A note on the eta function for quotients of \(PSL_ 2({\mathbb{R}})\) by co- compact Fuchsian groups
type: Item
instance of: scholarly article
title: A note on the eta function for quotients of \(PSL_ 2({\mathbb{R}})\) by co- compact Fuchsian groups
zbMATH Open document ID: 0664.58042
DOI: 10.1016/0040-9383(87)90023-1
author: Brian Steer
author: José Seade
published in: Topology
publication date: 1987-01-01T00:00:00Z
review text: The authors compute the value at 0 of the Atiyah-Patodi-Singer eta invariant associated to the Dirac operator on \(\Gamma \setminus PSL_ 2({\mathbb{R}})\), where \(\Gamma\) is a cocompact Fuchsian group. They do this by considering a family of metrics parameterized by \(t\in (0,\infty)\) and developing explicit formulae for the eigenvalues of the parameterized Dirac operator. Their approach makes use of a great deal of information concerning representations of \(SL_ 2({\mathbb{R}})\). Their main results are the following: Theorem 1: \(\Gamma \setminus PSL_ 2({\mathbb{R}})\) has no harmonic spinors for any cocompact Fuchsian group \(\Gamma\). Theorem 2: If \(\Gamma \subset PSL_ 2({\mathbb{R}})\) is a cocompact Fuchsian group of signature \(\{\) g; \(\alpha_ 1,\alpha_ 2,...,\alpha_ n\}\) then \[ \eta (0)=\frac{1}{48\pi}vol(F_{\Gamma})+\frac{1}{6}(\sum^{n}_{i=1}(\alpha _ i-1/\alpha_ i))=\frac{1}{24}\{2g-2+n+\sum^{n}_{i=1}(4\alpha_ i-5/\alpha_ i)\}, \] where \(F_{\Gamma}\) is a fundamental domain for the action of \(\Gamma\) on \({\mathbb{H}}\).
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 55Q45
zbMATH DE Number: 4086605
zbMATH Keywords: Atiyah-Patodi-Singer eta invariant
zbMATH Keywords: Dirac operator
zbMATH Keywords: cocompact Fuchsian group
zbMATH Keywords: representations of \(SL_ 2({\mathbb{R}})\)
zbMATH Keywords: harmonic spinors
MaRDI profile type: Publication
full work available at URL: 0040-9383(87)90023-1
OpenAlex ID: W2040542969
description: scientific article; zbMATH DE number 4086605