An index theorem for gauge-invariant families: The case of solvable groups

Victor Nistor

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family script G sign → B of Lie groups (these families are called "gauge-invariant families" in what follows). If the fibers of G → B are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah-Singer type formula that incorporates also topological information on the bundle script G sign → B. The algebras of invariant pseudodifferential operators that we study, ψinv(Y) and Ψinv(Y), are generalizations of "parameter dependent" algebras of pseudodifferential operators (with parameter in Rq). so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these results to study Fredholm boundary conditions on a simplex.

Original languageEnglish (US)
Pages (from-to)155-183
Number of pages29
JournalActa Mathematica Hungarica
Volume99
Issue number1-2
DOIs
StatePublished - Apr 1 2003

Fingerprint

Index Theorem
Solvable Group
Gauge
Pseudodifferential Operators
Invariant
Equivariant
Chern Character
Free Action
Invariant Operator
Algebra
Michael Francis Atiyah
Dependent
Elliptic Operator
Bundle
Fiber
Boundary conditions
Family

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

@article{c9176b586bf14bc98e3a6e251f9a9475,
title = "An index theorem for gauge-invariant families: The case of solvable groups",
abstract = "We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family script G sign → B of Lie groups (these families are called {"}gauge-invariant families{"} in what follows). If the fibers of G → B are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah-Singer type formula that incorporates also topological information on the bundle script G sign → B. The algebras of invariant pseudodifferential operators that we study, ψinv∞(Y) and Ψinv∞(Y), are generalizations of {"}parameter dependent{"} algebras of pseudodifferential operators (with parameter in Rq). so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these results to study Fredholm boundary conditions on a simplex.",
author = "Victor Nistor",
year = "2003",
month = "4",
day = "1",
doi = "10.1023/A:1024517714643",
language = "English (US)",
volume = "99",
pages = "155--183",
journal = "Acta Mathematica Hungarica",
issn = "0236-5294",
publisher = "Springer Netherlands",
number = "1-2",

}

An index theorem for gauge-invariant families : The case of solvable groups. / Nistor, Victor.

In: Acta Mathematica Hungarica, Vol. 99, No. 1-2, 01.04.2003, p. 155-183.

Research output: Contribution to journalArticle

TY - JOUR

T1 - An index theorem for gauge-invariant families

T2 - The case of solvable groups

AU - Nistor, Victor

PY - 2003/4/1

Y1 - 2003/4/1

N2 - We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family script G sign → B of Lie groups (these families are called "gauge-invariant families" in what follows). If the fibers of G → B are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah-Singer type formula that incorporates also topological information on the bundle script G sign → B. The algebras of invariant pseudodifferential operators that we study, ψinv∞(Y) and Ψinv∞(Y), are generalizations of "parameter dependent" algebras of pseudodifferential operators (with parameter in Rq). so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these results to study Fredholm boundary conditions on a simplex.

AB - We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family script G sign → B of Lie groups (these families are called "gauge-invariant families" in what follows). If the fibers of G → B are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah-Singer type formula that incorporates also topological information on the bundle script G sign → B. The algebras of invariant pseudodifferential operators that we study, ψinv∞(Y) and Ψinv∞(Y), are generalizations of "parameter dependent" algebras of pseudodifferential operators (with parameter in Rq). so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these results to study Fredholm boundary conditions on a simplex.

UR - http://www.scopus.com/inward/record.url?scp=0037396557&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037396557&partnerID=8YFLogxK

U2 - 10.1023/A:1024517714643

DO - 10.1023/A:1024517714643

M3 - Article

AN - SCOPUS:0037396557

VL - 99

SP - 155

EP - 183

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 1-2

ER -