[Paper Review] Fano mirror periods from the Frobenius structure conjecture

Travis Mandel|arXiv (Cornell University)|Mar 28, 2019

Molecular spectroscopy and chirality19 references3 citations

TL;DR

This paper proves that the regularized quantum period of a Fano orbifold, defined via descendant Gromov-Witten invariants, equals a classical period derived from a mirror Landau-Ginzburg potential under the Frobenius structure conjecture. It establishes that these quantum invariants count rational curves naively, providing a geometric realization of mirror symmetry for Fano varieties via log Gromov-Witten theory and toric transversality.

ABSTRACT

The Fano classification program proposed by Coates-Corti-Galkin-Golyshev-Kasprzyk is based on the mirror symmetry prediction that the regularized quantum period of a Fano should be equivalent to the classical period of its mirror Landau-Ginzburg potential. We prove that this mirror equivalence follows from versions of the Frobenius structure conjecture of Gross-Hacking-Keel. We also find that the regularized quantum period, which is defined in terms of descendant Gromov-Witten numbers, is in fact given by certain naive curve counts.

Motivation & Objective

• To establish mirror symmetry for Fano orbifolds by proving the regularized quantum period equals the classical period of the mirror Landau-Ginzburg potential.
• To show that descendant Gromov-Witten invariants—defining the quantum period—are equivalent to naive counts of rational curves with specified incidence and tangency conditions.
• To verify the mirror symmetry prediction in the Fanosearch program using the Frobenius structure conjecture and log Gromov-Witten theory.
• To demonstrate that the quantum period coefficients are positive integers by interpreting them as enumerative counts of rational curves.

Proposed method

• Uses the Frobenius structure conjecture and its naive-counting variant to construct the mirror Landau-Ginzburg potential for Fano pairs (Y,D).
• Applies log Gromov-Witten theory to the log Calabi-Yau pair (Y†, β) with a divisor D in |−KY|, using the moduli space Mlog0,Δ′qP(Y†,β) of log stable maps.
• Employs the forgetful map from log moduli spaces to M0,s+1 to relate ψ-classes and apply the projection formula, reducing virtual counts to classical counts.
• Relies on toric transversality and the assumption that D is a reduced normal crossings divisor with no orbifold points to ensure unobstructedness and finitely many solutions.
• Uses the fact that ψd+1d−2 in M0,d+1 is a point class, so its pullback specifies the domain curve generically, enabling enumeration of curves.
• Applies the projection formula to equate the virtual integral over M0,1(Y,β)vir to a sum over partitions P of dβ, weighted by multinomial coefficients.

Experimental results

Research questions

• RQ1 Does the regularized quantum period of a Fano orbifold equal the classical period of its mirror Landau-Ginzburg potential under the Frobenius structure conjecture?
• RQ2 Can descendant Gromov-Witten invariants be interpreted as naive counts of rational curves with prescribed tangency to a divisor D?
• RQ3 Under what conditions does the log Gromov-Witten theory of a Fano pair (Y,D) yield enumerative invariants matching the quantum period?
• RQ4 Is the quantum period of a Fano variety given by a sum over partitions of dβ, with coefficients counting torically transverse curves in log moduli spaces?

Key findings

• The regularized quantum period pβ is equal to the number of rational curves of class β in Y that pass through a generic point and meet a generic divisor D in dβ points, when D is smooth or under Conjectures 1.4 or 1.8.
• The coefficient pβ equals (dβ)! times the virtual integral of ψx^{dβ−2} evx*([pt]) over [M0,1(Y,β)]vir, and this integral computes a naive curve count.
• The quantum period  ĜY equals the classical period πW of the mirror potential W when the Frobenius structure conjecture holds, proving mirror symmetry for Fano orbifolds.
• For each β, the number of such curves is counted by Nβ^naive(qP) multiplied by the multinomial coefficient (dβ choose P(1),...,P(s)), where P is the partition of dβ induced by β·[Dpi].
• The result holds unconditionally when D is irreducible (toric transversality holds) or under Conjectures 1.4 and 1.8, which ensure toric transversality in the log setting.