On tree form-factors in (supersymmetric) Yang-Mills theory

{\it Perturbiner}, that is, the solution of field equations which is a generating function for tree form-factors in N=3 $(N=4)$ supersymmetric Yang-Mills theory, is studied in the framework of twistor formulation of the N=3 superfield equations. In the case, when all one-particle asymptotic states belong to the same type of N=3 supermultiplets (without any restriction on kinematics), the solution is described very explicitly. It happens to be a natural supersymmetrization of the self-dual perturbiner in non-supersymmetric Yang-Mills theory, designed to describe the Parke-Taylor amplitudes. In the general case, we reduce the problem to a neatly formulated algebraic geometry problem (see Eqs(\ref{5.15i}),(\ref{5.15ii}),(\ref{5.15iii})) and propose an iterative algorithm for solving it, however we have not been able to find a closed-form solution. Solution of this problem would, of course, produce a description of all tree form-factors in non-supersymmetric Yang-Mills theory as well. In this context, the N=3 superfield formalism may be considered as a convenient way to describe a solution of the non-supersymmetric Yang-Mills theory, very much in the spirit of works by E.Witten \cite{Witten} and by J.Isenberg, P.B.Yasskin and P.S.Green \cite{2}.Comment: 17 pages, Latex, the form of citation in the abstract have been corrected by xxx.lanl.gov reques

Geometry and Physics on w∞w_{\infty} Orbits

We apply the coadjoint orbit technique to the group of area preserving diffeomorphisms (APD) of a 2D manifold, particularly to the APD of the semi-infinite cylinder which is identified with $w_{\infty}$. The geometrical action obtained is relevant to both $w$ gravity and 2D turbulence. For the latter we describe the hamiltonian, which appears to be given by the Schwinger mass term, and discuss some possible developments within our approach. Next we show that the set of highest weight orbits of $w_{\infty}$ splits into subsets, each of which consists of highest weight orbits of $\bar{w}_N$ for a given N. We specify the general APD geometric action to an orbit of $\bar{w}_N$ and describe an appropriate set of observables, thus getting an action and observables for $\bar{w}_N$ gravity. We compute also the Ricci form on the $\bar{w}_N$ orbits, what gives us the critical central charge of the $\bar{w}_N$ string, which appears to be the same as the one of the $W_N$ string.Comment: 19 pages, LATEX, with notation $w_N$ changed to $\bar{w}_N$, with 3 more references and with note added in proo

On form-factors in Sin(h)-Gordon theory

We present here an explicit classical solution of the type of perturbiner in Sin(h)-Gordon model. This solution is a generating function for form-factors in the tree approximation.Comment: 10 pages, late

First Order Theories of Some Lattices of Open Sets

We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., $\mathbb{R}^n$, $n\geq1$, and the domain $P\omega$) this theory is $m$-equivalent to first order arithmetic