TY - JOUR

T1 - Analysis of gauged Witten equation

AU - Tian, Gang

AU - Xu, Guangbo

N1 - Funding Information:
G.T. is supported by NSF grant DMS-1309359 and an NSFC grant. G.X. is supported by AMS-Simons Travel Grant.
Publisher Copyright:
© 2015 De Gruyter.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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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U2 - 10.1515/crelle-2015-0066

DO - 10.1515/crelle-2015-0066

M3 - Article

AN - SCOPUS:84962944520

VL - 2015

SP - 187

EP - 274

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

ER -