Non-Linear Sigma Model on Conifolds

TL;DR

This paper constructs explicit Ricci-flat Kähler metrics on conifolds and resolved conifolds using complex coordinates, and develops associated non-linear sigma models, revealing their topological nature and relevance in string theory backgrounds.

Contribution

It provides explicit solutions for conifold metrics in complex coordinates and formulates topological non-linear sigma models with these spaces as targets, including the resolved conifold.

Findings

Explicit Ricci-flat Kähler metrics on conifolds and resolved conifolds.

Construction of topological non-linear sigma models with conifold targets.

Derivation of the warp factor and $AdS_{5}\times X_{5}$ geometry in string theory.

Abstract

Explicit solutions to the conifold equations with complex dimension $n=3,4$ in terms of {\it{complex coordinates (fields)}} are employed to construct the Ricci-flat K\"{a}hler metrics on these manifolds. The K\"{a}hler 2-forms are found to be closed. The complex realization of these conifold metrics are used in the construction of 2-dimensional non-linear sigma model with the conifolds as target spaces. The action for the sigma model is shown to be bounded from below. By a suitable choice of the 'integration constants', arising in the solution of Ricci flatness requirement, the metric and the equations of motion are found to be {\it{non-singular}}. As the target space is Ricci flat, the perturbative 1-loop counter terms being absent, the model becomes topological. The inherent U(1) fibre over the base of the conifolds is shown to correspond to a gauge connection in the sigma model. The same procedure is employed to construct the metric for the resolved conifold, in terms of complex coordinates and the action for a non-linear sigma model with resolved conifold as target space, is found to have a minimum value, which is topological. The metric is expressed in terms of the six real coordinates and compared with earlier works. The harmonic function, which is the warp factor in Type II-B string theory, is obtained and the ten-dimensional warped metric has the $AdS_{5}\times X_{5}$ geometry.