Symmetry protected topological (SPT) states are a new kind of gapped states that do not break any symmetry and have no topological order.
Despite having no symmetry breaking order and having no topological order (ie no fractionalized topological excitations and no robust gapless boundary excitations), SPT states can still be different non-trivial new states of matter, with different physics properties at the boundary. I will present
(1) a systematic theory of SPT order based on non-linear sigma model and its topological terms, which turns out to be a group cohomology theory
(2) ways to probe/measure SPT orders, and the connection to gauge/gravitational anomalies, which turns out to be an algebraic topology theory
(3) mechanism to generate SPT orders