Mysterious triality and the exceptional symmetry of loop spaces

IF 1.4 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Hisham Sati, Alexander A. Voronov
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引用次数: 0

Abstract

In previous work (Sati and Voronov in Commun Math Phys 400:1915–1960, 2023. https://doi.org/10.1007/s00220-023-04643-7, in Adv Theor Math Phys 28(8):2491–2601, 2024. https://doi.org/10.4310/atmp.241119034750), we introduced Mysterious Triality, extending the Mysterious Duality (Iqbal et al. in Adv Theor Math Phys 5:769–808, 2002. https://doi.org/10.4310/ATMP.2001.v5.n4.a5) 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 action of the Cartan subalgebra by symmetries of the equations of motion of \((11-k)\)d supergravity to a maximal parabolic subalgebra \(\mathfrak {p}_k^{k(k)}\) of the Lie algebra \(\mathfrak {e}_{k(k)}\) of the U-duality group. We do this by constructing the action on the rational homotopy model of the slightly more symmetric than \({\mathcal {L}}_c^k S^4\) toroidification \({\mathcal {T}}^k S^4\), which is another bookkeeping device for the equations of motion. To justify these results, 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.

神秘的三重性和环空间的特殊对称性
在以前的工作中(Sati和Voronov在公共数学物理400:1915-1960,2023)。[1] [https://doi.org/10.1007/s00220-023-04643-7] .数学与物理学报,28(8):2491 - 2601,2024。https://doi.org/10.4310/atmp.241119034750),我们介绍了神秘的三性,扩展了神秘的二元性(伊克巴尔等人在Adv理论数学物理5:769 - 808,2002)。https://doi.org/10.4310/ATMP.2001.v5.n4.a5)在物理和代数几何之间,以理性同伦理论的形式包括代数拓扑。从4球的合理Sullivan最小模型\(S^4\)开始,通过假设H捕获m理论的动力学,进而发展到m理论在环面上的降维\(T^k\), \(k \ge 1\),通过4球的迭代循环空间\({\mathcal {L}}_c^k S^4\)描述其动力学。由此,我们还提取了类型为\(E_k\)的例外李群/代数的极大环面/Cartan子代数和Weyl群对应的数据。本文利用\((11-k)\) d超引力运动方程的对称性,将Cartan子代数的作用推广到u对偶群的李代数\(\mathfrak {e}_{k(k)}\)的极大抛物子代数\(\mathfrak {p}_k^{k(k)}\),从而发现了更为丰富的对称性。我们通过在比\({\mathcal {L}}_c^k S^4\)环化\({\mathcal {T}}^k S^4\)稍微对称一点的有理同伦模型上构造作用来做到这一点,这是运动方程的另一种簿记装置。为了证明这些结果,我们确定了环化的最小模型\({\mathcal {T}}^k S^4\),推广了vigu - poirrier, Sullivan和Burghelea的结果,并建立了一个代数环化/总化共轭。
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来源期刊
Letters in Mathematical Physics
Letters in Mathematical Physics 物理-物理:数学物理
CiteScore
2.40
自引率
8.30%
发文量
111
审稿时长
3 months
期刊介绍: The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.
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