论文标题
尖端多项式Landau-Ginzburg Orbifold的镜像对称性
Mirror symmetry for a cusp polynomial Landau-Ginzburg orbifold
论文作者
论文摘要
对于任何三倍的正整数$ a'=（a_1'，a_2'，a_3'）$和$ c \ in \ mathbb {c}^*$，cusp polyenmial $ f_ {a'} = x_1^{a_1'}+x_2^{a_2'}+x_3^{a_3'} - c^{ - 1} x_1x_2x_3 $是镜子的镜子，是镜子到geigle-lenzing orbifold projective line line line line line line line line line line $ \ mathbb {p}^p}^1____________1'更确切地说，有了适合原始形式的选择，frobenius cusp多项式$ f_ {a'} $的弗罗贝尼斯（Frobenius）对Gromov-Witten $ \ Mathbb {p}^1_ {a_1'a_1'，A_1'，A_2'，A_3'} $的Frobenius歧管而言是同构的。 在本文中，我们将这种镜像现象扩展到了均衡的情况。也就是说，对于任何$ g $ - 一个尖的多项式$ f_ {a'} $的对称组，我们介绍了一对$（f_ {a'}，g）$的Frobenius歧管，并表明它与Geigle-lenz-lenzing Toppartive lowsive lowsive lowsive lowsive lowsive lowsive lofjective lofjective togeign frolobenius compotive。 $ \ mathbb {p}^1_ {a，λ} $，由另一个集合$ a $和$λ$索引，$ \ mathbb {c} \ setMinus \ setminus \ {0,1 \} $上的不同点。 对于$ a'$的某些特殊值，$ g $的特殊选择恰好是$ \ mathbb {p}^1_ {a'} \ cong \ mathbb {p}^1_ {a，λ} $。将我们对$ $（a，λ）$的镜像对称性同构结合在一起，将其与$ a'$的“常见”相结合，我们获得了Frobenius电位系数的某些身份。我们表明，这些身份等于雅各比·塞塔（Jacobi Theta）常数和dedekind eTa功能之间的身份。
For any triple of positive integers $A' = (a_1',a_2',a_3')$ and $c \in \mathbb{C}^*$, cusp polynomial $f_{A'} = x_1^{a_1'}+x_2^{a_2'}+x_3^{a_3'}-c^{-1}x_1x_2x_3$ is known to be mirror to Geigle-Lenzing orbifold projective line $\mathbb{P}^1_{a_1',a_2',a_3'}$. More precisely, with a suitable choice of a primitive form, Frobenius manifold of a cusp polynomial $f_{A'}$, turns out to be isomorphic to the Frobenius manifold of the Gromov-Witten theory of $\mathbb{P}^1_{a_1',a_2',a_3'}$. In this paper we extend this mirror phenomenon to the equivariant case. Namely, for any $G$ - a symmetry group of a cusp polynomial $f_{A'}$, we introduce the Frobenius manifold of a pair $(f_{A'},G)$ and show that it is isomorphic to the Frobenius manifold of the Gromov-Witten theory of Geigle-Lenzing weighted projective line $\mathbb{P}^1_{A,Λ}$, indexed by another set $A$ and $Λ$, distinct points on $\mathbb{C}\setminus\{0,1\}$. For some special values of $A'$ with the special choice of $G$ it happens that $\mathbb{P}^1_{A'} \cong \mathbb{P}^1_{A,Λ}$. Combining our mirror symmetry isomorphism for the pair $(A,Λ)$, together with the "usual" one for $A'$, we get certain identities of the coefficients of the Frobenius potentials. We show that these identities are equivalent to the identities between the Jacobi theta constants and Dedekind eta-function.
