Twisted homology jump loci, twisted Alexander polynomials, and $\Sigma$-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) $\Sigma$-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 K\"{a}hler manifolds, we prove that the $\Sigma^1$-invariant of $\pi_1(X)$ is controlled by the orbifold fibrations of $X$ for any representation $\sigma$, and that the twisted Alexander polynomial $\Delta^\sigma(X)$ must equal $0$ or $1$. Both results provide obstructions to geometric realizability that are strictly stronger than their classical untwisted counterparts.