We realise the Shatashvili-Vafa superconformal algebra for G 2 string compactifications by combining Odake and free conformal algebras following closely the recent mathematical construction of twisted connected sum G 2 holonomy manifolds. By considering automorphisms of this realisation, we identify stringy analogues of two mirror maps proposed by Braun and Del Zotto for these manifolds.

Recently, the infinitesimal moduli space of heterotic G 2 compactifications was described in supergravity and related to the cohomology of a target space differential. In this paper we identify the marginal deformations of the corresponding heterotic nonlinear sigma model with cohomology classes of a worldsheet BRST operator. This BRST operator is nilpotent if and only if the target space geometry satisfies the heterotic supersymmetry conditions. We relate this to the supergravity approach by showing that the corresponding cohomologies are indeed isomorphic. We work at tree-level in α ′ perturbation theory and study general geometries, in particular with non-vanishing torsion.

A new symmetry of (1, 0) supersymmetric non-linear σ-models in two dimensions with Fermi and mass sectors is introduced. It is a generalisation of the so-called special holonomy W-symmetry of Howe and Papadopoulos associated with structure group reductions of the target space M. Our symmetry allows in particular non-trivial flux and instanton-like connections on vector bundles over M. We also investigate potential anomalies and show that cohomologically non-trivial terms in the quantum effective action are invariant under a corrected version of our symmetry. Consistency with heterotic supergravity at first order in α ′ is manifest and discussed.

Gaussian Klauder coherent states are discussed in the context of the infinite well quantum model, otherwise known as the particle in a box. A supersymmetric partner system is also presented, as well as a construction of coherent states in this new system. We show that these states can be chosen, in both systems to have many properties usually expected for coherent states. In particular, they yield highly localised wave packets for a short period of time, which evolve in a quasi-classical manner and which saturate approximately Heisenberg uncertainty relation. These studies are elaborated in one-and two-dimensional contexts. Finally, some relations are established between the gaussian states being mostly used here and the generalised coherent states, which are more standardly found in the literature.

String backgrounds of the form 𝕄3× ℳ7 where 𝕄3 denotes 3-dimensional Minkowski space while ℳ7 is a 7-dimensional G2-manifold, are characterised by the property that the world-sheet theory has a Shatashvili-Vafa (SV) chiral algebra. We study the generalisation of this statement to backgrounds where the Minkowski factor 𝕄3 is replaced by AdS3. We argue that in this case the world-sheet theory is characterised by a certain $$ \mathcal{N} $$ N = 1 superconformal $$ \mathcal{W} $$ W -algebra that has the same spin spectrum as the SV algebra and also contains a tricritical Ising model $$ \mathcal{N} $$ N = 1 subalgebra. We determine the allowed representations of this $$ \mathcal{W} $$ W -algebra, and analyse to which extent the special features of the SV algebra survive this generalisation.