On gluing formulas for the spectral invariants of Dirac type operators

Abstract

Abstract. In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac type operators on compact manifolds and manifolds with cylindrical ends. We also explain the central ideas in their proofs. The gluing problem for the spectral invariantsSince their inception, the eta invariant and the ζ-determinant of Dirac type operators have influenced mathematics and physics in innumerable ways. Especially with the development of quantum field theory, the behavior of these spectral invariants under gluing of the underlying manifold has become an increasingly important topic. However, the gluing formula for the ζ-determinant of a Dirac Laplacian has remained an open question due to the nonlocal nature of this invariant. In fact, Bleecker and Booss-Bavnbek stated that [2, p. 89] "no precise pasting formulas are obtained but only adiabatic ones." In [23, 24], we give precise gluing formulas for ζ-determinants of Dirac type operators on compact manifolds and manifolds with cylindrical ends, respectively, and moreover we present new and unified derivations of the gluing formulas for both invariants. The purpose of this note is to announce these gluing formulas for the spectral invariants and to indicate the main ideas in their proofs. We also announce a relative invariant formula proved in [25, 24].We begin with describing the gluing problem for compact manifolds.

The zeta-determinants of Dirac Laplacians with boundary conditions on the smooth, self-adjoint Grassmannian

“…We next apply these results to the BFK-gluing formula to obtain the gluing formula for the zeta-determinants of Dirac Laplacians with respect to boundary conditions on Gr * ∞ (D). In fact, Loya and Park [18,19] have already obtained the same result but their method is different from the one that we present here. Moreover, it is an advantage of this approach to be able to see the relation between the result of this paper and the BFK-gluing formula.…”

“…The result of Theorem 1.3 was obtained earlier by Loya and Park [18,19] in a different way. We next apply Theorem 1.1 to Laplacians on a cylinder.…”

The gluing formula of the zeta-determinants of Dirac Laplacians for certain boundary conditions

“…The following result is due to P. Loya, J. Park ( [16], [17]) and the second author ( [15]), independently.…”

“…The following results are due to S. Scott and K. Wojciechowski ([19], [20], [26]), P. Loya and J. Park ( [16], [17]) and the second author ( [15]). Theorem 2.1.…”

“…Using their results and the BFKgluing formula, P. Loya, J. Park ( [16], [17]) and the second author ( [15]) studied independently the gluing formula of Dirac Laplacians with respect to boundary conditions belonging to the smooth self-adjoint Grassmannian on compact Riemannian manifolds.…”