Supersymmetry and harmonic theory

The next example comes from Hodge theory. On the differential forms in an n-dimensional oriented riemannian manifold M, we have two operations: the exterior derivative,

d : Ωp(M) → Ωp+1(M),

a first-order differential operator taking p-forms to (p+1)-forms, and the Hodge star

⋆ : Ωp(M) → Ωn-p(M).

Composing these two operations we can form new operations; for example, the exterior divergence

δ ≔ ⋆d⋆ : Ωp(M) → Ωp-1(M),

and the Hodge lapacian

Δ ≔ dδ + δd : Ωp(M) → Ωp(M).

A form ω satisfying Δω=0 is said to be harmonic. Clearly the harmonic condition is second order.

Let us now define the Dirac-like operator D ≔ d + δ which does not respect the degree of the form, but takes forms of odd degree to forms of even degree and viceverse. Clearly if a form obeys Dω=0 it is harmonic; so again we have a first-order condition implying a second-order condition. (Of course, on a compact manifold the two conditions are equivalent.)

The underlying supersymmetric theory in this case is the one-dimensional supersymmetric sigma model. This is a theory of harmonic maps from the line (or the circle) to the manifold M. Upon quantisation, the hamiltonian becomes the Hodge laplacian Δ and the supercharges play the role of D.

When M has further structure, e.g., when M is Kähler, the sigma model has "more" supersymmetry and this supersymmetry underlies much of the Hodge-Lefschetz theory. Something analogous happens when M is hyperkähler.

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