  The present paper is the second part of our project in which we describe
quantum field theories with instantons in a novel way by using the "infinite
radius limit" (rather than the limit of free field theory) as the starting
point. The theory dramatically simplifies in this limit, because the
correlation functions of all, not only topological (or BPS), observables may be
computed explicitly in terms of integrals over finite-dimensional moduli spaces
of instanton configurations. In Part I (arXiv:hep-th/0610149) we discussed in
detail the one-dimensional (that is, quantum mechanical) models of this type.
Here we analyze the supersymmetric two-dimensional sigma models and
four-dimensional Yang--Mills theory, using the one-dimensional models as a
prototype. We go beyond the topological (or BPS) sectors of these models and
consider them as full-fledged quantum field theories. We study in detail the
space of states and find that the Hamiltonian is not diagonalizable, but has
Jordan blocks. This leads to the appearance of logarithms in the correlation
functions. We find that our theories are in fact logarithmic conformal field
theories (theories of this type are of interest in condensed matter physics).
We define jet-evaluation observables and consider in detail their correlation
functions. They are given by integrals over the moduli spaces of holomorphic
maps, which generalize the Gromov--Witten invariants. These integrals generally
diverge and require regularization, leading to an intricate logarithmic mixing
of the operators of the sigma model. A similar structure arises in the
four-dimensional Yang--Mills theory as well.

High Energy Physics - Theory;Algebraic Geometry;Quantum Algebra