伊辛模型与特殊几何
The Ising model and Special Geometries
摘要：
我们证明了零化伊辛模型磁化率重积分$χ^{(n)}$的线性微分算子（$n \le 6$）的因子所对应的全局幂零G算子与它们的伴随算子是同态的.这种自伴到算子同态的性质等价于它们的对称平方或外平方有有理解的事实。微分伽罗瓦群属于特殊的正交群或辛群。这种自伴（直到算子等价）性质意味着我们已经知道的从几何导出的因子算子是特殊的全局幂零算子：它们对应于“特殊几何”。除了小的阶因子算子（出现在与$ χ^{(5)}$和$ χ^{(6)}$相关的线性微分算子中），特别是与模形式相关的那些算子之外，我们关注于相当大的12阶和23阶算子。我们证明了十二阶算子有一个外平方零化一个有理解。那么，它的微分伽罗瓦群就在辛群$ Sp(12, \mathbb{C})$中。23阶算子被证明可以分解为2阶算子和21阶算子。这个21阶算子的对称平方有一个有理解。因此，它的微分伽罗瓦群在正交群$ SO(21, \mathbb{C})$中。
英文摘要：
We show that the globally nilpotent G-operators corresponding to the factors of the linear differential operators annihilating the multifold integrals $χ^{(n)}$ of the magnetic susceptibility of the Ising model ($n \le 6$) are homomorphic to their adjoint. This property of being self-adjoint up to operator homomorphisms, is equivalent to the fact that their symmetric square, or their exterior square, have rational solutions. The differential Galois groups are in the special orthogonal, or symplectic, groups. This self-adjoint (up to operator equivalence) property means that the factor operators we already know to be Derived from Geometry, are special globally nilpotent operators: they correspond to "Special Geometries". Beyond the small order factor operators (occurring in the linear differential operators associated with $ χ^{(5)}$ and $ χ^{(6)}$), and, in particular, those associated with modular forms, we focus on the quite large order-twelve and order-23 operators. We show that the order-twelve operator has an exterior square which annihilates a rational solution. Then, its differential Galois group is in the symplectic group $ Sp(12, \mathbb{C})$. The order-23 operator is shown to factorize in an order-two operator and an order-21 operator. The symmetric square of this order-21 operator has a rational solution. Its differential Galois group is, thus, in the orthogonal group $ SO(21, \mathbb{C})$.
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