Exchange Relations for the q-Vertex Operators of Uq(sl2^)U_q(\widehat{sl2})

We consider the q-deformed Knizhnik-Zamolodchikov equation for the two point function of q-deformed vertex operators of $U_q(sl_2^)$. We give explicitly the fundamental solutions, the connection matrices and the exchange relations for the q-vertex operators of spin 1/2 and $j \in {1\over 2}{\bf Z}_{\geq 0}$. Consequently, we confirm that the connection matrices are equivalent to the elliptic Boltzman weights of IRF type obtained by the fusion procedure from ABF models.Comment: 20p

Screening Currents Ward Identity and Integral Formulas for the WZNW Correlation Functions

We derive, based on the Wakimoto realization, the integral formulas for the WZNW correlation functions. The role of the ``screening currents Ward identity'' is demonstrated with explicit examples. We also give a more simple proof of a previous result.Comment: 26 page

Five-dimensional AGT Relation and the Deformed beta-ensemble

We discuss an analog of the AGT relation in five dimensions. We define a q-deformation of the beta-ensemble which satisfies q-W constraint. We also show a relation with the Nekrasov partition function of 5D SU(N) gauge theory with N_f=2N.Comment: 38page. References and an appendix for 4D case added. Typos correcte

Many-body Dynamics of D0--Branes

We show that the growth of the size with the number of partons holds in a Thomas-Fermi analysis of the threshold bound state of D0--branes. Our results sharpen the evidence that for a fixed value of the eleven dimensional radius the partonic velocities can be made arbitrarily small as one approaches the large N limit.Comment: 9 pages, latex, minor change

On the Quantization of Nambu Brackets

We present several non-trivial examples of the three-dimensional quantum Nambu bracket which involve square matrices or three-index objects. Our examples satisfy two fundamental properties of the classical Nambu bracket: they are skew-symmetric and they obey the Fundamental Identity. We contrast our approach to the existing literature on the quantum deformations of Nambu mechanics. We also discuss possible applications of our results in M-theory.Comment: 18 pages, LaTeX fil