Asymptotics of lowlying Dirichlet eigenvalues of Witten Laplacians on domains in pinned path groups

Abstract Let G be a compact connected Lie group and P e , a ( G ) = C ([0, 1] → G | γ (0) = e , γ (1) = a ) be the pinned path space with a pinned Brownian motion measure ν λ , a defined by the heat kernel p ( λ −1 t , x , y ), where λ is a positive parameter. We consider a Witten Laplacian − L λ , D $-{L}_{\lambda ,\mathcal{D}}$ acting on functions with the Dirichlet boundary condition on a certain domain D ⊂ P e , a ( G ) $\mathcal{D}\subset {P}_{e,a}\left(G\right)$ which includes finitely many geodesics { l 1 , …, l N } between e and a . ν λ , a has the formal path integral expression ν λ , a ( d γ ) = Z λ − 1 ⁡ exp − λ E ( γ ) d γ ${\nu }_{\lambda ,a}\left(\mathrm{d}\gamma \right)={Z}_{\lambda }^{-1}\mathrm{exp}\left(-\lambda E\left(\gamma \right)\right)\mathrm{d}\gamma $ , where E ( γ ) = 1 2 ∫ 0 1 | γ ̇ ( t )