Surgery in colored tensor models

Abstract

Rooted in group field theory and matrix models, random tensor models are a recent background-invariant approach to quantum gravity in arbitrary dimensions. Colored tensor models (CTM) generate random triangulated orientable (pseudo)-manifolds. We analyze, in low dimensions, which known spaces are triangulated by specific CTM interactions. As a tool, we develop the graph-encoded surgery that is compatible with the structure of quantum field theory and use it to prove that a single model, the complex $\varphi^4$-interaction in rank-$2$, generates all orientable $2$-bordisms, thus, in particular, also all orientable, closed surfaces. We show that certain quartic rank-$3$ CTM, the $\varphi_3^4$-theory, has as boundary sector all closed, possibly disconnected, orientable surfaces. Hence all closed orientable surfaces are cobordant via manifolds generated by the $\varphi_3^4$-theory.