Approximate Yang-Mills-Higgs metrics on flat Higgs bundles over an affine manifold

Indranil Biswas, John Loftin, Matthias Stemmler

Research output: Contribution to journalArticle

Abstract

Given a flat Higgs vector bundle (E, ∇, φ) over a compact connected special affine manifold, we first construct a natural filtration of E, compatible with both ∇ and φ, such that the successive quotients are polystable flat Higgs vector bundles. This is done by combining the Harder-Narasimhan filtration and the socle filtration that we construct. Using this filtration, we construct a smooth Hermitian metric h on E and a smooth one-parameter family {At}t∈ℝ of ∞ automorphisms of E with the following property. Let ∇t and φt be the flat connection and flat Higgs field, respectively on E constructed from ∇ and φ using the automorphism At. If θt denotes the extended connection form on E associated to the triple h, ∇t and φt, then as t→ +infin;00, the connection form θt converges in the C Frechet topology to the extended connection form θ on E given by the affine Yang-Mills-Higgs metrics on the polystable quotients of the successive terms in the above mentioned filtration. In particular, as t→ +infin;00, the curvature of θt converges in the ∞00 Frechet topology to the curvature of θ.

Original languageEnglish (US)
Pages (from-to)737-754
Number of pages18
JournalCommunications in Analysis and Geometry
Volume22
Issue number4
DOIs
StatePublished - Jan 1 2014

Fingerprint

Higgs Bundles
Yang-Mills
Higgs
Filtration
Metric
Vector Bundle
Quotient
Curvature
Topology
Converge
Flat Connection
Socle
Automorphism
Automorphisms
Denote
Term
Form

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Geometry and Topology
  • Statistics, Probability and Uncertainty

Cite this

@article{61c3ddf0ae6547348d9fc7a040b02650,
title = "Approximate Yang-Mills-Higgs metrics on flat Higgs bundles over an affine manifold",
abstract = "Given a flat Higgs vector bundle (E, ∇, φ) over a compact connected special affine manifold, we first construct a natural filtration of E, compatible with both ∇ and φ, such that the successive quotients are polystable flat Higgs vector bundles. This is done by combining the Harder-Narasimhan filtration and the socle filtration that we construct. Using this filtration, we construct a smooth Hermitian metric h on E and a smooth one-parameter family {At}t∈ℝ of ∞∞ automorphisms of E with the following property. Let ∇t and φt be the flat connection and flat Higgs field, respectively on E constructed from ∇ and φ using the automorphism At. If θt denotes the extended connection form on E associated to the triple h, ∇t and φt, then as t→ +infin;00, the connection form θt converges in the C∞ Frechet topology to the extended connection form θ on E given by the affine Yang-Mills-Higgs metrics on the polystable quotients of the successive terms in the above mentioned filtration. In particular, as t→ +infin;00, the curvature of θt converges in the ∞00 Frechet topology to the curvature of θ.",
author = "Indranil Biswas and John Loftin and Matthias Stemmler",
year = "2014",
month = "1",
day = "1",
doi = "10.4310/CAG.2014.v22.n4.a5",
language = "English (US)",
volume = "22",
pages = "737--754",
journal = "Communications in Analysis and Geometry",
issn = "1019-8385",
publisher = "International Press of Boston, Inc.",
number = "4",

}

Approximate Yang-Mills-Higgs metrics on flat Higgs bundles over an affine manifold. / Biswas, Indranil; Loftin, John; Stemmler, Matthias.

In: Communications in Analysis and Geometry, Vol. 22, No. 4, 01.01.2014, p. 737-754.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Approximate Yang-Mills-Higgs metrics on flat Higgs bundles over an affine manifold

AU - Biswas, Indranil

AU - Loftin, John

AU - Stemmler, Matthias

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Given a flat Higgs vector bundle (E, ∇, φ) over a compact connected special affine manifold, we first construct a natural filtration of E, compatible with both ∇ and φ, such that the successive quotients are polystable flat Higgs vector bundles. This is done by combining the Harder-Narasimhan filtration and the socle filtration that we construct. Using this filtration, we construct a smooth Hermitian metric h on E and a smooth one-parameter family {At}t∈ℝ of ∞∞ automorphisms of E with the following property. Let ∇t and φt be the flat connection and flat Higgs field, respectively on E constructed from ∇ and φ using the automorphism At. If θt denotes the extended connection form on E associated to the triple h, ∇t and φt, then as t→ +infin;00, the connection form θt converges in the C∞ Frechet topology to the extended connection form θ on E given by the affine Yang-Mills-Higgs metrics on the polystable quotients of the successive terms in the above mentioned filtration. In particular, as t→ +infin;00, the curvature of θt converges in the ∞00 Frechet topology to the curvature of θ.

AB - Given a flat Higgs vector bundle (E, ∇, φ) over a compact connected special affine manifold, we first construct a natural filtration of E, compatible with both ∇ and φ, such that the successive quotients are polystable flat Higgs vector bundles. This is done by combining the Harder-Narasimhan filtration and the socle filtration that we construct. Using this filtration, we construct a smooth Hermitian metric h on E and a smooth one-parameter family {At}t∈ℝ of ∞∞ automorphisms of E with the following property. Let ∇t and φt be the flat connection and flat Higgs field, respectively on E constructed from ∇ and φ using the automorphism At. If θt denotes the extended connection form on E associated to the triple h, ∇t and φt, then as t→ +infin;00, the connection form θt converges in the C∞ Frechet topology to the extended connection form θ on E given by the affine Yang-Mills-Higgs metrics on the polystable quotients of the successive terms in the above mentioned filtration. In particular, as t→ +infin;00, the curvature of θt converges in the ∞00 Frechet topology to the curvature of θ.

UR - http://www.scopus.com/inward/record.url?scp=84907528433&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907528433&partnerID=8YFLogxK

U2 - 10.4310/CAG.2014.v22.n4.a5

DO - 10.4310/CAG.2014.v22.n4.a5

M3 - Article

VL - 22

SP - 737

EP - 754

JO - Communications in Analysis and Geometry

JF - Communications in Analysis and Geometry

SN - 1019-8385

IS - 4

ER -