In this paper two issues are addressed. First, we discuss renormalization properties of a class of gauged linear sigma models (GLSM), which reduce to $\mathbb{W}\mathbb{C}\mathbb{P}(\mathit{N},\tilde{\mathit{N}})$ nonlinear sigma models (NLSM) in the low-energy limit. Sometimes they are referred to as the Hanany-Tong models. If supersymmetry is $\mathcal{N}=(2,2)$ 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 $N=\tilde{N}$ 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 $\mathcal{N}=(0,2)$ GLSMs considered previously. In this case renormalization is not limited to one loop; all orders exact $\beta$ functions for GLSMs are known. We discuss logarithmically divergent loops at one- and two-loop levels.

• Received 30 October 2019

DOI:https://doi.org/10.1103/PhysRevD.101.025007

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Published by the American Physical Society