# Quantum statistics of branes

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Quantum statistics of particles (bosons, fermions, anyons) arises due to the possible topologies of curves in D-dimensional spacetime winding around each other

What happens if we replace particles by branes? It seems like their quantum statistics should be described by something like a generalization of TQFT in which the "spacetime" (worldbrane) 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 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 charactestic 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?

This post has been migrated from (A51.SE)
You are probably aware of this, but just for completeness: N coincident branes have a U(N) gauge symmetry, which is broken to $U(1)^N\times S_N$ when they are separated. The permutation symmetry $S_N$ is a discrete gauge symmetry which ensures branes are treated as identical particles. Your question seems related to which kind of identical particles they are (bosons, fermions, or anyons).
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