Week 24.03.2025 – 29.03.2025

Symmetry plays an important role in quantum systems: it can constrain the dynamics, give rise to selection rules, and provide computation methods in quantum computers. In recent years there are also new types of symmetries called generalized symmetries discovered in many quantum systems, including non-invertible symmetry and higher group symmetry. These lectures will be about symmetries in various quantum systems and their applications such as constraints on the low energy dynamics. Examples will be discussed in the lectures include quantum mechanics systems, gauge theories, lattice models, and the symmetry includes ordinary and higher form symmetry as well non-invertible symmetry.

I will discuss the M2-brane partition function for large class of asymptotically locally AdS_4 x S^7 spacetimes. I will show how supersymmetry localises the M2-brane position to a fixed point of an R-symmetry Killing vector. I will then discuss the one-loop partition function of instantonic M2-branes and show that it is assembled out of building blocks familiar to 3D supersymmetric quantum field theories. I will close out with a discussion of possible one-loop exactness of the answer and what it means for supersymmetric localisation of the M2-brane partition function.

14:00-14.30 Speaker: Koji Hashimoto (Kyoto university) Title: "Neural network representation of quantum systems" Abstract: We provide a novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters. Our simple idea is to use the universal approximation theorem of neural networks to generate arbitrary paths in the Feynman's path integral. The map can be applied to interacting quantum systems / field theories, even away from the Gaussian limit. Our findings bring machine learning closer to the quantum world. The talk is based on a collaboration with Y. Hirono, J. Maeda and J. Totsuka-Yoshinaka, https://arxiv.org/abs/2403.11420 14:30-15:00 Speaker: Akio Tomiya (Tokyo Woman's Christian University) Title: "CASK: A Gauge Covariant Transformer for Lattice Gauge Theory" Abstract: We introduce a Transformer architecture for lattice QCD that is designed to respect the gauge symmetry and the discrete rotational and translational symmetries of the lattice. The core innovation lies in defining the attention matrix via a Frobenius inner product between link variables and extended staples, ensuring gauge covariance. We apply this method to self-learning HMC and find that it surpasses existing gauge covariant neural networks in performance, demonstrating its potential to enhance lattice QCD computations. This talk is based on arXiv:2501.16955.

How well do we understand string theory? As one indicator can be thought of the degree of our understanding of its spectrum. Yet, other than comprising infinitely many physical states of arbitrarily high spin and mass, what does the string spectrum look like? Traditional methodologies can yield its state content on a level-by-level basis, a straightforward procedure which however becomes cumbersome as the level increases. In this seminar, we will discuss a new, covariant and efficient technology with which entire physical trajectories can be excavated. It is based on the observation that the Virasoro constraints in fact encode the generators of a bigger algebra, that is a symplectic algebra, which commutes with the spacetime Lorentz algebra. This enables constructing trajectories deeper inside the spectrum as clones of simpler ones, upon suitably dressing the latter, depending on the depth of the trajectory we aim to reach.

Orbifolding the N=4 SYM theory naively breaks its SU(4) R-symmetry group to a much smaller subgroup, such as SU(2)xU(1) if N=2 supersymmetry is preserved. I will discuss how, by extending our notion of symmetry to include Lie groupoids and their twists, one can recover the broken generators and show that, at the planar level, a version of the full SU(4) symmetry is still present. I will briefly discuss the implications of this hidden symmetry as far as the planar spectrum of the theory is concerned.