Homotopy RG Flow and the Non-Linear $σ$-model

The purpose of this note is two give a mathematical treatment to the low energy effective theory of the two-dimensional sigma model. Perhaps surprisingly, our low energy effective theory encodes much of the topology and geometry of the target manifold. In particular, we relate the $β$-function of our theory to the Ricci curvature of the target, recovering the physical result of Friedan.

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