Programme

Abstract: In 1941 Kramers and Wannier showed that the Ising model possesses an unusual “duality property” relating its high- and low-temperature phases. Since the latter phase has a spontaneously broken symmetry and the former does not, this mapping must be non-invertible. While straightforward generalisations have long been known, only in recent years have the underlying principles been understood. The key is to express the non-invertible symmetry and duality generators in terms of the same objects that give rise to knot invariants. In these lectures Fendley will provide an introduction to these beautiful connections. In particular, he will show how the fusion categories underlying knot invariants allow one to define topological defects in a wide variety of 2d classical lattice models and as well as quantum spin chains. These defects possess many interesting properties, and he will explain how to use them to compute universal physical quantities directly and exactly on the lattice.

Abstract: In the first lecture, Kong will review the main results in arXiv:1502.01690 (or arXiv:1702.00673). In particular, he will first review the notion of a morphism between topological orders (or QFTs). This notion reveals a deeper meaning of what is later called ‘sandwich construction’. Then he will use it to prove the so-called boundary-bulk relation, which says that bulk is the centre of a boundary. A rather complete formulation of this relation is given by the so-called “centre functor”, which provides a powerful formula to compute “sandwich”, or more generally, to compute the so-called factorisation homology. As a byproduct, the fact that boundary-bulk relation applies to both gapped and gapless quantum liquids also suggests that there is a unified mathematical theory of gapped and gapless quantum liquids.

Abstract: Is it possible to read off the quantum gravity dual of a CFT directly from its operator algebra? In this essay, a step-by-step recipe is presented, synthesising results and techniques from conformal bootstrap, topological symmetries, tensor networks, a novel symmetry-preserving real-space renormalisation algorithm devised originally in lattice models, and the asymptotics of quantum 6j symbols, thereby providing an answer in the affirmative. Quantum 2D Liouville theory serves as a simple and explicit example, illustrating how the quantum gravitational path integral can be built up from local pieces of BCFT correlation functions, which are called the “BCFT Legos”. The constructive map between gravity and CFT naturally and explicitly bridges local geometrical data, algebraic structures, and quantum entanglement, as envisaged by the It from Qubit motto.

Abstract: In the second lecture, Kong will review the unified mathematical theory of gapped and gapless boundaries of 2+1D topological orders developed in arXiv:1705.01087. As a consequence of this theory, a surprising result is discovered called ‘topological Wick rotation’, which was proposed to be generalisable to higher dimensions as a general principle for all quantum liquids (or QFTs) (arXiv:1905.04924).

Abstract: In 1941 Kramers and Wannier showed that the Ising model possesses an unusual “duality property” relating its high- and low-temperature phases. Since the latter phase has a spontaneously broken symmetry and the former does not, this mapping must be non-invertible. While straightforward generalisations have long been known, only in recent years have the underlying principles been understood. The key is to express the non-invertible symmetry and duality generators in terms of the same objects that give rise to knot invariants. In these lectures Fendley will provide an introduction to these beautiful connections. In particular, he will show how the fusion categories underlying knot invariants allow one to define topological defects in a wide variety of 2d classical lattice models and as well as quantum spin chains. These defects possess many interesting properties, and he will explain how to use them to compute universal physical quantities directly and exactly on the lattice.

Abstract: Is it possible to read off the quantum gravity dual of a CFT directly from its operator algebra? In this essay, a step-by-step recipe is presented, synthesising results and techniques from conformal bootstrap, topological symmetries, tensor networks, a novel symmetry-preserving real-space renormalisation algorithm devised originally in lattice models, and the asymptotics of quantum 6j symbols, thereby providing an answer in the affirmative. Quantum 2D Liouville theory serves as a simple and explicit example, illustrating how the quantum gravitational path integral can be built up from local pieces of BCFT correlation functions, which are called the “BCFT Legos”. The constructive map between gravity and CFT naturally and explicitly bridges local geometrical data, algebraic structures, and quantum entanglement, as envisaged by the It from Qubit motto.

Haiyi Luo (3:30), Yu-An Chen (3:55), Jianhao Zhang (4:20), Chenqi Meng (4:45), Weizhen Jia (5:10)

Abstract: Shao will review non-invertible symmetries in several physical systems, including the Ising model, the XX spin chain, and 3+1d QED. Shao will also discuss how these developments have led to new lattice chiral symmetries.

Abstract: In the third lecture, Kong will provide some concrete examples of gapless boundaries of 2+1D topological orders, and explain some generalisations and applications of topological Wick rotation, including a powerful formula to compute gapless sandwiches. He will also explain how topological Wick rotation includes what is later called `SymTFT’ or `topological holography’ as a special case.

Abstract: In Lecture 1, the classification of phases and notion of bulk-boundary correspondence in periodically driven (Floquet) systems will be reviewed. These are non-equilibrium systems with structures that closely resemble those in equilibrium contexts. In Lecture 2, recent work in open quantum systems will be explored, covering phases, phase transitions, symmetries/anomalies, and error correction.

Abstract: Shao will review non-invertible symmetries in several physical systems, including the Ising model, the XX spin chain, and 3+1d QED. He will also discuss how these developments have led to new lattice chiral symmetries.

Abstract: Phase transitions in quantum systems are intimately tied to how symmetry is realised in the ground state. Across a transition, the system may pass between phases distinguished by distinct patterns of spontaneous symmetry breaking, symmetry fractionalisation, or symmetry-protected topological order.
These lectures explore phase transitions governed by generalised symmetries as above in continuous gauge theories, with a particular focus on higher-form symmetries. These lectures begin by developing the framework of gauge theories enriched with higher-form symmetries, examining in detail how these symmetries can fractionalize. Then turning to the role of bosons and fermions in driving phase transitions that preserve the generalised symmetry structure, while realising the symmetry differently in the ground states on either side of the transition.

Coimbatore Balram Ajit (3:30), Bai Chen (3:55), Shuo Yang (4:20), Qing-Rui Wang (4:45), Weicheng Ye (5:10), Xingyu Ren (5:35) 

Abstract: Phase transitions in quantum systems are intimately tied to how symmetry is realised in the ground state. Across a transition, the system may pass between phases distinguished by distinct patterns of spontaneous symmetry breaking, symmetry fractionalisation, or symmetry-protected topological order.

These lectures explore phase transitions governed by generalised symmetries as above in continuous gauge theories, with a particular focus on higher-form symmetries. These lectures begin by developing the framework of gauge theories enriched with higher-form symmetries, examining in detail how these symmetries can fractionalize. Then turning to the role of bosons and fermions in driving phase transitions that preserve the generalised symmetry structure, while realising the symmetry differently in the ground states on either side of the transition.

Abstract: In Lecture 1, the classification of phases and the notion of bulk-boundary correspondence in periodically driven (Floquet) systems will be reviewed. These are non-equilibrium systems with structures that closely resemble those in equilibrium contexts. In Lecture 2, recent work in open quantum systems will be explored, specifically focusing on phases, phase transitions, symmetries/anomalies, and error correction.