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User: Tim Graefnitz

Tim Graefnitz has authored 1 sequences.

A353195 Coefficients of the open mirror map of P2.

2, 5, 32, 286, 3038, 35870, 454880, 6073311, 84302270, 1206291308, 17687468032, 264593385735, 4024945917314, 62101640836955, 969921269646560, 15309505269479942, 243897741785306000, 3917478255634975373, 63381933612745811168, 1032176017566352265886, 16907912684907490828614

Offset: 1

Tim Graefnitz, Apr 29 2022

nonn

The integers a[k] (k>0) defining this sequence are the coefficients of the open mirror map M(Q)=sum(k>0)a[k]Q^k, which is defined as follows:
Let F(z) = Sum_(k>0)((-1)^k*(3k)!/(k*(k!)^3)*z^k) be the holomorphic part of the logarithmic solution to the Picard-Fuchs type differential equation for P2 as defined by Lerche-Mayr (cf. A006480).
The inverse of the power series Q(z)=z*exp(F(z)) is defined as the closed mirror map z(Q) (c.f. A229451 and A061401).
The holomorphic part of the logarithmic solution to the open Picard-Fuchs equation for P2 is given by (1/3)*F(z).
The open mirror map M(Q) is obtained by inserting the closed mirror map z(Q) into the power series exp(1/3*F(z)).
The series M(Q) originally appeared as the open mirror map relating Aganagic-Vafa branes on the canonical bundle of P2 ("local P2") and its mirror.
The coefficients of the series M(Q) can be interpreted as curve counts in different ways:
(1) a[d] is the open Gromov-Witten invariant (counts of holomorphic disks) of moment fibers of local P2, of class d*H (H = hyperplane class) and winding w=1.
(2) a[d] is the closed local Gromov-Witten invariant of local F1 (F1 = Hirzebruch surface = blowup of P2) of class d*H-C (H = pullback of hyperplane class, C = exceptional line).
(3) a[d] is the relative (or log) Gromov-Witten invariant of the pair (F1,D) (D = smooth anticanonical divisor) of class d*H-C.
(4) a[d] is the 2-marked log Gromov-Witten invariant R_p,q of the pair (P2,D) (D = smooth anticanonical divisor) of class d*H, intersecting D in two points with multiplicity p and q, the former point is fixed.
(5) W = y + Sum_(d>0) a[d]*t^(3d)*y^(-3d+1) is the proper Landau-Ginzburg model of (P2,D) defined via broken lines.
There is no known recursion or closed formula for this sequence.
Conjecture: a(n) = (3*n - 1)*A364973(n). - - Kyler Siegel, Jul 06 2024

M. Aganagic, A. Klemm, and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web, Z. Naturforsch. A57 (2002) 1-28; arXiv:hep-th/0105045, 2011.
M. Aganagic and C. Vafa, Mirror Symmetry, D-Branes and Counting Holomorphic Discs, arXiv:hep-th/0012041, 2000.
M. Carl, M. Pumperla, B. Siebert, A tropical view on Landau-Ginzburg Models
K. Chan, A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry, Pacific J. Math. 254 (2011) 275-293; arXiv:1006.3827 [math.SG], 2010-2012.
K. Chan, S.-C. Lau, and H.-H. Tseng, Enumerative meaning of mirror maps for toric Calabi-Yau manifolds, Adv. Math. 244 (2013) 605-625; arXiv:1110.4439 [math.SG], 2011-2013.
M. van Garrel, T. Graber, and H. Ruddat, Local Gromov-Witten invariants are log invariants, Adv. Math. 350, (2019), 860-876; arXiv:1712.05210 [math.AG], 2017-2019.
T. Graber and E. Zaslow, Open-string Gromov-Witten invariants: calculations and a mirror “theorem”, Orbifolds in mathematics and physics, Contemp. Math. 310, AMS (2002), 107-121; arXiv:hep-th/0109075, 2001.
T. Graefnitz, H. Ruddat, and E. Zaslow, The proper Landau-Ginzburg potential is the open mirror map; arXiv:2204.12249 [math.AG], 2022.
M. Gross and B. Siebert, Local mirror symmetry in the tropics, Proc. Int. Congr. Math. Seoul (2014) Vol. II, 723-744; arXiv:1404.3585 [math.AG], 2014
S.-C. Lau, N. C. Leung, and B. Wu, A relation for Gromov-Witten invariants of local Calabi-Yau threefolds, Math.Res. Lett. 18 (5), (2011), 943-956; arXiv:1006.3828 [math.AG], 2010.
W. Lerche and P. Mayr, On N = 1 Mirror Symmetry for Open Type II Strings; arXiv:hep-th/0111113, 2001.
G. Mikhalkin and K. Siegel, Ellipsoidal superpotentials and stationary descendants, arXiv:2307.13252 [math.SG] (2023).

Cf. A006480, A229451, A061401, A364973.

def M(n):
    z,Q = var('z,Q')
    a = [var(f'a{k}') for k in range(n+1)]
    b = [0,1] + [0 for k in range(2,n+1)]
    F = sum([(-1)^k/k*factorial(3*k)/factorial(k)^3*z^k for k in range(1,n+1)])
    zQ = Q+sum([a[k]*Q^k for k in range(2,n+1)])
    Qz = (zQ*exp(F(zQ))).taylor(Q,0,n)
    for k in range(2,n+1):
        b[k] = a[k].substitute(solve(Qz.coefficient(Q^k).substitute([a[i]==b[i] for i in range(k)]) == 0,a[k]))
    return exp(1/3*F).substitute(z==sum([b[k]*Q^k for k in range(n+1)])).taylor(Q,0,n)

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User: Tim Graefnitz

A353195 Coefficients of the open mirror map of P2.

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