Let \((M,g)\) be a \(d\)-dimensional compact Riemannian manifold without boundary and let \(P\) be the nonminimal Laplace type operator on a smooth hermitean vector bundle \(V\) over \(M\) of fiber \(\mathbb{C}^N\) written locally as \[ P := - [g^{\mu\nu} u(x) \partial_\mu \partial_\nu + v^\nu(x) \partial_\nu + w(x)]. \] Here \(u(x)\in M_N(\mathbb{C})\) is a positive and invertible matrix valued function and \(v^\nu\) and \(w\) are \(M_N(\mathbb{C})\) matrix-valued functions. The operator is expressed in a local trivialization of \(V\) over an open subset of \(M\) which is also a chart on \(M\) with coordinates \((x^\mu).\) This trivialization is such that the adjoint for the hermitean metric corresponds to the adjoint of matrices and the trace on endomorphisms on \(V\) becomes the usual trace \(\text{tr}\) on matrices.
For any \(a\in \Gamma (\text{End}(V)),\) the authors consider the asymptotics of the heat-trace \[ \text{Tr}(a e^{- t P}) \underset{t \downarrow 0^+}{\sim} \sum_{r = 0}^\infty a_r(a, P) t^{(r - d)/2} \] where \(\text{Tr}\) is the operator trace, and each coefficient \(a_r(a, P)\) can be written as an integral of the functions \(a_r(a, P)(x) = \text{tr} [a(x) \mathcal{R}_r(x)].\)
The paper presents a way to compute \(\mathcal{R}_2\) by adapting the techniques developed by the authors in [ibid. 116, 90–118 (2017; Zbl 1373.58013)]. The idea behind this computation is to extract the real matrix content of the coefficient \(a_2\) which is related to the scalar curvature of the manifold \(M.\)