Continuation of Direct Products of Distributions

If, in some problems, one has to deal with the ``product'' of distributions $\rm f_i$ (also called generalized functions) $\rm\bar T = \Pi^m_{i=1} f_i$, this product has a priori no definite meaning as a functional $(\rm \bar T, \phi)$ for $\rm\phi \in S$. But if $\rm x^{\kappa +1} \Pi^m_{i=1} f_i$ exists, whatever the associativity is between some powers $\rm r_i$ of $\rm x$ ($\rm r_i \in \Bbb N, \sum_i r_i\leq \kappa +1, r_i \geq 0$) and the various $\rm f_i$, then a continuation of the linear functional $\rm \bar T$ from $\rm M$ onto $\rm S^{(N)}$ for some $\rm N$ is shown to exist in such a way that $\rm x^{\kappa +1} \bar T$ is defined unambiguously, and $\rm (\bar T, \phi), \phi \in S$, significant, though not unique.Comment: 3 pages, late

On the Four-Color-Map Theorem

Coloring planar Feynman diagrams in spinor quantum electrodynamics, is a non trivial model soluble without computer. Four colors are necessary and sufficient.Comment: 8 page

A search for exact superstring vacua

We investigate $2d$ sigma-models with a $2+N$ dimensional Minkowski signature target space metric and Killing symmetry, specifically supersymmetrized, and see under which conditions they might lead to corresponding exact string vacua. It appears that the issue relies heavily on the properties of the vector $M_{\mu}$, a reparametrization term, which needs to possess a definite form for the Weyl invariance to be satisfied. We give, in the $n = 1$ supersymmetric case, two non-renormalization theorems from which we can relate the $u$ component of $M_{\mu}$ to the $\beta^G_{uu}$ function. We work out this $(u,u)$ component of the $\beta^G$ function and find a non-vanishing contribution at four loops. Therefore, it turns out that at order $\alpha^{\prime 4}$, there are in general non-vanishing contributions to $M_u$ that prevent us from deducing superstring vacua in closed form.Comment: 9 pages, latex, CERN-TH.6946/9

Explicit supertring vacua in a background of gravitational waves and dilaton

We present an explicit solution of superstring effective equations, represented by gravitational waves and dilaton backgrounds. Particular solutions will be examined in a forthcoming note.Comment: 10 pages, latex, CERN-TH.7373/9