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

Abstract

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">D</m:mi> </m:mrow> </m:msub> </m:math> $-{L}_{\lambda ,\mathcal{D}}$ acting on functions with the Dirichlet boundary condition on a certain domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">D</m:mi> <m:mo>⊂</m:mo> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mi>e</m:mi> <m:mo>,</m:mo> <m:mi>a</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $\mathcal{D}\subset {P}_{e,a}\left(G\right)$ which includes finitely many geodesics { l 1 , …, l N } between e and a . ν λ , a has the formal path integral expression <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>ν</m:mi> </m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mo>,</m:mo> <m:mi>a</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi>γ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msubsup> <m:mrow> <m:mi>Z</m:mi> </m:mrow> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>⁡</m:mo> <m:mi>exp</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:mo>−</m:mo> <m:mi>λ</m:mi> <m:mi>E</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>γ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> <m:mi mathvariant="normal">d</m:mi> <m:mi>γ</m:mi> </m:math> ${\nu }_{\lambda ,a}\left(\mathrm{d}\gamma \right)={Z}_{\lambda }^{-1}\mathrm{exp}\left(-\lambda E\left(\gamma \right)\right)\mathrm{d}\gamma $ , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>E</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>γ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> <m:msubsup> <m:mrow> <m:mo movablelimits="false" form="prefix">∫</m:mo> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>γ</m:mi> </m:mrow> <m:mo>̇</m:mo> </m:mover> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow>