Gluing Formulae of Spectral Invariants and Cauchy Data Spaces

In this article, we survey the gluing formulae of the spectral invariants the ζ-regularized determinant of a Laplace type operator and the eta invariant of a Dirac type operator. After these spectral invariants had been originally introduced by Ray-Singer [30] and Atiyah-Patodi-Singer [1] respectively, these invariants have been studied by many people in many different parts of mathematics and physics. Here we discuss the gluing formulae of these spectral invariants. These formulae have been proved independently by several authors using different techniques. For nice introductions to this subject, we refer to Bleecker-Booß-Bavnbek [3] and Mazzeo-Piazza [21] where the reader can find many technical details and ideas of proofs. Therefore, instead of repeating the details of these introductions, we explain one principle which holds for all the known gluing formulae of the spectral invariants. This principle also enabled us to get a new proof of the gluing formulae of the eta invariant of a Dirac type operator and simultaneously to prove the gluing formula of the ζ-regularized determinant of a Dirac Laplacian [17], [18]. We hope that this article would be helpful in the understanding of gluing formulae of the spectral invariants and other related gluing problems in similar situations. Now let us review briefly the history of this subject: gluing problems