Emmy-Noether seminar – Elementary Particle Physics

Organizer: D. Cadamuro

The seminar covers topics related to the research of the Emmy Noether group at the Institute of Theoretical Physics at Leipzig. It will be held as a webinar until further notice.

Summer term 2024

Date	Speaker	Title (see below for abstract)
18.04	Guillem Perez-Nadal (University of Buenos Aires)	Entanglement of quantum fields on the cylinder

Abstracts:

(G. Perez-Nadal) Entanglement is an especially interesting quantum phenomenon, because it has no classical counterpart. In quantum field theory, however, very little is known about entanglement properties of typical states, even for free field theories. I will talk about some progress along these lines. More specifically, I will report on recent results for modular Hamiltonians and related objects for free fields on the cylinder (both fermionic and bosonic), in various global states (both ground and thermal states). These objects can all be expressed in terms of the resolvent of a certain operator. I will comment on the methods we use to compute that resolvent.

Winter term 2023/24

Date	Speaker	Title (see below for abstract)
16.11 (10:00)	João Penedones (EPFL)	Bootstrapping the a-anomaly in 4d QFTs
23.11	Paolo Meda (U. Trento)	Trace anomaly and evaporation of dynamical black holes
30.11	Horacio Casini (Centro Atómico Bariloche)	ABJ anomaly as a U(1) symmetry that mixes sectors and Noether’s theorem
25.01	Suneeta Vardarajan (IISER, Pune)	Generalized entropy in higher curvature gravity

Abstracts:

(J. Penedones) The space of 4d quantum field theories (QFTs) can be thought of as the Renormalization Group flows between Conformal Field Theories (CFTs). The a-theorem tells us that CFTs can be ordered by their a-anomaly value. I will explain how one can use the (numerical) S-matrix bootstrap methods to search for the minimal allowed value of the a-anomaly.

(P. Meda) We study the evaporation of a four-dimensional spherically symmetric black hole formed in a gravitational collapse. We analyze the back-reaction of a massless quantum scalar field conformally coupled to the scalar curvature by means of the semiclassical Einstein equations. We show that the evaporation is linked to an ingoing negative energy flux at the dynamical horizon and that this flux is induced by the quantum matter trace anomaly outside the black hole horizon whenever a suitable averaged energy condition is satisfied. For illustrative purposes, we evaluate the negative ingoing flux and the corresponding rate of evaporation in the case of a null radiating star described by the Vaidya spacetime. Based on Class. Quantum Grav. 38 (2021) 195022 [arXiv:2103.02057].

(H. Casini) We study the action of internal symmetries on the sectors that violate Haag duality (HDV sectors) for regions of non-trivial topology in quantum field theory. It is shown that if the HDV sectors are charged under a continuous symmetry, there cannot be a Noether current for the symmetry, though the local implementations (twist operators) still exist. This occurs in all known examples of a continuous symmetry without Noether current, and in particular, it gives a new understanding of Weinberg-Witten theorem. The simplest forms of non-trivial action of a continuous symmetry on the HDV classes can be classified into the case of non compact sectors, associated to free models, and an “anomalous” one, that we show is represented by models with ABJ anomaly. This gives a symmetry based perspective on the anomaly, providing a simple unified account of its main features: anomaly quantization, anomaly matching, non existence of Noether current, and applicability of Goldstone theorem.

(S. Vardarajan) In a recent development, the generalized entropy of the maximally extended Schwarzschild black hole at the bifurcation surface has been shown to be equal to the entropy of the von Neumann algebra of observables in the black hole exterior in Einstein gravity. A version of a generalized second law (GSL) has also been derived from a monotonicity property of the entropy of the algebra. We summarize these developments and generalize these results to a static black hole in an arbitrary diffeomorphism invariant theory of gravity. We also argue that a version of the GSL follows in such a theory.

Summer term 2023

Date	Speaker	Title (see below for abstract)
27.07	Wojciech Dybalski (Adam Mickiewicz University in Poznań)	Can infravacuum states be lightcone normal?

Abstracts:

(W. Dybalski) Infravacuum states were introduced by Kraus, Polley and Reents (KPR) to cure some aspects of the infrared problem. More precisely, the infravacuum property of a state means that it stabilizes the Dyson scattering matrix for the second quantized radiation interacting with an external current describing a charged particle in motion. One can ask, however, if replacing the vacuum by an infravacuum leads to an equivalent physical theory, at least in the future lightcone. It turns out that this property of lightcone normality breaks down for the original KPR states. We also considered a modification procedure for states, introduced by Doplicher, Figliolini and Guido (DFG). It enforces lightcone normality by suitably transporting the constituting modes backwards in time. We argue that the DFG-modified KPR-states become lightcone normal only after their infravacuum property is lost. We conjecture that the infravacuum property and lightcone normality are in conflict for a much larger class of states than discussed above. (Joint work in progress with Bartosz Biadasiewicz).

Winter term 2022/23

Date	Speaker	Title (see below for abstract)
17.11	Gian Michele Graf (ETH Zürich)	Infrared scattering states in the massless Nelson model
15.12	Wojciech Dybalski (Adam Mickiewicz University in Poznań)	The quantum stress-energy tensor and particle interpretation
21.12	Erik Tonni (SISSA, Trieste)	On entanglement Hamiltonians for the massless Dirac field
12.01	Rubens Longhi (Universität Potsdam)	On Maxwell’s Equations on Globally Hyperbolic Spacetimes with Timelike Boundary
23.02	Horacio Casini (Centro Atómico Bariloche)	(postponed)
09.03	João Penedones (EPFL)	Bootstrapping the a-anomaly in 4d QFTs (postponed)

Abstracts:

(G. Graf) The Nelson model describes a particle coupled to a (bosonic) field. In the case where the field is massless, the model is a simple caricature of QED, as both theories share some traits in the infrared, and in particular the difficulty of defining scattering states. A possible solution, to be discussed, is that the particle emits an ever growing number of ever softer bosons. A precise and asymptotically (in time) exact description of the process will be given, motivated, and compared to earlier work by others.

(W. Dybalski) The existence of the stress-energy tensor has long been postulated as a condition characterizing QFT with meaningful particle interpretation within the Haag-Kastler setting. In this talk I will outline some old and new results and ideas pointing in this direction. On the rigorous side, I will discuss the existence of particles in the sense of the theory of particle weights. A more heuristic part of the discussion will address the problem of conventional asymptotic completeness.

(E. Tonni) The reduced density matrix of a spatial subsystem can be written as the exponential of the entanglement (modular) Hamiltonian, hence this operator contains a lot of information about the entanglement of the corresponding spatial bipartition. For some particular models, states and bipartitions, this operator is local, but in general it is expected to be non-local. We study this feature for the massless Dirac field in one spatial dimension, either on the line or on the half-line. For one and two disjoint intervals on the line, we discuss the derivation of the local and bi-local terms of the entanglement Hamiltonian through the continuum limit of lattice results. On the half-line, we consider the bipartition given by an interval in generic position. Imposing the most general boundary conditions ensuring the global energy conservation leads to two inequivalent phases where either the vector or the axial symmetry is preserved. In these two phases, we find analytic expressions for the modular Hamiltonians of an interval on the half-line when the system is in its ground state, for the modular flows of the Dirac field and for the modular correlators. The continuum limit of lattice results is also discussed. The method can be employed to obtain analytic expressions for the modular Hamiltonians, the modular flows and the modular correlators when the bipartition is given by two disjoint equal intervals at the same distance from a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission.

(R. Longhi) We studied Maxwell’s equations $\delta dA=0$ on $k$-forms $A\in\Omega^k(M)$ on globally hyperbolic spacetimes $M$ with timelike boundary. In the boundaryless case every solution to $\delta dA=0$ identifies via a gauge transformation $A\to A+d\chi$, $\chi\in \Omega^{k-1}(M)$ an equivalence class of solutions among which there exists one satisfying $\Box A=0$ and $\delta A=0$ – i.e. the d’Alembert equation in the Lorenz gauge. Inspired by this, we identify two boundary conditions for $\Box$ (named $\Box$-tangential and $\Box$-normal) and two for $\delta d$ ($\delta d$-tangential and $\delta d$-normal) for which, provided advanced and retarded Green operators for $\Box$ exist, a similar feature appears. It turns out that every solution to $\delta dA=0$ with $\delta d$-tangential (resp. $\delta d$-normal) conditions is gauge equivalent to a solution to $\Box A=0$ with $\Box$-tangential (resp. $\Box$-normal) conditions satisfying the Lorenz gauge. Moreover, in view of possible applications in the framework of Algebraic Quantum Field Theory (AQFT), we construct the associated $*$-algebra of observables.

(J. Penedones) The space of 4d quantum field theories (QFTs) can be thought of as the Renormalization Group flows between Conformal Field Theories (CFTs).
The a-theorem tells us that CFTs can be ordered by their a-anomaly value. I will explain how one can use the (numerical) S-matrix bootstrap methods to search for the minimal allowed value of the a-anomaly.

Summer term 2022

Date	Speaker	Title (see below for abstract)
07.04	Angela Foerster (Federal University of Rio Grande do Sul)	Integrability in Ultracold Physics
30.06	Tobias Osborne (Universität Hannover)	Quantum Yang-Mills theory: an overview of a programme
28.07	Vieri Mastropietro (Università di Milano)	Effective Quantum Field theory and rigorous Renormalization Group

Abstracts:

(A. Foerster) Integrable systems can be found in different areas of Physics, such as statistical mechanics, quantum field theory, condensed matter, string theory, and more recently in cold atoms [1]. Here we provide an overview of the basic construction and the impact of quantum integrable models in ultracold atoms. We also discuss a general construction of integrable models for boson tunneling in multi-well systems. As applications, we show how to engineer an atomtronic switching device by exploring the triple well model [2], and we also demonstrate how the 4-well system can be controlled to generate and encode a phase into a NOON state [3]. We then discuss the physical feasibility of these systems through the use of ultracold dipolar atoms in BECs (3-wells) or optical superlattices (4-wells).
References:
[1] Yang-Baxter integrable models in experiments: from condensed matter to ultracold atoms, M. Batchelor, and A. Foerster, Journal of Physics A: Mathematical and Theoretical 49, 173001 (2016);
[2] Control of tunneling in an atomtronic switching device, K. Wittmann, L. Ymai, A. Tonel, J. Links and A. Foerster, Communications Physics (Nature) 1, 91 (2018);
[3] Protocol designs for NOON states, DS Grün, K Wittmann W, LH Ymai, J Links, A Foerster, Communications Physics (Nature) 5, 36 (2022).

(T. Osborne) I will present an overview of a programme to understand the low-energy physics of quantum Yang-Mills theory from a quantum-information perspective. The setting is that of the hamiltonian formulation of pure Yang-Mills theory in the temporal gauge on the lattice. Firstly, inspired by recent constructions for Z/2Z lattice gauge theory, in particular, Kitaev’s toric code, the gauge-invariant sector of hilbert space is described by introducing a primitive quantum gate: the quantum parallel transporter. A nonabelian generalisation of laplace interpolation to present an ansatz for the ground state of pure Yang-Mills theory which interpolates between the weak- and strong-coupling RG fixed points will then be described. The resulting state acquires the structure of a tensor network, namely, a multiscale entanglement renormalisation ansatz, and allows for the efficient computation of local observables and Wilson loops.

(V. Mastropietro) A non-perturbative formulation of the Standard Model could be achieved by a lattice, with a cut-off much higher than the experiments energy. The mathematical control of functional integrals is however increasingly difficult (if possible) with the size of cut-off. There is an expected connection between renormalizability and size of cut-off. The Standard Model is renormalizable; in principle a non-perturbative lattice definition is possible up to a cut-off exponentially large (much higher than any possible experiment; no problem for lack of asymptotic freedom in the electroweak sector). The perturbative renormalizability for a chiral gauge like EW is however not simply power counting but based on subtle properties like the reduction of the degree of divergence and the cancellation of the chiral anomalies. They have been established order by order with dimensional regularization; their validity at a non-perturbative level is a necessary prerequisite for a possible EW construction with exponentially small lattice. We investigate such properties in simpler but related models at a non perturbative level. In particular we consider:
a) A vector boson model in d=4 with lattice cut-off of the order of inverse coupling;
b) a similar model in $d=2$ (Sommerfield model) uniformly in the lattice;
c) an effective electroweak in $d=4$ on lattice with inverse coupling cut-off.
References:
V. Mastropietro, Physical Review D 105 (11), 114502;
A. Giuliani, V. Mastropietro, M. Porta, Communications in Mathematical Physics 384 (2), 997-1060;
V. Mastropietro, Physical Review D 103 (1), 013009+ (arXiv:2111.14442)

Winter term 2021/22

Date	Speaker	Title (see below for abstract)
28.10	Jens Mund (Universidade Federal de Juiz de Fora)	Perturbative construction of a string-localized Dirac field in a Hilbert space representation of QED
25.11	Patrick Dorey (Durham University)	Chaotic collisions of classical kinks
27.01	Raimar Wulkenhaar (Universität Münster)	From scalar fields on noncommutative geometries to blobbed topological recursion
24.02	Henning Bostelmann (University of York)	Fermionic integrable models and graded Borchers triples
24.03	Sebastiano Carpi (Università di Roma “Tor Vergata”)	Local energy bounds and strong locality for chiral conformal fields

Abstracts:

(J. Mund) The construction of charged physical states in QED has been a difficult task due to the infrared problems related to Gauss’ law. The latter implies that the electron is an infraparticle, i.e., it does not correspond to a discrete eigenvalue of the mass operator, and that the physical Dirac field cannot be pointlike localized.(Physical means acting in a Hilbert space, and locality is understood in the sense of commutators). One aspect of the IR problem is that the (LSZ) relation of the S-matrix to the Dirac field has not been understood. I report on an alternative strategy for a perturbative construction of the interacting physical Dirac field. It satisfies Gauss’ law and is not point- but ”string-localized”, i.e., localized on half rays extending to spacelike infinity. We conjecture that its (GNS reconstruction) Hilbert space describes the electron as an infraparticle, and allows for infrared finite matrix elements of the S-matrix. The “photon clouds” accompanying the electron are not put in by hand (like in the work of Faddeev and Kulish), but come out of the construction. Our only input is the usual interaction density with a string-localized vector potential. Then the Epstein-Glaser construction leads straightforwardly to an interacting Dirac field with the mentioned properties.

(P. Dorey) In this talk I will describe some of the surprising phenomena which are found when integrability is broken in field theories, both in the bulk and at the boundary. Complicated fractal-like structures emerge with sensitive dependence on initial conditions, but it turns out that some techniques from integrability can still be used to shed light on the situation.

(R. Wulkenhaar) Finite-dimensional approximations of noncommutative quantum field theories are matrix models. They often show rich mathematical structures. Many of them are exactly solvable or even related to integrability, or they generate numbers of interest in enumerative or algebraic geometry. For many matrix models it was possible to prove that they are governed by a universal combinatorial structure called Topological Recursion. The probably most beautiful example is Kontsevich’s matrix Airy function which computes intersection numbers on the moduli space of stable complex curves. The Kontsevich model arises from a $\lambda\Phi^3$-model on a noncommutative geometry. The talk addresses the question which structures are produced when replacing $\lambda\Phi^3$ by $\lambda\Phi^4$. The final answer will be that $\lambda \Phi^4$ obeys blobbed topological recursion, a systematic extension of topological recursion due to Borot and Shadrin.

(H. Bostelmann) The operator-algebraic construction of 1+1-dimensional integrable quantum field theories has received substantial attention over the past decade. These models are characterized by their asymptotic particle spectrum and their two-particle scattering matrix; so far, those particles have been bosonic. By contrast, we consider the case of asymptotic fermions. Abstractly, they arise from a grading of the underlying operator algebraic structures (Borchers triples); more concretely, one replaces the generating quantum fields fulfilling wedge-local commutation relations with a variant fulfilling anticommutation relations. Many of the technical methods required can be carried over from the bosonic case; most importantly, existing results on the technically hard part of the construction (i.e., establishing the modular nuclearity condition) do not require modification. Thus we are lead to a new family of rigorously constructed quantum field theories which are physically distinct from the bosonic case (with a different net of local algebras). Haag-Ruelle scattering theory confirms that they indeed describe fermions. Also, their local operators fulfill a modified version of the form factor axioms, consistent with the physics literature.

(S. Carpi) The notion of strong locality of quantum fields naturally appears in the study of the connection between the Wightman approach to QFT and the Haag-Kastler algebraic approach (AQFT). Roughly speaking, a family of quantum fields is strongly local if it generates a local Haag-Kastler net. The first proofs of strong locality for free field theory rely on the use of analytic vectors (Borchers-Zimmermann). A second tool, suitable also for interacting theories was found later and it is based on the so-called linear energy bounds (Nelson, Glimm-Jaffe, Driessler-Frohlich). The problem of strong locality has recently found new interest in the context of chiral two dimensional CFT in the study of a general connection between unitary vertex operator algebras (VOAs) and conformal nets on the circle recently the study of a general connection between unitary vertex operator algebras (VOAs) and conformal nets on the circle recently started by Carpi, Kawahigashi, Longo and Weiner. Many unitary vertex operator algebras have been shown to be strongly local. The proof depends more or less directly on linear energy bounds for suitable generating fields (vertex operators). In this talk I will present a new criterion for strong locality in chiral CFT based on local energy bounds. This applies to the W_3 VOAs with central charge c>2 giving rise to new conformal nets. The talk is based on recent joint works with Y. Tanimoto and M. Weiner.

Summer term 2021

Date	Speaker	Title (see below for abstract)
15.04	Sonia Mazzucchi (Università di Trento)	Mathematical theory of Feynman path integrals
27.05	Herbert Spohn (Technical University Munich)	Hydrodynamic equations for the classical Toda lattice
24.06	Olalla Castro Alvaredo (City, University of London)	An Introduction to Branch Point Twist Fields and Entanglement Measures in Integrable Quantum Field Theory
15.07	Ed Corrigan (University of York)	Aspects of boundaries, defects and integrability

Abstracts:

(S. Mazzucchi ) Since their first introduction in Feynman’s PhD thesis, Feynman path integrals have always been a powerful tool in theoretical physics on the one hand and a source of mathematical issues on the other hand. The talk will provide an overview of the mathematical theory of Feynman integration from its origins to the present day. Some applications and open problems will be also discussed.

(H. Spohn) The Toda lattice is one of the most famous integrable system of classical mechanics. For N lattice sites there are N+1 conserved quantities.We are interested in the hydrodynamic scale, which means to start with suitably adjusted random initial data. In my talk I will outline the resulting hydrodynamic Euler type equations. Their structure is very similar to the one obtained for the much more studied case of integrable many-body quantum systems.

(O. Castro Alvaredo) Over the past 15 years and enormous amount of work has been carried out looking at the properties of bipartite entanglement measures in the context of 1+1D quantum field theory. A lot of the work has looked specifically at conformal field theories where conformal invariance proves extremely powerful in constraining the functional form of many interesting quantities. However, entanglement measures also display interesting universal features beyond criticality. My focus will be on integrable quantum field theories and one of the leading techniques I have helped develop to study entanglement measures. This technique is based on the relationship between many entanglement measures (such as the von Neumann entropy) and a field known as branch point twist field. This field is very interesting from a QFT viewpoint as it is non-trivially semi-local with respect to other fields in the theory and this semi-locality leads to modified equations for its matrix elements (form factors). In this talk I will introduce entanglement measures and branch point twist fields and will show some of the results that may be obtained using this technique.

(E. Corrigan) The aim of the talk is to explain some of the ideas and features of two-dimensional integrable field theories when restricted to a half-line or adjusted by the inclusion of one or more ‘defects’. By design, the boundaries are purely reflecting while the defects are purely transmitting. This is a collection of ideas that is applicable to many types of integrable field theory but the illustrations used in this talk are framed mainly within the sin/sinh-Gordon model.

Winter term 2020/21

Date	Speaker	Title (see below for abstract)
22.10	Benjamin Doyon (King’s College London)	Thermalisation and the emergence of hydrodynamics in isolated many-body systems
12.11	Karol Kozlowski (ENS de Lyon)	Convergence of the form factor series in the Sinh-Gordon quantum field theory in 1+1 dimensions
14.01	Jonathan Dimock (University at Buffalo)	Ultraviolet Stability for Quantum Electrodynamics in d=3
28.01	Alessandra Frabetti (Université Lyon 1)	Closed BPHZ formulas with renormalization Hopf algebras
18.02	Alessandro Torrielli (University of Surrey)	Massless integrable scattering in the AdS/CFT correspondence
18.03	Eli Hawkins (University of York)	AQFT, Cohomology, and Deformations

Abstracts:

(B. Doyon) The passage from short-scale, microscopic motion to large-scale, emergent collective behaviours is at the heart of some of the deepest questions in modern theoretical physics. The problem may be framed as determining, from the intricate microscopic dynamics of a myriad constituents in interaction, the emergent degrees of freedom that are relevant for observations at large scales of space and time, and their own dynamics. Thermalisation and the emergence of hydrodynamics in reversible, isolated systems are the most fundamental examples. The conventional assumption that mass, momentum and energy are the only relevant quantities, leads to the standard Gibbs states and Euler equations (with natural relativistic generalisations). However, it is now well established that this assumption is broken in integrable models, where an infinity of conserved charges must be taken into account, such as in classical soliton gases and many-body quantum systems, as confirmed by modern experiments. In this talk, I will explain some of the recent ideas and rigorous results in these subjects.

(K. Kozlowski) Within the approach of the bootstrap program, the physically pertinent observables in a massive integrable quantum field theory in 1+1 dimensions are expressed by means of the so-called form factor series expansion. This corresponds to a series of multiple integrals in which the n^th summand is given by a n-fold integral. While being formally effective for various physical applications, so far, the question of convergence of such form factor series expansions was essentially left open. Still, convergence results are necessary so as to reach the mathematical well-definiteness of such construction and appear as necessary ingredients for the justification of numerous handlings that are carried out on such series. In this talk, I will describe the technique I recently developed that allows one to prove the convergence of the form factor series that arise in the context of the simplest massive integrable quantum field theory in 1+1 dimensions: the Sinh-Gordon model. The proof amounts to obtaining a sufficiently sharp estimate on the leading large-n behaviour of the n-fold integral arising in this context. This appeared possible by refining some of the techniques that were fruitful in the analysis of the large-n behaviour of integrals over the spectrum of n x n random Hermitian matrices.

(J. Dimock) We report on results for quantum electrodynamics on a finite volume Euclidean space-time in dimension d=3. The theory is formulated as a functional integral on a fine toroidal lattice involving both fermion fields and abelian gauge fields. The main result is that, after renormalization, the partition function is bounded uniformly in the lattice spacing. This is a first step toward the construction of the model. The result is obtained by renormalization group analysis pioneered by Balaban. A single renormalization group transformation involves block averaging, a split into large and small field regions, and an identification of effective actions in the small field regions via cluster expansions. This leads to flow equations for the parameters of the theory. Renormalization is accomplished by fine-tuning the initial conditions for these equations. Large field regions need no renormalization, but are shown to give a tiny contribution.

(A. Frabetti) In pQFT, the renormalization group can be formally seen as a group of formal diffeomorphisms in the powers of the coupling constant, acting on Green’s functions and on the Lagrangian by substitution of the bare coupling and multiplication by some renormalization factors built on the counterterms of divergent Feynman graphs by means of the BPHZ formula. For scalar theories, such groups are proalgebraic (functorial on the coefficients algebra) and are represented by Faà di Bruno types of Hopf algebras on graphs. In this talk I review how maths confirm that the BPHZ formula is universal and that it can be pushed to the sum of Feynman graphs describing the integral coefficients. The outcome is a closed BPHZ formula at each order of interaction. For non-scalar theories, thought, Feynman graphs have matrix-valued amplitudes: even if the counterterms are scalar-valued, the renormalization group cannot be represented by a Hopf algebra in a functorial way, because associativity fails for the composition of series with non-commutative coefficients. Maths show that a functorial extension to matrix-valued series can be done as a “non-associative group”. The talk is based on Connes-Kreimer’s results, on old joint works with Christian Brouder and on the recent paper https://doi.org/10.1016/j.aim.2019.04.053 with Ivan P. Shestakov.

(A. Torrielli) In recent years the framework of AdS/CFT integrability has seen one particular paradigm change due to the appearance of massless modes in the associated scattering problem. We shall briefly recap, following Zamolodchikov, ideas on massless integrable scattering. We shall then translate this scenario to the case of AdS3/CFT2, and demonstrate the appearance of a 2D critical theory in the so-called Berenstein-Maldacena-Nastase limit of the integrable scattering problem. This talk is primarily based on work with R. Borsato, A. Fontanella, O. Ohlsson Sax, A. Sfondrini, B. Stefanski and J. Stroemwall.

(E. Hawkins) An algebraic quantum field theory is, in particular, a functor from some category (of spacetimes, regions, intervals on a circle,…) to a category of algebras. I shall refer to such a structure as a diagram of algebras. Hochschild cohomology is fundamental to the theory of deformations of algebras, and there is a generalization of Hochschild cohomology for diagrams of algebras, constructed using a double complex. This double complex organizes the structures and symmetries of a diagram of algebras. It also motivates my generalizations of diagrams of algebras and their symmetries. Hochschild cohomology has the same algebraic structure as the set of multivector fields on a smooth manifold. This means that some interactions may be constructed algebraically from (generalized) symmetries, I will explain how to deform a diagram of von Neumann algebras using a symmetry and a generalized symmetry.