# Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

## Abstract

The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of Khavkine, Melati, and Moretti for vector valued free fields. As a by-product of such classification, we provide a mathematically rigorous proof that, at first order in perturbation theory, the renormalization group flow of the nonlinear Sigma model is the Ricci flow.

## Notes

### Acknowledgements

The work of C. D. was supported by the University of Pavia, while that of N. D. was supported in part by a research fellowship of the University of Pavia. We are grateful to Federico Faldino, Igor Khavkine, Alexander Schenkel and Jochen Zahn for the useful discussions. We are especially grateful to Klaus Fredenhagen for the enlightening discussions on the rôle of the algebra of functionals. This work is based partly on the MSc thesis of P. R. .

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