Peaks, Passes and Pits From Topography to Topology (via Quantum Mechanics)

James Clerk Maxwell,

1861 – “On Physical Lines of Force” 1864 – “On the Dynamical Theory of the Electromagnetic Field” 1870 – “On Hills and Dales” hilldale.pdf

Scottish examples

Peak Pass Pit

Critical points of a function on a surface Peaks (local maxima) Passes (saddle points) Pits (local minima) Can identify by looking at 2 nd derivative “Topology only changes when we pass through a critical point”.

Basic theorem (# Peaks) – (# Passes) + (# Pits) = Euler Characteristic (V-E+F) Euler Characteristic is a topological invariant; 2 for the sphere; 0 for the torus. Does not depend on which Morse function we choose!

The Hodge equations The Euler characteristic can also be obtained by counting solutions to certain partial differential equations – the “Hodge equations”. They are geometrical analogs of Maxwell’s equations! To see how PDE can relate to topology, think about vector fields and potentials…

The physics connection Ed Witten, Supersymmetry and Morse Theory, 1982

Witten’s method Consider the Hodge equation as a quantum mechanical Hamiltonian. Different types of ‘particle’ according to the Morse index (‘peakons, passons and pitons’). Euler characteristic given by counting the low energy states of these particles.

Perturbation theory Replace d by e sh d e -sh, where h is the Morse function and s is a real parameter. This perturbation does not change the number of low energy states. But it does change the Hodge equations!

In fact, it introduces a potential term, which forces our particles to congregate near the critical points of appropriate index. The potential is s 2 |  h| 2 + sX h where X h is a zero order vectorial term.

The term X h has a ‘zero point energy’ effect which forces each type of particle to congregate near the critical points of the appropriate index; ‘peakons’ near peaks, ‘passons’ near passes and so on.

Thus the number of low energy n-on modes approaches the number of critical points of index n, as the parameter s becomes large. Appropriately formulated, this proves the fundamental result of Morse theory; peaks – passes + pits = Euler characteristic.

James Clerk Maxwell,