A Gradient Flow for Worldsheet Nonlinear Sigma Models

TL;DR

This paper links Ricci flow mathematical advances to the renormalization group flows of nonlinear sigma models, introducing a potential functional whose eigenvalues track flow stability and fixed points.

Contribution

It generalizes Ricci flow techniques to include B-fields and dilaton decoupling, providing a gradient flow description of 1-loop beta-functions in sigma models.

Findings

The lowest eigenvalue functional is monotonic along the flow.

The eigenvalue functional reproduces the Weyl anomaly at fixed points.

Stability analysis of flat tori and K3 manifolds using the Hessian.

Abstract

We discuss certain recent mathematical advances, mainly due to Perelman, in the theory of Ricci flows and their relevance for renormalization group (RG) flows. We consider nonlinear sigma models with closed target manifolds supporting a Riemannian metric, dilaton, and 2-form B-field. By generalizing recent mathematical results to incorporate the B-field and by decoupling the dilaton, we are able to describe the 1-loop beta-functions of the metric and B-field as the components of the gradient of a potential functional on the space of coupling constants. We emphasize a special choice of diffeomorphism gauge generated by the lowest eigenfunction of a certain Schrodinger operator whose potential and kinetic terms evolve along the flow. With this choice, the potential functional is the corresponding lowest eigenvalue, and gives the order alpha' correction to the Weyl anomaly at fixed points of (g(t),B(t)). Since the lowest eigenvalue is monotonic along the flow and reproduces the Weyl anomaly at fixed points, it accords with the c-theorem for flows that remain always in the first-order regime. We compute the Hessian of the lowest eigenvalue functional and use it to discuss the linear stability of points where the 1-loop beta-functions vanish, such as flat tori and K3 manifolds.