Topological gravity, defects, and topological phases protected by both spatial and internal symmetries

A substantial attention of quantum gravity has been paid in different areas of physics, from high energy physics, condensed matter physics to quantum information science. In this paper, we explore the topological quantum gravity from the perspective of symmetry-protected topological (SPT) phases. To be specific, we study the gravitational effect on the classification and response of 3+1d topological phases protected simultaneously by both spatial and internal symmetries from the topological response theory. The latter describes how geometric defect (e.g. disclination) and internal symmetry defect (e.g. domain wall) are topologically entangled. New topological terms involving nontrivial geometries are proposed, which are extensions of purely internal topological theories discussed before. Physical observables derived from these response theories are discussed and compared with their analogs in 2+1d systems, some of which have been well-studied in quantum Hall (QH) systems, like the Wen-Zee (WZ) term. Generalizations to symmetry-enriched topological (SET) phases are also discussed (arXiv:1807.10844).