# Emmy-Noether seminar

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 2022

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.

Winter term 2021/22

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

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

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 nth 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.