##
**Differential equations, mirror maps and zeta values.**
*(English)*
Zbl 1118.14043

Yui, Noriko (ed.) et al., Mirror symmetry V. Proceedings of the BIRS workshop on Calabi-Yau varieties and mirror symmetry, December 6–11, 2003. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (ISBN 0-8218-4251-X/pbk). AMS/IP Studies in Advanced Mathematics 38, 481-515 (2006).

The proof of irrationality of \(\zeta(2)\) and \(\zeta(3)\) yielded linear differential equations of order \(2\) and order \(3\), respectively. Geometrically, \(\zeta(2)\) and \(\zeta(3)\) are respectively related to families of elliptic curves and \(K3\) surfaces.

The paper under review tries to extend this phenomenon to \(\zeta(4)\) and a \(4\)-th order linear differential equation arising from a Calabi–Yau threefold. The \(4\)-th order differential equation in question is obtained as the pull-back of the \(5\)-th order differential eqauation. This provides one instance of a geometric generalization relating \(\zeta(4)\) with a family of Calabi–Yau threefolds. Consider th \(4\)-th order Calabi–Yau differential equation of maximal unipotent monodromy. Let \(y_0,y_1,y_2,y_3\) be the Frobenius basis of solutions, and put \(t=y_1/y_0\). Then the inverse \(z=z(q)\) of \(q(z)=e^t=\exp({y_1\over{y_0}})\) is defined to be the mirror map and the Yukawa coupling is defined by \(K(q)=N_0\cdot {d^2\over{dz^2}} ({y_2\over{y_0}})\) where \(N_0\) is some integer \(\neq 0\). Write \[ K(q)=\sum_{n=0}^{\infty} C_nq^n=N_0+\sum_{\ell=1}^{\infty} N_{\ell}\ell {q^{\ell}\over{1-q^{\ell}}} \] with \(N_0=C_0\in{\mathbb Z}\) and \(N_{\ell}={1\over{\ell^3}}\sum_{d|\ell} \mu({\ell\over{d}})C_d\in{\mathbb Z}\) where \(\mu\) is the Möbius function. The \(5\)-th order linear differential equation of maximal unipotent monodromy is derived from the \(3\)-term polynomial recursion. The two linearly indepedent solutions \(\{A_n\}\) and \(\{B_n\}\) with the initial data \(A_0=1, A_1=12, B_0=0, B_2=13\) gives rise to the Apéry limit \({B_n\over{A_n}}\to \zeta(4) ={\pi^2\over{90}}\) as \(n\to\infty\). \(A_n\) is given explicitly in terms of binomial coefficients. Several observations are listed on the mirror map, the Yukawa coupling, and Kummer supercongruences, etc. arising from this \(5\)-th order equation.

Finding Calabi–Yau equations of maximal unipotent monodromy is one of main themes of this paper. The algorithm of Gosper and Zeilberger of creative telescoping is used to finding polynomials recursions and hence producing many examples of differential equations, and then computing mirror maps and Yukawa couplings. More than \(200\) of Calabi–Yau equations are tabulated together with mirror maps and Yukawa couplings.

For the entire collection see [Zbl 1104.14001].

The paper under review tries to extend this phenomenon to \(\zeta(4)\) and a \(4\)-th order linear differential equation arising from a Calabi–Yau threefold. The \(4\)-th order differential equation in question is obtained as the pull-back of the \(5\)-th order differential eqauation. This provides one instance of a geometric generalization relating \(\zeta(4)\) with a family of Calabi–Yau threefolds. Consider th \(4\)-th order Calabi–Yau differential equation of maximal unipotent monodromy. Let \(y_0,y_1,y_2,y_3\) be the Frobenius basis of solutions, and put \(t=y_1/y_0\). Then the inverse \(z=z(q)\) of \(q(z)=e^t=\exp({y_1\over{y_0}})\) is defined to be the mirror map and the Yukawa coupling is defined by \(K(q)=N_0\cdot {d^2\over{dz^2}} ({y_2\over{y_0}})\) where \(N_0\) is some integer \(\neq 0\). Write \[ K(q)=\sum_{n=0}^{\infty} C_nq^n=N_0+\sum_{\ell=1}^{\infty} N_{\ell}\ell {q^{\ell}\over{1-q^{\ell}}} \] with \(N_0=C_0\in{\mathbb Z}\) and \(N_{\ell}={1\over{\ell^3}}\sum_{d|\ell} \mu({\ell\over{d}})C_d\in{\mathbb Z}\) where \(\mu\) is the Möbius function. The \(5\)-th order linear differential equation of maximal unipotent monodromy is derived from the \(3\)-term polynomial recursion. The two linearly indepedent solutions \(\{A_n\}\) and \(\{B_n\}\) with the initial data \(A_0=1, A_1=12, B_0=0, B_2=13\) gives rise to the Apéry limit \({B_n\over{A_n}}\to \zeta(4) ={\pi^2\over{90}}\) as \(n\to\infty\). \(A_n\) is given explicitly in terms of binomial coefficients. Several observations are listed on the mirror map, the Yukawa coupling, and Kummer supercongruences, etc. arising from this \(5\)-th order equation.

Finding Calabi–Yau equations of maximal unipotent monodromy is one of main themes of this paper. The algorithm of Gosper and Zeilberger of creative telescoping is used to finding polynomials recursions and hence producing many examples of differential equations, and then computing mirror maps and Yukawa couplings. More than \(200\) of Calabi–Yau equations are tabulated together with mirror maps and Yukawa couplings.

For the entire collection see [Zbl 1104.14001].

Reviewer: Noriko Yui (Kingston)

### MSC:

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11B83 | Special sequences and polynomials |

32S40 | Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) |

33C20 | Generalized hypergeometric series, \({}_pF_q\) |

34M15 | Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain |

34M25 | Formal solutions and transform techniques for ordinary differential equations in the complex domain |

### Keywords:

Calabi-Yau equation; maximal unipotent monodromy; instanton number; binomial sum; recurence relation; Hadamard product; Lambert series; supercongruence
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\textit{G. Almkvist} and \textit{W. Zudilin}, AMS/IP Stud. Adv. Math. 38, 481--515 (2006; Zbl 1118.14043)

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