Non-Hermitian Statistical Geometry
Particle exchange in two dimensions gives rise to quantum statistics beyond the boson-fermion dichotomy, with central roles in fractional quantum Hall physics, topological quantum computation, and strongly correlated quantum matter. In conventional topological phases, microscopic exchange geometry is compressed into rigid braid and fusion data. We develop a theory of non-Hermitian statistical geometry in which relaxing Hermiticity softens these exchange data into a larger complex space of spectral and relative-geometric degrees of freedom. For two adjacent braid generators, a single projective exchange spectrum is insufficient: their relative eigenline geometry supplies an additional gauge-invariant modulus. On the generic irreducible two-channel branch, braid consistency locks these coordinates through an exact spectrum-geometry relation. Using Riesz projectors and Kato transport, we realize this geometry in a five-arm Kitaev-Majorana network and test spectral and relative-geometric responses in an independent two-dimensional four-vortex Kitaev model. Finite-time dynamics tests the exchange operation, microscopic fusion sewing tests its composition across fusion descriptions, and matched-spectrum controls isolate the role of relative geometry. Nonzero projective Kato curvature shows that braid-consistent, fusion-coherent exchange can depend on the control path. Non-Hermiticity thus exposes exchange geometry, braid consistency re-locks it, fusion coherence further constrains it, and projective flatness determines when geometric exchange becomes intrinsically topological.