Some observations on deformed Donaldson–Thomas connections
[Quelques observations sur les connexions de Donaldson–Thomas déformées]
Kawai, Kotaro1, 2
1Beijing Institute of Mathematical Sciences, and Applications, No. 544, Hefangkou Village, Huaibei Town, Huairou District, Beijing, 101408 (China)
2Department of Mathematics, Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585 (Japan)
Annales de l'Institut Fourier, Online first, 21 p.

A deformed Donaldson–Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a $G_2$-manifold $X$ satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. It can also be considered as an analogue of a $G_2$-instanton. In this paper, we see that some important observations that appear in other geometric problems are also found in the dDT case as follows.

(1) A dDT connection exists if a 7-manifold has full holonomy $G_2$ and the $G_2$-structure is “sufficiently large”. (2) The dDT equation is described as the zero of a certain multi-moment map. (3) The gradient flow equation of a Chern–Simons type functional of Karigiannis and Leung, whose critical points are dDT connections, agrees with the $\operatorname{Spin}(7)$ version of the dDT equation on a cylinder with respect to a certain metric on a certain space. This can be considered as an analogue of the observation in instanton Floer homology for 3-manifolds.

Une connexion déformée de Donaldson–Thomas (dDT) est une connexion hermitienne d’un fibré en ligne hermitien sur une variété $G_2$ $X$ satisfaisant une certaine EDP non linéaire. Ceci est considéré comme le miroir d’un cycle (co)associatif dans le contexte de la symétrie miroir. On peut également le considérer comme un analogue d’un $G_2$-instanton. Dans cet article, nous voyons que certaines observations importantes qui apparaissent dans d’autres problèmes géométriques se retrouvent également dans le cas dDT comme suit.

(1) Une connexion dDT existe si une variété 7 possède une holonomie complète $G_2$ et que la structure $G_2$ est « suffisamment grande ». (2) L’équation dDT est décrite comme le zéro d’une certaine application multi-moments. (3) L’équation de flux de gradient d’une fonctionnelle de type Chern–Simons de Karigiannis et Leung, dont les points critiques sont des connexions dDT, concorde avec la version $\operatorname{Spin}(7)$ de l’équation dDT sur un cylindre par rapport à une certaine métrique sur un certain espace. Ceci peut être considéré comme un analogue de l’observation en homologie de Floer instanton pour les variétés 3.

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Révisé le :
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Première publication :
DOI : 10.5802/aif.3753
Classification : 53C07, 58E15, 53D37
Keywords: mirror symmetry, gauge theory, $G_2$-manifold, deformed Donaldson–Thomas connections
Mots-clés : symétrie miroir, théorie de jauge, variété $G_2$, connections de Donaldson–Thomas déformées

Kawai, Kotaro  1 , 2

1 Beijing Institute of Mathematical Sciences, and Applications, No. 544, Hefangkou Village, Huaibei Town, Huairou District, Beijing, 101408 (China)
2 Department of Mathematics, Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585 (Japan) 
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Kawai, Kotaro. Some observations on deformed Donaldson–Thomas connections. Annales de l'Institut Fourier, Online first, 21 p.

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