The local index density of the perturbed de Rham complex

Jesús Álvarez López; Peter B. Gilkey

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 3, page 901-932
  • ISSN: 0011-4642

Abstract

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A perturbation of the de Rham complex was introduced by Witten for an exact 1-form Θ and later extended by Novikov for a closed 1-form on a Riemannian manifold M . We use invariance theory to show that the perturbed index density is independent of Θ ; this result was established previously by J. A. Álvarez López, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence on Θ . We establish similar results for manifolds with boundary imposing suitable boundary conditions and give an equivariant version for the local Lefschetz trace density. In the setting of the Dolbeault complex, one requires Θ to be a ∂ ¯ closed 1 -form to define a local index density; we show in contrast to the de Rham complex, that this exhibits a nontrivial dependence on Θ even in the setting of Riemann surfaces.

How to cite

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Álvarez López, Jesús, and Gilkey, Peter B.. "The local index density of the perturbed de Rham complex." Czechoslovak Mathematical Journal 71.3 (2021): 901-932. <http://eudml.org/doc/297970>.

@article{ÁlvarezLópez2021,
abstract = {A perturbation of the de Rham complex was introduced by Witten for an exact 1-form $\Theta $ and later extended by Novikov for a closed 1-form on a Riemannian manifold $M$. We use invariance theory to show that the perturbed index density is independent of $\Theta $; this result was established previously by J. A. Álvarez López, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence on $\Theta $. We establish similar results for manifolds with boundary imposing suitable boundary conditions and give an equivariant version for the local Lefschetz trace density. In the setting of the Dolbeault complex, one requires $\Theta $ to be a $\bar\{\partial \}$ closed $1$-form to define a local index density; we show in contrast to the de Rham complex, that this exhibits a nontrivial dependence on $\Theta $ even in the setting of Riemann surfaces.},
author = {Álvarez López, Jesús, Gilkey, Peter B.},
journal = {Czechoslovak Mathematical Journal},
keywords = {Witten deformation; local index density; de Rham complex; Dolbeault complex; equivariant index density},
language = {eng},
number = {3},
pages = {901-932},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The local index density of the perturbed de Rham complex},
url = {http://eudml.org/doc/297970},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Álvarez López, Jesús
AU - Gilkey, Peter B.
TI - The local index density of the perturbed de Rham complex
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 3
SP - 901
EP - 932
AB - A perturbation of the de Rham complex was introduced by Witten for an exact 1-form $\Theta $ and later extended by Novikov for a closed 1-form on a Riemannian manifold $M$. We use invariance theory to show that the perturbed index density is independent of $\Theta $; this result was established previously by J. A. Álvarez López, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence on $\Theta $. We establish similar results for manifolds with boundary imposing suitable boundary conditions and give an equivariant version for the local Lefschetz trace density. In the setting of the Dolbeault complex, one requires $\Theta $ to be a $\bar{\partial }$ closed $1$-form to define a local index density; we show in contrast to the de Rham complex, that this exhibits a nontrivial dependence on $\Theta $ even in the setting of Riemann surfaces.
LA - eng
KW - Witten deformation; local index density; de Rham complex; Dolbeault complex; equivariant index density
UR - http://eudml.org/doc/297970
ER -

References

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