  Quantum fluctuating loops in 2+1 dimensions give gapless many-body states
that are beyond current field theory techniques. Microscopically, these loops
can be domain walls between up and down spins, or chains of flipped spins
similar to those in the toric code. The key feature of their dynamics is that
the reconnection of a pair of strands is forbidden. This happens at
previously-studied multi-critical points between topologically nontrivial
phases. We show that this topologically constrained dynamics leads to
universality classes with unusual scaling properties. For example, scaling
operators at these fixed points are classified by topology, and not only by
symmetry. We introduce the concept of the topological operator classification,
provide universal scaling forms for correlation functions, and analytical and
numerical results for critical exponents. We use an exact correspondence
between the imaginary-time dynamics of the 2+1D quantum models and a classical
Markovian dynamics for 2D classical loop models with a nonlocal Boltzmann
weight (for which we also provide scaling results). We comment on open
questions and generalizations of the models discussed for both quantum
criticality and classical Markov processes.
