We extend the complex-valued analytic torsion, introduced by Burghelea and Haller on closed manifolds, to compact Riemannian bordisms. We do so by considering a flat complex vector bundle over a compact Riemannian manifold, endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian metric and the bilinear form are used to define non-selfadjoint Laplacians acting on vector-valued smooth forms under absolute and relative boundary conditions. In order to define the complex-valued analytic torsion in this situation, we study spectral properties of these generalized Laplacians. Then, as main results, we obtain so-called anomaly formulas for this torsion. Our reasoning takes into account that the coefficients in the heat trace asymptotic expansion associated to the boundary value problem under consideration, are locally computable. The anomaly formulas for the complex-valued Ray–Singer torsion are derived first by using the corresponding ones for the Ray–Singer metric, obtained by Brüning and Ma on manifolds with boundary, and then an argument of analytic continuation. In odd dimensions, our anomaly formulas are in accord with the corresponding results of Su, without requiring the variations of the Riemannian metric and bilinear structures to be supported in the interior of the manifold.

我们将Burghelea和Haller在封闭流形上引入的复值解析扭力扩展到紧致的黎曼坐标系。为此，我们考虑了紧凑的黎曼流形上的平面复矢量束，并赋予其纤维状的简并对称双线性形式。Riemmanian度量和双线性形式用于定义在绝对和相对边界条件下作用于矢量值光滑形式的非自伴拉普拉斯算子。为了定义这种情况下的复值解析扭力，我们研究了这些广义拉普拉斯算子的光谱特性。然后，作为主要结果，我们获得了这种扭转的所谓的异常公式。我们的推理考虑到与考虑中的边值问题相关的热迹线渐近扩展中的系数是局部可计算的。首先，使用Brüning和Ma在带边界的流形上获得的相应的Ray-Singer度量公式，来推导复值Ray-Singer扭转的异常公式。在奇数维上，我们的异常公式与Su的相应结果一致，而无需在歧管内部支撑黎曼度量和双线性结构的变化。