G-structures and superstrings from the worldsheet

Thesis

G-structures, where G is a Lie group, are a uniform characterisation of many differential geometric structures of interest in supersymmetric compactifications of string theories. Calabi–Yau n-folds for instance have torsion-free SU(n)-structure, while more general structures with non-zero torsion are required for heterotic flux compactifications. Exceptional geometries in dimensions 7 and 8 with G = G₂ and Spin(7) also feature prominently in this thesis.

We discuss multiple connections between such geometries and the world- sheet theory describing strings on them, especially regarding their chiral symmetry algebras, originally due to Odake and Shatashvili–Vafa in the cases where G is SU (n), G₂ and Spin(7). In the first part of the thesis, we describe these connections within the formalism of operator algebras. We also realise the superconformal algebra for G₂ by combining Odake and free algebras following closely the recent mathematical construction of twisted connected sum G₂ manifolds. By considering automorphisms of this realisation, we speculate on mirror symmetry in this context.

In the second part of the thesis, G-structures are studied semi-classically from the worldsheet point of view using (1, 0) supersymmetric non-linear σ- models whose target M has reduced structure. Non-trivial flux and instanton- like connections on vector bundles over M are also allowed in order to deal with general applications to superstring compactifications, in particular in the heterotic case. We introduce a generalisation of the so-called special holonomy W-symmetry of Howe and Papadopoulos to σ-models with Fermi and mass sectors. We also investigate potential anomalies and show that cohomologically non-trivial terms in the quantum effective action are invariant under a corrected version of this symmetry. Consistency with supergravity at first order in α' 0 is manifest and discussed. We finally relate marginal deformations of (1, 0) G 2 and Spin(7) σ-models with the cohomology of worldsheet BRST operators. We work at lowest order in α' 0 and study general heterotic backgrounds with bundle and non-vanishing torsion.

Authors: Fiset, M

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: String models; Geometry