Chern-Simons Term for BF Theory and Gravity as a Generalized Topological Field Theory in Four Dimensions

Using the Generalized Differential Calculus, we establish the generalized Chern-Weil homomormism, re-derive the geometric properties for both P (M4, G) and pseudo-Riemannian spacetime manifolds from BF theories, and obtain the generalized Chern-Simons term for BF theory in four dimensions. It gives a direct relation between these two types of topological field theories. We then obtain the BF gravity without matter and its action is the generalized second Chern class with gauge group SL(2, C). General Relativity in the absence of matter is derived either as a special generalized topological field theory or by adding constraints to the topological field theory. PACS number(s): 02.40, 04.20 Typeset using REVTEX E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]