We exploit the critical structure on the Quot scheme$\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$, in particular the associated symmetric obstruction theory, in order to study rankr K-theoreticDonaldson-Thomas (DT) invariants of the local Calabi-Yau$3$-fold${{\mathbb {A}}}^3$. We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if$r>1$, that the invariants do not depend on the equivariant parameters of the framing torus$({{\mathbb {C}}}^\ast )^r$. Reducing from K-theoretic tocohomologicalinvariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further toenumerativeDT invariants, we solve the higher rank DT theory of a pair$(X,F)$, whereFis an equivariant exceptional locally free sheaf on a projective toric$3$-foldX.As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to defineelliptic DT invariantsof${{\mathbb {A}}}^3$in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

中文翻译：

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高阶 K 理论 Donaldson-Thomas 点理论

我们利用 Quot 方案的关键结构$\text {报价}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$，特别是相关的对称障碍理论，以研究秩r K-理论局部 Calabi-Yau 的 Donaldson-Thomas (DT) 不变量$3$-折叠${{\mathbb {A}}}^3$. 我们将相关的配分函数计算为体积指数，证明了 Awata-Kanno 和 Benini-Bonelli-Poggi-Tanzini 在弦理论中提出的猜想。证明中的一个关键步骤是事实，如果$r>1$，不变量不依赖于框架环面的等变参数$({{\mathbb {C}}}^\ast )^r$. 从 K 理论减少到上同调不变量，我们计算相应的 DT 不变量，证明了 Szabo 的猜想。进一步减少到枚举的DT 不变量，我们解决了一对的高阶 DT 理论$(X,F)$， 在哪里F是投影复曲面上的等变异常局部自由束$3$-折叠X. 作为对 K 理论 DT 不变量的进一步改进，我们制定了物理学中研究的手性椭圆属的数学定义。这允许我们定义椭圆 DT 不变量的${{\mathbb {A}}}^3$在任意等级中，我们用它来解决贝尼尼-博内利-波吉-坦齐尼的猜想。