Some Examples of Family Floer Mirrors

Abstract

In this article, we give explicit calculations for the family Floer mirrors of some non-compact Calabi-Yau surfaces. We compare it with the mirror construction of Gross-Hacking-Keel for suitably chosen log Calabi-Yau pairs and the rank two cluster varieties of finite type. In particular, the analytifications of the later two give partial compactifications of the family Floer mirrors that we computed.

Figures (20)

• Figure 1: $\mathcal{A}$-scattering diagrams for rank 2 finite type vianna.
• Figure 2: ${\mathcal{X}}$-scattering diagrams for rank 2 finite type vianna.
• Figure 3: Fukaya's trick
• Figure 4: BPS rays on $B_t$ for the case discussed in Section \ref{['section: dp5']}
• Figure 5: BPS rays near the singularity.

Theorems & Definitions (66)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Lemma 4.1
• Theorem 4.2
• Definition 4.3
• Definition 4.4