Leading-order Gamma conjecture for del Pezzo mirror central charges

From papers

Let YY be a del Pezzo surface, let (X,W)(\mathcal X,W) be its Landau–Ginzburg mirror, let α:K(Y)H2(X,W1(+);Z)\alpha:K(Y)\to H_2(\mathcal X,W^{-1}(+\infty);\mathbb Z) be the map from the preceding central-charge conjecture, and let Ψ(E)\Psi(E) be the Gamma-modified Chern character of EK(Y)E\in K(Y). Leading-order Gamma conjecture. There exists an ϵ>0\epsilon>0 such that, as t0+t\to0^+,

α(E)eWΩ=YtωΨ(E)+O(tϵ).\int_{\alpha(E)}e^{-W}\Omega=\int_Y t^{-\omega}\cdot\Psi(E)+O(t^\epsilon).

This is the leading-order behavior of the preceding conjecture after setting z=1z=1; it isolates the Gamma-class contribution without involving Gromov–Witten invariants. Its resolution is not stated in the source.

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Sources & referencesView supporting material

Primary source

Bohan Fang, Junxiao Wang and Yan Zhou, “Mirror symmetric Gamma conjecture for del Pezzo surfaces”, arXiv:2309.02154 (2023).

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