  We introduce a generalization of the celebrated Lov\'asz theta number of a
graph to simplicial complexes of arbitrary dimension. Our generalization takes
advantage of real simplicial cohomology theory, in particular combinatorial
Laplacians, and provides a semidefinite programming upper bound of the
independence number of a simplicial complex. We consider properties of the
graph theta number such as the relationship to Hoffman's ratio bound and to the
chromatic number and study how they extend to higher dimensions. Like in the
case of graphs, the higher dimensional theta number can be extended to a
hierarchy of semidefinite programming upper bounds reaching the independence
number. We analyze the value of the theta number and of the hierarchy for dense
random simplicial complexes.
