We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction g = 2. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order.Calogero-Sutherland type models are a class of exactly solvable models. Starting from the inception [1], till today, these models continue to be of interest, especially due to their exact solvability [2]. Moreover, these models are related to (1 + 1)-dimensional conformal field theory, random matrices and a host of other things [3]. Some variants of the models have attracted attention in very recent past [3][4][5]. The original model of Calogero [1] described N particles in one dimension, interacting through an inverse square law and a two-body harmonic interaction. The latter interaction was added to the effect of complete discretization of the spectrum. This model gave in to an analytic treatment yielding an exact solution. Sutherland [6] considered a variant of this problem where the harmonic interaction was replaced by a harmonic well containing the N particles. It was shown that one can exactly solve for the ground state of the system. The norm of the ground state wave-function was shown to be of a form that coincides with the joint probability density function of the eigenvalues of random matrices from an orthogonal, unitary or symplectic Gaussian ensemble, provided, the * khare@iopb.ernet.in † koushik@iopb.ernet.in

In this note we study the resolution of conifold singularity by D-branes by considering compactification of D-branes on C 3 /(Z 2 × Z 2 ). The resulting vacuum moduli space of D-branes is a toric variety which turns out to be a resolved conifold, that is a nodal variety in C 4 . This has the implication that all the corresponding phases of Type-II string theory are geometrical and are accessible to the D-branes, since they are related by flops.In the aftermath of the second superstring revolution the role of D-branes in the study of space-time has been widely appreciated. It has become quite apparent that one can probe sub-stringy scales in spacetime using D-branes -the BPS states that carry Rammond-Rammond charges in string theory. D-branes have recently been used to understand the short-distance geometry of Calabi-Yau manifolds [1][2][3]. These studies discern rather directly, as compared to the studies with fundamental strings, the phase structure of Calabi-Yau manifolds [1,3]. It has been shown that the topological and geometric properties of Calabi-Yau manifolds studied using both D-branes and fundamental strings corroborate. More specifically, it has been shown [3] that the D0-branes do not allow for non-geometric phases to appear in the theory which is in keeping with the conclusion drawn earlier from the analysis of M-theory and F-theory [4].The analysis of [3] was for C 3 /Γ, where Γ is a discrete Abelian group in SU (3). Specific examples of Γ = Z 3 and Γ = Z 5 were worked out, which could be generalized for any Z n . It was concluded that the Dbranes probe resolved C 3 /Z n singularities. In this scheme one starts with a set of D-branes on C 3 , arranged according to the regular representation of Γ, followed by a truncation to the Γ-invariant sector of the action, thus ending up with a gauged sigma-model whose configuration space, in the low-energy limit, is interpreted as the sub-stringy space-time. This picture has a mathematical counterpart in terms of blowing up a singular Kähler quotient [5,6] in an orbifold, viewed as a toric variety.In this note we shall study another example of this construction, namely a blown-up C 3 /(Z 2 ×Z 2 ), following the lines of [3]. We shall find that, in this case, the D-brane probe senses a double-point singularity, which gets resolved for a generic value of the resolution parameter; and under a smooth variation of the resolution parameter one moves on from one resolution to another, which corresponds to a flop [7,8].

We explicitly obtain the generalization of the Ehlers transformation for stationary axisymmetric Einstein equations to string theory. This is accomplished by finding the twist potential corresponding to the moduli fields in the effective two dimensional theory. Twist potential and symmetric moduli are shown to transform under an O(d, d) which is a manifest symmetry of the equations of motion. The non-trivial action of this O(d, d) is given by the Ehlers transformation and belongs to the set O(d)×O(d) O(d) .

The bulk reconstruction formula for a Euclidean anti-de Sitter space is directly related to the inverse of the Gel'fand-Graev-Radon transform. Correlation functions of a conformal scalar field theory in the boundary are thereby related to correlation functions of a self-interacting scalar field theory in the bulk at different loop orders.