The сigar soliton and the Ricci flows perturbation solutions in the two-dimensional string $\sigma$ model

The Ricci flows perturbation equations and the Weyl anomaly coefficients are derived in the two-dimensional bosonic string $\sigma$ model. These equations correspond to the two-loop flow equations for the graviton field $g_{\mu\nu}$ and the dilaton field $\Phi$. The one-loop perturbation solution of the cigar soliton can be expressed in terms of the hypergeometric functions. The two-loop asymptotic perturbation solution of the cigar soliton is reduced by using a small parameter expansion method. Moreover, analytic solutions of the second basic form $l$ and $n$ are obtained in accordance with the perturbation Gauss–Codazzi equations. The modified expression of the deformed principal curvatures of a two-dimensional surface can then be given in terms of $l$ and $n$. The influence of quantum Ricci flows on the space–time geometry is analyzed and discussed, and the physical meaning of the Weyl anomaly coefficients varying with the momentum scale $\lambda$ is also explained.