Analysis of gauged Witten equation

Gang Tian, Guangbo Xu

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


The gauged Witten equation was essentially introduced by Witten in his formulation of the gauged linear σ-model (GLSM), which explains the so-called Landau-Ginzburg/Calabi-Yau correspondence. This is the first paper in a series towards a mathematical construction of GLSM, in which we set up a proper framework for studying the gauged Witten equation and its perturbations. We also prove several analytical properties of solutions and moduli spaces of the perturbed gauged Witten equation. We prove that solutions have nice asymptotic behavior on cylindrical ends of the domain. Under a good perturbation scheme, the energies of solutions are shown to be uniformly bounded by a constant depending only on the topological type. We prove that the linearization of the perturbed gauged Witten equation is Fredholm, and we calculate its Fredholm index. Finally, we define a notion of stable solutions and prove a compactness theorem for the moduli space of solutions over a fixed domain curve.

Original languageEnglish (US)
Pages (from-to)187-274
Number of pages88
JournalJournal fur die Reine und Angewandte Mathematik
StatePublished - 2015

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


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