论文标题
Rozansky-Witten理论，本地化然后倾斜
Rozansky-Witten theory, Localised then Tilted
论文作者
论文摘要
该论文有两个部分，在第一部分中，我们将本地化技术应用于Rozansky-Witten理论，这些理论是紧凑型Hyperkähler目标。我们通过首先将理论重新设计为一些超对称的Sigma模型来做到这一点。我们在$ s^1 \timesς_g$和镜头空间上获得了与Wilson Loops的分区功能的确切公式，结果与使用K3上的Feynman图的早期计算相匹配。第二部分是由非常好奇的预印\ cite {gukov：2020LQM}的动机，其中Hilbert空间维度的模棱两可的索引公式该理论被解释为一种Verlinde公式。在这种解释中，目标超卡勒几何形状的固定点对应于某些“状态”。在本文的后半部分，我们扩展了第一部分的形式主义，以纳入目标几何形状。对于某些非紧凑的Hyperkähler几何形状，我们可以将倾斜理论应用于连贯的滑轮的派生类别，其对象的对象标记了Wilson循环，从而使我们可以为后者选择“基础”。然后，我们可以在此基础上计算融合产品，并表明具有对角线融合规则的对象与几何学的固定点密切相关。使用这些对象作为计算希尔伯特空间的维度的基础，将其返回到Verlinde公式，从而回答了激励论文的问题。
The paper has two parts, in the first part, we apply the localisation technique to the Rozansky-Witten theory on compact HyperKähler targets. We do so via first reformulating the theory as some supersymmetric sigma-model. We obtain the exact formula for the partition function with Wilson loops on $S^1\timesΣ_g$ and the lens spaces, the results match with earlier computations using Feynman diagrams on K3. The second part is motivated by a very curious preprint \cite{Gukov:2020lqm}, where the equivariant index formula for the dimension of the Hilbert space the theory is interpreted as a kind of Verlinde formula. In this interpretation, the fixed points of the target HyperKähler geometry correspond to certain 'states'. In the second half of the paper we extend the formalism of part one to incorporate equivariance on the target geometry. For certain non-compact hyperKähler geometry, we can apply the tilting theory to the derived category of coherent sheaves, whose objects label the Wilson loops, allowing us to pick a 'basis' for the latter. We can then compute the fusion products in this basis and we show that the objects that have diagonal fusion rules are intimately related to the fixed points of the geometry. Using these objects as basis to compute the dimension of the Hilbert space leads back to the Verlinde formula, thus answering the question that motivated the paper.
