[Paper Review] A Correlator Formula for Quantum Sheaf Cohomology

Zhentao Lu|arXiv (Cornell University)|Nov 30, 2015

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper proves a conjecture by McOrist and Melnikov on quantum correlators in $(0,2)$ gauged linear sigma models for nef-Fano smooth projective toric varieties. Using quantum sheaf cohomology and an integral representation of correlators, the authors establish a summation formula that generalizes the Szenes-Vergne proof of the Toric Residue Mirror Conjecture, confirming the quantum correlator formula via homology equivalence of integration cycles.

ABSTRACT

For a class of monadic deformations of the tangent bundles over nef-Fano smooth projective toric varieties, we study the correlators using quantum sheaf cohomology. We prove a summation formula for the correlators, confirming a conjecture by McOrist and Melnikov in physics literature. This generalizes the Szenes-Vergne proof of Toric Residue Mirror Conjecture for hypersurfaces.

Motivation & Objective

• To prove a conjecture by McOrist and Melnikov on quantum correlators in $(0,2)$ theories for toric varieties.
• To extend the Szenes-Vergne approach to the $(0,2)$ setting using quantum sheaf cohomology.
• To establish a summation formula for quantum correlators that matches physical conjectures in the geometric phase.
• To demonstrate that the quantum correlator can be computed via a finite sum over lattice points, using properness and homology arguments.
• To lay groundwork for a $(0,2)$ mirror symmetry interpretation via a proposed $(0,2)$ toric residue.

Proposed method

• The authors define quantum correlators via quantum Stanley-Reisner ideals in quantum sheaf cohomology for deformed tangent bundles on nef-Fano toric varieties.
• They express the quantum correlator as an integral over a cycle $Z_{ ext{hom}}$ in the complement of the zero locus of a deformed Euler sequence.
• A key step involves rewriting the quantum correlator as an integral over a cycle $Z_{ ext{hom}}$, which is shown to be homologous to a sum of cycles $Z_S$ indexed by subsets $S \subset \{1,\dots,n\}$.
• Using a properness lemma for the deformed moment map $\tilde{v}^t$, they show that only the $S = \emptyset$ term contributes, while all $S \neq \emptyset$ terms integrate to zero.
• The proof relies on homology equivalence between the cycle $\sum (-1)^{|S|} Z_S$ and the standard cycle $Z_{\delta}(q)$, with vanishing of $\Lambda$-forms on $Z_S$ for $S \neq \emptyset$.
• The final equality is established by showing $\int_{Z_\phi} \Lambda = \int_{h_q} \Lambda$, matching the conjectured summation formula.

Experimental results

Research questions

• RQ1 Does the McOrist-Melnikov conjecture for quantum correlators in $(0,2)$ GLSMs hold in the geometric setting for toric varieties?
• RQ2 Can the quantum correlator be expressed as a finite summation formula over lattice points in the dual weight space?
• RQ3 Is the integration cycle for the quantum correlator homologous to a sum of cycles corresponding to different subsets of rays?
• RQ4 Can the vanishing of contributions from non-empty subsets $S$ be rigorously established via differential form analysis?
• RQ5 Can the summation formula be interpreted as a $(0,2)$ analog of toric residue, suggesting a new mirror symmetry statement?

Key findings

• The quantum correlator $\langle \sigma_{i_1}, \dots, \sigma_{i_s} \rangle^{\text{quantum}}$ is proven to equal the proposed summation formula from McOrist and Melnikov’s conjecture.
• The integration cycle $Z_\phi$ for the $S = \emptyset$ term is shown to be homologous to the standard cycle $Z_{\delta}(q)$, ensuring the correct value of the correlator.
• All terms with $S \neq \emptyset$ vanish under integration of the form $\Lambda$, due to the form being exact on the boundary $Z_S$.
• The properness of the deformed moment map $\tilde{v}^t$ ensures that the relevant preimages are compact and well-behaved for homotopy arguments.
• The result confirms the existence of a finite, explicit formula for quantum correlators in a broad class of $(0,2)$ theories on toric varieties.
• The proof technique generalizes the Szenes-Vergne approach from the $(2,2)$ case to the $(0,2)$ setting, providing a new bridge to mirror symmetry.