Examples of deformed G2-instantons/Donaldson–Thomas connections
Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 339-366.

In this note, we provide the first non-trivial examples of deformed G 2 -instantons, originally called deformed Donaldson–Thomas connections. As a consequence, we see how deformed G 2 -instantons can be used to distinguish between nearly parallel G 2 -structures and isometric G 2 -structures on 3-Sasakian 7-manifolds. Our examples give non-trivial deformed G 2 -instantons with obstructed deformation theory and situations where the moduli space of deformed G 2 -instantons has components of different dimensions. We finally study the relation between our examples and a Chern–Simons type functional which has deformed G 2 -instantons as critical points.

Dans cette note, nous fournissons les premiers exemples non triviaux de G 2 -instantons déformés, initialement appelés connexions Donaldson–Thomas déformées. En conséquence, on peut utiliser G 2 -instantons déformés pour faire la distinction entre des G 2 -structures presque parallèles et des G 2 -structures isométriques sur des 7-variétés 3-Sasakiennes. Nos exemples donnent des G 2 -instantons déformés non triviaux avec une théorie de la déformation obstruée et des situations où l’espace des modules des G 2 -instantons déformés a des composantes de dimensions différentes. Nous étudions enfin la relation entre nos exemples et une fonctionnelle de type Chern–Simons qui a les G 2 -instantons déformés en points critiques.

Published online:
DOI: 10.5802/aif.3465
Classification: 53C07, 53C25
Keywords: Deformed $\mathrm{G}_2$-instantons, deformed Donaldson–Thomas connections, nearly parallel $\mathrm{G}_2$-structures, isometric $\mathrm{G}_2$-structures, 3-Sasakian.
Lotay, Jason D. 1; Oliveira, Gonçalo 2

1 University of Oxford, U.K.
2 Universidade Federal Fluminense IME-GMA Niterói, Brazil
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Lotay, Jason D. and Oliveira, Gon\c{c}alo},
     title = {Examples of deformed {G\protect\textsubscript{2}-instantons/Donaldson{\textendash}Thomas} connections},
     journal = {Annales de l'Institut Fourier},
     pages = {339--366},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {1},
     year = {2022},
     doi = {10.5802/aif.3465},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3465/}
AU  - Lotay, Jason D.
AU  - Oliveira, Gonçalo
TI  - Examples of deformed G2-instantons/Donaldson–Thomas connections
JO  - Annales de l'Institut Fourier
PY  - 2022
SP  - 339
EP  - 366
VL  - 72
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3465/
UR  - https://doi.org/10.5802/aif.3465
DO  - 10.5802/aif.3465
LA  - en
ID  - AIF_2022__72_1_339_0
ER  - 
%0 Journal Article
%A Lotay, Jason D.
%A Oliveira, Gonçalo
%T Examples of deformed G2-instantons/Donaldson–Thomas connections
%J Annales de l'Institut Fourier
%D 2022
%P 339-366
%V 72
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3465
%R 10.5802/aif.3465
%G en
%F AIF_2022__72_1_339_0
Lotay, Jason D.; Oliveira, Gonçalo. Examples of deformed G2-instantons/Donaldson–Thomas connections. Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 339-366. doi : 10.5802/aif.3465. https://aif.centre-mersenne.org/articles/10.5802/aif.3465/

[1] Ball, Gavin; Oliveira, Goncalo Gauge theory on Aloff–Wallach spaces, Geom. Topol., Volume 23 (2019) no. 2, pp. 685-743 | DOI | MR | Zbl

[2] Boyer, C.; Galicki, K. 3-Sasaki manifolds, Surveys in Differential Geometry, Vol. 6; Essays on Einstein Manifolds, Volume 6, International Press, 2001

[3] Boyer, C. P.; Galicki, K.; Mann, B. M. The geometry and topology of 3-Sasakian manifolds, J. Reine Angew. Math., Volume 455 (1994), pp. 183-220 | MR | Zbl

[5] Chen, G. Supercritical deformed Hermitian-Yang–Mills equation (2020) (https://arxiv.org/abs/2005.12202)

[6] Clarke, A.; Garcia-Fernandez, M.; Tipler, C. T-Dual solutions and infinitesimal moduli of the G 2 -Strominger system (2020) (https://arxiv.org/abs/2005.09977)

[7] Clarke, Andrew; Oliveira, Gonçalo Spin (7)-instantons from evolution equations, J. Geom. Anal., Volume 31 (2021) no. 4, pp. 4328-4355 | DOI | MR | Zbl

[8] Donaldson, S. K. Two-forms on four-manifolds and elliptic equations, Inspired by S. S. Chern (Nankai Tracts Math.), Volume 11, World Sci. Publ., Hackensack, NJ, 2006, pp. 153-172 | DOI | MR | Zbl

[9] Fine, J.; Yao, C. Hypersymplectic 4-manifolds, the G 2 -Laplacian flow, and extension assuming bounded scalar curvature, Duke Math. J., Volume 167 (2018) no. 18, pp. 3533-3589 | DOI | MR | Zbl

[10] Friedrich, Th.; Kath, I.; Moroianu, A.; Semmelmann, U. On nearly parallel G 2 -structures, J. Geom. Phys., Volume 23 (1997) no. 3-4, pp. 259-286 | DOI | MR | Zbl

[11] Grove, K.; Wilking, B.; Ziller, W. Positively curved cohomogeneity one manifolds and 3-Sasakian geometry, J. Differential Geom., Volume 78 (2008) no. 1, pp. 33-111 | DOI | MR | Zbl

[12] Karigiannis, S.; Leung, N. C. Hodge theory for G 2 -manifolds: intermediate Jacobians and Abel–Jacobi maps, Proc. Lond. Math. Soc. (3), Volume 99 (2009) no. 2, pp. 297-325 | DOI | MR | Zbl

[13] Lectures and surveys on G 2 -manifolds and related topics (Karigiannis, S.; Leung, N. C.; Lotay, J. D., eds.), Fields Institute Communications, 84, Springer, New York, 2020, xxii+382 pages (Extended papers from the Minischool and Workshop held as part of the Major Thematic Program on Geometric Analysis at the Fields Institute, Toronto, August 19–25, 2017) | DOI | MR

[14] Kawai, K.; Yamamoto, H. Deformation theory of deformed Hermitian Yang–Mills connections and deformed Donaldson–Thomas connections (2020) (arxiv.org/abs/2004.00532)

[15] Lee, J.-H.; Leung, N. C. Geometric structures on G 2 and Spin(7)-manifolds, Adv. Theor. Math. Phys., Volume 13 (2009) no. 1, pp. 1-31 | DOI | MR | Zbl

[16] Leung, N. C.; Yau, S.-T.; Zaslow, E. From special Lagrangian to Hermitian-Yang–Mills via Fourier–Mukai transform, Adv. Theor. Math. Phys., Volume 4 (2000) no. 6, pp. 1319-1341 | DOI | MR | Zbl

[17] Mariño, M.; Minasian, R.; Moore, G.; Strominger, A. Nonlinear instantons from supersymmetric p-branes, J. High Energy Phys. (2000) no. 1, 5, 32 pages | DOI | MR | Zbl

[18] Wang, Yuanqi Moduli spaces of G 2 and Spin(7)-instantons on product manifolds, Ann. Henri Poincaré, Volume 21 (2020) no. 9, pp. 2997-3033 | DOI | MR | Zbl

Cited by Sources: