Geometric structures of the heterotic string

Abstract

This thesis investigates topological invariants attached to the moduli space of heterotic string compactifications on Calabi--Yau threefolds, with particular attention to the holomorphic sector of the Hull--Strominger system. After developing the necessary mathematical background, covering complex geometry, higher algebraic structures, and analytic torsion, the main results are presented chapter by chapter. Chapter 3 reviews the Hull--Strominger system, formulates it variationally through the heterotic superpotential, and singles out the F-term sector that controls the holomorphic deformation problem. Chapter 4 constructs the heterotic deformation complex. The physical moduli, namely complex structure deformations, gauge bundle deformations, and Hermitian metric deformations, are packaged into a single combined field, and an extended Dolbeault operator $\bar{D}$ is defined whose nilpotency precisely encodes the constraints of the Hull--Strominger system. A natural graded bracket then equips the field space with the structure of a differential graded Lie algebra, whose Maurer--Cartan equation parameterizes finite deformations of the heterotic background. Chapter 5 turns to the global geometry of $\bar{D}$. Its off-diagonal entries contain explicit holomorphic derivatives, which prevent it from defining a standard holomorphic bundle. We circumvent this by building explicit local trivializations, which are used to define a quasi-holomorphic extension sheaf $\tilde{Q}$. The main theorem is a Dolbeault theorem for $\bar{D}$, establishing a natural isomorphism between its cohomology and the \v{C}ech cohomology of $\tilde{Q}$.