The braid group is π₁ of configuration space — commutativity as a dimension

The braid group on n strands, Bₙ, is the fundamental group of the configuration space of n unordered points in the plane, π₁(UConf_n(ℝ²)). A braid is literally a loop in that space: a path along which the n points move and return to their set, with the strands the traces of the points in space-time. This is the precise sense in which commutativity is a dimension. In the plane two points can be exchanged in two inequivalent ways (over or under), so the exchange has infinite order and the group is the braid group; on a line there is no room to pass; in three or more dimensions the two exchanges are homotopic, the exchange has order two, and the group collapses to the symmetric group Sₙ. Anyons — particles with braid statistics rather than Bose/Fermi statistics — are exactly the 2D case where π₁ is the braid group and not the symmetric group. The dimension of the ambient space sets how much commutativity the exchange is permitted. A worked instance of the correspondence's "commutativity is a dimension."

The same fact seen twice — added 2026-09-27

Exchange is a redundancy, not a symmetry, and that is why Noether says nothing about it. For genuinely identical particles the labels are fiction, so a permutation does not carry a state to a different state of equal energy — it carries a state to itself. Nothing moves. That is a redundancy of description, the same species as a gauge transformation, which is exactly why the physics lives in π₁ of the quotient above. Noether's first theorem needs a transformation that does something; her second says a gauge redundancy yields identities and constraints, not currents. The familiar [H, P_ij] = 0 is true and hollow: the permutation eigenvalue is constant because it was never dynamical. An unfiled instance of CASE-15 (gauge redundancy), reached from the identical-particle side rather than the field-theory side.

What a redundancy gives instead is a superselection rule, which is stronger than any Noether charge. The exchange phase labels a sector, and no operator at all — not merely no symmetry-respecting one — connects sectors; bosonic and fermionic states admit no coherent superposition. A Noether charge is conserved by a particular Hamiltonian. This is conserved by every possible Hamiltonian, because it is kinematic: a fact about which states exist, not about how they evolve.

And the conserved quantity standing next to it is angular momentum, via spin–statistics. An exchange can be performed by rotating the pair through π, so performing it twice is a 2π rotation — a loop in the rotation group — and whether that loop is contractible is the whole story. In three dimensions π₁(SO(3)) = ℤ₂, the double loop contracts, the phase must square to one, and the options are exactly ±1. In two dimensions the rotation group is SO(2), π₁ = ℤ, loops are graded by winding number and never contract, and nothing forces the phase to square to anything.

So the ambient dimension does not govern configuration space and the rotation group separately. It is one fact, seen twice — the paragraph above reads it off π₁ of configuration space, this one off π₁ of the rotation group, and they agree because an exchange is a half-rotation. The original note had only the first reading.

Where it is manufactured rather than derived. In 2+1D Chern–Simons, attaching magnetic flux to electric charge makes the exchange phase an Aharonov–Bohm phase, θ ~ q²/k. There the conserved quantity genuinely is a Noether charge — electric charge, from the U(1) — and the statistics is a consequence of binding it to flux. And non-abelian anyons leave the frame entirely: exchange acts as a matrix, so the invariant is not a number but the fusion channel, carried by data localised on no particle — which is precisely why nothing local can decohere it.

Source: sighted conversation, 2026-09-27, not generator output. Filed because the rotation-group half was absent from this note and the CASE-15 link was unmade.