The Adiabatic b-Calculus and the Global Anomaly Formula on Manifolds with Boundary

Abstract

We study the determinant line bundle for a family of elliptic operators on compact manifolds with boundary with the associated Quillen metric and Bismut-Freed connection, and we prove the global anomaly formula in this context. This extends Witten's holonomy theorem to manifolds with boundary. Witten studied the limiting behavior of the eta invariant of a Dirac operator on the total space of a fibration over 𝕊¹ when the metric in the base 𝕊¹ is blown up in a certain manner, a process known as taking the adiabatic limit. In this paper, we construct the calculus of adiabatic b-pseudodifferential operators, which naturally synthesizes the b-calculus and the adiabatic calculus of pseudodifferential operators. Using this calculus, and assuming the fiber operators to be invertible, we compute the adiabatic limit of the natural eta invariant in this setting and relate it to the integral of the Bismut-Freed connection and new correction terms coming from the boundary of the fibers.

OFFER (One-Formula-Facilitates-Every-Relation) for Calculus
Samuel S. P. Shen
arXiv19 Oct 2026PreprintCC BY-NC-SA 4.0Full textGeneral mathematicsarXiv
Differential calculus on quantized irreducible flag manifolds
Toshiyuki Tanisaki
arXiv13 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Higher-order variation and pathwise Ito calculus on manifolds
Rama Cont
arXiv6 Sep 2026PreprintarXiv licenceFull textGeometry & topology ProbabilityarXiv
Trace formula for billiard flow on surfaces with concave boundary
Alain Kangabire
arXiv18 Sep 2026PreprintarXiv licenceFull textDynamical systemsarXiv
A note on the failure of Payne's inequality on manifolds with boundary
Davide Buoso, Vladimir Lotoreichik
arXiv14 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEs Geometry & topologyarXiv

Cite

BibTeX RIS

BibTeX

@misc{nelson2026adiabatic,
  title = {{The Adiabatic b-Calculus and the Global Anomaly Formula on Manifolds with Boundary}},
  author = {Chad Nelson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16938},
  eprint = {2610.16938},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16938},
}

RIS

TY  - UNPB
TI  - The Adiabatic b-Calculus and the Global Anomaly Formula on Manifolds with Boundary
AU  - Chad Nelson
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16938
UR  - https://arxiv.org/abs/2610.16938
AB  - We study the determinant line bundle for a family of elliptic operators on compact manifolds with boundary with the associated Quillen metric and Bismut-Freed connection, and we prove the global anomaly formula in this context. This extends Witten's holonomy theorem to manifolds with boundary. Witten studied the limiting behavior of the eta invariant of a Dirac operator on the total space of a fibration over 𝕊^1 when the metric in the base 𝕊^1 is blown up in a certain manner, a process known as taking the adiabatic limit. In this paper, we construct the calculus of adiabatic b-pseudodifferential operators, which naturally synthesizes the b-calculus and the adiabatic calculus of pseudodifferential operators. Using this calculus, and assuming the fiber operators to be invertible, we compute the adiabatic limit of the natural eta invariant in this setting and relate it to the integral of the Bismut-Freed connection and new correction terms coming from the boundary of the fibers.
ER  -

Abstract

Related

Cite