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

  • Charles Misner, Harmonic maps as models for physical theories, Phys. Rev. D 18, 4510 (1978) (spire)

Discussion in the context of integrable systems includes

Last revised on November 19, 2015 at 08:59:07. See the history of this page for a list of all contributions to it.