy the universal coefficient theorem. 

Unfortunately, simple C*-algebras suffer from a phenomenon of dimension reduction: every simple C*-algebra with finite nuclear dimension must have nuclear dimension at most one. In order to overcome this phenomenon, we (together with Liao and Winter) have introduced the notion of diagonal dimension for an inclusion of C*-algebras, where D is a commutative sub-C*-algebra of A so that this new dimension theory generalizes Lebesgue covering dimension of D and nuclear dimension of A simultaneously. In this talk, I will explain its future impact on the classification of simple nuclear C*-algebras and its connection to dynamic asymptotic dimension introduced by Guentner, Willett and Yu. 

Speaker: Karl-Hermann Neeb (FAU Erlangen-Nürnberg)

Title: Local nets on causal flag manifolds

Abstract: We are interested in obtaining local nets in the sense of Haag--Kastler from unitary representations of a connected Lie group G. A natural set of axioms  leads to a causal structure on M. We focus on the case where M = G/P  is a flag manifold of a simple Lie group G, or a covering space thereof.  Then G must be a hermitian Lie group and M a conformal compactification of a euclidean Jordan algebra V. It's simply connected covering is a simple space-time manifold in the sense of Mack--de Riese. 

We show that the unitary representations permitting non-trivial nets are the positive energy representations  (direct integrals of lowest weight representations). These nets have several interesting features. One is that the ``wedge regions'' that link the geometry of M to the modular theory of the algebras involved are given by the intervals (double cones) of W. Bertram's cyclic order on M. Another is that locality properties of the net can be specified in terms of open G-orbits in the space of pairs, which is most interesting for covering spaces because the number of these orbits corresponds to the number of sheets in the covering.

Speaker: Vedran Sohinger (University of Warwick)

Title: Invariant measures as probabilistic tools in the analysis of nonlinear ODEs and PDEs

Abstract: Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In the first part of the talk, we will discuss the connection of Gibbs measures with the Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equilibrium measures of the Liouville equation. In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of nonlinear Hamiltonian PDEs. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari.
In the second part of the talk, we will explain a general principle that allows us to obtain almost sure global solutions for Hamiltonian PDEs provided that one has a stationary probability measure. In this context, stationarity refers to a solution of the associated Liouville equation. This more general notion replaces the invariance from before. The second part of the talk is joint work with Zied Ammari and Shahnaz Farhat.

Speaker: Ko Sanders (FAU Erlangen-Nürnberg)

Title: On distributions of positive type and applications to QFT

Abstract: When quantum fields are represented as operators on a Hilbert space, their two-point distributions naturally give rise to distributions of positive type. A number of basic results on such distributions, especially for translation invariant two-point distributions, have been known for a long time. E.g., the Bochner-Schwartz Theorem fully characterises translation invariant distributions of positive type. In this talk I will present two apparently new results on distributions of positive type, one pertaining to pointwise products and the other to  methods for cutting and pasting. Both results were motivated by physical questions and extend the toolbox of theoretical physics. I will present the results in a general mathematical context before discussing their applications to quantum energy inequalities and to separable states for a free scalar QFT.

Speaker: Stefaan Vaes (KU Leuven) 

Title: Ergodic states on type III_1 factors and ergodic actions

Abstract: I will report on a joint work with Amine Marrakchi. Since the early days of Tomita-Takesaki theory, it is known that a von Neumann algebra that admits a state with trivial centralizer must be a type III_1 factor, but the converse remained open. I will present a solution of this problem, proving that such ergodic states form a dense G_\delta set among all normal states on any III_1 factor with separable predual. Through Connes' Radon-Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which I will discuss in the second half of the talk.

Speaker: Giulio Codogni (University of Rome Tor Vergata)

Title: Vertex algebras and Teichmüller modular forms

Abstract: Vertex algebras are algebraic structures coming from two dimensional conformal field theory. This talk is about their relation with moduli spaces of Riemann surfaces.

I will first review some background material. In particular, I will recall that a vertex algebra is a graded vector space V with additional structures, and these structures force the Hilbert-Poincaré series of V, conveniently normalized, to be a modular form.

I will then associate to any holomorphic vertex algebra a collection of Teichmüller modular forms (= sections of powers of the lambda class on the moduli space of Riemann surfaces), whose expansion near the boundary gives back some information about the correlation functions of the vertex algebra. This is a generalization of the Hilbert-Poincaré series of V, it uses moduli spaces of Riemann surfaces of arbitrarily high genus, and it is sometime called partition function of the vertex algebra. I will also explain some partial results towards the reconstruction of the vertex algebra out of these Teichmüller modular forms.

Using the above mentioned construction, we can use vertex algebras to study problems about the moduli space of Riemann surfaces, such as the Schottky problem, the computation of the slope of the effective cone, and the computation of the dimension of the space of sections of powers of the lambda class. On the other hand, this construction allows us to use the geometry of the moduli space of Riemann surfaces to classify vertex algebras; in particular, I will discuss how conjectures and known results about the slope of the effective cone can be used to study the unicity of the moonshine vertex algebras.

This is a work in progress with Sebastiano Carpi.

Speaker: Roberto Conti (Sapienza Università di Roma)

Title: Positive definite Fell bundle maps

Abstract: C*-algebraic bundles (nowadays simply called Fell bundles) were introduced by Fell at the end of the sixties as yet another tool to deal with the representation theory of locally compact groups. The reason why the related literature is still growing is probably due to the fact that they fit pretty well with various aspects of the theory of (twisted) actions (and coactions) of groups (and groupoids) on C*-algebras. For instance, many familiar constructions like group C*-algebras and crossed products can be viewed as cross sectional C*-algebras of suitable Fell bundles.

In the talk we will introduce the concept of positive definite "multiplier" between Fell bundles and discuss some consequences and applications. Especially, a notion of amenability for Fell bundles naturally appears. Other applications are concerned with the construction of certain functors from the category of positive definite multipliers to the category of completely positive maps between C*-algebras and with the existence of certain C*-correspondences associated to left actions of Fell bundles on right

Hilbert bundles. (Joint work with E. Bedos)

Speaker: Valeriano Aiello (Sapienza Università di Roma)

Title: Colorazioni, sottogruppi del gruppo di Thompson e rappresentazioni.

Abstract: Circa dieci anni fa, V. Jones introduceva diverse rappresentazioni unitarie dei gruppi di Thompson e vari sottogruppi interessanti sono emersi come stabilizzatori di vettori in queste rappresentazioni. In questo seminario presenterò il lavoro svolto su questo argomento.

Speaker: Rostislav I. Grigorchuk (University of Texas A&M)

Title: Self-replicating, liftable and scale groups

Abstract:  Scale groups are closed subgroups of the group of isometries of a regular tree that fix an end of the tree and are vertex-transitive. They play an important role in the study of locally compact totally disconnected groups as was recently observed by P-E.Caprace and G.Willis. In the 80’s they were studied by A. Figa-Talamanca and C. Nebbia in the context of abstract harmonic analysis and amenability. It is a miracle that they are closely related to self-replicating (or fractal) groups, a special subclass of self-similar groups.
In my talk I will discuss two ways of building scale groups. One is based on the use of scale-invariant groups studied by V. Nekrashevych and G. Pete, and a second is based on the use of liftable groups - a special class of self-replicating groups. The examples based on both approaches will be demonstrated including the lamplighter group and a torsion 2-group G of intermediate growth between polynomial and exponenttial.  It will be shown that its finitely presented non elementary amenable relative G' gives the example of a scale group acting 2-transitively on a punctured boundary.
Additionally, the group of isometries of the ring of p-adic integers and the group of dilations of the field of p-adics will be mentioned in relation with the discussed topics.

Speaker: Yasuyuki Kawahigashi (The University of Tokio)

Title: Quantum 6j-symbols and braiding

Abstract: Alpha-induction is a tensor functor producing a new fusion category from a modular tensor category and a Q-system.  This can be formulated in terms of quantum 6j-symbols and braiding and gives alpha-induced bi-unitary connections. Last year, we showed that locality of the Q-system implies flatness of the alpha-induced connections.  We now prove that the converse also holds.

Speaker: Boris Bolvig Kjær (University of Copenhagen)

Title: The double semion model in infinite volume

Abstract: According to physics literature, topologically ordered gapped ground states of 2-dimensional spin systems can be described by a topological quantum field theory. Many examples arise from microscopic models with local commuting projector Hamiltonians, namely Levin-Wen models. In this talk, I will describe the general framework for classifying infinite volume gapped ground states (by Naaijkens, Ogata, et.al.) in the simple context of abelian Levin-Wen models. This framework is heavily inspired by the DHR analysis in relativistic quantum field theory. It applies to the doubled semion model whose anyon theory is a braided fusion category equivalent to the representation category of the twisted Drinfeld double of Z_2. Based on joint work with Alex Bols and Alvin Moon, https://arxiv.org/abs/2306.13762.

Speaker: Ricardo Correa da Silva (FAU Erlangen-Nürnberg)

Title: Crossing Symmetry and Endomorphisms of Standard Subspaces

Abstract: 

This seminar aims to introduce the "crossing map", a transformation of operators in Hilbert spaces defined in terms of modular theory and inspired by "crossing symmetry" from elementary particle physics, and discuss the strong connection between crossing-symmetric twists and endomorphisms of standard subspaces. Crossing symmetry has many interesting connections, including T-twisted Araki-Woods algebras, q-Systems, and algebraic Fourier transforms.

Speaker: Christopher J Fewster (University of York)

Title: Exact measurement schemes for local observables and the preparation of physical local product states

Abstract: 

For a long time, quantum field theory (QFT) lacked a clear and consistent measurement framework, a gap that was described as "a major scandal in the foundations of quantum physics" [1]. I will review the recent framework put forward by Verch and myself [2], which is consistent with relativity in flat and curved spacetimes and has resolved the long-standing problem of "impossible measurements" put forward by Sorkin [3]. The central idea in this framework is that the "system" QFT of interest is measured by coupling it to a "probe" QFT, in which the system, probe, and their coupled variant, all obey axioms of AQFT in curved spacetime. It has been shown that every local observable of the free scalar field has an asymptotic measurement scheme, i.e., can be measured to arbitrary accuracy by a sequence of probes and couplings [4]. I will describe new results that (a) show that there are exact measurement schemes for all local observables in a class of free theories, (b) provide a protocol for the construction of Hadamard local product states in curved spacetime. The latter is complementary to a recent existence result of Sanders [5]. 

[1] Earman, J., and Valente, G. Relativistic Causality in Algebraic Quantum Field Theory, International Studies in the Philosophy of Science, 28:1, 1-48, (2014)  

[2] Fewster, C.J., Verch, R. Quantum Fields and Local Measurements. Commun. Math. Phys. 378, 851–889 (2020).  

[3] Bostelmann, H.,  Fewster, C.J., and  Ruep, M.H. Impossible measurements require impossible apparatus Phys. Rev. D 103, 025017 (2021)

[4] Fewster, C.J., Jubb, I. & Ruep, M.H. Asymptotic Measurement Schemes for Every Observable of a Quantum Field Theory. Ann. Henri Poincaré 24, 1137–1184 (2023).  

[5] Sanders, K. On separable states in relativistic quantum field theory, J. Phys. A: Math. Theor. 56 505201 (2023)

Speaker: Yuto Moriwaki (Riken (Wako, Japan))

Title: Operator product expansion in two-dimension conformal field theory

Abstract: Conformal field theory can be defined using the associativity and the commutativity of the product of quantum fields (operator product expansion). An important difference between conformal field theory and classical commutative associative algebra is "the divergence" arising from the product of quantum fields, a difficulty that appears in quantum field theory in general. In this talk we will explain that in the two-dimensional case this algebra can be controlled by "the representation theory" of a vertex operator algebra and that the convergence of quantum fields is described by the operad structure of the configuration space.

Speaker: Timothy Rainone, (Occidental College)

Title:  The Matricial Field property in cross-sectional C*-algebras

Abstract: Blackadar and Kirchberg introduced the notion of a Matricial Field (MF) C*-algebra; an algebra is MF if it can be embedded into a corona of matrix algebras. We will discuss this property in crossed-products arising from C*-dynamical systems and more generally in cross-sectional C*-algebras constructed from Fell bundles. The C*-analogue of Connes' embedding conjecture is the Blackadar-KIrchberg Question (BKQ) which asks whether every stably-finite algebra admits the MF property. Using K-theoretic techniques we can answer this question in the affirmative for certain crossed products of classificable algebras by free groups. 

Speaker: Claudio Dappiaggi (University of Pavia)

Title: Stochastic Partial Differential Equations and Renormalization à la Epstein-Glaser

Abstract:  We present a novel framework for the study of a large class of nonlinear stochastic partial differential equations, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use specific techniques proper of microlocal analysis.

These allow us to deal with renormalization using an Epstein-Glaser perspective, hence without resorting to any specific regularization scheme.  As a concrete example we shall use this method to discuss the stochastic \Phi^3_d model and we shall comment on its applicability to the stochastic nonlinear Schrödinger equation.

Speaker: Mizuki Oikawa (The University of Tokyo)

Title: New center construction and α-induction for equivariantly braided tensor categories

Abstract: α-induction is known to be a construction of a modular invariant full Q-system from a chiral Q-system of a conformal net. In this talk, I would like to introduce the equivariant version of this procedure. Indeed, we need a new construction of tensor categories, the neutral double construction, introduced by the speaker. Moreover, I would like to explain the relationship between the neutral double construction and an equivariant version of the center construction, which is also introduced by the speaker.

Speaker: Roberto Conti (Sapienza University of Rome)

Title: Heat properties for groups

Abstract:

Somewhat motivated by the original approach of J.-B. Fourier to solve the heat equation on a bounded domain, we formulate some new properties of countable discrete groups involving certain completely positive multipliers of the reduced group C*-algebra and norm-convergence of Fourier series. The stronger "heat property" implies the Haagerup property, while the "weak heat property" is satisfied by a much larger class of groups. Examples will be provided to illustrate the various aspects. In perspective,  a challenging goal would be to obtain yet another characterization of groups with Kazhdan's property (T). (Based on joint work with E. Bédos.)

Speaker: Fabio Cipriani (Politecnico di Milano)

Title: Spectral triples, Dirichlet spaces, and discrete groups.

Abstract:
We study natural conditions on extended spectral triples $(A,h,D)$ by which the quantum differentials $da$ of elements $a \in A$, belong to the ideal generated by the line element $ds = D^{-1}$. We also study upper and lower bounds on the singular values of the $da$'s to form a conformally invariant energy functional. We apply the general framework to study natural spectral triples of Dirichlet spaces and in particular those on duals of discrete groups arising on negative definite functions.

Speaker:Karl-Henning Rehren (University of Göttingen) 

Title: LV formalism in perturbative AQFT.

Abstract:

pAQFT defines nets of local algebras by a limiting construction with relative S-matrices. The latter can be constructed perturbatively from an interaction Lagrangian.

In many instances, the construction can be improved by adding a total derivative to the interaction Lagrangian (which would have no effect in classical field theory). The LV formalism controls whether and how this modification affects the (relative) S-matrices and provides a tool to identify the local observables of the model.

Speaker: Luca Giorgetti (Università di Roma Tor Vergata)

Title: A planar algebraic description of conditional expectations

Abstract: Jones’ notion of index was introduced for II_1 subfactors and soon after generalized to arbitrary inclusions of von Neumann algebras, not necessarily tracial nor with trivial centers, in several ways. A unital inclusion of von Neumann algebras N < M is said to have finite Jones index if it admits at least one normal faithful conditional expectation

of M onto N with finite index. In the talk, I will report on a representation formula for such finite index expectations and their dual expectations (as defined by Haagerup and Kosaki) by means of the solutions of the conjugate equations for the inclusion morphism of N into M and its conjugate morphism. In particular, this provides a 2-categorical formulation of the theory of index in this general setting. Another consequence is that an arbitrary inclusion of von Neumann algebras with a prescribed finite index expectation can be described by a Q-system. These results are both originally due to Longo in the subfactor case.

Based on https://arxiv.org/abs/2111.04488

Supported by EU MSCA-IF beyondRCFT grant n. 795151 and by MIUR

Excellence Department Project awarded to the Department of Mathematics of the University of Rome Tor Vergata, CUP E83C18000100006

This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006.

Speaker: Jean-Luc Sauvageot, (Institut de Mathématiques de Jussieu)

Title: Misurabilità, densità spettrali e tracce residuali in geometria non commutativa

Abstract:We introduce, in the dual Macaev ideal of compact operators of a Hilbert space, the spectral weight rho(L) of a positive, self-adjoint operator L having discrete spectrum away from zero. We provide criteria for its measurability and unitarity of its Dixmier traces ( rho(L) is then called a spectral density) in terms of the growth of the spectral multiplicities of L and in terms of the asymptotic continuity of the eigenvalue counting function NL. Existence of meromorphic extensions and residues of the zeta-function zeta L of a spectral density are provided, under summability conditions on the spectral multiplicities. The hypertrace property of the states Omega L(·) = Tr omega(· rho(L)) on the norm closure of the Lipschitz algebra AL follows if the relative multiplicities of L vanish faster then its spectral gaps or if, at least, NL is asymptotically regular.

Speaker: Christian Gaß (University of Göttingen)

Title: Renormalization in string-localized field theories: a microlocal analysis

Abstract: String-localized quantum field theory (SL QFT) provides an alternative to gauge theoretic approaches to QFT. In the last one-and-a-half decades, many conceptual benefits of SL QFT have been discovered. However, a renormalization recipe for loop graphs with internal SL fields was not at hand until now.

In this talk, I present a proof that the problem of renormalization remains a pure short distance problem in SL QFT. This happens in spite of the delocalization of SL fields and the analytic complexity of their propagators – provided that one takes care in how to set up perturbation theory in SL QFT. As a result, the improved short-distance behavior of SL fields remains a meaningful notion, which indicates that there can exist renormalizable models in SL QFT whose point-localized counterparts are non-renormalizable.

The talk is based on arXiv:2107.12834.

Speaker: Aleksei Bykov (Univ. Rome Tor Vergata)

Title: Hamiltonian approach to QFT on quantum spacetime and its adiabatic limits

Abstract: It is a common place that in quantum gravity at high energies the smooth manifold of spacetime must be replaced by its quantized version, and non-commutative quantum spacetimes are among the most promising candidates for this replacement. However, the construction of a quantum field theory on such a spacetime seems to be a non-trivial and ambiguous task. One of the approaches to this problem was proposed and developed by Bahns, Doplicher, Fredenhagen and Piacitelli for the Doplicher-Fredenhagen-Roberts quantum spacetime. I will start by briefly reviewing this approach in a somewhat more general perspective and presenting its main advantages and disadvantages. I will expose my results on the existence of the weak adiabatic limit and my ongoing work on the strong adiabatic limit. If time permits, I will also discuss the

ambiguities that arise in the process of finite renormalization required to fulfill the necessary conditions for the existence of the strong adiabatic limit.

Speaker: Nicola Pinamonti (University of Genova)

Title: Sine-Gordon fields with non vanishing mass on Minkowski spacetime and equilibrium states

Abstract: During this talk we shall discuss the construction of the massive Sine-Gordon field in the ultraviolet finite regime when the background is a two-dimensional Minkowski spacetime. The correlation functions of the model in the adiabatic limit will be obtained combining recently developed methods of perturbative algebraic quantum field theory with techniques developed in the realm of constructive quantum field theory over Euclidean spacetimes. More precisely, perturbation theory is used to represent interacting fields as power series in the coupling constant over the free theory. Adapting techniques like conditioning and inverse conditioning to spacetimes with Lorentzian

signature, we shall see that these power series converge if the interaction Lagrangian has generic compact support. Finally, adapting the cluster expansion technique to the Lorentzian case, we shall see that the adiabatic limit of the correlation functions of the interacting equilibrium state at finite temperature is finite.
The talk is based on a joint work with D. Bahns and K. Rejzner [arxiv.org:2103.09328]

Speaker: Fabio Cipriani (Politecnico Milano)

Title: Densities and their measurability in NCG

Abstract: The aim of the talk is to provide conditions ensuring Connes' measurability of the canonical, $(1,\infty)$-summable, spectral weight $\rho(D)$, we associate to any self-adjoint operator $D$ with discrete spectrum and of its associated state $\phi_D(T):={\rm Tr\,}_\omega (T\rho(D))$. The framework allows to discuss measurability for  discrete groups with sub-exponential growth and $\theta$-summable spectral triples. Examples also include 1. the volume density of Euclidean domains of infinite volume 2. densities on the C$^*$-algebra of pseudo-differential operators of 0-order on a compact Riemannian manifold 3. densities on Toeplitz extensions.

Speaker: Gerardo Morsella (University of Rome Tor Vergata)

Title: UV and IR finiteness on quantum spacetime through perturbative algebraic QFT

Abstract: Local QFT on Doplicher-Fredenhagen-Roberts quantum spacetime is equivalent to a non-local QFT on ordinary (commutative) spacetime, which has quite generally a better UV behavior in perturbation theory, but for which the control of the adiabatic limit is problematic. I will show that the adaptation of the methods of perturbative QFT, originally developed for local QFT on ordinary spacetimes, to QFT on quantum spacetime yields a UV-finite theory in which enjoys some remnants of causality. This is sufficient to prove the existence of the adiabatic limit of vacuum expectation values of interacting observables.

Speaker: Alexander Stottmeister (University of Hannover)

Title: Operator-algebraic renormalization, fermions and conformal field theory

Abstract: We will discuss the application of the recently introduced framework of operator-algebraic renormalization to fermionic lattice systems. Focussing on the 1+1-dimensional situation, we will illustrate how the scaling limit of the massless free fermion can be obtained. Moreover, we will show how conformal symmetry can be approximated in terms of a proposal by Koo & Saleur.

Speaker: Yoh Tanimoto (University of Rome Tor Vergata)

Title: Towards integral perturbation of two-dimensional CFT

Abstract: We present some ideas to construct new Haag-Kastler nets starting with a two-dimensional conformal field theory. This is a natural variation of the Barata-Jaekel-Mund construction. It is suggested in physics literature that a similar constructions should lead to integrable models, and we will discuss possible extensions of this programme.