Past Quantum Field Theory Seminar

Recent years have seen many new ideas appearing in the solution theories of singular stochastic partial differential equations. An exciting application of SPDEs that is beginning to emerge is to the construction and analysis of quantum field theories. In this talk, I will describe how stochastic quantisation of Parisi–Wu can be used to study QFTs, especially those arising from gauge theories, the rigorous construction of which, even in low dimensions, is largely open.

I will discuss a formulation of an Abelian Chern-Simons theory on the lattice employing the modified Villain formalism. The theory suffers from a well-known problem of having extra zero modes in the Gaussian operator. I will argue that these zero modes are associated with a kind of subsystem symmetry which projects out almost all naive Wilson loops. The operators which survive are framed Wilson loops. These turn out to be topological charges of the associated one-form symmetry, and it has the correct topological spin and correlation functions.

Supersymmetric localization allows one to reduce the computation of the partition function of a supersymmetric theory to a finite-dimensional integral, but the space over which one integrates is often singular. In this talk I will explain how one can use shifted symplectic geometry to get rigorous definitions of partition functions and state spaces in theories with extended supersymmetry. For instance, this gives a field-theoretic origin of DT invariants of CY4 manifolds. This is a report on joint work with Brian Williams.

6d N=(2,0) superconformal field theories have natural surface operators similar in many ways to Wilson lines in gauge theories. In this talk, I will discuss how they can be studied using conformal bootstrap techniques, including connection to W-algebras and the so-called inversion formula, focusing on the limit of large central charge.

I will report on a research program which uses ideas from stochastic analysis in the context of constructive Euclidean quantum field theory. Stochastic analysis is the study of measures on path spaces via push-forward from Gaussian measures. The foundational example is the map, introduced by Itô, which sends Brownian motion to a diffusion process solution to a stochastic differential equation. Parisi–Wu's stochastic quantisation is the stochastic analysis of an Euclidean quantum field, in the above sense. In this introductory talk, I will put these ideas in context and illustrate various stochastic quantisation procedures and some of the rigorous results one can obtain from them.

Standard twistor theory involves a complex projective
3-space PT which naturally divides into two halves PT+
and PT–, joined by their common 5-real-dimensional
boundary PN. However, this splitting has two quite
different basic physical interpretations, namely
positive/negative helicity and positive/negative
frequency, which ought not to be confused in the
formalism, and the notion of “bi-twistors” is introduced
to resolve this issue. It is found that quantized bi-
twistors have a previously unnoticed G2* structure,
which enables the split-octonion algebra to be directly
formulated in terms of quantized bi-twistors, once the
appropriate complex structure is incorporated.

Starting from the principle of locality in quantum field theory, which
states that an object is influenced directly only by its immediate

surroundings, I will first briefly review some features of the notion of
locality arising in physics and mathematics. These are then encoded
in  locality relations, given by symmetric binary relations whose graph
consists of pairs of "mutually independent elements".

I will mention challenging questions that arise from  enhancing algebraic
structures to their locality counterparts, such as i) when  is the quotient
of a locality vector space by a linear subspace, a locality vector space, if
equipped with the quotient locality relation,  ii) when does  the locality
tensor product of two locality vector spaces  define a locality vector
space. These are discussed in recent joint work  with Pierre Clavier, Loïc
Foissy and Diego López.

Locality morphisms, namely maps that factorise on   products of  pairs of
"mutually independent" elements, play a key role in the context of
renormalisation in
multiple variables. They include "locality evaluators", which we use to

consistently evaluate meromorphic germs in several variables at
their poles. I will  also report on recent joint work with Li Guo and Bin
Zhang. which gives a classification of locality evaluators on certain
classes of algebras of meromorphic germs.

In this talk, I review some recent results obtained for black holes using
effective field theory methods applied to quantum gravity, in particular the
unique effective action. Black holes are complex thermodynamical objects
that not only have a temperature but also have a pressure. Furthermore, they
have quantum hair which provides a solution to the black hole information
paradox.

In this talk I will review some of the key ideas behind
the study of entanglement measures in 1+1D quantum field theories employing
the so-called branch point twist field approach. This method is based on the
existence of a one-to-one correspondence between different entanglement
measures and different multi-point functions of a particular type of
symmetry field. It is then possible to employ standard methods for the
evaluation of correlation functions to understand properties of entanglement
in bipartite systems. Time permitting, I will then present a recent
application of this approach to the study of a new entanglement measure: the
symmetry resolved entanglement entropy.

In the search for a complete description of quantum mechanical and
gravitational phenomena, we are inevitably led to consider observables on
boundaries at infinity. This is the common mantra that there are no local
observables in quantum gravity and gives rise to the tantalising possibility
of a purely boundary--or holographic--description of physics in the
interior. The AdS/CFT correspondence provides an important working example
of these ideas, where the boundary description of quantum gravity in anti-de
Sitter (AdS) space is an ordinary quantum mechanical system-- in particular,
a Lorentzian Conformal Field Theory (CFT)--where the rules of the game are
well understood. It would be desirable to have a similar level of
understanding for the universe we actually live in. In this talk I will
explain some recent efforts that aim to understand the rules of the game for
observables on the future boundary of de Sitter (dS) space. Unlike in AdS,
the boundaries of dS space are purely spatial with no standard notion of
locality and time. This obscures how the boundary observables capture a
consistent picture of unitary time evolution in the interior of dS space. I

will explain how, despite this difference, the structural similarities
between dS and AdS spaces allow to forge relations between boundary
correlators in these two space-times. These can be used to import
techniques, results and understanding from AdS to dS.

Can the S-matrix be complexified in a way consistent with causality? Since the 1960's, the affirmative answer to this question has been well-understood for 2→2 scattering of the lightest particle in theories with a mass gap at low momentum transfer, where the S-matrix is analytic everywhere except at normal-threshold branch cuts. We ask whether an analogous picture extends to realistic theories, such as the Standard Model, that include massless fields, UV/IR divergences, and unstable particles. Especially in the presence of light states running in the loops, the traditional iε prescription for approaching physical regions might break down, because causality requirements for the individual Feynman diagrams can be mutually incompatible. We demonstrate that such analyticity problems are not in contradiction with unitarity. Instead, they should be thought of as finite-width effects that disappear in the idealized 2→2 scattering amplitudes with no unstable particles, but might persist at higher multiplicity. To fix these issues, we propose an iε-like prescription for deforming branch cuts in the space of Mandelstam invariants without modifying the analytic properties. This procedure results in a complex strip around the real part of the kinematic space, where the S-matrix remains causal. To help with the investigation of related questions, we introduce holomorphic cutting rules, new approaches to dispersion relations, as well as formulae for local behavior of Feynman integrals near branch points, all of which are illustrated on explicit examples.

A topological insulator has anomalous boundary spectrum which completely fills up gaps in the bulk spectrum. This ``topologically protected’’ spectral property is a physical manifestation of coarse geometry and index theory ideas. Special examples involve spectral flow and gerbes, related to Hamiltonian anomalies, and they arise experimentally in quantum Hall systems, time-reversal invariant mod-2 insulators, and shallow-water waves.

I’ll review a new, simpler explanation for the large-scale properties of the
cosmos, presented with L. Boyle in our recent preprint arXiv:2201.07279. The
basic ingredients are elementary and well-known, namely Einstein’s theory of
gravity and Hawking’s method of computing gravitational entropy. The new
twist is provided by the boundary conditions we proposed for big bang-type
singularities, allowing conformal zeros but imposing CPT symmetry and

analyticity at the bang. These boundary conditions, which have significant
overlap with Penrose’s Weyl curvature hypothesis, allow gravitational
instantons for universes with Lambda, massless radiation and space
curvature, of either sign, from which we are able to infer a gravitational
entropy. We find the gravitational entropy can exceed the de Sitter entropy
and that, to the extent that it does, the most probable large-scale geometry
for the universe is flat, homogeneous and isotropic. I will briefly
summarise our earlier work showing how the gauge-fermion Lagrangian of the
standard model may be reconciled with Weyl symmetry and a small cosmological
constant, at leading order, provided there are precisely three generations
of fermions. The same mechanism generates scale-invariant primordial
perturbations. The cosmic dark matter consists of a right-handed neutrino.
In summary, we have taken significant steps towards a new, highly principled
and testable theory of cosmology.

Symmetry protected topological (SPT) phases have recently attracted a lot of
attention from physicists and mathematicians as a topological classification
scheme for gapped ground states. In this talk I will briefly introduce the
operator algebraic approach to SPT phases in the infinite-volume limit. In
particular, I will focus on the quasifree (free-fermionic) setting, where we

can adapt tools from algebraic quantum field theory to describe phases of
gapped ground states using K-homology and the coarse index.

I shall present joint work with Maxim Kontsevich describing an interesting
domain of complex metrics on a smooth manifold. It is a complexification of
the space of ordinary Riemannian metrics, and has the Lorentzian metrics
(but not metrics of other signatures) on its boundary. Use of the domain
leads to a modified axiom system for QFT which illuminates not only the
special role of Lorentz signature, but also of features such as local
commutativity, unitarity, and global hyperbolicity.

In his very first note on noncommutative differential geometry, Connes
showed that the position and momentum operators on the line could be used to
construct constant curvature connections over an irrational noncommutative

2-torus $\mathcal{A}_\theta$. When $\theta$ is a quadratic irrationality,
this yields, in particular, constant curvature connections on non-trivial
noncommutative line bundles---is there an underlying monopole on some
non-trivial noncommutative principal $U(1)$-bundle? We use this case study
to illustrate how approaches to quantum principal bundles introduced by
Brzeziński–Majid and Đurđević, respectively, can be fruitfully synthesized
to reframe classical gauge theory on quantum principal bundles in terms of
synthesis of total spaces (as noncommutative manifolds) from vertical and
horizontal geometric data.

I will introduce asymptotic safety, which is a quantum field theoretic
paradigm providing a predictive ultraviolet completion for quantum field
theories. I will show examples of asymptotically safe theories and then
discuss the search for asymptotically safe models that include quantum
gravity.
In particular, I will explain how asymptotic safety corresponds to a new
symmetry principle - quantum scale symmetry - that has a high predictive
power. In the examples I will discuss, asymptotic safety with gravity could
enable a first-principles calculation of Yukawa couplings, e.g., in the
quark sector of the Standard Model, as well as in dark matter models.

The non-compact exceptional simple group G_2* turns out to be the symmetry group of quantized twistor theory. Certain implications of this remarkable fact will be explored in this talk.

This is a joint GR-QFT seminar, to celebrate in advance the 90th birthday of Roger Penrose later in the summer, comprising 9 talks on conformal cyclic cosmology.  The provisional schedule is as follows:

11:00 Roger Penrose (Oxford, UK) : The Initial Driving Forces Behind CCC

11:10 Paul Tod (Oxford, UK) : Questions for CCC

11:20 Vahe Gurzadyan (Yerevan, Armenia): CCC predictions and CMB

11:30 Krzysztof Meissner (Warsaw, Poland): Perfect fluids in CCC

11:40 Daniel An (SUNY, USA) : Finding information in the Cosmic Microwave Background data

11:50 Jörg Frauendiener (Otago, New Zealand) : Impulsive waves in de Sitter space and their impact on the present aeon

12:00 Pawel Nurowski (Warsaw, Poland and Guangdong Technion, China): Poincare-Einstein expansion and CCC

12:10 Luis Campusano (FCFM, Chile) : (Very) Large Quasar Groups

12:20 Roger Penrose (Oxford, UK) : What has CCC achieved; where can it go from here?

Dark matter is known to exist, but no-one knows what it is or where it came
from.  We describe a new production mechanism of particle dark matter, which
hinges on a first-order cosmological phase transition.  We then show that
this mechanism can be slightly modified to produce primordial black holes.

While solar mass and supermassive black holes are now known to exist,
primordial black holes have not yet been seen but could solve a number of
problems in cosmology.  Finally, we demonstrate that if an evaporating
primordial black hole is observed, it will provide a unique window onto
Beyond the Standard Model physics.

I will be first introducing the Polyakov loop space formalism to
gauge theories. I will also discuss how the Polyakov loop space is modified
by Planck scale corrections.  The gauge theory will be deformed by the
Planck length as the minimum measurable length in the background spacetime.
This deformation will in turn deform the Polyakov loops space. It will be
observed that this deformation can have important consequences for
non-abelian monopoles in gauge theories.