Twisted homology jump loci, twisted Alexander polynomials, and $\mathrm{\Sigma }$-invariants

The twisted Alexander polynomials of a space, associated to a linear representation $\sigma$ of the fundamental group, are non-abelian refinements of the classical Alexander polynomial from knot theory. In this paper, we show that they arise naturally from a new family of invariants – the twisted homology jump loci – which extend the rank-one characteristic varieties to higher-rank local systems. Using the tropical geometry of these twisted loci, we obtain sharper upper bounds for the Bieri–Neumann–Strebel–Renz (BNSR) $\mathrm{\Sigma }$-invariants. For compact orientable $3$-manifolds with toroidal or empty boundary, we use a theorem of Friedl–Vidussi to show that the closure of the union of these twisted tropical bound is sharp: it recovers the fibered faces of the Thurston norm ball exactly, a result that fails without twisting. For compact Kähler manifolds, we prove that the ${\mathrm{\Sigma }}^1$-invariant of ${\pi }_1(X)$ is controlled by the orbifold fibrations of $X$ for any representation $\sigma$, and that the twisted Alexander polynomial ${\mathrm{\Delta }}^{\sigma }(X)$ must equal $0$ or $1$. Both results provide obstructions to geometric realizability that are strictly stronger than their classical untwisted counterparts.