The Burghelea-Friedlander-Kappeler–gluing formula for zeta-determinants on a warped product manifold and a product manifold

Abstract

The Burghelea-Friedlander-Kappeler (BFK)-gluing formula for the regularized zeta-determinants of Laplacians contains a constant which is expressed by the constant term in the asymptotic expansion of the regularized zeta-determinants of a one-parameter family of the Dirichlet-to-Neumann operators. When the dimension of a cutting hypersurface is odd or the metric is a product one near a cutting hypersurface, this constant is well known. In this paper, we discuss this constant in two cases: one is when a warped product metric is given near a cutting hypersurface, and the other is when a manifold is a product manifold. Especially in the first case, we use the result of Fucci and Kirsten [Commun. Math. Phys. 317, 635-665 (2013)] in which the regularized zeta-determinant of the Laplacian defined on a warped product manifold is computed.

The quasilocal degrees of freedom of Yang-Mills theory

“…=1 of Lie(I A ) and of a connection ν ∈ Ω 1 (N A , gl(n)). The curvature of this connection, if geometrically constrained, would provide yet another source of obstruction to the non-Abelian generalization of the flow equation (91).…”

“…In [89], a computation of the contact term is proposed which starts from a comparison between a globally gauge-fixed path integral and its regional counterparts. The main ingredient of this computation is the Forman-BFK formula for the factorization of (zeta-regularized, Faddeev-Popov) functional determinants of Laplacians [67,90,91] (the relevance of this ingredient to calculations of black-hole entropy was already identified 86 by Carlip [93]). This formula features precisely the Abelian analogue of the operator (R −1 + + R −1 − ) that is central to our gluing formula.…”

Extended actions, dynamics of edge modes, and entanglement entropy

“…By defining the theory separately on the subregions, where one can compute Z M and ZM , it is however not possible to reconstruct the total Z by simply multiplying the results for the subregions. This is because the functional determinants do not factorize, but instead satisfy a relation of the form Z = K Z M ZM , sometimes referred to as the Forman-Burghelea-Friedlander-Kappeler (FBFK) gluing formula [56][57][58][59][60][61]. Although this argument was initially formulated for a scalar field theory, 4…”

“…This is the Maxwell analogue of the Poisson kernel integral obtained in [55] in the case of a scalar field. There, it was argued that properly splitting and sewing scalar field theory path integrals on manifolds with boundaries requires "scalar edge modes" in order to reproduce the FBFK gluing formula for functional determinants 18 [56][57][58][59]89], and that the corresponding edge scalar partition function on each side of the boundary comes from the boundary term needed in order to have a well-defined variational principle for the bulk scalar field action. As such, this argument would be puzzling when transposed to Maxwell theory, since in Maxwell the bulk action already has a well-defined variational principle without the need to add a boundary term, and one does not see where the Poisson kernel contributions of [55] could come from.…”

“…Importantly, one should recognize that this factor is not here in the radial gauge path integral computed in appendix B. Crucially, this determinant does not simply split between the two subregions. Instead, according to the FBFK gluing formula [56][57][58][59], we have that…”