In this paper, we provide state sum path integral definitions of exotic invertible topological phases proposed in the recent paper by Hsin, Ji, and Jian [arXiv:2105.09454 [cond-mat.str-el]. The exotic phase has time-reversal (T) symmetry, and depends on a choice of the space-time structure called the Wu structure. The exotic phase cannot be captured by the classification of any bosonic or fermionic topological phases and thus gives a novel class of invertible topological phases. When the T symmetry defect admits a spin structure, our construction reduces to a sort of the decorated domain wall construction, in terms of a bosonic theory with T symmetry defects decorated with a fermionic phase that depends on a spin structure of the T symmetry defect. By utilizing our path integral, we propose a lattice construction for the exotic phase that generates the Z8 classification of the (3+1)d invertible phase based on the Wu structure. This generalizes the Z8 classification of the T-symmetric (1+1)d topological superconductor proposed by Fidkowski and Kitaev. On oriented space-time, this (3+1)d invertible phase with a specific choice of Wu structure reduces to a bosonic Crane-Yetter TQFT which has a topological ordered state with a semion on its boundary. Moreover, we propose a subclass of G-SPT phases based on the Wu structure labeled by a pair of cohomological data in generic space-time dimensions. This generalizes the Gu-Wen subclass of fermionic SPT phases.
Symmetry, Structure, and Spacetime, Volume 3 Repost
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2. Introducing Differential Geometry: PDF Manifolds: Topological spaces, differentiable manifolds and maps between manifolds. Tangent Spaces: tangent vectors, vector fields, integral curves and the Lie derivative. Tensors, covectors and one-forms. Differential Forms: the exterior derivative, de Rahm cohomology, integration and Stokes' theorem.
3. Introducing Riemannian Geometry: PDF The metric; Riemannian and Lorentzian manifolds, the volume form and the Hodge dual. The Maxwell action. Hodge theory. Connections and the covariant derivative, curvature and torsion, the Levi-Civita connection. The divergence theorem. Parallel transport, normal coordinates and the exponential map, holonomy, geodesic deviation. The Ricci tensor and Einstein tensor. Connection 1-forms and curvature 2-forms.
4. The Einstein Equations: PDF The Einstein-Hilbert action, the cosmological constant; diffeomorphisms and the Bianchi identity; Minkowski, de Sitter and anti-de Sitter spacetimes; Symmetries and isometries, Killing vectors, conserved quantities; Asymptotics of spacetime, conformal transformations and Penrose diagrams; Coupling matter, the energy-momentum tensor, perfect fluids, spinors, energy conditions; Cosmology.
5. When Gravity is Weak: PDF The Linearised theory, gauge symmetry, the Newtonian limit; Gravitational waves, de Donder gauge, transverse traceless gauge, LIGO; Gravitational wave production, binary systems, the quadrupole formula, gravitational wave sources.
6. Black Holes: PDF The Schwarzschild solution, Birkhoff's theorem, Eddington-Finkelstein Coordinates, Kruskal diagrams and Penrose diagrams, weak cosmic censorship; The Reissner-Nordstrom solution, Cauchy horizons and strong cosmic censorship, Extremal black holes; The Kerr solution, global structure, the ergoregion, the Penrose process and superradiance, no hair theorems.
Problem SheetsJoão Melo has put together a preparatory worksheet, based on Chapter 1 of the lectures notes, to help refresh your understanding of geodesics before the course begins. It can be downloaded here. 2ff7e9595c
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