Breaking the permutation character of diffeomorphisms on spinor structures
J. M. Hoff da Silva
TL;DR
The paper analyzes how diffeomorphisms act on spinor structures on manifolds with nontrivial topology, showing that multiple nonequivalent spinor structures cause diffeomorphisms to permute spinor sectors rather than act trivially, a bijection that is tied to the topology via $\\check{H}^1(M,\\mathbb{Z}_2)$. It introduces a topological compensating field $B$ and an invertible map $\\rho$ linking exotic and standard spinors, yielding a modified Dirac operator $D_E$ whose spectrum becomes nondegenerate and directionally biased by a momentum-topology coupling $\\mathbf{k}$. This leads to a dynamical preference for a particular spinor structure and a departure from pure permutation symmetry, which the author encodes with altered composition laws in the joint spinor-structure space $P\\times\\tilde{P}$. The work further discusses implications for diffeomorphism-related anomalies and suggests viewing diffeomorphisms as maps between theory phases, with potential connections to gauge-fixing ideas and nonunitary transformations, thereby motivating further study of topology-induced effects in spinor dynamics.
Abstract
We investigate the impact of diffeomorphisms where more than one nonequivalent spinor structure is built upon a given base manifold endowed with nontrivial topology. We call attention to the fact that a relatively straightforward construction evinces a lack of symmetry between fermionic modes from different spinor bundle sections, leading to a dynamic preference breaking the permutation character of diffeomorphisms on spinor structures.