Quantum cohomology of a Pfaffian Calabi-Yau variety: Verifying mirror symmetry predictions (Q2721345)

\textit{E. A. Rødland} [Compos. Math. 122, 135-149 (2000; Zbl 0974.14026)] constructed a Calabi-Yau threefold \(M^3\) as the rank \(4\) locus of a general skew-symmetric \(7\times 7\) matrix with coefficients in \(\mathbb{P}^6\). \(M^3\) is a non-complete intersection Calabi-Yau threefold with \(h^{1,1}(M^3)=1\). A mirror family \(W_q\) is constructed as the orbifold \(M_q^3/{\mathbb{Z}}_7\), where \(M_q^3\) is a one-parameter family of invariants of a natural \(\mathbb{Z}_7\)-action on the space of all skew-symmetric \(7\times 7\)-matrices. In this paper, it is shown that \((M^3, W_q)\) is a (topological) mirror pair, that is, \(h^{2,1}(W_q)=1\). Further, at a point of maximal unipotent monodromy, the mirror symmetry prediction of Rødland posed in the above article is proved in the affirmative. The main result is formulated as follows. NEWLINENEWLINENEWLINETheorem: At a point of maximal unipotent monodromy, the Picard-Fuchs operator for the periods (with \(D=qd/dq\)):NEWLINENEWLINENEWLINE(a) \((1-289q-57q^2+q^3)(1-3q)^2D^4+4q(3q-1)(143+57q-87q^2+3q^3)D^3+2q(-212-473q+725q^2-435q^3+27q^4)D^2 +2q(-69-481q+159q^2-171q^3+18q^4)D+q(-17-202q-8q^2-54q^3+9q^4)\)NEWLINENEWLINENEWLINEis equivalent to the operatorNEWLINENEWLINENEWLINE(b) \(D^2{1\over K}D^2,\quad\text{where}\quad K(q)=14+\sum_{d\geq 1} n_d d^3{q^d\over{1-q^d}}\). Here \(n_d\), the instanton number of degree \(d\) rational curves on \(M^3\), is defined using Gromov-Witten invariants by NEWLINE\[NEWLINE\langle p,p,p\rangle^{M^3}_d=\sum_{k\mid d} k^3 n_k.NEWLINE\]NEWLINE More precisely, if \(I_0,I_1,I_2,I_3\) is a basis of solutions to (a) with holomorphic solution \(I_0=1+\sum_{d\geq 1} a_d q^d\) and logarithmic solution \(I_1=\ln(q) I_0+\sum_{d\geq 1} b_d q^d\). Then \({I_0\over{I_0}},{I_1\over{I_0}},{I_2\over{I_0}},{I_3\over{I_0}}\) is a basis of solutions for (b) after change of coordinates \(q= \exp(I_1/I_0)\). NEWLINENEWLINENEWLINEProof is along the line of \textit{A. Givental} on complete intersections on toric manifolds [in: ``Topological field theory, primitive forms and related topics'' (Kyoto 1996), 141-175 (1998; Zbl 0936.14031) and Int. Math. Res. Not. 13, 613-663 (1996; Zbl 0881.55006)], and \textit{B. Kim} on quantum hyperplane principle [J. Korean Math. Soc. 37, 455-461 (2000; Zbl 0986.14035)]. Specifically it is built on the following observations:NEWLINENEWLINENEWLINE(i) The degeneracy locus \(M^3\) is identified with the vanishing locus of a section of a vector bundle on a Grassmannian manifold. This vector bundle decomposes into a direct sum of vector bundles \(E\oplus H\), where \(H\) is again a direct sum of line bundles.NEWLINENEWLINENEWLINE(ii) The quantum hyperplane principle of Kim extends to relate the \(E\)-restricted quantum cohomology with the \(E\oplus H\)-restricted one.NEWLINENEWLINENEWLINE(iii) The \(E\)-restricted quantum cohomology can be effectively computed using localization techniques and WDVV-relations.