神秘三重性与循环空间的例外对称性
Mysterious Triality and the Exceptional Symmetry of Loop Spaces
AI总结：
本文通过扩展Cartan子代数到最大抛物子代数，揭示了循环空间的更丰富对称性，并建立了代数托roidification/totalization的伴随关系。
摘要：
在先前的工作中，我们引入了神秘三重性，将Iqbal、Neitzke和Vafa之间的神秘双射扩展到代数拓扑，形式为有理同调论。从4维球面S^4的有理Sullivan最小模型出发，通过假设H捕捉M理论的动力学，进而将M理论在T^k（k≥1）上的维度约简，其动力学由迭代循环空间L_c^k S^4描述。从这里，我们还提取了对应于最大torus/Cartan子代数和Weyl群的数据。在本文中，我们通过展示在有理同调范畴中，抛物子代数p_k^{k(k)}对稍微比L_c^k S^4更对称的托roidification T^k S^4的作用，发现更丰富的对称性。这一作用普遍代表了将M理论约简到11-k维时超引力方程的动力学对称性。在过程中，我们确定了托roidification T^k S^4的最小模型，推广了Vigué-Poirrier、Sullivan和Burghelea的结果，并建立了代数托roidification/totalization的伴随关系。
英文摘要：
In previous work, we introduced Mysterious Triality, extending the Mysterious Duality of Iqbal, Neitzke, and Vafa between physics and algebraic geometry to include algebraic topology in the form of rational homotopy theory. Starting with the rational Sullivan minimal model of the 4-sphere $S^4$, capturing the dynamics of M-theory via Hypothesis H, this progresses to the dimensional reduction of M-theory on torus $T^k$, $k \ge 1$, with its dynamics described via the iterated cyclic loop space $\mathcal{L}_c^k S^4$ of the 4-sphere. From this, we also extracted data corresponding to the maximal torus/Cartan subalgebra and the Weyl group of the exceptional Lie group/algebra of type $E_k$. In this paper, we discover much richer symmetry by extending the data of the Cartan subalgebra to a maximal parabolic subalgebra $\mathfrak{p}_k^{k(k)}$ of the split real form $\mathfrak{e}_{k(k)}$ of the exceptional Lie algebra of type $E_k$ by exhibiting an action, in rational homotopy category, of $\mathfrak{p}_k^{k(k)}$ on the slightly more symmetric than $\mathcal{L}_c^k S^4$ toroidification $\mathcal{T}^k S^4$. This action universally represents symmetries of the equations of motion of supergravity in the reduction of M-theory to $11-k$ dimensions. Along the way, we identify the minimal model of the toroidification $\mathcal{T}^k S^4$, generalizing the results of Vigué-Poirrier, Sullivan, and Burghelea, and establish an algebraic toroidification/totalization adjunction.
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