We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra g^, from g-module homomorphisms. When g=sl_2, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to g^, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.

We propose a polynomial time computable graph invariant, which comes out directly from a modified version of discrete Greens function on a graph. It is our hope that it can help to construct fast algorithms for the graph isomorphism testing. We explain the physics behind this discrete Greens function, which is a very basic idea applied to graphs.

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For the SU (N) invariant supersymmetric matrix model related to membranes in 4 space-time dimensions we argue that< Psi, chi>= 0 for the previously obtained solution of Q chi= 0, Q^{dagger} Psi= 0.

We classify three dimensional isolated weighted homogeneous rational complete intersection singularities, which define many new four dimensional N= 2 superconformal field theories. We also determine the mini-versal deformation of these singularities, and therefore solve the Coulomb branch spectrum and Seiberg-Witten solution.