Geometric Transitions and New Dualities in Heterotic Compactifications

Abstract

In this thesis, we investigate a new set of dualities between 4-dimensional Calabi-Yau threefold compactifications of the heterotic string, where the base manifolds are linked by a conifold transition. In the N = 1 heterotic context, unlike Type II theories, dualities are more intricate due to the presence of an additional, non-trivial gauge bundle. In recent work, a geometric mechanism was proposed detailing how 5-branes and gauge bundles traverse consistently through the heterotic conifold transition. It was observed that compactifications connected in this way lead to 4-dimensional effective theories with the identical massless gauge singlet spectrum, dictated by geometric moduli. This was originally seen in the context of Target Space Dual theories obtained in the geometric phase of (0,2) Gauged Linear Sigma Models (GLSMs), sharing a common Landau-Ginzburg orbifold phase. The geometric connection of such GLSM target space dual theories was previously unclear, but they are now found to be connected by a conifold transition. In our study, we go much beyond the simple counting of moduli cohomology dimensions to construct a duality map between the field spaces of the two compactifications by explicitly tracking complex structure, Kähler, and bundle degrees of freedom. As a strong verification of the obtained moduli map, we match holomorphic terms in the 4D action, such as gauge kinetic functions and 147,440 Yukawa couplings, in one example. This provides compelling evidence for the proposed duality. Our study establishes these GLSM target space dualities as a subset of a larger duality between a series of configurations connected by a conifold. In the second part of the thesis, we complete this moduli mapping framework by extracting the Kähler moduli map through an algorithmic process, utilizing a highly restrictive ansatz, and simultaneously matching gauge kinetic functions, stability walls, and non-perturbative instanton superpotential terms. Additionally, we show that loci of enhanced gauge symmetry, introduced in the bundle via ker(F)/im(E) and equivalently viewed in the (0,2) GLSM as additional fermionic gauge symmetry, can be preserved across the heterotic conifold by an elegant extension of the recently proposed geometric mechanism.