Ricci Flow and Weyl Transformations

[nigelscott]

I am trying to my head around these two things in the context of string theory. The Polyakov action becomes simpler to solve in the conformal gauge which, as I understand it, makes the manifold locally Ricci flat in 2D. In Professor Susskind's lectures on String Theory he introduces the concept of Ricci flow (approx minute 35).. Both seem to achieve the same result. Is the difference related to sub-manifold versus background (ambient)?

 

[AndyW]

Hello,

Thank you for your post. I understand your confusion about the relationship between the Polyakov action and Ricci flow in the context of string theory.

To start, the Polyakov action is a mathematical expression that describes the dynamics of a string in spacetime. It is used to calculate the equations of motion for the string, which determine its behavior and shape as it moves through spacetime. The conformal gauge is a specific choice of coordinates that simplifies the calculations of the Polyakov action. In this gauge, the manifold (spacetime) is locally Ricci flat in 2D, meaning that the Ricci curvature is zero in two dimensions.

On the other hand, Ricci flow is a concept introduced by Professor Susskind to describe how a manifold evolves over time. It is a geometric process in which the metric of a manifold is changed according to the Ricci curvature. This can be thought of as a way to smooth out any curvature or deformations in the manifold. In the context of string theory, Ricci flow is used to study how the geometry of the string worldsheet (a 2D surface) changes over time.

So, in summary, the difference between the Polyakov action and Ricci flow is that the former is a mathematical expression used to describe the dynamics of a string, while the latter is a geometric process used to study the evolution of a manifold. They both play important roles in understanding string theory, but they are not achieving the same result.

I hope this helps clarify the difference between these two concepts. If you have any further questions, please feel free to ask. Thank you for your interest in string theory.