Convergence of the Hesse–Koszul flow on compact Hessian manifolds

Abstract

We study the long time behavior of the Hesse-Koszul flow on compact Hessian manifolds. When the first affine Chern class is negative, we prove that the flow converges to the unique Hesse-Einstein metric. We also derive a convergence result for a twisted Hesse-Koszul flow on any compact Hessian manifold. These results give alternative proofs for the existence of the unique Hesse-Einstein metric by Cheng-Yau and Caffarelli-Viaclovsky as well as the real Calabi theorem by Cheng-Yau, Delanoë and Caffarelli-Viaclovsky.

Renormalization group flows from the Hessian geometry of quantum effective actions

“…where t is the proper time parameter of the trajectory. The implications of this equation can be seen by using K α = δΓ δ∆ α and taking a second derivative of (45). Taking note of ( 27) and ( 19), this yields the requirement…”

“…In [44,45], a geometrical flow for the metric was constructed, which preserves its Hessian property. This was achieved using the second Koszul form…”

“…Note also that the expression for the one-loop effective action is equivalent to the second Koszul form if we take two functional derivatives. Instead of equation (A2), we might also consider a flow for the potential itself using the Monge-Ampére operator [45] M [Γ] ≡ det g ab .…”

“…Further technical details are provided in the appendices. Appendix A reviews the Hesse-Koszul flow [44,45] of the metric of the Hessian manifold, which bears some similarity to the Wetterich equation [39][40][41]. The closure of the 2PI RG is described in appendix B, and subtleties related to the non-commutativity of various derivatives are described in appendix C…”