Finite elements for symmetric and traceless tensors in three dimensions

We construct a family of finite element sub-complexes of the conformal complex on tetrahedral meshes and show their exactness on contractible domains. This complex includes vector fields and symmetric and traceless tensor fields, connected through the conformal Killing operator, the linearized Cotton-York operator, and the divergence operator, respectively. This leads to discrete versions of transverse traceless (TT) tensors, i.e., symmetric, traceless and divergence-free matrix fields, in continuum mechanics and general relativity. We also show the inf-sup stability of the $H(\mathrm{div})$-conforming finite element symmetric and traceless tensors paired with discontinuous vectors.