Toric residue mirror conjecture for Calabi-Yau complete intersections

Abstract

The toric residue mirror conjecture of Batyrev and Materov [2,3] expresses a toric residue as a power series whose coefficients are certain integrals over moduli spaces. This conjecture for Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties was proved independently by Szenes and Vergne [10] and Borisov [5]. We build on the work of these authors to generalize the residue mirror map to not necessarily reflexive polytopes. Using this generalization we prove the toric residue mirror conjecture for Calabi-Yau complete intersections in Gorenstein toric Fano varieties [3].We start by introducing notation and explaining the main idea of the generalization. We work over the field K = Q. Let M ≃ Z d , let ∆ ⊂ M K be a d-dimensional lattice polytope, and let T be a coherent triangulation of ∆, defined by a convex piecewise linear integral function on ∆. All lattice points in ∆ are assumed to be vertices of the simplices in T . We place ∆ in M K = (M × Z) K as ∆ × {1} and let C ∆ ⊂ M K be the cone over ∆ with vertex 0. Then T defines a subdivision of C ∆ into a fan Σ.The idea of the toric residue mirror conjecture is to relate the semigroup ring S ∆ = K[C ∆ ∩ M] to the cohomology of the fan Σ. Let I ∆ ⊂ S ∆ be the ideal generated by monomials t m where m ∈ M lies in the interior of C ∆ . Given general elements f 0 , . . . , f d ∈ S

(0,2) deformations of linear sigma models

“…To fix the Γ ρ rotations, we set some of the coefficients in J ρ to their (2, 2) values, taking, 14) and similarly for the pairs J 3 , J 4 and J 5 , J 6 . Finally, we fix the Σ a rotations by setting E a11 , E a22 , and E a33 to their (2,2) values.…”

“…Remarkably, however, these subspaces are preserved by the mirror isomorphism: toric deformations of M map to polynomial deformations of M • and vice versa. The resulting "algebraic gauge" coordinates combined with the monomial divisor mirror map [9] are natural for explicit GLSM computations [10] and may be used to prove mirror symmetry-at least at the level of topological theory-without relying on special Kähler coordinates [11][12][13][14].…”

Non-local observables in theA-model

“…For local observables this can be seen by comparing the form of the local correlators in eqn. (4.2) with the Horn uniformization formula of GKZ[21] and the toric residue formulas found in[22][23][24]. We thank E. Materov, K. Karu and M. Vergne for discussions on this point.…”

Asymptotic Curvature of Moduli Spaces for Calabi–Yau Threefolds

“…9), proved in the toric complete intersection case in [13]. Once we have defined the mirror map, the limits of the Yukawa couplings may be calculated with respect to natural toric coordinates z i at the relevant boundary point defined by the choice of framing, where z i = z i (q 1 , .…”