We consider deformations of torsion-free G
 2 structures, defined by the G
 2-invariant 3-form φ and compute the expansion of $${\ast \varphi }$$
 to fourth order in the deformations of φ. By considering M-theory compactified on a G
 2 manifold, the G
 2 moduli space is naturally complexified, and we get a Kahler metric on it. Using the expansion of $${\ast \varphi }$$
 , we work out the full curvature of this metric and relate it to the Yukawa coupling.

Local Geometry of the G 2 Moduli Space

We consider deformations of torsion-free G2 structures, defined by the G2-invariant 3-form $\phi$ and compute the expansion of the Hodge star of $\phi$ to fourth order in the deformations of $\phi$. By considering M-theory compactified on a G2 manifold, the G2 moduli space is naturally complexified, and we get a Kahler metric on it. Using the expansion of the Hodge star of $\phi$ we work out the full curvature of this metric and relate it to the Yukawa coupling.

Local geometry of the G2 moduli space

In Berger’s classification of Riemannian holonomy groups there are several infinite families and two exceptional cases: the groups Spin(7) and G2. This thesis is mainly concerned with 7-dimensional manifolds with holonomy G2. A metric with holonomy contained in G2 can be defined in terms of a torsion-free G2-structure, and a G2-manifold is a 7-dimensional manifold equipped with such a structure. There are two known constructions of compact manifolds with holonomy exactly G2. Joyce found examples by resolving singularities of quotients of flat tori. Later Kovalev found different examples by gluing pairs of exponentially asymptotically cylindrical (EAC) G2-manifolds (not necessarily with holonomy exactlyG2) whose cylinders match. The result of this gluing construction can be regarded as a generalised connected sum of the EAC components, and has a long approximately cylindrical neck region. We consider the deformation theory of EAC G2-manifolds and show, generalising from the compact case, that there is a smooth moduli space of torsion-free EACG2-structures. As an application we study the deformations of the gluing construction for compact G2-manifolds, and find that the glued torsion-free G2-structures form an open subset of the moduli space on the compact connected sum. For a fixed pair of matching EAC G2-manifolds the gluing construction provides a path of torsion-free G2-structures on the connected sum with increasing neck length. Intuitively this defines a boundary point for the moduli space on the connected sum, representing a way to ‘pull apart’ the compact G2-manifold into a pair of EAC components. We use the deformation theory to make this more precise. We then consider the problem whether compact G2-manifolds constructed by Joyce’s method can be deformed to the result of a gluing construction. By proving a result for resolving singularities of EAC G2-manifolds we show that some of Joyce’s examples can be pulled apart in the above sense. Some of the EAC G2-manifolds that arise this way satisfy a necessary and sufficient topological condition for having holonomy exactly G2. We prove also deformation results for EAC Spin(7)-manifolds, i.e. dimension 8 manifolds with holonomy contained in Spin(7). On such manifolds there is a smooth moduli space of torsion-free EAC Spin(7)-structures. Generalising a result of Wang for compact manifolds we show that for EAC G2-manifolds and Spin(7)-manifolds the special holonomy metrics form an open subset of the set of Ricci-flat metrics.

/pdf/deformations-and-gluing-of-asymptotically-cylindrical-2df1rzcqhc.pdf

Deformations and gluing of asymptotically cylindrical manifolds with exceptional holonomy

Heterotic string compactifications on integrable $G_2$ structure manifolds $Y$ with instanton bundles $(V,A), (TY,\tilde{\theta})$ yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative $\cal D$ and show that it is equivalent to a heterotic $G_2$ system encoding the geometry of the heterotic string compactifications. This operator $\cal D$ acts on a bundle ${\cal Q}=T^*Y\oplus{\rm End}(V)\oplus{\rm End}(TY)$ and satisfies a nilpotency condition $\check{\cal D}^2=0$, for an appropriate projection of $\cal D$. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group $\check H^1_{\check{\cal D}}(\cal Q)$. We comment on the similarities and differences of our result with Atiyah's well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the $\alpha'$ expansion.

The infinitesimal moduli space of heterotic $G_2$ systems

Heterotic string compactifications on integrable G
 2 structure manifolds Y with instanton bundles $${(V,A), (TY,\tilde{\theta})}$$
 yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative $${\mathcal{D}}$$
 and show that it is equivalent to a heterotic G
 2 system encoding the geometry of the heterotic string compactifications. This operator $${\mathcal{D}}$$
 acts on a bundle $${\mathcal{Q}=T^*Y \oplus {\rm End}(V) \oplus {\rm End}(TY)}$$
 and satisfies a nilpotency condition $${\check{{\mathcal{D}}}^2=0}$$
 , for an appropriate projection of $${\mathcal D}$$
 . Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group $${\check H^1_{\check{{\mathcal{D}}}}(\mathcal{Q})}$$
 . We comment on the similarities and differences of our result with Atiyah’s well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the $${\alpha'}$$
 expansion.

/pdf/the-infinitesimal-moduli-space-of-heterotic-g-2-systems-lp6nw10t29.pdf

The Infinitesimal Moduli Space of Heterotic G 2 Systems

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 moduli spaces of these structures can be isotropically immersed in J by means of G2-analogues of Abel-Jacobi maps.

Hodge theory for G2-manifolds: intermediate Jacobians and Abel–Jacobi maps

There are two natural classes of Einstein metrics on bounded domains. (I) metrics which extend smoothly to the boundary, and (II) complete metrics which conformally extend to the boundary, (conformally compact metrics). We will discuss similarities and differences on the structure of these spaces of Einstein metrics, in particular in regard to the “natural” boundary value problems. Speaker: Adrian Butscher (Stanford University) Title: “Gluing Constructions for Constant Mean Curvature Surfaces” Abstract: I will review the now classical Kapouleas gluing construction for CMC surfaces in Euclidean space and present some results and work in progress concerning the extensions of this theory to general ambient manifolds. An important feature which emerges is that the ambient Riemannian curvature seems to play a significant role in the existence of such surfaces; and exploiting this, it seems possible to construct examples of CMC surfaces having properties very different from their Euclidean analogues. I will review the now classical Kapouleas gluing construction for CMC surfaces in Euclidean space and present some results and work in progress concerning the extensions of this theory to general ambient manifolds. An important feature which emerges is that the ambient Riemannian curvature seems to play a significant role in the existence of such surfaces; and exploiting this, it seems possible to construct examples of CMC surfaces having properties very different from their Euclidean analogues. Speaker: Benoit Charbonneau (Duke University) Title: “Existence of periodic instantons” Abstract: Yang–Mills instantons on S1 × R3 (often called calorons) are in correspondence, via the Nahm transform, to solutions to Nahm’s equations on the circle. In joint work with Jacques Hurtubise, we completed Nye and Singer’s proof of this Nahm transform correspondence. We also proved that the solutions on the circle are in correspondence, by a twistor transform, to certain classes of vector bundles on an associated twistor space. Those correspondence allow us to compute the moduli space of these objects, settling some very natural existence questions. Yang–Mills instantons on S1 × R3 (often called calorons) are in correspondence, via the Nahm transform, to solutions to Nahm’s equations on the circle. In joint work with Jacques Hurtubise, we completed Nye and Singer’s proof of this Nahm transform correspondence. We also proved that the solutions on the circle are in correspondence, by a twistor transform, to certain classes of vector bundles on an associated twistor space. Those correspondence allow us to compute the moduli space of these objects, settling some very natural existence questions.

/pdf/special-structures-in-riemannian-geometry-1vvd9ldmhb.pdf

Special Structures in Riemannian Geometry

Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps

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Hodge theory for G2-manifolds: intermediate Jacobians and Abel–Jacobi maps

Special Structures in Riemannian Geometry

Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps