Axion-dilaton cosmology, Ricci flows and integrable structures

In this work, based on [Ioannis Bakas, Domenico Orlando, and P. Marios Petropoulos. Ricci flows and expansion in axion-dilaton cosmology. JHEP 01 (2007) 040], we study renormalization-group flows by deforming a class of conformal sigma-models. At leading order in α', renormalization-group equations represent a Ricci flow. In the three-sphere background, the latter is described by the Halphen system, which is exactly solvable in terms of modular forms. The round sphere is found to be the unique perturbative infra-red fixed point at one loop order