  Chern-Simons topological field theories TFTs are the only TFTs that have
already found application in the description of some exotic strongly-correlated
electron systems and the corresponding concept of topological quantum
computing. Here, we show that TFTs of another type, specifically the
gauge-field-less Witten-type TFTs known as topological sigma models, describe
the recently proposed digital memcomputing machines (DMMs) - engineered
dynamical systems with point attractors being the solutions of the
corresponding logic circuit that solves a specific task. This result derives
from the recent finding that any stochastic differential equation possesses a
topological supersymmetry, and the realization that the solution search by a
DMM proceeds via an instantonic phase. Certain TFT correlators in DMMs then
reveal the presence of a transient long-range order both in space and time,
associated with the effective breakdown of the topological supersymmetry by
instantons. The ensuing non-locality and the low dimensionality of instantons
are the physical reasons why DMMs can solve complex problems efficiently,
despite their non-quantum character. We exemplify these results with the
solution of prime factorization.
