The stringy geometry of integral cohomology in mirror symmetry

TL;DR

The paper investigates how integral torsion in CY3 cohomology, captured by $A(X)={H^2(X,\mathbb{Z})}_{tor}$ and $B(X)={H^3(X,\mathbb{Z})}_{tor}$, enriches closed-string mirror symmetry. It identifies $A(X)$ as a universal abelian symmetry of the SCFT and $B(X)$ as a space of flat gerbes for the NS-NS B-field, and shows that their inclusion leads to a generalized duality where topology of flat gerbes is treated on par with manifold topology. Through K-theory, Wall’s invariants, and explicit constructions via orbifolds and Greene–Plesser/LG orbifolds, the work derives isomorphisms such as $igl\{K^0(X)\bigr\}_{tor} \simeq A(X)\oplus B(X)^*$ and $igl\{K^1(X)\bigr\}_{tor} \simeq A(X)^*\oplus B(X)$, and demonstrates how mirror symmetry maps these torsion structures between a pair $(X,X^\circ)$. The analysis of quintic quotients and discrete torsion illustrates how $B(X)$ and discrete worldsheet phases interact to produce or constrain deformations, often aligning with flat gerbes in the geometric phase. Overall, the paper proposes a refined, topology-aware framework for mirror symmetry with explicit examples, and outlines numerous avenues for extending these ideas to higher dimensions and richer categorical structures.

Abstract

We examine the physical significance of torsion co-cycles in the cohomology of a projective Calabi-Yau three-fold for the (2,2) superconformal field theory (SCFT) associated to the non-linear sigma model with such a manifold as a target space. There are two independent torsion subgroups in the cohomology. While one is associated to an orbifold construction of the SCFT, the other encodes the possibility of turning on a topologically non-trivial flat gerbe for the NS-NS B-field. Inclusion of these data enriches mirror symmetry by providing a refinement of the familiar structures and points to a generalization of the duality symmetry, where the topology of the flat gerbe enters on the same footing as the topology of the underlying manifold.