Holomorphic Anomaly Equations for the Formal Quintic

Abstract

We define a formal Gromov-Witten theory of the quintic 3-fold via localization on P 4 . Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov-Witten theory of the quintic 3-fold. The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations. Such a relationship has been recently found by Q. Chen, S. Guo, F. Janda, and Y. Ruan via the geometry of new moduli spaces.

Stable quotients and the holomorphic anomaly equation

“…the expected equations in Conjecture 10 exactly matches 2 the conjectural holomorphic anomaly equation [1, (2.52)] for the true quintic theory and this was the main result in [17]. Also the ring (13) can be reduced to the Yamaguchi-Yau ring introduced in [20] for the true quintic theory only with the choice of specialization (14).…”

“…Our main results are the proof of holomorphic anomaly equations for the equivariant Gromov-Witten theories of local P 2 and local P 3 . We also state the generalization to full equivariant formal quintic theory of the result in [17].…”

“…For Calabi-Yau 3-folds, holomorphic anomaly equations were first discovered in physics ( [2]). Also there were many studies to understand holomorphic anomaly equations mathematically ( [12,16,17,19]). But less is known for higher dimensional Calabi-Yau manifolds in physics.…”

“…Holomorphic anomaly for equivariant formal quintic invariants. A particular twisted theory on P 4 related to the quintic 3fold was introduced in [17]. Let the algebraic torus…”

Crepant resolution and the holomorphic anomaly equation for [C3/Z3]

“…Our study of the orbifold Gromov–Witten theory of in all genera yields two main results. (i)We prove the holomorphic anomaly equations for in the precise form predicted by ‐model physics . (ii)We prove an exact crepant resolution correspondence in all genera relating the Gromov–Witten theories of and . For (i), our approach follows the path of the higher genus study in . For (ii), our correspondence is simple, explicit, and carries no unevaluated analytic continuation.…”