An integrable evolution equation in geometry

We introduce an integrable Hamiltonian system which Lax deforms the Dirac operator D=d+d* on a finite simple graph or compact Riemannian manifold. We show that the nonlinear isospectral deformation always leads to an expansion of the original space, featuring a fast inflationary start. The nonlinear evolution leaves the Laplacian L=D^2 invariant so that linear Schroedinger or wave dynamics is not affected. The expansion has the following effects: a complex structure can develop and the nonlinear quantum mechanics asymptotically becomes the linear relativistic Dirac wave equation u''=Lu. While the later is not aware of the expansion of space and does not see the emerged complex structure, nor the larger non-commutative geometric setup, the nonlinear flow is affected by it. The natural Noether symmetries of quantum mechanics introduced here force to consider space as part of a larger complex geometry. The nonlinear evolution equation is a symmetry of quantum mechanics which still features super-symmetry, but it becomes clear why it is invisible: while the McKean-Singer formulas str(exp(i D(t) t)) = str(exp(-L t))=chi(G) still hold, the super-partners f,Df are orthogonal only at t=0 and become parallel or anti-parallel for |t| to infinity.

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