AI 中文总结
该研究针对固定荷下全息扩展复杂度的短时行为问题，提出转入固定荷 sector 的劳斯函数的普适方案，经多类探针验证后将结果转化为Krylov链数据，为场论多种子全息复杂度构建提供路径。
AI 中文摘要
全息扩展（Krylov）复杂度将边界态复杂度的增长与落入体空间的探针的固有径向动量关联起来，幺正演化使扩展复杂度成为时间的偶函数。我们证明，当探针携带守恒诺特定理荷且由其未约化拉格朗日量描述时，该要求不成立。解决方法简单且普适：转入固定荷 sector 的劳斯函数（Routhian）可恢复复杂度的正确短时行为。我们从第一性原理确立该方案，并在大量探针体系中测试：带电粒子、张力与荷失谐的非BPS D-膜、沿内部等距方向激发的膜、世界体积规范场、AdS₄×CP³中涨落的D0-膜，以及AdS₅×S⁵中结合缠绕与转动的基本弦，还补充了AdS₃×S³×T⁴、ABJM和带电Anabalón-Ross背景的例子。随后我们将这些结果转化为Krylov链数据，提取 Lanczos 系数与Krylov数关联函数，并提出带电扩展探针的复杂度自然分解为集体、涨落、荷及混合贡献。该分解为构建真正场论的多种子全息复杂度开辟了具体路径。
英文摘要
Holographic spread (Krylov) complexity relates the growth of a boundary state's complexity to the proper radial momentum of a probe falling into the bulk. Unitary evolution makes spread complexity an even function of time. We show that this requirement fails whenever a probe carries a conserved Noether charge and is described by its unreduced Lagrangian. The cure is simple and universal: passing to the Routhian of the fixed-charge sector restores the correct short-time behaviour of the complexity. We establish this prescription from first principles and test it across an extensive family of probes: charged particles, non-BPS D-branes with detuned tension and charge, branes excited along internal isometries, worldvolume gauge fields, a fluctuating D0-brane in AdS$_4\times \mathbb{CP}^3$ and fundamental strings combining winding with rotation in AdS$_5\times \mathrm{S}^5$ complemented by further examples in AdS$_3\times \mathrm{S}^3\times T^4$, ABJM, and the charged Anabalón-Ross background. We then translate these results into Krylov-chain data, extracting Lanczos coefficients and Krylov-number correlators, and propose that complexity for charged, extended probes organises naturally into collective, fluctuation, charge and mixed contributions. This decomposition opens a concrete path toward a genuinely field-theoretic, multi-seed construction of holographic complexity.
