70th North British Mathematical Physics Seminar

Abstract: I will give a general overview of the symmetries of renormalization group fixed points, including scale, conformal and Weyl invariance in context of quantum field theory. I will discuss the distinction between these symmetries, as well as what is generally known about when scale invariant theories can be assumed to be conformally invariant, and when conformally invariant theories can be assumed to be Weyl invariant. Time permitting, I will also discuss recent work on understanding these symmetries in theories without a stress-energy tensor, using linearized gravity as an example.

Abstract: As first described in detail by Allcock in 1969, there exist solutions to the free Schroedinger equation in 1 dimension with non-negative momentum whose probability current, counterintuitively, achieves arbitrarily negative values. The integral of the current however, over any given fixed time period [t_1,t_2] is bounded from below by a value \approx -0.04. This phenomenon of a state exhibiting probability flow in the opposite direction to its momentum is named quantum backflow. In 1994, Bracken and Melloy converted the question of understanding quantum backflow into that of an eigenvalue problem of an associated bounded operator on $L^2(\mathbb{R})$. The spectrum of this operator coincides with the possible probability flow that a state can exhibit. What has not yet been considered is what this probability flow looks like over multiple time periods. I will show, given times parametrizing the backflow periods (t_1,...,t_{2M}), that there is a corresponding bounded self-adjoint operator whose spectrum corresponds to the possible values of the backflow contributing from the intervals [t_1,t_2],[t_3,t_4],...,[t_{2M-1},t_{2M}]. Further, I will give both interesting and counterintuitive results regarding the spectra and convergence of these operators in a variety of limits.

Abstract: The double copy is a powerful tool connecting gauge theoretic and gravitational scattering amplitudes. It was originally derived from string theory, relating the tree level amplitudes of closed string amplitudes to two copies of open string amplitudes. In the field theory limit, this reduces to being able to obtain tree-level graviton amplitudes from the "square" of tree-level gluon amplitudes. At the same time, these field theory amplitudes have miraculously simple expressions as integrals over the moduli space of rational maps to twistor space for any number of external legs and helicity configuration of gluons and gravitons. However, the double copy relation between these formulae has historically been extremely non-obvious. In this talk, I will use concepts from graph theory to demonstrate the derivation of a double copy based in twistor space, and explore what this can teach us about the relation between gauge theory and gravity.

Abstract: Line operators in 3d topological QFT's are expected to have the structure of a braided tensor category. In turn, braided tensor categories tend to arise mathematically as modules for quasi-triangular Hopf algebras, a.k.a. "quantum groups." Nevertheless, it has been a notoriously difficult problem to explicitly locate these quantum groups in 3d physics. I will propose a general approach to solving this problem in the case of QFT's that admit topological boundary conditions, inspired by recent work of N. Aamand for Chern-Simons theory. Simple applications include finding quantum groups in Dijkgraaf-Witten theory (a.k.a. finite-group gauge theory) and topologically twisted 3d N=4 gauge theories. (Joint work in progress with T. Creutzig and W. Niu.)

Abstract: It is well-known that local operators in quantum field theory transform in linear representations of a global symmetry group. It is natural to ask how this extends to generalised global symmetries, which have been an active area of research over the past decade. In this talk, I will review how twisted-sector local operators in two dimensions transform in representations of the tube algebra associated to some generalised symmetry, and describe generalisations of this construction to twisted-sector local and line operators in three dimensions. If time permits, I will also discuss how the notion of unitarity can be extended to generalised (and in particular non-invertible) symmetries.