Vafa and Witten predicted that the partition functions of topologically twisted N=4 supersymmetric Yang-Mills theory on a four-manifold transform modularly under S-duality, exchanging a gauge group G with its Langlands dual LG. For a smooth projective surface S and G=SU(r), Tanaka and Thomas gave an algebro-geometric definition of Vafa-Witten invariants by virtual localization on moduli spaces of Higgs pairs (E, φ) with fixed determinant and trace-free Higgs field. The spectral construction identifies these Higgs pairs with compactly supported sheaves on the local Calabi-Yau threefold KS , making the deformation theory closely analogous to Donaldson-Thomas theory. On the Langlands-dual side, Jiang introduced twisted Vafa-Witten invariants using μ r -gerbes and twisted Higgs sheaves, and Jiang-Kool reformulated the theory using Yoshioka's twisted sheaves and rational B-fields. I will sketch these constructions, explain how SU(r)/ℤr partition function is obtained by summing over discrete fluxes, and then sketch basic calculations for K3 surfaces.

报告人简介：

普罗米特·昆杜，上海科技大学博士后。他于印度贾达普尔大学获得数学学士学位，后于印度统计糖心视频 取得数学硕士学位，并于美国堪萨斯大学获得博士学位。其主要研究领域为代数几何，尤其关注 Deligne-Mumford 栈上的模空间理论、Bogomolov–Gieseker 不等式、Vafa-Witten 不变量以及 BPS 不变量等前沿课题。他与合作者已在《Mathematische Zeitschrift》《Pure and Applied Mathematics Quarterly》《European Journal of Mathematics》等国际学术期刊发表多篇论文。