The type IIA/IIB effective actions compactified on T d are known to be invariant under the T -duality group SO(d, d; Z), although the invariance of the R-R sector cannot be seen so directly. Inspired by a work of Brace, Morariu and Zumino, we introduce new potentials, which are mixtures of R-R potentials and the NS-NS 2-form, in order to make the invariant structure of R-R sector more transparent. We give a simple proof that if these new potentials transform as a Majorana-Weyl spinor of SO (d, d; Z), the effective actions are indeed invariant under the T -duality group. The argument is made in such a way that it can apply to Kaluza-Klein forms of arbitrary degree. We also demonstrate that these new fields simplify all the expressions, including the Chern-Simons term. * )

It is shown that the Dirac equations in general higher dimensional Kerr-NUT-de Sitter spacetimes are separated into ordinary differential equations.

We explicitly calculate the Riemannian curvature of D-dimensional metrics recently discussed by Chen, Lü and Pope. We find that they can be concisely written by using a single function. The Einstein condition which corresponds to the Kerr-NUT-de Sitter metric is clarified for all dimensions. It is shown that the metrics are of type D.

We observe that, at β-deformed matrix models for the four-point conformal block, the point q = 0 is the point where the three-Penner type model becomes a pair of decoupled two-Penner type models and where, in the planar limit, (an array of) two-cut eigenvalue distribution(s) coalesce into (that of) one-cut one(s). We treat the Dotsenko-Fateev multiple integral, with their paths under the recent discussion, as perturbed double-Selberg matrix model (at q = 0, it becomes a pair of Selberg integrals) to construct two kinds of generating functions for the q-expansion coefficients and compute some. A formula associated with the Jack polynomial is noted. The second Nekrasov coefficient for SU(2) with N f = 4 is derived. A pair of Young diagrams appears naturally. The finite N loop equation at q = 0 as well as its planar limit is solved exactly, providing a useful tool to evaluate the coefficients as those of the resolvents.The planar free energy in the q-expansion is computed to the lowest non-trivial order. A free field representation of the Nekrasov function is given. *

No abstract