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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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Volume 2018, Issue 740

# Analysis of gauged Witten equation

Gang Tian
• Beijing International Center for Mathematical Research, Peking University, 100871 Beijing, P. R. China; and Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA
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/ Guangbo Xu
• Department of Mathematics, University of California, Irvine, 340 Rowland Hall, Irvine, CA 92697, USA; and Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA
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Published Online: 2015-11-04 | DOI: https://doi.org/10.1515/crelle-2015-0066

## Abstract

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.

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Revised: 2015-07-09

Published Online: 2015-11-04

Published in Print: 2018-07-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1309359

G.T. is supported by NSF grant DMS-1309359 and an NSFC grant. G.X. is supported by AMS-Simons Travel Grant.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 740, Pages 187–274, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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