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arXiv 2003.13565math.AGhep-thmath-phmath.MP
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高阶K理论Donaldson-Thomas点论

Higher rank K-theoretic Donaldson-Thomas theory of points

Nadir Fasola, Sergej Monavari, Andrea T. Ricolfi

备注：
v2. Replaced appendix with Theorem 6.5. Minor changes and corrections following referee's comments. 50 pages. Accepted for publication in Forum Math. Sigma
发表信息：
Forum of Mathematics, Sigma 9 (2021) e15
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摘要：

我们利用关键轨迹结构的报价计划$\mathrm{Quot}_{\mathbb A^3}(\mathscr O^{\oplus r},n)$，特别是相关的对称阻塞理论，以定义秩$r$ K理论唐纳森-托马斯不变量的卡拉比-丘$3$折叠$\mathbb A^3$。我们计算了相关的分拆函数作为一个plethystic指数，证明了由Awata Kanno和Benini-Bonelli-Poggi-Tanzini在弦理论中提出的一个猜想。证明中的一个关键步骤是不变量不依赖于框架环面$(\mathbb C^\ast)^r$的等变参数。从K-理论的上同调不变量，我们计算相应的DT不变量，证明了Szabo的一个猜想。进一步减少计数DT不变量，我们解决了一对$(X,F)$，其中$F$是一个等变例外向量丛上的一个投影复曲面$3$倍$X$的高阶DT理论。最后给出了Benini-Bonelli-Poggi-Tanzini在物理学中研究的手征椭圆亏格的数学定义。这使我们能够定义椭圆DT不变量的$\mathbb A^3$在任意秩，并研究其第一性质。

英文摘要：

We exploit the critical locus structure on the Quot scheme $\mathrm{Quot}_{\mathbb A^3}(\mathscr O^{\oplus r},n)$, in particular the associated symmetric obstruction theory, in order to define rank $r$ K-theoretic Donaldson-Thomas invariants of the 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 that the invariants do not depend on the equivariant parameters of the framing torus $(\mathbb C^\ast)^r$. Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$, where $F$ is an equivariant exceptional vector bundle on a projective toric $3$-fold $X$. Finally, we give a mathematical definition of the chiral elliptic genus studied in physics by Benini-Bonelli-Poggi-Tanzini. This allows us to define elliptic DT invariants of $\mathbb A^3$ in arbitrary rank, and to study their first properties.

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