The superconformal index of half-BPS states in $\mathcal{N}=4$ supersymmetric Yang-Mills with gauge group $\mathrm{U}(N)$ admits an expansion in terms of giant gravitons, ${\mathcal{I}}_N(q)={\mathcal{I}}_{\infty }(q){\sum }_{m=0}^{\infty }{q}^{mN}{\widehat{\mathcal{I}}}_m(q)$, where $m$ is the number of giant gravitons and ${\mathcal{I}}_{\infty }(q)$ is the graviton index. The expansion can be viewed as the implementation of trace relations for finite $N$. We derive this expansion directly in supergravity from the class of half-BPS solutions due to Lin, Lunin, and Maldacena in type IIB supergravity. The moduli space of these configurations can be quantized using covariant quantization methods. We show how this quantization leads to the precise expression for the expansion in terms of giant gravitons. Our proposal provides a derivation of the giant graviton expansion directly in terms of quantized supergravity degrees of freedom, and it recovers discrete data via quantum geometries that are classically nonsmooth.

中文翻译：

---

冒泡几何的巨引力子展开：量子化几何的离散性


具有规范组 $\mathrm{U}(N)$ 的 $\mathcal{N}=4$ 超对称杨米尔斯中半 BPS 态的超共形指数承认巨引力子方面的扩展， ${\mathcal{I}}_N(q)={\mathcal{I}}_{\infty }(q){\sum }_{m=0}^{\infty }{q}^{mN}{\widehat{\mathcal{I}}}_m(q)$ ，其中 $m$ 是巨引力子数， ${\mathcal{I}}_{\infty }(q)$ 是引力子指数。扩展可以被视为有限 $N$ 追踪关系的实现。我们直接从 IIB 型超重力中的 Lin、Lunin 和 Maldacena 的半 BPS 解类中导出超重力中的这种展开式。这些配置的模空间可以使用协变量化方法来量化。我们展示了这种量子化如何导致巨引力子膨胀的精确表达。我们的建议直接根据量​​子化超重力自由度提供了巨引力子展开式的推导，并通过经典非光滑的量子几何恢复离散数据。