论文标题
八世代神奇的超级重力，尼梅尔晶格以及异常和希尔伯特模块化形式
Octonionic Magical Supergravity, Niemeier Lattices, and Exceptional and Hilbert Modular Forms
论文作者
论文摘要
我们研究了五个维度的八度离子魔法超级重力的BPS黑洞的量子脱位，这是由特殊的Jordan代数定义的。我们从理论上将量子退化纯粹数字定义为电荷空间中不同状态的数量，并具有一组离散U-二元组的一组不变标签。我们认为，五个维度的八度魔法超级重力的球形对称固定式bps黑洞的量子变性是由特殊组$ e_ {7（-25）} $的模块化形式的傅立叶系数给出的。黑洞的费用在由整体八元$ \ MATHCAL {r} $上的特殊Jordan Algebra $ J_3^{\ Mathcal {R}} $定义的晶格中。排名第一和等级的两个BPS黑洞（零面积）的量子变性由单数模块的傅立叶系数$ e_4（z）$和$ e_8（z）=（e_4（z））^2 $ of $ e_ $ e_ {7（z））^2 $。等级3（大）BPS黑洞将在其他地方进行研究。在N. Elkies和B. Gross的工作之后由$ sl（2，a）$的希尔伯特模块化形式的傅立叶系数给出。如果立方环$ a $的判别为$ d = p^2 $，$ p $ a prime数字，那么24维二次空间中的各向同性线定义了一对Niemeier lattices，可以将其视为一些BPS黑洞的电荷晶格。对于$ p = 7 $，它们是没有根的水ech晶格，而晶格$ a_6^4 $，带有168个根矢量。我们还回顾了搜索M/SuperString理论起源的当前状态。
We study the quantum degeneracies of BPS black holes of octonionic magical supergravity in five dimensions that is defined by the exceptional Jordan algebra. We define the quantum degeneracy purely number theoretically as the number of distinct states in the charge space with a given set of invariant labels of the discrete U-duality group. We argue that the quantum degeneracies of spherically symmetric stationary BPS black holes of octonionic magical supergravity in five dimensions are given by the Fourier coefficients of the modular forms of the exceptional group $E_{7(-25)}$. The charges of the black holes take values in the lattice defined by the exceptional Jordan algebra $J_3^{\mathbb{O}}(\mathcal{R})$ over integral octonions $\mathcal{R}$. The quantum degeneracies of charge states of rank one and rank two BPS black holes (zero area) are given by the Fourier coefficients of singular modular forms $E_4(Z)$ and $E_8(Z)=(E_4(Z))^2$ of $E_{7(-25)}(Z)$. The rank 3 (large) BPS black holes will be studied elsewhere. Following the work of N. Elkies and B. Gross on the embeddings of cubic rings $A$ into the exceptional Jordan algebra and their actions on the 24 dimensional orthogonal quadratic subspace of $J_3^{\mathbb{O}}(\mathcal{R})$, we show that the quantum degeneracies of rank one black holes described by such embeddings are given by the Fourier coefficients of the Hilbert modular forms of $SL(2,A)$. If the discriminant of the cubic ring $A$ is $D=p^2$ with $p$ a prime number then the isotropic lines in the 24 dimensional quadratic space define a pair of Niemeier lattices which can be taken as charge lattices of some BPS black holes. For $p=7$ they are the Leech lattice with no roots and the lattice $A_6^4$ with 168 root vectors. We also review the current status of the searches for the M/superstring theoretic origins of the octonionic magical supergravity.
