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arXiv 2109.14580hep-thmath.DG
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超对称真空纯旋量方程的Pin（d，d）协方差和非阿贝尔T-对偶

Pin(d,d) covariance of pure spinor equations for supersymmetric vacua and Non-Abelian T-duality

Aybike Catal-Ozer, Emine Dirioz

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摘要：

在II型超引力的超对称紧化中，保持四维${\cal{N}} = 1$超对称性要求六维内流形$M$的广义切丛$TM \oplus T^*M$的结构群从$SO(6,6)$约化为$SU(3) \times SU(3)$ .这种拓扑条件意味着存在两个全局相容的非零模纯旋量$Φ_1$和$Φ_2$.此外，这些纯旋量应满足一定的一阶微分方程。在本文中，我们证明了非阿贝尔T-对偶（NATD）是这些纯旋量方程的解生成变换。我们首先证明了纯旋量方程在$Pin(d,d)$变换下是协变的。然后，我们使用的事实，NTD是由一个坐标相关的$Pin(d,d)$变换。关键点在于，由该变换产生的通量与等距群相关的几何通量相同，相对于等距群实现NATA。我们通过研究具有$SU(2)$等距和$SU(3)$结构的IIB型超引力解的NATD证明了我们的方法。

英文摘要：

In a supersymmetric compactification of Type II supergravity, preservation of ${\cal{N}} = 1$ supersymmetry in four dimensions requires that the structure group of the generalized tangent bundle $TM \oplus T^*M$ of the six dimensional internal manifold $M$ is reduced from $SO(6,6)$ to $SU(3) \times SU(3)$. This topological condition on the internal manifold implies existence of two globally defined compatibe pure spinors $Φ_1$ and $Φ_2$ of non-vanishing norm. Furthermore, these pure spinors should satisfy certain first order differential equations. In this paper, we show that Non-Abelian T-duality (NATD) is a solution generating transformation for these pure spinor equations. We first show that the pure spinor equations are covariant under $Pin(d,d)$ transformations. Then, we use the fact NATD is generated by a coordinate dependent $Pin(d,d)$ transformation. The key point is that the flux produced by this transformation is the same as the geometric flux associated with the isometry group, with respect to which one implements NATD. We demonstrate our method by studying NATD of certain solutions of Type IIB supergravity with $SU(2)$ isometry and $SU(3)$ structure.

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