(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

• Main Author: Bianca Dittrich
• Format: Article
• Language: English
• Published: SpringerOpen 2017-05-01
• Series: Journal of High Energy Physics
• Subjects: Models of Quantum Gravity; Topological States of Matter
• Online Access: http://link.springer.com/article/10.1007/JHEP05(2017)123

Abstract We apply the recently suggested strategy to lift state spaces and operators for (2 + 1)-dimensional topological quantum field theories to state spaces and operators for a (3 + 1)-dimensional TQFT with defects. We start from the (2 + 1)-dimensional TuraevViro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects. This work has important applications for quantum gravity as well as the theory of topological phases in (3 + 1) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably. We in particular show that the fusion bases of the (2 + 1)-dimensional theory lead to a rich set of bases for the (3 + 1)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian. Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.