Week 04.12.2023 – 10.12.2023

I will discuss some of the recent developments in the Krylov complexity. In particular, I will focus on the applications of the Krylov basis techniques to the modular Hamiltonian evolution and I will discuss a new angle on entanglement entropy in QCD at high energies. Based on arXiv:2306.14732 [hep-th] and work in progress.

Topological quantum field theory can emerge in gapped many-body quantum systems at low energies. In 2+1D systems, anyons can emerge, and in 3+1D, emergent excitations, including point-particles and loops-like excitations, possibly knotted or linked. In this lecture, we introduce an ongoing effort to understand (in fact, derive) laws of the emergent theory in 2+1D, 3+1D, (and higher D) gapped systems from a few axioms about the entanglement of a many-body ground state wave function. This research program, referred to as entanglement bootstrap, is an approach independent of quantum field theory, and it uses nontrivial quantum information and topology ideas. We explain the axioms and key concepts. We sketch the proof of several main theorems, including the definition of superselection sectors (anyons in 2+1D, point and loop excitations in 3+1D), the fusion spaces, and their constraints. We explain why immersion (i.e., local embedding) is valuable for, e.g., putting systems on closed space manifolds and what we hope to learn next. (see https://www.london-tqft.co.uk for more details)

When do two different looking quantum field theories describe the same physics? This is essentially asking when the quantum field theories are isomorphic. In the case of topological quantum field theories, there are sometimes a way to determine them via topological invariants. For a superconformal field theory, what would be the minimal set of “invariants” to determine when they are isomorphic? I will discuss some approaches to this question in the context of superconformal field theories in four and six dimensions. Utilizing 4d class S theories that also admits 6d (1,0) SCFT origins, I will explain how a certain class of 4d N=2 SCFTs, which a priori look like distinct theories, can be shown to describe the same physics. I will further explain how the 6d (1,0) origin sheds light on the 3d duality.

We analyze a model of qubits which we argue has an emergent quantum gravitational description similar to the fermionic Sachdev-Ye-Kitaev (SYK) model. The model is generic in that it includes all possible q-body couplings, lacks most symmetries, and has no spatial structure, so our results can be construed as establishing a certain ubiquity of quantum holography in systems dominated by many-body interactions. We will discuss implications for Hamiltonian complexity, the factorization problem in quantum gravity, and quantum simulations of holography. Based on 2311.01516 with Mike Winer.

I will explain how to describe form factors of single-trace half-BPS operators in planar N=4 super Yang Mills theory using the T-dual Wilson loop picture. After reviewing earlier results for operators in the stress-tensor multiplet, I will present the dual Wilson loop description for the so-called MHV form factors of half-BPS operators. The general proposal relates these form factors to the matrix elements of a null periodic super Wilson loop with outgoing states composed of zero-momentum scalars. I will present perturbative tests of this description at weak coupling. I will then explain how to obtain exact result at finite coupling in the collinear limit using the Wilson loop Operator Product Expansion. I will conclude with general comments and speculations about form factors of unprotected operators such as the Konishi operator.

In the calculation of the perturbative amplitude of superstring theory at one loop, modular graph functions (MGFs) emerge as notable mathematical constructs. These MGFs, representing Feynman diagrams on the surface of a torus, must be integrated over the fundamental domain. My talk will introduce MGFs, elucidate their generating series, and delve into the concept of equivariance, playing a key role in classifying MGFs. Additionally, I will cover recent advancements in understanding the generating series of MGFs and the integration of MGFs over the fundamental domain.

In the 2nd lecture, we dig into the underlying logic of entanglement bootstrap. Illustrative examples are aimed to be simple but nontrivial. The following will be included: (1) We explain a few basic uses of strong subadditivity and quantum Markov states; we explain why axiom A0 is crucial to protect coherence. We derive the information convex set of the sphere as an application. (2) Related to the information convex set of the annulus, we explain the definition of quantum dimensions, why the vacuum has the smallest entropy, and why a certain "merged state" has the maximum entropy. (3) We classify immersed annuli on a sphere and explain why some puzzles of figure-8 annulus are not solved in naive ways. (4) In the context the reference state has a 0-form symmetry, we sketch a way to create a symmetry defect line, which (in some models) permutes anyons. This lecture is given using an ipad and is, thus, flexible. We also discuss topics from questions (feedbacks) during the 1st (and 2nd) lecture. See https://www.london-tqft.co.uk for details.