Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds
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2024
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| author | Doran, Charles Pioline, Boris Schimannek, Thorsten |
| author_facet | Doran, Charles Pioline, Boris Schimannek, Thorsten |
| contents | Motivated in part by the modular properties of enumerative invariants of K3-fibered Calabi-Yau threefolds, we introduce a family of 39 Calabi-Yau mirror pairs $(X,Y)$ with $h_{1,1}(X)=h_{2,1}(Y)=2$, labelled by certain integer quadruples $(m,i,j,s)$ with $m\leq 11$. On the A-model side, $X$ arises as a complete intersection in a projective bundle over a Fano fourfold $V_m^{[i,j]}$, and admits a Tyurin degeneration into a pair of degree $m$ Fano threefolds $F_m^{[i]}\cup F_m^{[j]}$ intersecting on an anticanonical K3 divisor of degree $2m$. On the B-model side, $Y$ is fibered by $M_{m}$-polarized K3-surfaces of Picard rank 19, and determined by a branched covering of $\mathbb{P}^1$, consistent with the Doran-Harder-Thompson mirror conjecture. When $s=0$, $Y$ itself acquires a Tyurin degeneration, and correspondingly $X$ acquires a fibration by degree $2m$ K3 surfaces, such that the two Kähler moduli control the size of the K3-fiber and base $\mathbb{P}^1$. While the mirror pairs with $m\leq 4$ can be realized as complete intersections in products of projective spaces or as hypersurfaces in toric varieties, the examples with $m\geq 5$ are intrinsically non-toric. We obtain uniform formulae for the genus 0 and 1 topological free energies near the Tyurin degeneration (mirror to the large base limit), exhibiting modular properties under the Fricke-extended congruence group $Γ_0(m)^+$. We use these results to compute the vertical Gopakumar-Vafa and Noether-Lefschetz invariants and check that their generating functions satisfy the expected modular properties. We also compute generating series of Gopakumar-Vafa invariants with fixed non-zero base degree and exhibit their modular properties. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_02994 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds Doran, Charles Pioline, Boris Schimannek, Thorsten High Energy Physics - Theory Algebraic Geometry Motivated in part by the modular properties of enumerative invariants of K3-fibered Calabi-Yau threefolds, we introduce a family of 39 Calabi-Yau mirror pairs $(X,Y)$ with $h_{1,1}(X)=h_{2,1}(Y)=2$, labelled by certain integer quadruples $(m,i,j,s)$ with $m\leq 11$. On the A-model side, $X$ arises as a complete intersection in a projective bundle over a Fano fourfold $V_m^{[i,j]}$, and admits a Tyurin degeneration into a pair of degree $m$ Fano threefolds $F_m^{[i]}\cup F_m^{[j]}$ intersecting on an anticanonical K3 divisor of degree $2m$. On the B-model side, $Y$ is fibered by $M_{m}$-polarized K3-surfaces of Picard rank 19, and determined by a branched covering of $\mathbb{P}^1$, consistent with the Doran-Harder-Thompson mirror conjecture. When $s=0$, $Y$ itself acquires a Tyurin degeneration, and correspondingly $X$ acquires a fibration by degree $2m$ K3 surfaces, such that the two Kähler moduli control the size of the K3-fiber and base $\mathbb{P}^1$. While the mirror pairs with $m\leq 4$ can be realized as complete intersections in products of projective spaces or as hypersurfaces in toric varieties, the examples with $m\geq 5$ are intrinsically non-toric. We obtain uniform formulae for the genus 0 and 1 topological free energies near the Tyurin degeneration (mirror to the large base limit), exhibiting modular properties under the Fricke-extended congruence group $Γ_0(m)^+$. We use these results to compute the vertical Gopakumar-Vafa and Noether-Lefschetz invariants and check that their generating functions satisfy the expected modular properties. We also compute generating series of Gopakumar-Vafa invariants with fixed non-zero base degree and exhibit their modular properties. |
| title | Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds |
| topic | High Energy Physics - Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2408.02994 |