齐次阿贝尔丛上的非微扰热核渐近性
Non-perturbative Heat Kernel Asymptotics on Homogeneous Abelian Bundles
摘要：
研究了无边界黎曼流形$M$上作用在复向量丛光滑截面上的Laplace型偏微分算子的热核，其结构群为$G\times U(1)$.向量丛上的总联络自然地分成$G$ -联络和U（1）-联络，假设U（1）-联络具有平行曲率$F$。我们发现了一个新的局部短时间渐近展开的非对角热核$U(t|x,x')$接近的对角线$M\times M$假设曲率$F$的顺序$t^{-1}$。这个展开式的系数是黎曼曲率张量（和$G$ -连接的曲率）及其导数中的多项式函数，其通用系数以非多项式但解析的方式取决于曲率$F$，更确切地说，取决于$tF$。这些功能产生的所有条款二次和线性的黎曼曲率和任意顺序的$F$在通常的热内核系数。在这个意义上，我们有效地总结了通常的短时间热核渐近展开的曲率$F$的所有订单。我们计算这个新的渐近展开的前三个系数（对角和非对角）。
英文摘要：
We study the heat kernel for a Laplace type partial differential operator acting on smooth sections of a complex vector bundle with the structure group $G\times U(1)$ over a Riemannian manifold $M$ without boundary. The total connection on the vector bundle naturally splits into a $G$-connection and a U(1)-connection, which is assumed to have a parallel curvature $F$. We find a new local short time asymptotic expansion of the off-diagonal heat kernel $U(t|x,x')$ close to the diagonal of $M\times M$ assuming the curvature $F$ to be of order $t^{-1}$. The coefficients of this expansion are polynomial functions in the Riemann curvature tensor (and the curvature of the $G$-connection) and its derivatives with universal coefficients depending in a non-polynomial but analytic way on the curvature $F$, more precisely, on $tF$. These functions generate all terms quadratic and linear in the Riemann curvature and of arbitrary order in $F$ in the usual heat kernel coefficients. In that sense, we effectively sum up the usual short time heat kernel asymptotic expansion to all orders of the curvature $F$. We compute the first three coefficients (both diagonal and off-diagonal) of this new asymptotic expansion.
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