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

## TL;DR

This paper investigates the renormalization properties of gauged linear sigma models (GLSMs) and their low-energy non-linear sigma model (NLSM) limits, highlighting differences in UV divergences and renormalization behavior for models with different supersymmetries.

## Contribution

It clarifies the renormalization differences between GLSMs and NLSMs, especially for ${
m N}=(2,2)$ and ${
m N}=(0,2)$ supersymmetries, and explains the origin of UV divergences.

## Key findings

- UV logarithms vanish at N=tilde N in GLSMs
- One-loop renormalizations differ between GLSMs and NLSMs
- All-orders beta functions are discussed for N=(0,2) models

## Abstract

In this paper two issues are addressed. First, we discuss renormalization properties of a class of gauged linear sigma models (GLSM) which reduce to $\mathbb{WCP}(N,\tilde{N})$ non-linear sigma models (NLSM) in the low-energy limit. Sometimes they are referred to as the Hanany-Tong models. If supersymmetry is ${\cal N} =(2,2)$ the ultraviolet-divergent logarithm in LGSM 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 logarithms in the parent GLSM, while they do appear on the corresponding NLSM does not vanish. In the second part of the paper we discuss the same problem for a class of ${\cal N} =(0,2)$ GLSMs considered previously. In this case renormalization is not limited to one loop; all-orders exact $\beta$ functions for GLSMs are known. We discuss divergent loops at one and two-loop levels.