| Abstract: |
In this talk I will give an overview of Tanaka-Thomas's algebro-geometric formulation of Vafa-Witten theory for complex surfaces. We will explain the basic set-up: counting trace-free Higgs pairs (E,\varphi) with a C^*-action scaling \varphi, the resulting symmetric obstruction theory, and the use of virtual localization to define the invariants (including the treatment of strictly semistable objects). As a concrete illustration, we specialise to ruled surfaces S=\mathbb{P}(\mathcal{O}*C\oplus L), showing how Higgs data descends to the base curve via \pi* and how this aids stability and deformation calculations. The goal is to provide the core ideas and tools needed to start computing in examples, with ruled surfaces as the running example. This is an ongoing work with Amin Gholampour.
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