Lapse singularities, caustics and entanglement

We study diffraction catastrophes of wave functions in diffeomorphism invariant quantum theories, for which $\hat H\Psi=0$. These wave functions can be represented in terms of integrations over cycles in a complexified lapse variable $N$. The integrand $\exp(i{\mathbb S}(N))$ may have multiple essential singularities at finite values of $N$ and at infinity. A basis set for Greens functions and solutions of the wave equation is represented by Lefschetz thimbles connecting these singularities. The finite $N$ singularities are shown to be directly related to $A_{n\ge 3}$ caustics. We give an example similar to a minisuperspace cosmological model constructed by Halliwell and Myers, to which we add a scalar field. We show that caustics with codimension $d\ge 2$ exhibit strong entanglement with respect to partitions of their unfolding degrees of freedom. If an unfolding direction corresponds to a physical clock in a solution of the Wheeler-DeWitt equation, the caustic bears some resemblance to a quantum measurement. The R\'enyi entanglement entropy ${\cal R}_n$ is expressed in terms of integrals over $2n$ lapse variables $N_i$. Writing the integrand as $\exp(i\Gamma)$, we find that the finite $N$ essential singularities of $\exp(i{\mathbb S})$ are replaced with non-essential singularities of $\exp(i\Gamma)$ at cyclically related $N_i = N_j$ , which the Lefschetz thimbles evade. The relative homology classes to which the integration cycles belong are higher dimensional variants of links.