The stringy geometry of integral cohomology in mirror symmetry

1. Problem Statement

Standard mirror symmetry for Calabi-Yau (CY) three-folds is typically formulated in terms of Hodge numbers ($h^{1,1}$ and $h^{1,2}$), which characterize the massless spectrum of the associated $(2,2)$ superconformal field theory (SCFT). However, the integral cohomology of a CY manifold contains additional topological data: torsion subgroups.

The paper addresses the physical significance of these torsion subgroups in the context of string compactification and mirror symmetry. Specifically, it investigates:

• $A(X) = \{H^2(X, \mathbb{Z})\}_{\text{tor}}$: Associated with the fundamental group $\pi_1(X)$ and global symmetries in the SCFT.
• $B(X) = \{H^3(X, \mathbb{Z})\}_{\text{tor}}$: Associated with the possibility of turning on topologically non-trivial flat gerbes for the NS-NS $B$-field.

The central question is how these torsion groups modify the mirror correspondence. Does mirror symmetry simply exchange $A(X)$ and $B(X)$, or does it require a more generalized framework involving orbifolds and discrete torsion?

2. Methodology

The authors employ a multi-faceted approach combining algebraic topology, string theory worldsheet physics, and Landau-Ginzburg (LG) orbifold constructions:

• Topological Analysis:

  • They utilize the Universal Coefficient Theorem and Poincaré duality to characterize the integral cohomology of CY 3-folds, identifying the torsion subgroups $A(X)$ and $B(X)$.
  • They analyze the relationship between these cohomology groups and K-theory torsion subgroups $\{K^0(X)\}_{\text{tor}}$ and $\{K^1(X)\}_{\text{tor}}$, which classify D-brane charges.
  • They apply Wall's theorem and the Atiyah-Hirzebruch spectral sequence to establish isomorphisms between K-theory torsion and cohomology torsion, specifically under the assumption that elements of $A(X)$ have odd order (though they argue this holds generally).

• Worldsheet SCFT & Orbifolds:

  • They interpret $A(X)$ as the quantum symmetry of an orbifold SCFT. If $X$ is a quotient $\tilde{X}/\Gamma$, the SCFT on $X$ is an orbifold $C[\tilde{X}]/\Gamma$, and $A(X)$ corresponds to the dual group $\Gamma^*$.
  • They interpret $B(X)$ as the classification of flat gerbes (discrete $B$-field backgrounds). Turning on a flat gerbe $\beta \in B(X)$ modifies the path integral phases.

• Mirror Constructions:

  • Toric Geometry & GLSM: They use the Batyrev construction and Gauged Linear Sigma Models (GLSM) to relate the quotient geometry to its mirror.
  • Gepner Models: They utilize Greene-Plesser duality, realizing mirror pairs as Landau-Ginzburg (LG) orbifolds of minimal models. They explicitly compute partition functions (Poincaré polynomials) for these orbifolds with discrete torsion to identify mirror duals.

3. Key Contributions and Results

A. Generalized Mirror Symmetry Isomorphism

The paper establishes a refined isomorphism for the total cohomology of mirror pairs $X$ and $X^\circ$:
$$H_{\text{even}}(X, \mathbb{Z}) \simeq H_{\text{odd}}(X^\circ, \mathbb{Z})$$
This implies a relationship between the torsion subgroups:
$$A(X) \oplus B(X)^* \simeq B(X^\circ) \oplus A(X^\circ)^*$$
where $G^*$ denotes the Pontryagin dual.

• Result: The authors prove that if $A(X)$ consists only of odd-order elements, this isomorphism follows from the splitting of the Atiyah-Hirzebruch spectral sequences for K-theory. They provide evidence that this splitting holds even for even-order torsion in specific examples.

B. Physical Interpretation of $A(X)$ and $B(X)$

• $A(X)$ (Orbifold Symmetry): The presence of non-trivial $A(X)$ indicates that the SCFT $C[X]$ possesses a global symmetry (the quantum symmetry of the orbifold). The mirror of a manifold with $A(X) \neq 0$ is a manifold where this symmetry acts geometrically (or as a quantum symmetry on a singular space).
  • Example: A $\mathbb{Z}_5$ quotient of the quintic has $A(X) = \mathbb{Z}_5$. Its mirror is a smooth quintic where the $\mathbb{Z}_5$ symmetry acts as a geometric isometry, but the mirror of the quotient involves a specific orbifold construction.
• $B(X)$ (Flat Gerbes): Non-trivial $B(X)$ allows for flat, topologically non-trivial $B$-field gerbes. The paper argues that turning on a flat gerbe $\beta \in B(X)$ corresponds to choosing a specific discrete torsion in the orbifold construction of the mirror theory.

C. Discrete Torsion vs. Flat Gerbes

A major technical contribution is the detailed analysis of the relationship between discrete torsion in LG orbifolds and flat gerbes on smooth geometries.

• The Conjecture: Discrete torsion in the Gepner point construction corresponds to the choice of a flat gerbe $\beta_t \in B(X)$ in the large-radius limit.
• The Nuance: The authors demonstrate that the group of allowed discrete torsions ($G_{dt}$) at the Gepner point is often larger than the torsion subgroup $B(X)$ of the smooth geometry.
  • When discrete torsion affects twisted sectors associated with resolving surface singularities in the ambient toric variety, the Hodge numbers jump (changing the geometry).
  • When discrete torsion only affects sectors associated with curve singularities (or trivial sectors), the Hodge numbers remain unchanged. These "safe" choices of discrete torsion correspond exactly to the flat gerbes $B(X)$.
• Explicit Calculation: They compute spectra for various Gepner models (e.g., $Z_5 \times Z_5$ quotients, $Z_3 \times Z_9$, etc.). They show that for specific subgroups of discrete torsion, the Hodge numbers $(h^{1,1}, h^{1,2})$ remain constant, identifying these subgroups with $B(X)$. For other choices, the Hodge numbers jump, indicating a transition to a different geometric phase.

D. New Dual Pairs

The inclusion of these data leads to new mirror pairs.

• A theory $C[X, \beta]$ (SCFT on $X$ with a flat gerbe) is not necessarily mirror to a theory on the standard mirror manifold $X^\circ$.
• Instead, the mirror may be a theory $C[Y^\circ, \beta^\circ]$ where $Y^\circ$ is topologically distinct from $X^\circ$, or the mirror operation involves a non-geometric symmetry.
• They construct explicit mirror pairs for Gepner models with non-trivial discrete torsion, showing that the mirror of a quotient with discrete torsion is another quotient with a specific dual discrete torsion.

4. Significance and Implications

1. Refinement of Mirror Symmetry: The paper argues that mirror symmetry is not merely an exchange of Hodge numbers but an isomorphism of the full integral cohomology (including torsion) and the associated K-theory structures. This requires broadening the definition of mirror pairs to include orbifolds and flat gerbes.
2. Stringy Geometry: It highlights that "stringy" effects (like flat gerbes and discrete torsion) are not just perturbations but fundamental components of the geometry. The moduli space of SCFTs includes discrete sectors labeled by $B(X)$ that are invisible in classical differential geometry but crucial for the full string theory.
3. Duality Symmetry Generalization: The work suggests a generalization of duality symmetries where the topology of the flat gerbe enters on the same footing as the topology of the underlying manifold.
4. Connection to D-branes: The results link the torsion in cohomology to the classification of torsional D-branes via K-theory, providing a deeper understanding of the spectrum of BPS and non-BPS states in these backgrounds.
5. Future Directions: The paper opens avenues for studying non-abelian fundamental groups using categorical symmetries, the relationship between discrete $\theta$-angles in GLSMs and flat gerbes, and generalizations to $G_2$ manifolds and CY 4-folds.

In summary, the paper provides a rigorous topological and physical framework for understanding how torsion in integral cohomology enriches mirror symmetry, demonstrating that flat gerbes and orbifold discrete torsion are dual aspects of the same underlying stringy geometry.