Geometric structures on G2 and Spin(7)-manifolds

This paper is a review of current developments in the study of moduli spaces of G2 manifolds. G2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G2. Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi-Yau manifolds in 7 dimensions. They play an important role in physics as natural candidates for supersymmetric vacuum solutions of M-theory compactifications. Despite the physical motivation, many of the resul...

We first review the notion of a $G_2$-manifold, defined in terms of a principal $G_2$ ("gauge") bundle over a $7$-dimensional manifold, before discussing their relation to supergravity. In a second thread, we focus on associative submanifolds and present their deformation theory. In particular, we elaborate on a deformation problem with coassociative boundary condition. Its space of infinitesimal deformations can be identified with the solution space of an elliptic equation w...

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 connectio...

The main purpose of this paper is to give a mathematical definition of ``mirror symmetry'' for Calabi-Yau and G_2 manifolds. More specifically, we explain how to assign a G_2 manifold (M,\phi,\Lambda), with the calibration 3-form \phi and an oriented 2-plane field \Lambda, a pair of parametrized tangent bundle valued 2 and 3-forms of M. These forms can then be used to define various different complex and symplectic structures on certain 6-dimensional subbundles of T(M). When ...

The real Fourier-Mukai transform sends a section of a torus fibration to a connection over the total space of the dual torus fibration. By this method, Leung, Yau and Zaslow introduced deformed Hermitian Yang-Mills (dHYM) connections for K\"ahler manifolds and Lee and Leung introduced deformed Donaldson-Thomas (dDT) connections for $G_2$- and ${\rm Spin}(7)$-manifolds. In this paper, we suggest an alternative definition of a dDT connection for a manifold with a ${\rm Spin}(...

A deformed Donaldson-Thomas connection for a manifold with a ${\rm Spin}(7)$-structure, which we call a ${\rm Spin}(7)$-dDT connection, is a Hermitian connection on a Hermitian line bundle $L$ over a manifold with a ${\rm Spin}(7)$-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 indic...

We study the moduli space of torsion-free G2-structures on a fixed compact manifold, and define its associated universal intermediate Jacobian J. We define the Yukawa coupling and relate it to a natural pseudo-Kahler structure on J. We consider natural Chern-Simons type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang-Mills connections), and also deformed Donaldson-Thomas connections. We show that the modu...

We study deformations of associative submanifolds $Y^3\subset M^7$ of a $G_2$ manifold $M^7$. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative submanifolds of $M$. The local equations at each associative $Y$ are restrictions of a global equation on a certain associated Grassmann bundle over $ M$.

Here we study the deformations of associative submanifolds inside a G_2 manifold M^7 with a calibration 3-form \phi. A choice of 2-plane field \Lambda on M (which always exits) splits the tangent bundle of M as a direct sum of a 3-dimensional associate bundle and a complex 4-plane bundle TM= E\oplus V, and this helps us to relate the deformations to Seiberg-Witten type equations. Here all the surveyed results as well as the new ones about G_2 manifolds are proved by using onl...

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 compon...