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arXiv 1602.05415hep-th
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关于通量真空、SU(n)-结构和广义复几何

On flux vacua, SU(n)-structures and generalised complex geometry

Daniël Prins

摘要:

要将弦论与可观测物理联系起来,理解超对称通量真空是必不可少的。本论文利用SU(n)-结构和广义复几何这两个数学框架来研究通量真空。具有SU(n)-结构的流形是Calabi-Yau流形的推广。广义复几何是同时推广复几何和辛几何的几何框架。在SU(4)结构流形上给出了II型超引力和M理论的流真空类。N=(1,1)的IIA型真空提升为N=1的M理论真空,其四通量不一定是(2,2)和本原的。给出了非紧Calabi-Yau空间Stenzel空间上的显式真空。然后通过构造非CY SU(4)-结构的族来推广这些,以在非辛SU(4)-变形的Stenzel空间上找到真空。它表明,N =(2,0)型IIB的超对称性条件可以改写的广义复几何的语言,部分在广义几乎复杂结构的可积性条件。d=2的这种改写超出了校准方程,与d= 4,6相反,其中校准方程等价于超对称性。最后,欧几里德II型理论研究SU(5)-结构流形,发现广义方程是必要的,但不足以满足超对称方程。这里也提供了明确的解决方案类别。通过将这些解提升到d=1,N=1的M理论,就可以与洛伦兹物理学联系起来。

英文摘要:

Understanding supersymmetric flux vacua is essential in order to connect string theory to observable physics. In this thesis, flux vacua are studied by making use of two mathematical frameworks: SU(n)-structures and generalised complex geometry. Manifolds with SU(n)-structure are generalisations of Calabi-Yau manifolds. Generalised complex geometry is a geometrical framework that simultaneously generalises complex and symplectic geometry. Classes of flux vacua of type II supergravity and M-theory are given on manifolds with SU(4)-structure. The N= (1,1) type IIA vacua uplift to N=1 M-theory vacua, with four-flux that need not be (2,2) and primitive. Explicit vacua are given on Stenzel space, a non-compact Calabi-Yau. These are then generalised by constructing families of non-CY SU(4)-structures to find vacua on non-symplectic SU(4)-deformed Stenzel spaces. It is shown that the supersymmetry conditions for N = (2,0) type IIB can be rephrased in the language of generalised complex geometry, partially in terms of integrability conditions of generalised almost complex structures. This rephrasing for d=2 goes beyond the calibration equations, in contrast to d=4,6 where the calibration equations are equivalent to supersymmetry. Finally, Euclidean type II theory is examined on SU(5)-structure manifolds, where generalised equations are found which are necessary but not sufficient to satisfy the supersymmetry equations. Explicit classes of solutions are provided here as well. Contact with Lorentzian physics can be made by uplifting such solutions to d=1, N=1 M-theory.

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