The metric of S7 can be written as an SU(2)-instanton bundle over S4. It is also possible to write it differently as an anti-instanton bundle. We use this observation to construct an instanton–anti-instanton, SU(2)×SU(2), bundle over S4. We show that this 10d manifold admits two Einstein metrics. We then rewrite the metric to isolate two U(1) directions of the fibre and dimensionally reduce along them to get an eight dimensional metric describing an S2×S2 bundle over S4. This metric allows a harmonic 4-form which we use to derive new supergravity solutions in eleven and ten dimensions as AdS3×M8 and AdS2×M8, respectively.

中文翻译：

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S4 上的 S3 × S3 捆绑包和新的超重力解决方案


S7 的度量可以写成 S4 上的 SU(2)-瞬时束。也可以将其以不同的方式编写为反瞬时包。我们利用这一观察结果在 S4 上构建瞬子-反瞬子 SU(2)×SU(2) 束。我们证明这个 10d 流形承认两个爱因斯坦度量。然后，我们重写度量以隔离光纤的两个 U(1) 方向，并沿它们进行降维，以获得描述 S4 上的 S2×S2 束的八维度量。该度量允许调和 4 形式，我们用它来导出十一维和十维的新超重力解，分别为 AdS3×M8 和 AdS2×M8。