# PhD thesis defence

Thursday, July 29, 2021 — 9:30 AM EDT

Ragini Singhal, Department of Pure Mathematics, University of Waterloo

"Deformation theory of nearly $\rm{G}_2$-structures and nearly $\rm{G}_2$ instantons"

We study two different deformation theory problems on manifolds with a nearly $\rm{G}_2$-structure. The first involves studying the deformation theory of nearly $\rm{G}_2$ manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly $\rm{G}_2$-structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly $\rm{G}_2$-structure on the Aloff--Wallach space are all obstructed to second order. We also completely describe the de Rham cohomology of nearly $\rm{G}_2$ manifolds.

In the second problem we study the deformation theory of $\rm{G}_2$ instantons on nearly $\rm{G}_2$ manifolds. We make use of the one-to-one correspondence between nearly parallel $\rm{G}_2$-structures and real Killing spinors to formulate the deformation theory in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to explicitly describe the deformation space of the canonical connection on the four normal homogeneous nearly $\rm{G}_2$ manifolds.

We also describe the infinitesimal deformation space of the $\rm{SU}(3)$ instantons on Sasaki--Einstein $7$-folds which are nearly $\rm{G}_2$ manifolds with two Killing spinors. A Sasaki--Einstein structure on a $7$-dimensional manifold is equivalent to a $1$-parameter family of nearly $\rm{G}_2$-structures. We show that the deformation space can be described as an eigenspace of a twisted Dirac operator.

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