AI 中文总结
该研究针对Hirzebruch曲面上的精炼SU(N) VW不变量，通过简化模反常方程、计算模完备化变分、利用不定θ级数推导壁处不变量，得到了具有通用性的相关结果。
AI 中文摘要
我们证明了关于Hirzebruch曲面上的精炼SU(N) VW不变量的三个结果。首先，我们通过将生成级数满足的模反常方程用互补广义误差函数表示，显著简化了该方程，这需要将这些误差函数延拓到其奇异轨迹。其次，我们计算了生成级数的模完备化在极化变化下的变分。最后，利用不定θ级数表示生成函数，我们可以计算边际稳定性壁处的VW不变量，并将其表示为壁两侧的不变量以及低秩不变量的组合。所有结果均展现出通用结构，表明其具有更广泛的适用性。
英文摘要
We prove three results about refined $SU(N)$ VW invariants on Hirzebruch surfaces. First, we crucially simplify the modular anomaly equation satisfied by the generating series of these invariants by expressing it in terms of complementary generalized error functions, which requires extending these error functions to their singular loci. Second, we evaluate the variation of the modular completions of the generating series upon change of polarization. Finally, a representation of the generating functions in terms of indefinite theta series allows us to compute VW invariants at the walls of marginal stability, and to express them in terms of the invariants on the two sides of the wall and those of lower ranks. All the results exhibit a universal structure suggesting their wider applicability.
Comments25+12 pages
