Hodge star operator

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.

The naturalness of the star operator means it can play a role in differential geometry, when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms. This allows the definition of the codifferential as the Hodge adjoint of the exterior derivative, leading to the Laplace–de Rham operator. This generalizes the case of 3-dimensional Euclidean space, in which divergence of a vector field may be realized as the codifferential opposite to the gradient operator, and the Laplace operator on a function is the divergence of its gradient. An important application is the Hodge decomposition of differential forms on a closed Riemannian manifold.

A general k-vector is a linear combination of decomposable k-vectors, and the definition of the Hodge star is extended to general k-vectors by defining it as being linear.

In two dimensions with the normalized Euclidean metric and orientation given by the ordering (x, y), the Hodge star on k-forms is given by

On the complex plane regarded as a real vector space with the standard sesquilinear form as the metric, the Hodge star has the remarkable property that it is invariant under holomorphic changes of coordinate. If z = x + iy is a holomorphic function of w = u + iv, then by the Cauchy–Riemann equations we have that x/u = y/v and y/u = − x/v. In the new coordinates

The Hodge star relates the exterior and cross product in three dimensions:

and hence deserve the name self-dual and anti-self-dual two-forms. Understanding the geometry, or kinematics, of Minkowski spacetime in self-dual and anti-self-dual sectors turns out to be insightful in both mathematical and physical perspectives, making contacts to the use of the two-spinor language in modern physics such as spinor-helicity formalism or twistor theory.

One can also obtain the Laplacian Δ f = div grad f in terms of the above operations:

where s is the parity of the signature of the inner product on V, that is, the sign of the determinant of the matrix of the inner product with respect to any basis. For example, if n = 4 and the signature of the inner product is either (+ − − −) or (− + + +) then s = −1. For Riemannian manifolds (including Euclidean spaces), we always have s = 1.

If n is odd then k(nk) is even for any k, whereas if n is even then k(nk) has the parity of k. Therefore:

The codifferential is not an antiderivation on the exterior algebra, in contrast to the exterior derivative.

The codifferential is the adjoint of the exterior derivative with respect to the square-integrable inner product:

The Hodge star sends harmonic forms to harmonic forms. As a consequence of Hodge theory, the de Rham cohomology is naturally isomorphic to the space of harmonic k-forms, and so the Hodge star induces an isomorphism of cohomology groups

which in turn gives canonical identifications via Poincaré duality of H k(M) with its dual space.