Cut-Out Wedges in $H_{3}$ and the Borel-Resurgent Chern-Simons Matrix Integrals

Abstract

In this paper, we present a systematic study of the Chern--Simons theory with gauge group \(\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})\) restricted to a wedge-identified manifold in the hyperbolic upper-half-space. The wedge geometry is created by imposing an angular cutoff in the \((x,y)\) plane and identifying two boundary lines, which introduces a single noncontractible loop in the manifold. By imposing the flat-connection condition of the Chern--Simons gauge fields, the path integral reduces to a finite-dimensional matrix integral in \(\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})\) . Although Chern-Simons theory is a topological theory, the resulting matrix integral remains nontrivial due to noncompact directions and boundary constraints. The large-\(k\) expansion of the matrix integral is carried out by selecting a classical configuration in the space of holonomies and expanding around it in inverse powers of \(k\). The resulting coefficients of the asymptotic series exhibit factorial growth, enabling us to apply the Borel resummation techniques. Summation over these subleading sectors removes potential ambiguities in the Borel integral and clarifies the emergence of a resurgent transseries structure. In the Borel-resurgent analysis, we show that, despite the apparent simplicity of the reduced action, the wedge geometry yields a rich interplay of perturbative and non-perturbative phenomena. This work presents an explicit example of how a finite-dimensional matrix integral in its expansions is physically meaningful through Borel resummation.