Deformation Theory of Deformed Donaldson–Thomas Connections for Spin(7)-manifolds

• Published in: The Journal of geometric analysis Vol. 31; no. 12; pp. 12098 - 12154
• Main Authors: Kawai, Kotaro, Yamamoto, Hikaru
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Deformation; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Instantons; Mathematics; Mathematics and Statistics; Special holonomy; Primary: 53C07; 58D27; Calibrated submanifold; Deformed Donaldson–Thomas; Moduli space; Deformation theory; 53C38; 53C25; 58H15; Mirror symmetry; Secondary: 53D37
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-021-00712-2

A deformed Donaldson–Thomas connection for a manifold with a Spin ( 7 ) -structure, which we call a Spin ( 7 ) -dDT connection, is a Hermitian connection on a Hermitian line bundle L over a manifold with a Spin ( 7 ) -structure defined by fully nonlinear PDEs. It was first introduced by Lee and Leung as a mirror object of a Cayley cycle obtained by the real Fourier–Mukai transform and its alternative definition was suggested in our other paper. As the name indicates, a Spin ( 7 ) -dDT connection can also be considered as an analogue of a Donaldson–Thomas connection ( Spin ( 7 ) -instanton). In this paper, using our definition, we show that the moduli space M Spin ( 7 ) of Spin ( 7 ) -dDT connections has similar properties to these objects. That is, we show the following for an open subset M Spin ( 7 ) ′ ⊂ M Spin ( 7 ) . (1) Deformations of elements of M Spin ( 7 ) ′ are controlled by a subcomplex of the canonical complex defined by Reyes Carrión by introducing a new Spin ( 7 ) -structure from the initial Spin ( 7 ) -structure and a Spin ( 7 ) -dDT connection. (2) The expected dimension of M Spin ( 7 ) ′ is finite. It is b 1 , the first Betti number of the base manifold, if the initial Spin ( 7 ) -structure is torsion-free. (3) Under some mild assumptions, M Spin ( 7 ) ′ is smooth if we perturb the initial Spin ( 7 ) -structure generically. (4) The space M Spin ( 7 ) ′ admits a canonical orientation if all deformations are unobstructed.