nLab
harmonic map

Contents

Idea

A harmonic map is a smooth function f:XYf \colon X \longrightarrow Y between two Riemannian manifolds (X,g X)(X,g_X), (X,g Y)(X,g_Y) which is a critical point of the Dirichlet kinetic energy functional

E(f) Xdf 2dVol X. E(f) \coloneqq \int_{X} \Vert d f\Vert^2 dVol_X \,.

where dfΓ(T *Xϕ *TY)d f \in \Gamma (T^\ast X \otimes \phi^\ast T Y) is the derivative, where the norm is given jointly by the metrics of XX and YY and where the volume form is that of XX.

This is a standard kinetic action action functional for sigma models, the Polyakov action.

References

Discussion in the context of action functionals for theories of physics includes

Discussion in the context of integrable systems includes