双场论中Vaisman代数体的二重态与规范对称性
Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory
摘要：
由Vaisman提出的度规代数体（Vaisman代数体）支配着双场论（DFT）中由C-括号生成的规范对称性代数。我们表明，Vaisman代数胚是由一个类似的Drinfel'd双李代数胚。基于二重时空作为仿厄米流形的几何实现，我们研究了DFT上的外代数和仿Dolbeault上同调，并讨论了DFT规范对称性背后的Drinfel'd二重时空的结构.与广义几何中的Courant代数胚相似，仿厄米流形中的Lagrange子丛$(L,\tilde{L})$在Vaisman代数胚中具有Dirac-like结构.我们发现DFT中强约束的代数根源可以追溯到$(L,\tilde{L})$是李双代数胚所需的相容性条件。这一分析为解决密度泛函理论中规范对称性的“coquecigrue问题”提供了基础。
英文摘要：
The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analogue of the Drinfel'd double of Lie algebroids. Based on a geometric realization of doubled space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure of the Drinfel'd double behind the DFT gauge symmetry. Similar to the Courant algebroid in the generalized geometry, Lagrangian subbundles $(L,\tilde{L})$ in a para-Hermitian manifold play Dirac-like structures in the Vaisman algebroid. We find that an algebraic origin of the strong constraint in DFT is traced back to the compatibility condition needed for $(L,\tilde{L})$ be a Lie bialgebroid. The analysis provides a foundation toward the "coquecigrue problem" for the gauge symmetry in DFT.
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