Bosonic topological insulators (BTI) in three dimensions are symmetry-protected topological phases (SPT) protected by time-reversal and boson number conservation symmetries. BTI in three dimensions were first proposed and classified by the group cohomology theory which suggests two distinct root states, each carrying a Z2 index. Soon after, surface anomalous topological orders were proposed to identify different root states of BTI, which even leads to a new BTI root state beyond the group cohomology classification. In this paper, we propose a universal physical mechanism via vortex-line condensation from a 3d superfluid to achieve all three root states. It naturally produces bulk topological quantum field theory (TQFT) description for each root state. Topologically ordered states on the surface are rigorously derived by placing TQFT on an open manifold, which allows us to explicitly demonstrate the bulk-boundary correspondence. Finally, we generalize the mechanism to ZN symmetries and discuss potential SPT phases beyond the group cohomology classification.

We consider a boundary value problem for the Dirac equation in a smooth, asymptotically flat Lorentzian manifold admitting a Killing field which is timelike near and tangential to the boundary. A self-adjoint extension of the Dirac Hamiltonian is constructed. Our results also apply to the situation that the spacetime includes horizons, where the Hamiltonian fails to be elliptic.

The fermionic signature operator is constructed in Rindler space-time. It is shown to be an unbounded self-adjoint operator on the Hilbert space of solutions of the massive Dirac equation. In two-dimensional Rindler space-time, we prove that the resulting fermionic projector state coincides with the Fulling-Rindler vacuum. Moreover, the fermionic signature operator gives a covariant construction of general thermal states, in particular of the Unruh state. The fermionic signature operator is shown to be well-defined in asymptotically Rindler space-times. In four-dimensional Rindler space-time, our construction gives rise to new quantum states.

A modified theory of general relativity is proposed, where the gravitational constant is replaced by a dynamical variable in space-time. The dynamics of the gravitational coupling is described by a family of parametrized null geodesics, implying that the gravitational coupling at a space-time point is determined by solving transport equations along all null geodesics through this point. General relativity with dynamical gravitational coupling (DGC) is introduced. We motivate DGC from general considerations and explain how it arises in the context of causal fermion systems. The underlying physical idea is that the gravitational coupling is determined by microscopic structures on the Planck scale which propagate with the speed of light. In order to clarify the mathematical structure, we analyze the conformal behaviorn and prove local existence and uniqueness of the time evolution. The differences to Einstein’s theory are worked out in the examples of the Friedmann-RobertsonWalker model and the spherically symmetric collapse of a shell of matter. Potential implications for the problem of dark matter and for inflation are discussed. It is shown that the effects in the solar system are too small for being observable in present-day experiments.

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