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In this talk I will discuss a novel mechanism  that couples matter fields to three-dimensional de Sitter quantum gravity. This construction is based on the Chern-Simons formulation of three-dimensional Euclidean gravity, and it centers on a collection of Wilson loops winding around Euclidean de Sitter space. We coin this object a Wilson spool.  To construct the spool, we build novel representations of su(2). To evaluate the spool, we adapt and exploit several known exact results in Chern-Simons theory. Our proposal correctly reproduces the one-loop determinant of a free massive scalar field on S^3 as G_N->0. Moreover, allowing for quantum metric fluctuations, it can be systematically evaluated to any order in perturbation theory.   

We propose new G2-holonomy manifolds, which geometrize the Gaiotto-Kim 4d N = 1 duality
domain walls of 5d N = 1 theories. These domain walls interpolate between different extended
Coulomb branch phases of a given 5d superconformal field theory. Our starting point is the
geometric realization of such a 5d superconformal field theory and its extended Coulomb
branch in terms of M-theory on a non-compact singular Calabi-Yau three-fold and its Kahler
cone. We construct the 7-manifold that realizes the domain wall in M-theory by fibering the
Calabi-Yau three-fold over a real line, whilst varying its Kahler parameters as prescribed by
the domain wall construction. In particular this requires the Calabi-Yau fiber to pass through
a canonical singularity at the locus of the domain wall. Due to the 4d N = 1 supersymmetry
that is preserved on the domain wall, we expect the resulting 7-manifold to have holonomy G2.
Indeed, for simple domain wall theories, this construction results in 7-manifolds, which are
known to admit torsion-free G2-holonomy metrics. We develop several generalizations to new
7-manifolds, which realize domain walls in 5d SQCD theories.

We argue that the existence of solitons in theories in which local symmetries are spontaneously broken requires spacetime to be enlarged by additional coordinates that are associated with large local transformations. In the context of gravity theories the usual coordinates of spacetime can be thought of arising in this way. E theory automatically contains such an enlarged spacetime. We propose that spacetime appears in an underlying theory when the local symmetries are spontaneously broken.

There has been recent advances in understanding the microscopic origin of the Bekenstein-Hawking entropy of supersymmetric ant de Sitter (AdS) black holes using holography and localization applied to the dual quantum field theory. In this talk, after a brief overview of the general picture, I will propose a BPS partition function -- based on gluing elementary objects called gravitational blocks -- for known AdS black holes with arbitrary rotation and generic magnetic and electric charges. I will then show that the attractor equations and the Bekenstein-Hawking entropy can be obtained from an extremization principle.

Random unitary circuits (RUCs) have served as important sources of insights in studying operator dynamics. While the simplicity of RUCs allows us to understand the nature of operator growth in a quantitative way, randomness of the dynamics in time prevents them to capture certain aspects of operator dynamics. To explore these aspects, in this talk, I consider the operator dynamics of a minimal Floquet many-body circuit whose time-evolution operator is fixed at each time step. In particular, I compute the partial spectral form factor of the model and show that it displays nontrivial universal physics due to operator dynamics. I then discuss the out-of-ordered correlator of the system, which turns out to capture the main feature of it in a generic chaotic many-body system, even in the infinite on-site Hilbert space dimension limit.

I will discuss various possible ways a global symmetry can act on operators in a quantum field theory. The possible actions on q-dimensional operators are referred to as q-charges of the symmetry. Crucially, there exist generalized higher-charges already for an ordinary global symmetry described by a group G. The usual charges are 0-charges, describing the action of the symmetry group G on point-like local operators, which are well-known to correspond to representations of G. We find that there is a neat generalization of this fact to higher-charges: i.e. q-charges are (q+1)-representations of G. I will also discuss q-charges for generalized global symmetries, including not only invertible higher-form and higher-group symmetries, but also non-invertible categorical symmetries. This talk is based on a recent (arXiv: 2304.02660) and upcoming works with Sakura Schafer-Nameki.

Let $V$ be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let $[V]$ be the vector space of complex valued functions on $V$ and let $[V]_{\mathbb Z}$ be the subgroup of $[V]$ consisting of integer valued functions. We show that there exists a Z-basis of $[V]_{\mathbb Z}$ consisting of characteristic functions of certain explicit isotropic subspaces of $V$ such that the matrix of the Fourier transform from $[V]$ to $[V]$ with respect to this basis is triangular. This continues the tradition started by Hermite who described eigenvectors for the Fourier transform over real numbers.

Recently there is a rising interest in the research of mean-field optimization, in particular because of its role in analyzing the training of neural networks. In this talk, by adding the Fisher Information (in other word, the Schrodinger kinetic energy) as the regularizer, we relate the mean-field optimization problem with a so-called mean field Schrodinger (MFS) dynamics. We develop a free energy method to show that the marginal distributions of the MFS dynamics converge exponentially quickly towards the unique minimizer of the regularized optimization problem. We shall see that the MFS is a gradient flow on the probability measure space with respect to the relative entropy. Finally we propose a Monte Carlo method to sample the marginal distributions of the MFS dynamics. This is a joint work with Giovanni Conforti, Zhenjie Ren and Songbo Wang.

The Anderson operator is a perburbation of the Laplace-Beltrami operator by a space white noise potential. I will explain how to get a short self-contained functional analysis construction of the operator and how a sharp description of its heat kernel leads to useful quantitative estimates on its eigenvalues and eigenfunctions. One can associate to Anderson operator a (doubly) random field called the Anderson Gaussian free field. The law of its (random) partition function turns out to characterize the law of the spectrum of the operator. The square of the Anderson Gaussian free field turns out to be related to a probability measure on paths built from the operator, called the polymer measure.