The \(\zeta\)-determinant and Quillen determinant for a Dirac operator on a manifold with boundary
The proof of the coincidence of the \(\zeta\)-determinant and the Quillen determinant of a Dirac operator on an odd dimensional manifold with boundary, which was announced in [\textit{S. G. Scott} and \textit{K. P. Wojciechowski}, C. R. Acad. Sci., Paris, Sér. I, Math. 328, No.~2, 139-144 (1999; Zbl 0957.58026)] is presented. Precisely, the operator \({\mathcal D} : C^\infty(M,S)\to C^\infty(M, S)\) is defined on a compact connected manifold \(M\) with boundary \(Y\), where \(S\) is a bundle of Clifford modules. It is assumed \(Y\) has a color neighborhood \(N = [0, 1]\times Y\) and \(\mathcal D\) has the form \({\mathcal D}_{|N}= G(\partial_u + B)\), where \(G : S|Y \to S|Y\) is a unitary bundle isomorphism, \(B\) the corresponding Dirac operator on \(Y\). Let \(\Pi_\geq\) be the projection from \(L^2(Y; S|Y)\) to the space spanned by the eigenvectors corresponding to the nonnegative eigenvalues of \(B\). The Grassmannian \(\text{Gr}({\mathcal D})\) is defined to be the space of all pseudodifferential projections which differ from \(\Pi_\geq\) by an operator of order \(-1\). The elliptic boundary value problem \({\mathcal D}_P\) is the operator \(\mathcal D\) with domain determined by a boundary condition \(P\), which is extensively studied in [\textit{B. Booss-Bavnbek} and \textit{K. P. Wojciechowski}, `Elliptic boundary problems for Dirac operators', Boston (1993; Zbl 0797.58004), hereafter referred to as [1]]. In that book, the selfadjoint Grassmannian \(\text{Gr}^*_\infty({\mathcal D})\) is introduced. In [\textit{B. Booss-Bavnbek} and \textit{K. P. Wojciechowski}, Commun. Math. Phys. 121, 1-9 (1989; Zbl 0683.58044)], \(H^2(\text{Gr}^*_\infty({\mathcal D});\mathbb{Z}) = 0\) is shown. Existence of \(\zeta\)-determinants for \({\mathcal D}_P\) is proved in [\textit{K. P. Wojciechowski}, Commun. Math. Phys. 201, No.~2, 423-444 (1999; Zbl 0948.58022)]. Strictly saying, the Quillen determinant is a section of the determinant bundle of \(\text{Gr}({\mathcal D})\). If the boundary condition is taken from \(\text{Gr}^*_\infty({\mathcal D})\), then we can select a non-zero cross section of the determinant bundle (restricted on \(\text{Gr}^*_\infty({\mathcal D}))\) and we can define the canonical determinant \(\text{det}_{\mathcal C}{\mathcal D}_P\) assuming \(\text{ker} B = \{0\}\) [\textit{S. G. Scott}, Commun. Math. Phys. 173, 43-76 (1995; Zbl 0840.58049)]. Then the main result of this paper is the following: Theorem 0.1. The following equality holds over \(\text{Gr}^*_\infty({\mathcal D})\) \[ \text{det}_\zeta{\mathcal D}_P=\text{det}_\zeta {\mathcal D}_{P({\mathcal D})}\cdot\text{det}_{\mathcal C}{\mathcal D}_P. \] Here \(P({\mathcal D})\) is the Calderon projection, the projection onto the \(L^2\)-closure of the Cauchy data space, which is shown to belong to \(\text{Gr}^*_\infty({\mathcal D})\) ([1]). The proof of Theorem 0.1 is done by computing a variation of determinants [cf. \textit{R. Forman}, Invent. Math. 88, 447-493 (1987; Zbl 0602.58044)]. The difficulty to treat a family of unbounded operators with varying domain is overcome by applying ``Unitary Trick'' of \textit{R. G. Douglas} and \textit{K. P. Wojciechowski} [Commun. Math. Phys. 142, 139-168 (1991; Zbl 0746.58074)]. The outline of the paper is as follows: In the Introduction, motivations of definitions of determinants and boundary conditions are precisely explained. The definition of canonical determinants is given in Section 1. Section 2 deals with boundary problems defined by \(\text{Gr}^*_\infty({\mathcal D})\) and the explicit formula of \({\mathcal D}^{-1}_P\) is given, when \(\text{ker }{\mathcal D}_P = \{0\}\) (Theorem 2.4). This formula plays a key role in the proof of the main result. The Poisson operator, the second term of \({\mathcal D}^{-1}_P{\mathcal D}_P\) is also investigated and its relation to the parametrix in [1] is remarked. Variations of \(\zeta\)-determinants and canonical determinants are computed in Sections 3 and 4 (Theorems 3.2 and 4.5). In Section 5, after showing pathconnectedness of \(\widetilde{\text{Gr}}^*_\infty({\mathcal D})\), the space of \(P \in \text{Gr}^*_\infty({\mathcal D})\) such that \({\mathcal D}_P\) is invertible (Proposition 5.1), Theorem 0.1 is proved by integrating variational formulae in Sections 3 and 4 (Theorem 5.4). In Section 6, the Calderon projection is characterized as the only critical point of the function \(|\text{det}_\zeta|\) on \(\text{Gr}^*_\infty({\mathcal D})\) (Theorem 6.1). By virtue of Theorem 0.1, to show this fact, it is sufficient to discuss on \(\text{det}_{\mathcal C}\), which is the key of the proof. In Section 5, Theorem 0.1 is proved assuming \(\text{ker} B = \{0\}\). But this is not necessary. It is shown in Section 7 among other things such as to remark weak contractibleness of \(\text{Gr}^*_\infty({\mathcal D})\) [\textit{L. Nicolaescu}, `Generalized Symplectic Geometries and the Index of Families of Elliptic Problems', Mem. Am. Math. Soc. 609, (1997; Zbl 0903.35015)], which gives a sharper result than Proposition 5.1. The authors remark that because of the reduction to the boundary integral ([1]), to discuss the relation between the \(\zeta\)-determinant and the Quillen determinant on a manifold with boundary is easier than the corresponding problem on a closed manifold. As for the discussion of related topics in the even-dimensional case, the authors refer to \textit{B. Booss-Bavnbek, G. Morchio, F. Stroochi} and \textit{K. P. Wojciechowski} [J. Geom. Phys. 22, 219-244 (1997; Zbl 0884.58086)].
scientific article; zbMATH DE number 1562879