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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 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.

In their proposed compactification of superstrings [4], Candelas, Horowitz, Strominger and Witten took the matric product of a maximal symmetric four dimensional spacetime M with a six dimensional CalabiYau vacua X as the ten dimensional spacetime; they identified the YangMills connection with the SU (3) connection of the CalabiYau metric and set the dilaton to be a constant. To make this theory compatible with the standard grand unified field theory, Witten [28] and HoravaWitten [20] proposed to use higher rank bundles for strong coupled heterotic string theory so that the gauge groups can be SU (4) or SU (5). Mathematically, this approach relies on UhlenbeckYaus theorem on constructing HermitianYangMills connections on stable bundles [27]. Many authors, including Friedman, Morgan and Witten [18]; Donagi, Ovrat, Pantev and Reinbacher [12]; Andreas [1], Kachru [21] and others, have worked

Let M be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary M of positive scalar curvature. We show that under these conditions, M and in particular M must be connected. These results resolve some puzzles concerning the AdS/CFT correspondence.