A comparison of the absolute and relative real analytic torsion forms (Q6611892)

Consider a smooth proper fibration \(M_1\to S\) with typical fibre \(Z_1\) and boundary \(X:=\partial M_1\) such that \(X\to S\) is a smooth fibration with typical fibre \(Y\). Assume a product structure near the boundary. Let \((F,\nabla^F)\) be a flat vector bundle over \(M_1\) and choose metrics on \(F\) and along the fibres. \textit{J.-M. Bismut} and \textit{J. Lott} [J. Am. Math. Soc. 8, No. 2, 291--363 (1995; Zbl 0837.58028)] constructed real analytic torsion forms associated to \(F\to X\to S\) and to a long exact sequence of cohomology groups associated to \(Y\hookrightarrow Z_1\). Furthermore the author [Int. Math. Res. Not. 2015, No. 16, 6793--6841 (2015; Zbl 1326.58018)] constructed real analytic torsion forms associated to \(M_1\to S\) with absolute and with relative boundary conditions.\N\NThe author proves a formula which relates these four torsion forms to each other, up to exact differential forms on \(S\). From this formula he then derives another version of a gluing formula for torsion forms, which generalises a formula given for its degree-0-part by \textit{J. Brüning} and \textit{X. Ma} [Math. Z. 273, No. 3--4, 1085--1117 (2013; Zbl 1318.58018)]. The proofs rely heavily on the gluing formula for torsion forms given in the preprint by Puchol, Zhang and the author [\textit{M. Puchol} et al., ``Adiabatic limit, Witten deformation and analytic torsion forms'', Preprint, \url{arXiv:2009.13925}].