In this paper two issues are addressed. First, we discuss renormalization properties of a class of gauged linear sigma models (GLSM), which reduce to WCP(N,Ñ) nonlinear sigma models (NLSM) in the low-energy limit. Sometimes they are referred to as the Hanany-Tong models. If supersymmetry is N=(2,2) the ultraviolet-divergent logarithm in GLSM appears, in the renormalization of the Fayet-Iliopoulos parameter, and is exhausted by a single tadpole graph. This is not the case in the daughter NLSMs. As a result, the one-loop renormalizations are different in GLSMs and their daughter NLSMs. We explain this difference and identify its source. In particular, we show why at N=Ñ there is no UV logarithm in the parent GLSM, while they do appear in the corresponding NLSM. In the second part of the paper we discuss the same problem for a class of N=(0,2) GLSMs considered previously. In this case renormalization is not limited to one loop; all orders exact β functions for GLSMs are known. We discuss logarithmically divergent loops at one- and two-loop levels.
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The authors are grateful to Jin Chen, Sasha Gorsky, Sergey Ketov, Andrei Losev, David Tong, and Alexey Yung for very useful discussions and correspondence. This work is supported in part by DOE Grant No. DE-SC0011842.
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