Revisiting Gamma conjecture I: counterexamples and modifications

2024/05/27 by Galkin, Sergey, Hu, Jianxun, Iritani, Hiroshi +3
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2405.16979

Abstract

We continue investigation of asymptotics of quantum differential equation for Fano manifolds, with a special regard to Gamma conjecture I and its underlying Conjecture O. We introduce the A-model conifold value, a symplectic invariant of a Fano manifold, and propose modifications for Gamma conjecture I based on this new definition. We discuss an interplay of birational transformations with an extension of Gamma conjecture I over the Kähler moduli space. These heuristics are applied to rigorously identify the principal asymptotic class in the case of ℙ¹-bundles X_n=ℙ_ℙⁿ(O\oplusO(n)). We observe, in particular, that for X_n of dimension at least four, the Conjecture O holds just for even values of n, and in these cases we falsify the original non-modified Gamma conjecture I.

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