# Wales MPPM Zoom Seminar

#### Mathematical Physics -Physical Mathematics

Information about zoom links, password and for regular email updates can be obtained by joining the associated google group. To do so please send an email request to one of the organisers listed below.

#### Upcoming seminars

We are taking a summer break – back on Tuesday 28th September 2021.

##### Jones’ actions: from conformal field theory to Richard Thompson’s group.

In his quest in constructing conformal field theories from subfactors Vaughan Jones found an efficient machine to produce actions of groups such as the celebrated Richard Thompson’s group. I will tell this beautiful story and present a general overview of this novel technology. Examples and applications toward group theory will be discussed. Some of the results presented come from joint works with Vaughan Jones and with Dilshan Wijesena. Moreover, I may cite joint works with Valeriano Aiello, Roberto Conti and Alexander Stottmeister.

##### TBA

Previous talks in 2021:

##### Claudia Scheimbauer (TU Munich)

The AKSZ construction as a fully extended topological field theory

Classical Poincaré-Lefschetz duality is a starting point of understanding what an orientation on a derived stack given by a homotopy type of a cobordism M is. If furthermore we have an (n-)symplectic “target’’ X, we obtain a (shifted) symplectic on Map(M,X) which is given by pulling back along an evaluation and then integrating. This explains the Atiyah-Bott symplectic form on G-local systems, arising from the Killing form on the Lie algebra of a reductive group G, via  derived symplectic geometry in the sense of Pantev-Toen-Vaquié-Vezzosi. In this talk I will sketch these ideas and explain how this leads to a fully extended oriented topological field theory with values in a suitable higher category of Lagrangians, explaining the difficulties with coherence in defining such a functor. In essence, the difficulty with coherence stems from integration of differential forms not being functorial for cobordisms on the nose, but only up to homotopy. In mathematical physics, this TFT is a reinterpretation/analog of the classical AKSZ construction for certain $\sigma$-models and describes “semi-classical TFTs”. This is joint work with Damien Calaque and Rune Haugseng and fully extends a construction by the former.

##### Ivan Cherednik (University of North Carolina, Chapel Hill)

Riemann hypothesis for plane curve singularities

We will begin with a mini-review of the classical theories of
zeta-functions, and then switch to the zeta and L-functions
of plane curve singularities, conjecturally coinciding with the
motivic super-polynomials, defined in terms of compactified
Jacobians of these singularities. The focus is on the functional
equation, matching the DAHA super-duality and, presumably,
the physics S-duality in M-theory. DAHA provides here powerful
operator methods, closely related to refined Verlinde algebras.
We will calculate the DAHA super-polynomial and the motivic
one for trefoil. Generally, they are expected to coincide with
the stable Khovanov-Rozansky polynomials of algebraic knots.

##### Andrew Schopieray (PIMS)

Number fields and the quantum subgroup problem Slides

There was a long-standing conjecture that, stated very loosely, there exist finitely-many exceptional quantum subgroups for each of the affine Lie algebras.  This conjecture can be understood as a question about commutative algebra objects in modular tensor categories.  It was announced a few years ago that this conjecture shall become a theorem, the proof of which relies on number-theoretic aspects of modular tensor categories including the Galois symmetry of modular data.  In this talk, I will discuss both results and future directions of research that have arisen from my own contributions to resolving this conjecture, including Witt group relations, non-pseudounitary fusion rules, and minimal modular closure conjectures.

##### Bin Gui (Rutgers)

Categorical extensions of conformal nets

Wightman QFT and Haag-Kastler QFT are two major mathematically rigorous axiomatizations of quantum field theory. For 2d chiral conformal field theory, these two are respectively the vertex operator algebra approach (describing the chiral fields) and the conformal net approach (describing bounded observables on the circle). To construct and understand full and boundary CFT, and to investigate the representation tensor categories for chiral CFT, one also needs charged fields (known as intertwining operators or conformal blocks in the literature of vertex algebras) which map between different representations of the chiral fields. In this talk, I will introduce categorical extensions of conformal nets as a Haag-Kastler theory for charged fields. I will argue that this approach provides a natural framework of describing the locality property for charged fields. I will use Connes fusion (Connes relative tensor product) to motivate the definition.

##### Mayuko Yamashita (RIMS Kyoto)

The classification problem of non-topological invertible QFT’s and a “physicists-friendly” model for the Anderson duals

Freed and Hopkins conjectured that the deformation classes of non-topological invertible quantum field theories are classified by a generalized cohomology theory called the Anderson dual of bordism theories. The main difficulty of this problem lies in the fact that we do not have the axioms for QFT’s. In this talk, I will explain the ongoing work to give a new approach tothis conjecture. We construct a new, “physicists-friendly” model for the Anderson duals.This model is constructed so that it abstracts a certain property of invertible QFT’s which physicists believe to hold in general. I will start from basic motivations for the classification problem, report the progress of our work and explain future directions. This is the joint work with Yosuke Morita (Kyoto, math) and Kazuya Yonekura (Kyushu, physics).

##### Nilanjana Datta (DAMTP Cambridge)

Perfect discrimination of unitary channels and novel quantum speed limits

Discriminating between unknown objects in a given set is a fundamental task in experimental science. Suppose you are given a quantum system which is in one of two given states with equal probability. Determining the actual state of the system amounts to doing a measurement on it which would allow you to discriminate between the two possible states. It is known that unless the two states are mutually orthogonal, perfect discrimination is possible only if you are given arbitrarily many identical copies of the state. In this talk we consider the task of discriminating between quantum channels, instead of quantum states. In particular, we discriminate between a pair of unitary channels acting on a quantum system whose underlying Hilbert space is infinite-dimensional. We prove that in contrast to state discrimination, one only needs a finite number of uses of these channels in order to discriminate perfectly between them. Furthermore, no entanglement is needed in the discrimination task. The measure of discrimination is given in terms of the energy-constrained diamond norm, and a key ingredient of the proofs of these results is a generalization of the Toeplitz-Hausdorff Theorem of convex analysis . Moreover, we employ our results to study a novel type of quantum speed limits which apply to pairs of quantum evolutions. This work was done jointly with Simon Becker (Cambridge), Ludovico Lami (Ulm) and Cambyse Rouze (Munich).

##### Reinhard Werner (Leibniz Universität, Hannover)

Topological classification of spin chains and quantum walks

I will consider two types of one-dimensional discrete time lattice systems: On the one hand, there are spin chains, infinite tensor products of finite quantum systems. With a discrete time evolution automorphism, they are called quantum cellular automata (QCAs). On the other, there are quantum walks, direct sums of finite quantum systems, whose one-step dynamics is given by a unitary operator. In both cases, the discreteness of time leads to an interesting interplay between unitarity and locality, which is absent in the continuous time (Hamiltonian) analogues. It is measured by an index, which assumes rational values in the QCA case and integers in the walk case. I will illustrate it by the index theorem for juggling patterns (these are special walks) and the memory invariant for reversible qubit-stream processors (special QCAs). The geometric features of the lattice are encoded in a “coarse structure”, which will be important for analogues of this theory in higher dimensions. But even in the translation invariant one-dimensional case there are different choices, which I will briefly describe.

I will then consider quantum walks with discrete symmetries and a spectral gap condition. No translation invariance and only approximate locality are assumed. In the Hamiltonian case this leads to the so-called tenfold way, of which I will give a characterization justifying the number ten. In the unitary case, the same assumptions once again lead to more possibilities. Topological classification is in terms of an index group, and requires three indices: one for the asymptotic behaviour on the left, one on the right, and one to classify non-gentle perturbations, i.e., local modifications of the walk that cannot be deformed away.

##### Cain Edie-Mitchell (Vanderbilt)

Type II quantum subgroups for sl_n

Quantum subgroups are module categories, which encode the “higher representation theory” of the Lie algebras. They appear naturally in mathematical physics, where they correspond to extensions of the Wess-Zumino-Witten models. The classification of these quantum subgroups has been a long-standing open problem. The main issue at hand being the possible existence of exceptional examples. Despite considerable attention from both physicists and mathematicians, full results are only known for sl_2 and sl_3.

In this talk I will discuss recent progress in the classification of type II quantum subgroups for sl_n. Our results finish off the classification for n = 4,5,6,7, and pave the way for higher ranks. In particular we discover several exceptional examples.

##### Zhengwei Liu (Tsinghua University, Beijing)

Quantum Fourier Analysis
Quantum Fourier Analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. In this talk, we will introduce its background, development and perspectives, based on selected examples, results and applications.

There is a youtube channel for some earlier talks from this link.

##### Yoshiko Ogata (Tokyo U)

Classification of gapped ground state phases in quantum spin systems

I would like to explain about classification of gapped ground state phases in quantum spin systems, in the operator algebraic framework of quantum statistical mechanics. The talk consists of two parts. The first part is about classification problem of so called SPT (symmetry protected topological) phases.We see that it is classified by group cohomology, in one and two dimension.In the second part (joint work with Pieter Naaijkens), we see that super selection sector is a topological invariant.

##### Emily Riehl (Johns Hopkins)

Elements of ∞-Category Theory

Confusingly for the uninitiated, experts in weak infinite-dimensional category theory make use of different definitions of an ∞-category, and theorems in the ∞-categorical literature are often proven “analytically”, in reference to the combinatorial specifications of a particular model. In this talk, we present a new point of view on the foundations of ∞-category theory, which allows us to develop the basic theory of ∞-categories — adjunctions, limits and colimits, co/cartesian fibrations, and pointwise Kan extensions — “synthetically” starting from axioms that describe an ∞-cosmos, the infinite-dimensional category in which ∞-categories live as objects. We demonstrate that the theorems proven in this manner are “model-independent”, i.e., invariant under change of model. Moreover, there is a formal language with the feature that any statement about ∞-categories that is expressible in that language is also invariant under change of model, regardless of whether it is proven through synthetic or analytic techniques. This is joint work with Dominic Verity.

##### Yuki Arano (Kyoto U)

Ergodic theory of affine isometric actions on Hilbert spaces

I introduce nonsingular actions of groups on probability spaces constructed from affine isometric actions on Hilbert spaces, which we call the non-singular Gausian action. Then I discuss the ergodicity and the type determination for groups acting on trees.

##### Jonathan Rosenberg (U of Maryland)

Gap Labeling and the Noncommutative Bloch Theorem Slides

Noncommutative geometry provides powerful mathematical tools for studying thephysical properties of non-periodic but still somewhat regular solids.  Such materials, often called “quasi-crystals”, do occur in nature and are based on non-periodic tilings.  (These were recognized in Schechtman’s 2011 Nobel Prize.) Attempts to predict the Bragg peaks and spectral properties of such materials led to a program pioneered by Jean Bellissardand pursued by many others, of studying “gap labeling”, predicting spectral gaps based on thedynamics of the tiling.  This is closely related to the Čech cohomology of the “tiling space” andthe K-theory of a certain C*-algebra associated to the tiling.  We explain how this comes about,how various people gave false proofs of the “Gap Labeling Conjecture”, and how to obtain somenew results.  This is joint work with Claude Schochet, Eric Akkermans, and Yaroslav Don.

##### Johannes Kellendonk (Université Claude Bernard Lyon 1)

Bragg spectrum and gap-labelling, a topological approach

A classical result in solid state physics tells us that the gaps in the electronic spectrum of a one dimensional crystal are located at those values for the quasi momentum k which belong to the Bragg spectrum of the crystal. The result is proven by perturbation theory. If one labels a gap with the integrated density of states of energies up to that gap one obtains an order preserving map from the positive Bragg spectrum to the set of gap labels. The purpose of the talk is to explain a similar map for aperiodic solids which is based on methods from non-commutative topology, namely we discuss an order preserving homomorphism between the group of topological Bragg peaks and the gap labelling group. A comparison with work by theoretical physicists from the late ‘80s, which is based on a perturbation expansion of the trace map for hierarchical systems, finds disagreement. Not all gaps predicted by these perturbative expansions are there. K-theory provides selection rules on the opening of gaps.

##### Matthew Buican (Queen Mary, London)

aXb=c in 2+1D TQFT

I will discuss fusion rules of the form aXb=c in 2+1D topological quantum field theories. Here a, b, and c are simple objects of the associated modular tensor categories (MTC). As I will explain, when a and b have categorical dimension larger than one, such fusion rules imply interesting global constraints on the MTC. The simplest possibility is that the MTC factorizes, but there are other, more interesting, possibilities as well. Some of these latter possibilities may potentially be relevant for understanding various problems in mathematical physics that I will describe.