Integrality and divisibility arithmetic of mirror maps and instanton numbers for Calabi-Yau threefolds: the open core of Lian-Yau (1996)

Lian and Yau's 1996 paper 'Arithmetic properties of mirror map and quantum coupling' proved divisibility and congruence properties of the instanton numbers $n_d$ of Calabi-Yau mirror families conditionally on the integrality of the mirror map $z(q)$ and of the $n_d$, and posed five open questions: whether K3 mirror maps from orbifold constructions are Thompson series; whether the threefold integrality assumptions hold; whether the experimentally observed divisibility $5^3\mid n_d$ extends to all degrees $d$ on the quintic; what conceptually explains the observed Monster and $M_{24}$ connections; and how sharp the resulting mod-$p$ bounds are. The K3-Thompson-series conjecture is refuted (Verrill-Yui counterexample; Doran's Picard-Fuchs uniformization explains the observed modularity without the Monster), and the $M_{24}$ thread grew into Mathieu and Umbral Moonshine as an independent field. Integrality of the one-variable hypergeometric mirror maps is now unconditional (Zudilin; Krattenthaler-Rivoal; Delaygue), while integrality of the instanton numbers is approached by partially conditional $p$-adic and symplectic routes (Kontsevich-Schwarz-Vologodsky; Vologodsky; Beukers-Vlasenko; Ionel-Parker) and by arithmetic homological mirror symmetry for Greene-Plesser mirrors (Ganatra-Hanlon-Hicks-Pomerleano-Sheridan), which remains conditional on widely expected foundational results; the naive-versus-true mirror map distinction (Bleau-Sheridan) further marks the unconditional general statement as open. Whether $5^3\mid n_d$ holds for all degrees on the quintic, and the exact $p$-adic valuation program in general, remain open. The surviving open core - unconditional integrality of $z(q)$ and $n_d$ for one-parameter Calabi-Yau threefold mirror families at the source's generality, all-degree divisibility refinements, and sharpness of the $N_e$-based bounds - remains open.

Background

Mirror symmetry pairs a Calabi-Yau threefold - a smooth projective three-dimensional variety with trivial canonical bundle and no holomorphic forms in degrees one and two - with a mirror partner in which deformations of complex structure encode the curve-counting geometry of the original. Its archetypal success was the prediction by Candelas, de la Ossa, Green and Parkes of the numbers of rational curves of every degree on the quintic threefold (the zero locus of a degree-five homogeneous polynomial in projective four-space $\mathbb{P}^4$) from period data on its mirror. The mirror partner is constructed by the orbifold construction of Greene and Plesser: quotient a Fermat-type hypersurface or complete intersection by a finite abelian group of phase symmetries and resolve the quotient. Low-degree counts were known independently by classical algebraic geometry - Katz computed the corresponding line counts on complete-intersection Calabi-Yau threefolds, and the source paper records that Katz's work in particular implies $5\mid n_d$ for the quintic.

The computation runs through the Picard-Fuchs equation - the linear differential equation satisfied by the periods (integrals of the holomorphic three-form over a basis of three-cycles) of a family of Calabi-Yau manifolds as the complex-structure parameter varies. At a point of maximal unipotent monodromy (MUM) - a degeneration around which the period solutions acquire a full tower of logarithmic behavior - two distinguished solutions $F$ and $G$ define the flat coordinate $t=G/F$ and the canonical coordinate $q=e^{2\pi i t}$. The mirror map $z(q)$ is the resulting power series expressing the algebraic modulus in terms of $q$, normalized so that $z(q)=q+\cdots$. Lian and Yau observed that in all numerically checked threefold examples $z(q)$ has integer Fourier coefficients (checked to order at least $q^{30}$), and they proved this integrality for elliptic-curve and K3 families in numerous cases; for threefold families it remained a hypothesis in their 1996 paper.

The quantum coupling (also called the Yukawa coupling) is the third $t$-derivative of the genus-zero prepotential; mirror symmetry identifies it with the classical intersection number plus an instanton sum

$$K(q)=K_{\mathrm{cl}}+\sum_{d\ge 1}d^3\,\frac{n_d\,q^d}{1-q^d},$$

where $K_{\mathrm{cl}}$ is the classical cubic intersection number and the instanton number $n_d$ is predicted to count rational curves of degree $d$ on the original threefold (for the quintic, $n_1=23\cdot5^3=2875$, the number of lines). The $n_d$ are the genus-zero BPS (Gopakumar-Vafa) invariants: genus-zero Gromov-Witten invariants - curve-counting invariants defined by integration over moduli spaces of stable maps - determine them through the multiple-cover relation $N_{0,d}=\sum_{k\mid d}n_{d/k}/k^3$, and Gopakumar and Vafa argued on string-theoretic grounds that the $n_d$ are integers; nevertheless, an unconditional arithmetic-geometric proof of the integrality of the $n_d$ for specific families such as the quintic became, and remains, a subtle problem.

Under these integrality hypotheses (the 'mirror hypothesis'), Lian and Yau derived mod-$p$ structure from two invariants attached to the family: the classical intersection number $K_{\mathrm{cl}}$, and $N_e$ - the least common multiple of the coefficients of the auxiliary polynomial $e(z)$, with integer coefficients, that together with the discriminant polynomial $\Delta(z)$ appears in the nonlinear ordinary differential equation with integer coefficients satisfied by the mirror map and the quantum coupling. Their theorems imply, for example, that if $p^r$ divides $N_e$ and $p$ does not divide $d$, then $p^{s+r-1}$ (for $p=2,5$) or $p^{s+r}$ (for $p=3,7$) divides $n_d$, where $p^s$ is the largest power of $p$ dividing the content $m(K)$ of the quantum coupling - the greatest common divisor of its Fourier coefficients, which in their worked examples coincides with the corresponding power of $p$ dividing $K_{\mathrm{cl}}$. For the Fermat quintic, $N_e=-5^3$ and $K_{\mathrm{cl}}=5$, giving $5^3\mid n_d$ for all $d$ with $5\nmid d$; the power $5^3$ is optimal, since $n_1=23\cdot5^3$. Experimentally, $5^3$ appeared to divide all $n_d$ (as tabulated by Candelas et al.), and the source records that this all-degree statement had been conjectured earlier, in Clemens's 1986 International Congress address 'Curves on higher-dimensional complex projective manifolds', which the paper cites. Analogous sharp bounds are worked out there for a quartic-sextic complete intersection in $\mathbb{P}^5[1,1,1,2,2,3]$ ($N_e=-2^6\cdot3^5$, $K_{\mathrm{cl}}=2$) and for the degree-ten hypersurface in $\mathbb{P}^4[1,1,1,2,5]$.

The same paper asked five linked questions: (i) whether K3 mirror maps from orbifold constructions are, up to translation and inversion, Thompson series - the modular generator functions attached to elements of the Monster finite simple group in the Conway-Norton 'monstrous moonshine' correspondence; (ii) whether the integrality assumptions on threefold mirror maps hold; (iii) whether all-degree divisibility such as $5^3\mid n_d$ holds; (iv) what the conceptual origin of the observed Monster and $M_{24}$ (a sporadic finite simple group) connections is; and (v) how sharp the $N_e$-based congruence bounds are. Questions (i) and (iv) have since been answered - negatively for the Monster, and by the development of Mathieu and Umbral Moonshine as an independent field - while (ii), (iii) and (v) survive, in refined form, as the open core.

Problem Statement

The problem is the open arithmetic core of Lian-Yau (1996), Sections 6-7, at the source's generality: one-parameter mirror families of Calabi-Yau threefolds obtained by the Greene-Plesser orbifold construction (Fermat-type hypersurfaces and complete intersections in weighted projective spaces, quotiented by finite abelian phase-symmetry groups), together with their one-parameter toric (Batyrev-Borisov) instances - mirror pairs built from dual reflexive polytopes (lattice polytopes whose polar duals are again lattice polytopes) - as admissible cases. For such a family, let $z(q)$ be the mirror map at the point of maximal unipotent monodromy (the naive, period-based mirror map of Bleau-Sheridan; the true mirror map is the one appearing in mirror symmetry theorems, and the two coincide under a combinatorial criterion that holds for the quintic but not in general), and $K(q)=K_{\mathrm{cl}}+\sum_{d\ge1}d^3 n_d q^d/(1-q^d)$ the quantum coupling with instanton numbers $n_d$. The problem has three jointly stated components, which are the arithmetic faces of a single objective - an unconditional account of the integrality and congruence structure that mirror symmetry predicts for these families:

(1) Integrality (the source's Section 7 hypotheses). Prove or refute, unconditionally, that the Fourier coefficients of $z(q)$ are integers and that the $n_d$ are integers for the class above - beyond the hypergeometric and weighted-projective one-variable cases where integrality of the naive mirror map is now unconditional (Zudilin; Krattenthaler-Rivoal; Delaygue), beyond the existing $p$-adic results (integrality for $p>3$ and $p>5$; membership in $\mathbb{Z}[N^{-1}]$), and beyond the conditional arithmetic-homological-mirror-symmetry theorem of Ganatra et al; in particular, resolve the distinction between the naive and the true mirror map (Bleau-Sheridan) for the families where the two differ.

(2) All-degree divisibility. On the quintic threefold, prove or refute that $5^3$ divides $n_d$ for every $d\ge1$ (known for $5\nmid d$ under the source's hypotheses; $5^3$ is the optimal power since $n_1=23\cdot5^3$); and determine the analogous all-degree divisibility for the source's other examples - the quartic-sextic complete intersection in $\mathbb{P}^5[1,1,1,2,2,3]$ (the analogues $3^5\mid n_d$ and $2^6\mid n_d$, beyond the known cases $d\not\equiv0\bmod 3$ and $d$ odd) and the degree-ten hypersurface in $\mathbb{P}^4[1,1,1,2,5]$.

(3) Exact valuations and sharpness. Determine the exact $p$-adic valuations $v_p(n_d)$ for all degrees $d$ and primes $p$ - in particular the behavior at degrees divisible by $p$, where the source's method degenerates - and thereby the sharpness of the $N_e$-based bounds $p^{s+r-1}$ (for $p=2,5$) and $p^{s+r}$ (for $p=3,7$).

Component (3) is the strongest of the three: determining the valuations $v_p(n_d)$ for all primes $p$ and degrees $d$ yields the divisibility statements of (2) for the corresponding families and the integrality of the $n_d$ in (1), so a resolution of (3) discharges those parts; the integrality of the coefficients of $z(q)$ in (1) remains a separate requirement.

The source's Section 6 conjecture that K3 mirror maps are Thompson series is excluded (refuted by the Verrill-Yui counterexample and superseded by Doran's uniformization theory), as is the Monster/$M_{24}$-mechanism question (transformed into Mathieu and Umbral Moonshine).

References

1. Bong H. Lian and Shing-Tung Yau, 'Arithmetic properties of mirror map and quantum coupling', Communications in Mathematical Physics 176 (1996) 163-191, DOI 10.1007/bf02099367, https://doi.org/10.1007/bf02099367; preprint arXiv hep-th/9411234, https://arxiv.org/abs/hep-th/9411234
2. Philip Candelas, Xenia C. de la Ossa, Paul S. Green and Linda Parkes, 'A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory', Nuclear Physics B 359 (1991) 21-74, DOI 10.1016/0550-3213(91)90292-6, https://doi.org/10.1016/0550-3213(91)90292-6
3. B. R. Greene and M. R. Plesser, 'Duality in Calabi-Yau moduli space', Nuclear Physics B 338 (1990) 15-37, DOI 10.1016/0550-3213(90)90622-K, https://doi.org/10.1016/0550-3213(90)90622-K
4. J. H. Conway and S. P. Norton, 'Monstrous Moonshine', Bulletin of the London Mathematical Society 11 (1979) 308-339, DOI 10.1112/blms/11.3.308, https://doi.org/10.1112/blms/11.3.308
5. Sheldon Katz, 'Lines on complete intersection threefolds with K=0', Mathematische Zeitschrift 191 (1986) 293-296, DOI 10.1007/BF01164033, https://doi.org/10.1007/BF01164033
6. Herbert Clemens, 'Curves on higher-dimensional complex projective manifolds', in Proceedings of the International Congress of Mathematicians, Berkeley, California, 1986, Vol. 1, American Mathematical Society, Providence, 1987, pp. 634-640, ISBN 0-8218-0110-4, Zbl 0682.14024, https://zbmath.org/0682.14024.
7. Charles Doran, 'Picard-Fuchs uniformization: Modularity of the mirror map and mirror-moonshine', in The Arithmetic and Geometry of Algebraic Cycles (CRM Proceedings and Lecture Notes 24), American Mathematical Society, Providence, 2000, pp. 257-281, DOI 10.1090/crmp/024/13, https://doi.org/10.1090/crmp/024/13; preprint arXiv math/9812162, https://arxiv.org/abs/math/9812162
8. Charles F. Doran, 'Picard-Fuchs uniformization and modularity of the mirror map', Communications in Mathematical Physics 212 (2000) 625-647, DOI 10.1007/S002200000228, https://doi.org/10.1007/S002200000228
9. Helena Verrill and Noriko Yui, 'Thompson series, and the mirror maps of pencils of K3 surfaces', in The Arithmetic and Geometry of Algebraic Cycles (CRM Proceedings and Lecture Notes 24), American Mathematical Society, Providence, 2000, pp. 399-432, DOI 10.1090/CRMP/024/20, https://doi.org/10.1090/CRMP/024/20
10. Terry Gannon, 'Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics', Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2006, ISBN 0-521-83431-3, DOI 10.1017/CBO9780511535116, https://doi.org/10.1017/CBO9780511535116 (the book form of the survey 'Monstrous Moonshine: The first twenty-five years', arXiv math/0402345, https://arxiv.org/abs/math/0402345)
11. Bong H. Lian and Shing-Tung Yau, 'The nth root of the mirror map', in Calabi-Yau Varieties and Mirror Symmetry (Fields Institute Communications 38), American Mathematical Society, Providence, 2003, pp. 195-199, DOI 10.1090/fic/038/10, https://doi.org/10.1090/fic/038/10
12. Bong H. Lian and Shing-Tung Yau, 'Integrality of certain exponential series', in Ming-Chang Kang (ed.), Lectures on Algebra and Geometry, Proceedings of the International Conference on Algebra and Geometry, Taipei 1995, International Press, Cambridge, MA, 1998, pp. 215-227, Zbl 0998.12009, https://zbmath.org/0998.12009 (the prime-n integrality result for Dwork-family mirror maps; also circulated as 'Mirror maps, modular relations and hypergeometric series I')
13. V. V. Zudilin, 'Integrality of power expansions related to hypergeometric series', Mathematical Notes 71 (2002) 604-616, DOI 10.1023/A:1015827602930, https://doi.org/10.1023/A:1015827602930
14. Christian Krattenthaler and Tanguy Rivoal, 'On the integrality of the Taylor coefficients of mirror maps', Duke Mathematical Journal 151 (2010), no. 2, 175-218, DOI 10.1215/00127094-2009-063, https://doi.org/10.1215/00127094-2009-063; preprint arXiv 0907.2577, https://arxiv.org/abs/0907.2577
15. Eric Delaygue, 'A criterion for the integrality of the Taylor coefficients of mirror maps in several variables', Advances in Mathematics 234 (2013) 414-452, DOI 10.1016/j.aim.2012.09.028, https://doi.org/10.1016/j.aim.2012.09.028; preprint arXiv 1108.4352, https://arxiv.org/abs/1108.4352
16. Maxim Kontsevich, Albert Schwarz and Vadim Vologodsky, 'Integrality of instanton numbers and p-adic B-model', Physics Letters B 637 (2006) 97-101, DOI 10.1016/j.physletb.2006.04.012, https://doi.org/10.1016/j.physletb.2006.04.012; preprint arXiv hep-th/0603106, https://arxiv.org/abs/hep-th/0603106
17. Vadim Vologodsky, 'Integrality of instanton numbers', preprint arXiv 0707.4617, https://arxiv.org/abs/0707.4617
18. Frits Beukers and Masha Vlasenko, 'On p-integrality of instanton numbers', Pure and Applied Mathematics Quarterly 19 (2023), no. 1, 7-44, DOI 10.4310/PAMQ.2023.v19.n1.a2, https://doi.org/10.4310/PAMQ.2023.v19.n1.a2; preprint arXiv 2109.10427, https://arxiv.org/abs/2109.10427
19. Eleny-Nicoleta Ionel and Thomas H. Parker, 'The Gopakumar-Vafa formula for symplectic manifolds', Annals of Mathematics (2) 187 (2018), no. 1, 1-64, DOI 10.4007/annals.2018.187.1.1, https://doi.org/10.4007/annals.2018.187.1.1; preprint arXiv 1306.1516, https://arxiv.org/abs/1306.1516
20. Sheel Ganatra, Andrew Hanlon, Jeff Hicks, Daniel Pomerleano and Nick Sheridan, 'Integrality of mirror maps and arithmetic homological mirror symmetry for Greene-Plesser mirrors', Advances in Mathematics 481 (2025) 110535, DOI 10.1016/j.aim.2025.110535, https://doi.org/10.1016/j.aim.2025.110535; preprint arXiv 2312.01949, https://arxiv.org/abs/2312.01949
21. Sophie Bleau and Nick Sheridan, 'On the positivity and integrality of coefficients of mirror maps', arXiv 2409.07601, https://arxiv.org/abs/2409.07601
22. Tohru Eguchi, Hirosi Ooguri and Yuji Tachikawa, 'Notes on the K3 Surface and the Mathieu group M_24', arXiv 1004.0956, https://arxiv.org/abs/1004.0956; published in Experimental Mathematics 20 (2011), no. 1, 91-96, DOI 10.1080/10586458.2011.544585, https://doi.org/10.1080/10586458.2011.544585
23. Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A. Harvey, 'Umbral Moonshine', arXiv 1204.2779, https://arxiv.org/abs/1204.2779; published in Communications in Number Theory and Physics 8 (2014), no. 2, 101-242, DOI 10.4310/cntp.2014.v8.n2.a1, https://doi.org/10.4310/cntp.2014.v8.n2.a1
24. Noriko Yui, 'Modularity of Calabi-Yau varieties: 2011 and beyond', arXiv 1212.4308, https://arxiv.org/abs/1212.4308
25. Rajesh Gopakumar and Cumrun Vafa, 'M-Theory and Topological Strings--I', arXiv hep-th/9809187, https://arxiv.org/abs/hep-th/9809187
26. Jim Bryan, Sheldon Katz and Naichung Conan Leung, 'Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds', Journal of Algebraic Geometry 10 (2001), no. 3, 549-568, arXiv math/9911056, https://arxiv.org/abs/math/9911056 (per-curve result: multiple-cover contributions of a nodal generic or a smoothly embedded contractible rational curve are integer amounts, via Kollar's length invariant)
27. John Pardon, 'Universally counting curves in Calabi-Yau threefolds', arXiv 2308.02948, https://arxiv.org/abs/2308.02948
28. You-Cheng Chou, 'Integrality of genus zero Gopakumar-Vafa type invariants of semi-positive varieties', arXiv 2312.01413, https://arxiv.org/abs/2312.01413; published in International Journal of Modern Physics A 39 (2024), no. 33, DOI 10.1142/S0217751X2446014X, https://doi.org/10.1142/S0217751X2446014X