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In this paper two issues are addressed. First, we discuss renormalizationproperties of a class of gauged linear sigma models (GLSM) which reduce to$\mathbb{WCP}(N,\tilde{N})$ non-linear sigma models (NLSM) in the low-energylimit. Sometimes they are referred to as the Hanany-Tong models. Ifsupersymmetry is ${\cal N} =(2,2)$ the ultraviolet-divergent logarithm in LGSMappears, in the renormalization of the Fayet-Iliopoulos parameter, and isexhausted by a single tadpole graph. This is not the case in the daughterNLSMs. As a result, the one-loop renormalizations are different in GLSMs andtheir daughter NLSMs We explain this difference and identify its source.  In particular, we show why at $N=\tilde N$ there is no UV logarithms in theparent GLSM, while they do appear on the corresponding NLSM does not vanish. Inthe second part of the paper we discuss the same problem for a class of ${\calN} =(0,2)$ GLSMs considered previously. In this case renormalization is notlimited to one loop; all-orders exact $\beta$ functions for GLSMs are known. Wediscuss divergent loops at one and two-loop levels.

From Gauged Linear Sigma Models to Geometric Representation of $\mathbb{WCP}(N,\tilde{N})$ in 2D