The Geometry of Lattice Field Theory

Using some tools of algebraic topology a general formalism for lattice field theory is presented. The lattice is taken to be a simplicial complex that is also a manifold and is referred to as a simplicial manifold. The fields on this lattice are cochains, that are called lattice forms to emphasize the connections with differential forms in the continuum. This connection provides a new bridge between lattice and continuum field theory. When a continuum field theory is written in terms of differential forms, the translation to a lattice is usually straightforward. A metric can be put onto this simplicial manifold by assigning lengths to every link or l-simplex of the lattice. Regge calculus is a way of defining general relativity on this lattice. A geometric discussion of Regge calculus is presented. The Regge action, which is a discrete form of the Hilbert action, is derived from the Hilbert action using distribution valued forms. This is a new derivation that emphasizes the underlying geometry. Kramers-Wannier duality in statistical mechanics is discussed in this general setting. The notation borrowed from algebraic topology provides a powerful tool for discussing this duality transformation in a way that displays the underlying geometry. Kramers-Wannier duality is related to electromagnetic duality in gauge theories. This is discussed along with a discussion of topological charge, which treats magnetic monopoles and vortices in an analogous manner. Nonlinear field theories, which include gauge theories and nonlinear sigma models are discussed in the continuum and then are put onto a lattice. The main new result here is the generalization to curved spacetime, which consists of making the theory compatible with Regge calculus.