Quantum statistics of particles (bosons, fermions, anyons) arise due to the possible topologies of curves in $D$-dimensional spacetime winding around each other
What happens if we replace particles with branes? It seems like their quantum statistics should be described by something like a generalization of TQFT in which the "spacetime" (world brane) is equipped with an embedding into an "ambient" manifold (actual spacetime). The inclusion of non-trivial topology for the "ambient" manifold introduces additional effects, to 1st approximation describable by the inclusion of k-form fluxes coupling to the brane. To 2nd approximation, however, there is probably non-trivial coupling between these fluxes and the "generalized quantum statistics"
A simple example of non-trivial "brane quantum statistics" is the multiplication of quantum amplitudes of strings by the exponential of the Euler characteristic times a constant. In string theory, this corresponds to changing the string coupling constant/dilaton background.
Were such generalized TQFTs studied? Which non-trivial examples are there for branes in string theory?