A353195 - OEIS

%I #48 Jul 25 2024 14:25:57

%S 2,5,32,286,3038,35870,454880,6073311,84302270,1206291308,17687468032,

%T 264593385735,4024945917314,62101640836955,969921269646560,

%U 15309505269479942,243897741785306000,3917478255634975373,63381933612745811168,1032176017566352265886,16907912684907490828614

%N Coefficients of the open mirror map of P2.

%C 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:

%C 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).

%C 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).

%C The holomorphic part of the logarithmic solution to the open Picard-Fuchs equation for P2 is given by (1/3)*F(z).

%C 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)).

%C 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.

%C The coefficients of the series M(Q) can be interpreted as curve counts in different ways:

%C (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.

%C (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).

%C (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.

%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.

%C (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.

%C There is no known recursion or closed formula for this sequence.

%C Conjecture: a(n) = (3*n - 1)*A364973(n). - - _Kyler Siegel_, Jul 06 2024

%H M. Aganagic, A. Klemm, and C. Vafa, <a href="https://arxiv.org/abs/hep-th/0105045">Disk Instantons, Mirror Symmetry and the Duality Web</a>, Z. Naturforsch. A57 (2002) 1-28; arXiv:hep-th/0105045, 2011.

%H M. Aganagic and C. Vafa, <a href="https://arxiv.org/abs/hep-th/0012041">Mirror Symmetry, D-Branes and Counting Holomorphic Discs</a>, arXiv:hep-th/0012041, 2000.

%H M. Carl, M. Pumperla, B. Siebert, <a href="https://www.math.uni-hamburg.de/home/siebert/preprints/LGtrop.pdf">A tropical view on Landau-Ginzburg Models</a>

%H K. Chan, <a href="https://arxiv.org/abs/1006.3827">A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry</a>, Pacific J. Math. 254 (2011) 275-293; arXiv:1006.3827 [math.SG], 2010-2012.

%H K. Chan, S.-C. Lau, and H.-H. Tseng, <a href="https://arxiv.org/abs/1110.4439">Enumerative meaning of mirror maps for toric Calabi-Yau manifolds</a>, Adv. Math. 244 (2013) 605-625; arXiv:1110.4439 [math.SG], 2011-2013.

%H M. van Garrel, T. Graber, and H. Ruddat, <a href="https://arxiv.org/abs/1712.05210">Local Gromov-Witten invariants are log invariants</a>, Adv. Math. 350, (2019), 860-876; arXiv:1712.05210 [math.AG], 2017-2019.

%H T. Graber and E. Zaslow, <a href="https://arxiv.org/abs/hep-th/0109075">Open-string Gromov-Witten invariants: calculations and a mirror “theorem”</a>, Orbifolds in mathematics and physics, Contemp. Math. 310, AMS (2002), 107-121; arXiv:hep-th/0109075, 2001.

%H T. Graefnitz, H. Ruddat, and E. Zaslow, <a href="https://arxiv.org/abs/2204.12249">The proper Landau-Ginzburg potential is the open mirror map</a>; arXiv:2204.12249 [math.AG], 2022.

%H M. Gross and B. Siebert, <a href="https://arxiv.org/abs/1404.3585">Local mirror symmetry in the tropics</a>, Proc. Int. Congr. Math. Seoul (2014) Vol. II, 723-744; arXiv:1404.3585 [math.AG], 2014

%H S.-C. Lau, N. C. Leung, and B. Wu, <a href="https://arxiv.org/abs/1006.3828">A relation for Gromov-Witten invariants of local Calabi-Yau threefolds</a>, Math.Res. Lett. 18 (5), (2011), 943-956; arXiv:1006.3828 [math.AG], 2010.

%H W. Lerche and P. Mayr, <a href="https://arxiv.org/abs/hep-th/0111113">On N = 1 Mirror Symmetry for Open Type II Strings</a>; arXiv:hep-th/0111113, 2001.

%H G. Mikhalkin and K. Siegel, <a href="https://arxiv.org/abs/2307.13252">Ellipsoidal superpotentials and stationary descendants</a>, arXiv:2307.13252 [math.SG] (2023).

%o (SageMath)

%o def M(n):

%o     z,Q = var('z,Q')

%o     a = [var(f'a{k}') for k in range(n+1)]

%o     b = [0,1] + [0 for k in range(2,n+1)]

%o     F = sum([(-1)^k/k*factorial(3*k)/factorial(k)^3*z^k for k in range(1,n+1)])

%o     zQ = Q+sum([a[k]*Q^k for k in range(2,n+1)])

%o     Qz = (zQ*exp(F(zQ))).taylor(Q,0,n)

%o     for k in range(2,n+1):

%o         b[k] = a[k].substitute(solve(Qz.coefficient(Q^k).substitute([a[i]==b[i] for i in range(k)]) == 0,a[k]))

%o     return exp(1/3*F).substitute(z==sum([b[k]*Q^k for k in range(n+1)])).taylor(Q,0,n)

%Y Cf. A006480, A229451, A061401, A364973.

%K nonn

%O 1,1

%A _Tim Graefnitz_, Apr 29 2022