Year: 2026
Type: preprint
Abstract: We determine all possible signatures of flat Hermitian bundles over compact, connected, oriented surfaces of positive genus with nonempty boundary. If $Σ_{g,n}$ has genus $g\geq 1$ and $n\geq 1$ boundary components, then the signatures arising from representations $π_1(Σ_{g,n})\to\mathrm{U}(p,q)$ are exactly the integers $m$ satisfying \[ |m| \leq (p+q)(2g+n-2) -(2g-2)|p-q| -\min\{2,n|p-q|\}. \] Every such integer, including the two extremal values, is realized by a block-diagonal representation whose image is contained, after possibly interchanging $p$ and $q$, in $\mathrm{U}(1,1)^{\times\min\{p,q\}}\times\mathrm{U}(|p-q|)$.
Source: arXiv (Cornell University)
Authors: Kim, Inkang, Pansu, Pierre, Wan, Xueyuan
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: Groups acting on fractals
OpenAlex ID: W7213301320