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Intrinsic mirror symmetry and Frobenius structure theorem via Gromov-Witten theory of root stacks

Samuel Johnston

Abstract

Using recent results of Battistella, Nabijou, Ranganathan and the author, we compare candidate mirror algebras associated with certain log Calabi-Yau pairs constructed by Gross-Siebert using log Gromov-Witten theory and Tseng-You using orbifold Gromov- Witten theory of root stacks. Although the structure constants used to defined these mirror algebras do not typically agree, we show that any given structure constant involved in the construction the algebra of Gross and Siebert can be computed in terms of structure constants of the algebra of Tseng and You after a sequence of log blowups. Using this relation, we provide another proof of associativity of the log mirror algebra, and a proof of the weak Frobenius Structure Theorem in full generality. Along the way, we introduce a class of twisted punctured Gromov-Witten invariants of generalized root stacks induced by log étale modifications, and use this to study the behavior of log Gromov-Witten invariants under ramified base change.

Intrinsic mirror symmetry and Frobenius structure theorem via Gromov-Witten theory of root stacks

Abstract

Using recent results of Battistella, Nabijou, Ranganathan and the author, we compare candidate mirror algebras associated with certain log Calabi-Yau pairs constructed by Gross-Siebert using log Gromov-Witten theory and Tseng-You using orbifold Gromov- Witten theory of root stacks. Although the structure constants used to defined these mirror algebras do not typically agree, we show that any given structure constant involved in the construction the algebra of Gross and Siebert can be computed in terms of structure constants of the algebra of Tseng and You after a sequence of log blowups. Using this relation, we provide another proof of associativity of the log mirror algebra, and a proof of the weak Frobenius Structure Theorem in full generality. Along the way, we introduce a class of twisted punctured Gromov-Witten invariants of generalized root stacks induced by log étale modifications, and use this to study the behavior of log Gromov-Witten invariants under ramified base change.
Paper Structure (12 sections, 24 theorems, 65 equations, 3 figures)

This paper contains 12 sections, 24 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Future Work
  3. Acknowledgements
  4. Tropical and Logarithmic preliminaries
  5. Tropicalization and Artin Fans
  6. Tropical maps and punctured log maps
  7. Construction of virtual classes on $\mathscr{M}(X,\beta)$
  8. Intrinsic Mirror constructions
  9. Counterexample to equality of theta basis
  10. Log Gromov-Witten theory of root stacks
  11. Comparing Invariants
  12. WDVV, TRR, and intrinsic mirror symmetry for log Calabi-Yau varieties

Key Result

Theorem 1.1

For $(X,D)$ a log Calabi-Yau variety in the sense of int_mirror, with log and orbifold mirror algebras $R^{log}_{(X,D)}$ and $R^{orb}_{(X,D)}$ respectively, then any structure constants $N_{p,q,r}^\textbf{A}$ defining the product on $R^{log}_{(X,D)}$ in the theta basis is equal to a sum of orbifold

Figures (3)

  • Figure 5.1: The broken lines contributing to $\vartheta_{p_2}$ and $\vartheta_{p}$, which all have a unique bending vertex, colored violet if it contributes to $\vartheta_{p_2}$ and colored blue if it contributes to $\vartheta_{p}$. We color the unique ray in the scattering diagram red, and final rays of the broken lines by blue, violet, yellow and green in order to differentiate.
  • Figure 8.1: The tropical curve associated with the generic point of $\mathscr{M}_2$
  • Figure 8.2: Tropical curves associated to lower dimensional strata of $\mathscr{M}_2$ which contribute to $\vartheta_{p_1}\vartheta_{p_2}\vartheta_{p_3}[\vartheta_{D_2}z^A]$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3: Conjecture $9.2$ int_mirror
  • Lemma 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 4.2
  • Corollary 4.3
  • Corollary 4.4
  • Proposition 4.5
  • ...and 36 more