Scientific Series

Applications of physics to geometry have deep historical roots going back at least to Archimedes, while recent decades have seen structures of classical and quantum gauge theory lead to enormous advances in the theory of three and four manifolds. Meanwhile, geometry provides conceptual tools for physics as foundational  as variational principles, the kinematics of spacetime, and the topological  classification of matter. This talk will describe some possibilities for bringing number theory, more specifically, *arithmetic geometry* into this interaction. After giving some examples of the suggestive interplay of  number theory and physics, I will speculate somewhat on how ideas of quantum field theory may elucidate some of the deepest and hardest problems of number theory, especially the effective Mordell conjecture and the Hasse-Weil conjecture.

I will give a holographic argument in favor of the AdS Penrose inequality, which conjectures a lower bound on the total mass in terms of the area of apparent horizons. This inequality is often viewed as a test of cosmic censorship. Time permitting, I’ll also discuss a generalization to charged black holes and connections with a quasi-local energy and the second law for apparent horizons.  

I will introduce my recent works on the phenomenology of solutions to the strong CP problem, QCD axion and Parity. I will first describe the production of the QCD axion in the early universe and show that the dark matter abundance is naturally reproduced for a wide range of the parameter space. I will then show a tight relation between the Parity solution, dark matter direct detection, the proton decay, and the precise measurements of the standard model parameters.

Quasars are among the most powerful light sources in the universe and, as such, can be seen at cosmological distances. Is some rare occasions (although not that rare), a massive galaxy on their line of sight can act as a gravitational lens and produce multiple images of distant quasars. These can be used both for cosmology and astrophysics by measuring the so-called time delays between the lensed images from photometric monitoring, a quantity directly related to the Hubble-Lemaître parameter H0. The same data, affected by microlensing by stars in the lensing galaxy, can be used to measure the size and energy profile of quasar accretion disks on spatial scales of the order of the light-day, i.e. inaccessible to even the largest interferometers, including the SKA. The results in this talk are on behalf of the COSMOGRAIL and H0LiCOW collaborations. 

In the first part of the talk, I will mention the ongoing numerical efforts using lattice calculations to understand the holographic dualities relating the super Yang-Mills (SYM) theories in various dimensions and their conjectured Type II supergravity theories in the decoupling limit. In the second part, I will discuss the tensor renormalization group study of the SU(2) gauge-Higgs model in two dimensions using the higher-order tensor renormalization group (HOTRG) algorithm and compare the results with the Monte Carlo simulations. 

Quantum gravity may be viewed as a bipartite matter-geometry system. Evolution of matter-geometry entanglement entropy is an interesting question for issues such as the emergence of QFT on curved spacetime from quantum gravity: emergence would require initial states to evolve to product states. We study this question in a cosmological model. We give numerical evidence that matter-geometry entanglement entropy increases indefinitely (with respect to a relational time) for apparently arbitrary initial states. This result raises the question of whether matter and geometry remain entangled in a large Universe.

By applying a parabolic-hyperbolic formulation of the constraints and superposing Kerr-Schild black holes, a simple method is introduced to initialize time evolution of binary systems. As the input parameters are essentially the same as those used in the post-Newtonian (PN) setup the proposed method interrelates various physical expressions applied in PN and in fully relativistic formulations. The global ADM charges are also determined by the input parameters, and no use of boundary conditions in the strong field regime is made.

The Leggett-Garg (LG) inequalities were introduced, as a temporal parallel of the Bell inequalities, to test macroscopic realism -- the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The talk will begin with a review of the LG framework. Unlike the Bell inequalities, the original LG inequalities are only a necessary condition for macrorealism, and are therefore not a decisive test. I argue, for the case of measurements of a single dichotomic variable Q, that when the original four three-time LG inequalities are augmented with a set of twelve two-time inequalities also of the LG form, Fine's theorem applies and these augmented conditions are then both necessary and sufficient [1]. A comparison is carried out with the alternative necessary and sufficient conditions for macrorealism  based on no-signaling in time conditions which ensure that all probabilities for Q at one and two times are independent of whether earlier or intermediate measurements are made. I argue that the two tests differ in their implementation of the key requirement of non-invasive measurability so are testing different notions of macrorealism, and these notions are elucidated.  I also describe some alternative protocols which achieve non-invasiveness, one involving continuous measurement of the velocity conjugate to Q [2], which was recently implemented in an experiment at IQC, the other involving a modification of the standard ideal negative measurement protocol [3].

[1] J.J.Halliwell, Phys Rev A96, 012121 (2017); A93, 022123 (2016); arxiv:1811.10408.

[2]  J.J.Halliwell, Phys. Rev. A94, 052114 (2016).

[3] J.J.Halliwell, Phys. Rev. A99, 022119 (2019).

Many of the rich interactions between mathematics and physics arise using general mathematical frameworks that describe a host of physical phenomena: from differential equations, to algebra, to topology and geometry. On the other hand, mathematics also possesses many examples of "exceptional objects": they constitute the finite set of leftovers that appear in numerous classification problems. For example, groups of symmetries in three dimensions appear in two infinite families (cyclic groups and dihedral groups of n-sided polygons) and the symmetry groups of the five Platonic solids--- the 'exceptional' structures. 

The mathematical subject of moonshine refers to surprising relationships between other kinds of special/exceptional objects that arise from the theory of finite groups and from number theory. Increasingly, string theory has been a source of insights in and explanations for moonshine. It is even the source of new examples of moonshine that further implicate special objects in geometry. We will review moonshine, survey these developments, and highlight some of the (many!) exciting mysteries that remain.

It is known that there is a relationship between conformal Carroll transformations and BMS symmetry. In this talk I will explore the geometry of generic Carroll structures which may be thought of as the basic underlying geometric structure on null hypersurfaces. A Carroll structure can be thought of as a fibre bundle with Ehresmann connection, and one finds that (generalized) BMS symmetry emerges as the conformal symmetry of this bundle and connection. I’ll briefly also describe how this story fits into the physics of ’soft modes.’

Mirror sectors -- hidden sectors that are approximate copies of the Standard Model -- are a generic prediction of many models, notably the Mirror Twin Higgs model. Such models can have a rich cosmology and many interesting detection signatures beyond the realm of colliders. In this talk, I will focus on the possibility that mirror matter can form stars which undergo mirror nuclear fusion in their cores. I will discuss the mechanisms by which these objects can emit Standard Model light and estimate their luminosity and prospects for their detection.

An assessment of the particle standard model and an alternative formulation of
the model are presented. An ultraviolet complete particle model is constructed
for the observed particles of the standard model. The quantum field theory
associates infinite derivative entire functions with propagators and vertices, which
make perturbative quantum loops finite and maintain Poincaré invariance and
unitarity of the model. The electroweak model SU(2) X U(1) group is treated as a
broken symmetry group with non-vanishing experimentally determined boson
and fermion masses. A spontaneous symmetry breaking of the vacuum by a
scalar Higgs field is not invoked to restore boson and fermion masses to the
initially massless SU(2) X U(1) Lagrangian of the standard model. The hierarchy
naturalness problem of the Higgs boson mass is resolved and the model
contains only experimentally observed masses and coupling constants. The
renormalization group features of the scalar Higgs field are investigated and the
model is shown to be asymptotically safe and free from the triviality problem. It is
demonstrated that the finite nonlocal quantum field theory satisfies
microcausality. The model can predict a stable vacuum evolution. Experimental
tests to distinguish the standard model from the alternative model are proposed.
The finite quantum field theory can be extended to quantum gravity.