Complex Topological Invariants of Three-Hyperbolic Manifolds

We derive invariants associated with compact real three-dimensional hyperbolic manifolds which can be combined to form a component of Thurston complex invariant. Explicit formulas for the Chern-Simons invariant of irreducible U(n)-flat connections on hyperbolic fibered manifolds are obtained. We discuss supersymmetry surviving for supergravity solutions involving real hyperbolic factors, and briefly review the determinant line bundle with its connection to the eta invariant, the global anomaly formula, and the vanishing theorems for type (0,q) cohomology of locally symmetric spaces.