Sebastien Picard

We study the moduli space of the heterotic system, which holds significant importance in physics. Fixing the complex structure, we explore the moduli space by considering two different yet “dual” deformation paths starting from a Kähler solution. They correspond to deformations along the Bott-Chern cohomology class and the Aeppli cohomology class respectively. Together with the deformation of the gauge bundle, we prove the existence of heterotic solutions along these two paths using the implicit function theorem. Hence, we construct local coordinates in the neighborhood of a Kähler solution along the submanifold of fixed complex structure on the full heterotic moduli. This is an initial step in constructing the full metric of heterotic moduli.

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In this thesis, we study the continuity of a family of operators Et acting on functions or sections. First we study two easier cases. 1. The operator is a matrix At; 2. The operator or a Laplacian −Δt which acts on a function. These cases can be solved using min-max techniques coming from linear algebra, which gives us the explicit expression of each eigenvalue. But generally, for elliptic operators Et which acts on a section of a vector bundle, min-max techniques does work. Compared to the Laplacian case, one can not use integration by parts to cancel the Laplacian. And therefore the explicit expression of each eigenvalue can not be obtained. We will introduce Kodaira-Spencer theory for general cases, which is much more powerful and complicated. We’ll introduce the continuity theorem and give detailed proofs for the main theorem.

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