Stable Singularities in String Theory

TL;DR

The paper investigates stable singularities in Calabi–Yau targets within $N=(2,2)$ theories and identifies a stringy obstruction arising from torsion in $H_2(X)$ (the Brauer group) and discrete torsion. It develops the A-model framework to incorporate this torsion via a discrete parameter $oldsymbol{ hisalpha}$ when $ ext{Tors}(H^3(X))\cong\mathbb{Z}_m$, showing instanton contributions acquire $oldsymbol{ hisalpha}$-dependent phases and that the moduli space splits into sheets connected at the large-radius limit. Through explicit nodal examples (double covers of $\mathbb{P}^3$ and $(\mathbb{P}^1)^3$) and their blow-ups, the authors demonstrate how a nontrivial Brauer group can obstruct smoothing to a Calabi–Yau when $oldsymbol{ hisalpha}=1$, while $oldsymbol{ hisalpha}=0$ may allow such deformations; this provides a concrete mechanism linking discrete torsion, worldsheet instantons, and global topology. The results illuminate how string theory encodes global topological data beyond classical geometry, with connections to orbifold 2-cocycles and mirror symmetry, and suggest a broader framework for understanding singularities via Brauer-group data and irrelevant perturbations in the IR.

Abstract

We study a topological obstruction of a very stringy nature concerned with deforming the target space of an $N=2$ non-linear \sm. This target space has a singularity which may be smoothed away according to the conventional rules of geometry but when one studies the associated conformal field theory one sees that such a deformation is not possible without a discontinuous change in some of the correlation functions. This obstruction appears to come from torsion in the homology of the target space (which is seen by deforming the theory by an irrelevant operator). We discuss the link between this phenomenon and orbifolds with discrete torsion as studied by Vafa and Witten.

Figures (3)

• Figure 1: The A-model moduli space for target space with ${\hbox{Z}}_3\subset H_2(X)$.
• Figure 2: The infra-red limit of the quintic threefold with 5 points blown up.
• Figure 3: The relationship between $X$, $X^\prime$ and $\tilde{X}$.