Code and Data
This page accompanies the preprint "Gauss-Manin Connection in Disguise: Holomorphic Anomaly Equations and Higher Genus Real Gromov-Witten Invariants" and contains the codes used in the paper and further data obtained.
All computations were done in Mathematica (tested with version 14.3). Note that we need the complex GV invariants as input. That was taken from A. Klemm's data page (the BPSc files below; keep each one in the same folder as its notebook). In order to compute degree d real invariants, we need degree d/2 complex invariants.
The last cell is an example; RealGVUpTo[chiMax, dmax] gives the invariants of genus g ≤ chiMax + 1 and degree d ≤ dmax. For the quintic at χ = 5 and the bicubic at χ = 9 the boundary conditions leave one coefficient free, so the notebooks take nℝ6,5 or nℝ10,9 as input (BPSr[6, 5] = 2 or BPSr[10, 9] = 3, or the negative values).
Quintic Threefold
Real Gopakumar-Vafa invariants up to genus 5, and up to genus 8 for both choices nℝ6,5 = ±2.
- AlgebraicHAE_quintic.nb: Mathematica notebook computing the moduli space S, the Gauss-Manin connection and the vector fields R0, …, R8, the dictionary, the solution of the algebraic anomaly equation by direct integration, and the fixing of the ambiguity to obtain the real Gopakumar-Vafa invariants.
- BPSc_quintic.wl: complex Gopakumar-Vafa invariants of the quintic (genus ≤ 64, degree ≤ 44) from A. Klemm's data page, read by the notebook.
- RealGV_quintic.csv: real Gopakumar-Vafa invariants nℝg,d for g ≤ 8 (odd d ≤ 99, even d ≤ 88), for nℝ6,5 = 2 and −2.
The Bicubic
Real Gopakumar-Vafa invariants up to genus 9, and up to genus 12 for both choices nℝ10,9 = ±3. Includes the solutions of the Picard-Fuchs equation and the extra real period.
- AlgebraicHAE_bicubic.nb: the same computation for the bicubic, including the periods at z = 0, the mirror map and the extra real period τ.
- BPSc_bicubic.wl: complex Gopakumar-Vafa invariants of the bicubic (genus ≤ 33, degree ≤ 66) from A. Klemm's data page, read by the notebook.
- RealGV_bicubic.csv: real Gopakumar-Vafa invariants nℝg,d for g ≤ 8 with d ≤ 40 and for g = 9, …, 12 with d ≤ 30, for nℝ10,9 = 3 and −3.