论文标题
较高等级K理论唐纳森 - 托马斯的观点
Higher rank K-theoretic Donaldson-Thomas theory of points
论文作者
论文摘要
我们在引号方案上利用关键的基因座结构$ \ MATHRM {quot} _ {\ Mathbb a^3}（\ Mathscr O^{\ oplus r}，n）$，特别是相关的对称性阻塞理论，以定义$ $ k theoretic donaldson-thomas-thomas novariant $ $ $ $ $ $ $ $ $ 33 a^3 $。我们将相关的分区函数计算为一个多种指数，证明了Awata-Kanno和Benini-Bonelli-Poggi-Tanzini在弦理论中提出的一种猜想。证明的关键步骤是，不变性不依赖于框架圆环$（\ Mathbb c^\ ast）^r $的均等参数。从K理论降低到共同体不变，我们计算了相应的DT不变性，证明了Szabo的猜想。为了进一步减少列举DT不变性，我们解决了一对$（x，f）$的更高等级的DT理论，其中$ f $是一个均等的extifial vector bundle，上面是投射的福利$ 3 $ fold $ x $。最后，我们给出了贝尼尼· - 尼里·帕吉·塔齐尼（Benini-Bonelli-Poggi-Tanzini）在物理学中研究的手性椭圆属的数学定义。这使我们能够定义$ \ mathbb a^3 $的椭圆形DT不变，并研究其第一个属性。
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.
