# AQFTUK

## Meeting 5-6/6/2023, Cardiff

### 5-6/6/2023 at Cardiff University

(Location: Abacws/3.38)

We are pleased to announce the sixth workhsop of the AQFT in the UK research network. All are welcome to attend, but please do register below. The programme and more information can be found below.

## Schedule

We will start on Monday at 13:00. On Tuesday, the last talk finishes around 16:00. This should give enough time for most UK participants to arrive on Monday, and leave on Tuesday.

### Monday 5 June

12:30 - 13:00 Welcome

13:00 - 14:00 Alastair Grant-Stuart (Nottingham)

Modules for factorization algebras, by comparison with AQFT

14:00 - 15:00 Henning Bostelmann (York)

Tomita-Takesaki theory for double cones; or, the Curse of the Massive Modular Mummy

15:00 - 15:30 Coffee Break

15:30 - 16:00 Berend Visser (York)

Extending the time-slice property to the functional formalism

16:00 - 17:00 Simen Bruinsma (York)

The universal first-order Massey product for minimal models of locally constant factorization algebras

### Tuesday 6 June

09:30 - 10:30 Roberta Iseppi (Göttingen)

Towards a BV formalism for noncommutative manifolds

10:30 - 11:00 Arne Hofmann (Göttingen)

Smooth Wick polynomials and tadpole renormalisation

11:00 - 11:30 Coffee Break

11:30 - 12:30 Tiziano Gaudio (Lancaster)

Q-systems, Schellekens’ list and superconformal vertex operator superalgebras

12:30 - 14:00 Lunch Break

14:00 - 15:00 Amanda Young (Munich)

On Gapped Ground State Phases of Decorated AKLT Models

15:00 - 15:30 Dimitrios Ampelogiannis (King's College)

Quantum lattice models: notions of ergodicity and universality of the long-time dynamics

15:30 - 16:00 Mahdie Hamdan (Cardiff)

Non-frustration free ground states of non-abelian quantum double models

16:00 - Coffee and informal discussion

## Registration

Registration is free, but please do register here so we can get an estimate of the number of attendees.

## Abstracts

Alastair Grant-Stuart

(University of Nottingham)

Title:

Modules for factorization algebras, by comparison with AQFT

Abstract:

AQFT and factorization algebras each give mathematical formalisms for the description of quantum field theory. The factorization algebra formalism is the younger of the two, and its literature is as yet missing a detailed account of states and representations. I will discuss work in progress with Alexander Schenkel to build a suitable notion of modules (representations) for factorization algebras, by comparison with representations in AQFT. We approach this by extending recent results [Benini, Perin & Schenkel (2020), https://link.springer.com/article/10.1007/s00220-019-03561-x showing equivalence of AQFTs and factorization algebras in the Lorentzian context under appropriate niceness conditions. Our extension requires that we introduce new structure on factorization algebras to parallel the involutions (*-operations) on AQFT algebras of observables.

Henning Bostelmann

(University of York)

Title:

Tomita-Takesaki theory for double cones; or, the Curse of the Massive Modular Mummy

Abstract:

The Tomita-Takesaki modular operator for local algebras has an important structural role in quantum field theory and in relativistic quantum information. However, beyond the case of wedge algebras, describing this operator more concretely is a longstanding open problem. Even for the massive free field in a double cone, we do not know a closed expression for the modular generator, and several attempts at finding one have failed in the past. In this talk, we instead present a numerical approach to finding the generator's kernel. This method is applied to the double cone in the free massive Bose field in (1+1) and (3+1) dimensions, but has potential to yield insight in a much larger range of cases.

Berend Visser

(University of York)

Title:

Extending the time-slice property to the functional formalism

Abstract:

In the functional approach to constructing (classical and quantum) field theories, the space of observables is the homology of a complex of smooth functionals. By using smooth functionals, this method seeks to go beyond perturbative treatments, in which usually only polynomials or local functionals are needed.

The algebra of quantum observables is constructed using a *-product of functionals, but this is only defined if we restrict the singular structure of the functionals. The usual approach is to use microcausal functionals.

We have been investigating the time-slice axiom in the functional approach. It is straightforward to prove that it is satisfied with arbitrary smooth functionals, but the proof does not work with microcausal functionals, because this class is not closed under integration.

We discuss how this problem can be remedied by sharpening the requirements on the singular structure of the microcausal functionals. We will define in this talk the class of equicausal functionals. With these functionals, we can prove the time-slice axiom and show that the *-product closes on this subspace, giving it the structure of a differential graded algebra.

Simen Bruinsma

(University of York)

Title:

The universal first-order Massey product for minimal models of locally constant factorization algebras

Abstract:

Like algebraic quantum field theories, factorization algebras are a functorial formalism of quantum field theory. We investigate how the structure of a locally constant factorization algebra gets transferred to its cohomology, yielding a weaker structure: a homotopy coherent factorization algebra. A first example is studied: the factorization envelope of local Lie algebras on R^m. In dimension one, we retrieve the universal enveloping algebra, as was found before by Costello and Gwilliam. In dimension two, we find higher structures, so-called Massey products, which are shown to be nontrivial.

Roberta Iseppi

(University of Göttingen)

Title:

Towards a BV formalism for noncommutative manifolds

Abstract:

The Batalin-Vilkovisky (BV) formalism was first discovered in the context of the quantisation of gauge theories via the path integral approach. Since then, this formalism developed, revealing a particularly rich mathematical structure. In this talk, we will present recent results in the direction of the development of a BV formalism in the context of noncommmutative geometry. In particular, we will focus on finite spectral triples, which not only can be viewed as a noncommutative generalization of the classical notion of spin manifold but which also play a key role in the description of the particle physics content of gauge theories in the framework of noncommutative geometry.

Arne Hofmann

(University of Göttingen)

Title:

Smooth Wick polynomials and tadpole renormalisation

Abstract:

The relationship of the Wick polynomial construction to the renormalisation of Feynman diagrams with short loops (tadpoles) is discussed. Using a version of analytic renormalisation due to Guillemin, the smooth dependence on parameters of the Wick polynomials is shown. This is an essential input to the classification of the finite renormalisation freedom of Wick polynomials. A crucial ingredient is the precise microlocal singularity structure of the 2-point function, which is a classical Fourier integral operator.

Tiziano Gaudio

(Lancaster University)

Title:

Q-systems, Schellekens’ list and superconformal vertex operator superalgebras

Abstract:

Are the simple CFT type vertex operator algebra (VOA) extensions of a completely unitary VOA unitary? In this talk, we answer this question providing an independent tensor categorical result, that is: a haploid associative algebra in a $C^*$-tensor category $\mathcal{C}$ is equivalent to a Q-system (a special $C^*$-Frobenius algebra) in $\mathcal{C}$ if and only if it is rigid. This in turn implies the unitarity of all the 70 strongly rational holomorphic VOAs with central charge $c=24$ and non-zero weight-one subspace, corresponding to entries 1--70 of the so called Schellekens list. Furthermore, we obtain some new holomorphic conformal nets associated to every entry of the list by two different as well as equivalent ways, relying also on the notion of strong locality introduced by Carpi-Kawahigashi-Longo-Weiner. Finally, we show how to get the complete classification of the simple CFT type vertex operator superalgebra (VOSA) extensions of the unitary discrete series of the $N=1$ and $N=2$ super-Virasoro VOSAs from the known classification of the corresponding superconformal nets.

This talk is based on a joint work with S. Carpi, L. Giorgetti and R. Hillier.

Amanda Young

(TU Munich)

Title:

On Gapped Ground State Phases of Decorated AKLT Models

Abstract:

In their seminal work, Affleck, Kennedy, Lieb and Tasaki introduced a family of SU(2)-invariant quantum spin models and investigated their ground state properties - ncluding the existence of positive spectral gap above the ground state energy in the thermodynamic limit. They also conjectured that if the coordination number of a regular, translation invariant lattice was sufficiently small, the associated AKLT model would have a positive spectral gap. Otherwise, the model would exhibit Neel order and, hence, be gapless. Decorated versions of AKLT models obtained from replacing edges of lattices with chains of spin-1 particles have also been of interest, e.g., as their ground states constitute a universal quantum computation resource. A natural question, then, is whether or not decorated AKLT models belong to gapped or gapless ground state phases.

In this talk, we consider AKLT models defined on decorated versions of infinite, simple, connected graphs and show that for sufficiently large decoration, the resulting model belongs to a gapped phase. The proof of this result is based off a modified version of the Tensor Network State approach introduced by Abdul-Rahman et. al. which produces tighter estimates on the minimal decoration required to prove a positive spectral gap. We close with a brief discussion on the stability of this gap under small perturbations of the Hamiltonian. For the AKLT model on the decorated hexagonal model, cluster expansion techniques can be used to show that the ground states are sufficiently indistinguishable so spectral gap stability results in the spirit of Bravyi, Hastings and Michalakis hold.

This talk is based of joint works with A. Lucia (arXiv:2212.11872), and with A. Lucia and A. Moon (arXiv:2209.01141).

Dimitrios Ampelogiannis

(King's College London)

Title:

Quantum lattice models: notions of ergodicity and universality of the long-time dynamics

Abstract:

We rigorously examine the ergodic properties of quantum lattice models in the C* algebra formulation of statistical mechanics. A well-known result is the Lieb-Robinson bound which states that time-evolved local operators have an exponentially small effect outside a light-cone defined by the Lieb-Robinson velocity, implying a notion of ergodicity for this space-like region. But what happens within the Lieb-Robinson light-cone? We will discuss the recent results of “almost everywhere” ergodicity: the average of any observable over a space-time ray tends (in the strong operator topology of the Gelfand-Naimark-Segal representation) to its expectation value in the state, along almost every ray within the light-cone. We will at the same time discuss many-body ergodicity and contrast these notions with von Neumann’s quantum ergodic theorem. Finally, we will touch on the topic of emergence of large-scale behaviors and how “almost everywhere” ergodicity is indeed physically relevant in this context: it captures the idea that macroscopically a lot of information encoded in the initial microscopic dynamics is lost.

Mahdie Hamdan

(Cardiff University)

Title:

Non-frustration free ground states of non-abelian quantum double models

Abstract:

Quantum spin models are widely studied for their potential use in quantum computing, where they can serve as building blocks for quantum algorithms. Kitaev's quantum double model is of particular interest, as its properties could allow for fault-tolerance quantum computation. In two-dimensional quantum spin systems on the infinite lattice $\mathbb{Z}^{2}$, this model is known to exhibit a unique frustration-free ground state, although other ground states may exist, and it is theorized that these ground states always correspond to objects in a modular tensor category, deeply connected to the representation theory of the quantum double of an underlying group $G$. Using an operator algebraic approach, I will introduce Kitaevs quantum double model for a finite non-abelian group $G$ on a lattice $\mathbb{Z}^{2}$ and explore its ground states. I conjecture that these non-frustration free ground states, called anyons, correlate to non-trivial irreducible representations of the quantum double $D(G)$ and that they form a complete set of ground states. This is a joint work with Pieter Naaijkens.

## How to reach the venue

The workshop will take place in Abacws in Seminar room 3.38. The building is on Senghennydd Road, Cardiff, and is directly adjacent to Cathays railway station. If you've been to the School of Mathematics before the pandemic: our new building is about a 200m down the road form the old building. The building is within walking distance (10 to 25 minutes) of the city centre. You can also take the train to Cathays station from either Cardiff Queen street (one stop) or Cardiff Central Station (2 stops).

## More information

Please contact the organisers Pieter Naaijkens or Mahdie Hamdan if you have any questions.