For this after dinner talk I intersperse images of real lattices with a discussion of the motivations for lattice gauge theory and some current unresolved issues.

Gauge invariance is used to establish an equivalence relation between solutions of the time-independent and time-dependent Ginzburg-Landau equations that describe the same physical state of a superconductor. The equivalence relation shows how equilibrium conngurations are obtained as large-time asymptotic limits of solutions of the time-dependent Ginzburg-Landau equations.

We describe how and to what extent the noncommutative two-torus can be approximated by a tower of finite-dimensional matrix geometries. The approximation is carried out for both irrational and rational deformation parameters by embedding the C∗-algebra of the noncommutative torus into an approximately finite algebra. The construction is a rigorous derivation of the recent discretizations of non...

In a previous publication [hep-lat/9707022] I showed that the fermion determinant for strictly massless quarks can be written on the lattice as detD, where D is a certain finite square matrix explicitly constructed from the lattice gauge fields. Here I show that D obeys the Ginsparg-Wilson relation Dγ5D = Dγ5 + γ5D.

We consider recently-constructed solutions of three dimensional SL(N,R)×SL(N,R) Chern-Simons theories with non-relativistic symmetries. Solutions of the Chern-Simons theories can generically be mapped to solutions of a gravitational theory with a higherspin gauge symmetry. However, we will show that some of the non-relativistic solutions are not equivalent to metric solutions, as this mapping f...

We propose a large N dual of 4d, N = 1 supersymmetric, SU(N) Yang-Mills with adjoint field Φ and arbitrary superpotential W (Φ). The field theory is geometrically engineered via D-branes partially wrapped over certain cycles of a non-trivial Calabi-Yau geometry. The large N , or low-energy, dual arises from a geometric transition of the Calabi-Yau, where the branes have disappeared and have bee...

Spontaneous breaking of local Lorentz symmetry occurs when a local vector or tensor field acquires a nonzero vacuum expectation value. The effects of such breaking are examined in the context of gravity theory. These include an associated spontaneous breaking of diffeomorphism symmetry and generation of massless Nambu-Goldstone modes. The possibility of a Higgs mechanism is examined as well, an...