发表机构
University of Southern California（南加州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在一族量子化K-理论库仑分支上构造量子簇结构，证明相关分次Koszul-偏反心给出其幺正范畴化，明确了初始种子变量数、量子簇代数性质及主箭图的突变类型。
AI 中文摘要
我们在一族量子化的K-理论库仑分支上构造量子簇结构，并证明相关的分次Koszul-偏反心给出这些代数的幺正范畴化。对每个r≥1，规范群是GL₂×(ℂ*)^r在GL((ℂ²)^r)中的像，其中(g,z₁,…,zᵣ)通过gzₐ作用在第a个直和项上。每个初始种子有r+2个可变变量和r个可逆冻结变量。在ℤ[v±¹]上，量子簇代数与其上量子簇代数重合，且同构于对应心的Grothendieck环。每个量子簇单项式由一个单对象表示，每个突变由一个短正合序列实现。证明使用与突变兼容的二次细化及产生修饰单极子生成元的显式突变序列。主箭图在r=4时具有四孔球面的突变类型，在r≥5时为突变无限型。
英文摘要
We construct quantum cluster structures on a family of quantized $K$-theoretic Coulomb branches and prove that the associated loop-graded Koszul-perverse hearts give monoidal categorifications of these algebras. For each $r\geq1$, the gauge group is the image of $GL_2\times(\mathbb{C}^*)^r$ in $GL((\mathbb{C}^2)^r)$, where $(g,z_1,\ldots,z_r)$ acts on the $a$th summand by $gz_a$. Each initial seed has $r+2$ mutable and $r$ invertible frozen variables. Over $\mathbb{Z}[v^{\pm1}]$, the quantum cluster algebra coincides with its upper quantum cluster algebra and is isomorphic to the Grothendieck ring of the corresponding heart. Every quantum cluster monomial is represented by a simple object, and every mutation is realized by a short exact sequence. The proof uses a quadratic refinement compatible with mutation and explicit mutation sequences that produce the dressed monopole generators. The principal quiver has the four-punctured-sphere mutation type for $r=4$ and is mutation-infinite for $r\geq5$.
