This is a review text file submitted electronically to MR. Reviewer: Thomas Schick Reviewer number: 024016 Address:

FB Mathematik Uni Göttingen Bunsenstr. 3 D-37073 Göttingen Germany schick@uni-math.gwdg.de

Author: Loya, P. and Park, J. Short title: On the glueing problem for the spectral invariants of the Dirac operator Control number: 2222357 Primary classification: 58J28, 58J52 Secondary classification(s): Review text: Let M be a closed manifold, split along a codimension 1 submanifold Y as union M = M_{−} ∪_{Y} M_{+} of two manifolds with boundary. Let D be a Dirac-type operator on the bundle S over M which is of product type D = G(∂_{u} + D_{Y}) near Y. If {λ_{k}} are the eigenvalues of D, the eta- and zeta-functions are defined by η_{D}(s) = ∑_{λ_{k}≠0} sgn(λ_{k})|λ_{k}|^{−s} and ζ_{D^{2}}(s) = ∑_{λ_{k}≠0}(λ_{k}^{2})^{−s}, respectively, extended meromorphically to ℂ. They are regular at s = 0. Then η(D) = η_{D}(0) and the ζ-determinant is det_{ζ} D^{2} = exp(−\frac{d}{ds}|_{s=0}ζ_{D^{2}}(s)).

Let next C_{+}, C_{−} be the orthogonalized Calderon projectors in L^{2}(Y, S|_{Y}) (onto the Cauchy data space), and let more generally P_{+}, P_{−} be similar orthogonal projections such that P_{±} − C_{±} is smoothing and GP_{±} = (1 − P_{±})G. Examples of such P_{±} are generalized APS projections.

Then D_{P_{±}}: dom(D_{P_{±}}) → L^{2}(M_{±}, S) are Fredholm operators, and η(D_{P_{±}}) as well as det_{ζ}(D_{P_{±}}^{2}) can be defined as in the closed case.

Write C_{±} with respect to the ±i eigenspace decomposition of G in block form

as C_{±} = \frac{1}{2} (\begin{matrix}1 & κ_{C_{±}}^{−1} \\ κ_{C_{±}} & 1 \end{matrix}) and set U_{C_{−}C_{+}} := κ_{C_{−}}κ_{C_{+}}^{−1}, Û_{C_{−}C_{+}} its restriction

to the complement of its (−1)-eigenspace. κ_{P_{±}} are defined in the same way. Let γ_{0} be the restriction to the boundary, v_{1}, …, v_{h_{M}} an orthonormal basis of ker(D) and L := ∑ γ_{0}v_{k} ⊗ γ_{0}v_{k}, a positive operator on the finite dimensional space γ_{0}(ker(D)). Set L_{1,2} := −Q_{±}GR_{−}^{−1}GQ_{±}, where Q_{±} is the projection onto im(C_{−}) ∩ im(1 − P_{±}), and R_{−} is a sum of Dirichlet-Neumann operators on the double of M_{−}. This is another positive operator on a finite dimensional vector space. Define U_{1} := κ_{C_{−}}κ_{P_{+}}^{−1}, U_{2} := κ_{P_{−}}κ_{C_{+}}^{−1}

The main result of the paper is the following gluing formula for ζ-determinants:

\frac{det_{ζ} D^{2}}{det_{ζ} D_{P_{+}}^{2} det_{ζ} D_{P_{−}}^{2}} =2^{−ζ_{D^{2}_{Y}} (0)−h_{Y}} det L^{−2} det_{F} (2 − Û_{C_{−}C_{+}} − Û_{C_{−}C_{+}}^{−1})

· ∏_{i=1}^{2} det L_{i}^{2} det_{F} (2 + Û_{i} + U_{i}^{−1})^{−1}

The proof also yields a new derivation of the gluing formula (mod ℤ) for the eta-invariant, the error term involving U_{12} := κ_{P_{+}}κ_{P_{−}}^{−1}. The authors also derive a formula for the integer defect. With different methods, such formulas have been derived by many different authors.

The proof develops new methods which rely on the theory of elliptic boundary problems, the analysis of resolvents and of a new model problem on the finite cylinder [−1, 1] × Y. In particular, the family K(λ) of Calderon projectors of (D_{±} − λ), describing how the Cauchy data spaces of (D_{±} − λ) over M_{±} match into the global data of (D − λ) over M, plays an essential role.