Minimal $D=5$ supergravity admits asymptotically globally AdS$_5$ gravitational solitons (strictly stationary, geodesically complete spacetimes with positive mass). We show that, like asymptotically flat gravitational solitons, these solutions satisfy mass and mass variation formulas analogous to those satisfied by AdS black holes. A thermodynamic volume associated to the non-trivial topology of the spacetime plays an important role in this construction. We then consider these solitons within the holographic ``complexity equals action'' and ``complexity equals volume'' conjectures as simple examples of spacetimes with nontrivial rotation and topology. We find distinct behaviours for the volume and action, with the counterterm for null boundaries playing a significant role in the latter case. For large solitons we find that both proposals yield a complexity of formation proportional to a power of the thermodynamic volume, $V^{3/4}$. In fact, up to numerical prefactors, the result coincides with the analogous one for large black holes.

中文翻译：

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AdS5 中孤子的化学和复杂性

最小 $D=5$ 超引力在全局上渐近地承认 AdS$_5$ 引力孤子（严格静止的，具有正质量的测地线完整时空）。我们表明，就像渐近平坦的引力孤子一样，这些解满足质量和质量变化公式，类似于 AdS 黑洞满足的那些公式。与时空的非平凡拓扑相关的热力学体积在此构造中起着重要作用。然后，我们将全息“复杂性等于动作”和“复杂性等于体积”猜想中的这些孤子视为具有非平凡旋转和拓扑的时空的简单示例。我们发现音量和动作的不同行为，空边界的反项在后一种情况下起着重要作用。对于大型孤子，我们发现这两个提议都会产生与热力学体积的幂成正比的复杂性，$V^{3/4}$。事实上，直到数值前因数，结果与大黑洞的类似结果一致。