It is well known that space-time coordinates and corresponding Dp-brane worldvolume become non-commutative, if open string ends on Dp-brane with Neveu-Schwarz background field B µν . In this paper we extend these considerations including the dilaton field Φ, linear in coordinates x µ . In that case the coordinate in direction orthogonal to the hyper plane Φ = const, commutes with other Dp-brane coordinates.

We present a procedure for constructing actions describing propagation of W-strings on group manifolds by using the Hamiltonian canonical formalism and representations of W-algebras in terms of Kac-Moody currents. An explicit construction is given in the case of the W 3 string.

In this article we offer the interpretation of the fermionic T-duality of the type II superstring theory in double space. We generalize the idea of double space doubling the fermionic sector of the superspace. In such doubled space fermionic T-duality is represented as permutation of the fermionic coordinates $\theta^\alpha$ and $\bar\theta^\alpha$ with the corresponding fermionic T-dual ones, $\vartheta_\alpha$ and $\bar\vartheta_\alpha$, respectively. Demanding that T-dual transformation law has the same form as inital one, we obtain the known form of the fermionic T-dual NS-R i R-R background fields. Fermionic T-dual NS-NS background fields are obtained under some assumptions. We conclude that only symmetric part of R-R field strength and symmetric part of its fermionic T-dual contribute to the fermionic T-duality transformation of dilaton field and analyze the dilaton field in fermionic double space. As a model we use the ghost free action of type II superstring in pure spinor formulation in approximation of constant background fields up to the quadratic terms.Comment: Four paragraphs in the Introduction added in order to better motivate the subject, explained the choice of action (detailed derivation

We introduce the 2D dimensional double space with the coordinates Z M = (x µ , y µ ) which components are the coordinates of initial space x µ and its T-dual y µ . We shall show that in this extended space the T-duality transformations can be realized simply by exchanging places of some coordinates x a , along which we want to perform Tduality and the corresponding dual coordinates y a . In such approach it is evident that T-duality leads to the physically equivalent theory and that complete set of T-duality transformations form subgroup of the 2D permutation group. So, in the double space we are able to represent the backgrounds of all T-dual theories in unified manner.