  We introduce a representation theory for risk operations on locally compact
groups in a partition of unity on a topological manifold for
Markowitz-Tversky-Kahneman (MTK) reference points. We identify (1) risk torsion
induced by the flip rate for risk averse and risk seeking behaviour, and (2) a
structure constant or coupling of that torsion in the paracompact manifold. The
risk torsion operator extends by continuity to prudence and maxmin expected
utility (MEU) operators, as well as other behavioural operators introduced by
the Italian school. In our erstwhile chaotic dynamical system, induced by
behavioural rotations of probability domains, the loss aversion index is an
unobserved gauge transformation; and reference points are hyperbolic on the
utility hypersurface characterized by the special unitary group SU(n). We
identify conditions for existence of harmonic utility functions on paracompact
MTK manifolds induced by transformation groups. And we use those mathematical
objects to estimate: (1) loss aversion index from infinitesimal tangent
vectors; and (2) value function from a classic Dirichlet problem for first exit
time of Brownian motion from regular points on the boundary of MTK base
topology.
