OFFSET
1,1
COMMENTS
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
LINKS
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).
PROG
(SageMath)
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)
CROSSREFS
KEYWORD
nonn
AUTHOR
Tim Graefnitz, Apr 29 2022
STATUS
approved