  Group field theories (GFT) are higher dimensional generalizations of matrix
models whose Feynman diagrams are dual to triangulations. Here we propose a
modification of GFT models that includes extra field indices keeping track of
the bubbles of the graphs in the Feynman evaluations. In dimension three, our
model exhibits new symmetries, interpreted as the action of the vertex
translations of the triangulation. The extra field indices have an elegant
algebraic interpretation: they encode the structure of a semi-simple algebra.
Remarkably, when the algebra is chosen to be associative, the new structure
contributes a topological invariant from each bubble of the graph to the
Feynman amplitudes.
