Equal-mass sunset integrals and genus-zero local Gromov--Witten theory of Calabi--Yau n-folds
Abstract
We prove an exact identity, as convergent logarithmic germs at large Euclidean momentum, between the equal-mass two-dimensional (n−1)-loop sunset Feynman integral, with n≥4, and the genus-zero one-point descendant invariants of the local Calabi--Yau n-fold Xn=Tot(KFn), where Fn⊂(ℙ1)n is a smooth (1,…,1) hypersurface. Writing s for the momentum variable and y=−1/s, the identity reads I(n−1)circleddash(s)=(−1)n−1yA [nRn−1−θQ𝒟0Δ Rn−2 +∑j=0n−3cjRj], where R=log y+O(y) is the local mirror coordinate, A=θyR, Q=eR, 𝒟0Δ(Q) is the primary one-point potential, and each cj has the form cj=cj(0)+∑a=0n−2(cj,a 𝒟aΔ(Q) +c'j,a θQ𝒟aΔ(Q)) with the coefficients in ℚ[ζ(3),ζ(5),…] the odd zeta values coming from the Gamma class. The descendant series 𝒟aΔ(Q) are given explicitly by the ambient hypergeometric series and the inverse mirror map. We also prove global univalence of the principal mirror map, with explicit upper and lower bounds on the Q-Taylor radii, extract the full primary evaluation divisor and its integral cover transform, and reduce the ambient endpoint constants to a finite universal Gamma-class part and ⌊(n−1)/2⌋ real parameters.
Related
- The three- and four-loop equal-mass sunset integrals and local Calabi--Yau fourfolds and fivefolds
- Gromov-Witten theory of the local model of the standard flop
- Structure of higher-genus open-closed Gromov--Witten theory of 𝒪ℙ1(p−1)⊕𝒪ℙ1(−p−1)
- Conifold factorization in topological recursion and Gromov--Witten theory
- Punctured log Gromov-Witten theory of log modifications and double ramification cycles with target log variety
BibTeX
@misc{maennel2026equalmass,
title = {{Equal-mass sunset integrals and genus-zero local Gromov--Witten theory of Calabi--Yau $n$-folds}},
author = {Hartmut Maennel and Pierre Vanhove},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10828},
eprint = {2610.10828},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05788335v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Equal-mass sunset integrals and genus-zero local Gromov--Witten theory of Calabi--Yau n-folds
AU - Hartmut Maennel
AU - Pierre Vanhove
PY - 2026
DA - 2026/10/07
PB - HAL
DO - 10.48550/arxiv.2610.10828
UR - https://hal.science/hal-05788335v1
AB - We prove an exact identity, as convergent logarithmic germs at large Euclidean momentum, between the equal-mass two-dimensional (n−1)-loop sunset Feynman integral, with n≥4, and the genus-zero one-point descendant invariants of the local Calabi--Yau n-fold X_n=Tot(K_F_n), where F_n⊂(ℙ^1)^n is a smooth (1,…,1) hypersurface. Writing s for the momentum variable and y=−1/s, the identity reads I^(n−1)_circleddash(s)=(−1)^n−1yA [nR^n−1−θ_Q𝒟_0^Δ R^n−2 +∑_j=0^n−3c_jR^j], where R=log y+O(y) is the local mirror coordinate, A=θ_yR, Q=e^R, 𝒟_0^Δ(Q) is the primary one-point potential, and each c_j has the form c_j=c_j^(0)+∑_a=0^n−2(c_j,a 𝒟_a^Δ(Q) +c'_j,a θ_Q𝒟_a^Δ(Q)) with the coefficients in ℚ[ζ(3),ζ(5),…] the odd zeta values coming from the Gamma class. The descendant series 𝒟_a^Δ(Q) are given explicitly by the ambient hypergeometric series and the inverse mirror map. We also prove global univalence of the principal mirror map, with explicit upper and lower bounds on the Q-Taylor radii, extract the full primary evaluation divisor and its integral cover transform, and reduce the ambient endpoint constants to a finite universal Gamma-class part and ⌊(n−1)/2⌋ real parameters.
ER -