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.
Highlights
In 1979 Witten suggested [1] an ultraviolet (UV) completion for CPðN − 1Þ, one of the most popular nonlinear sigma models (NLSM), with the aim of large-N solution of the latter
Unlike the N 1⁄4 ð2; 2Þ case in the (0,2) models the second loop does not vanish in generic cases, resulting in “new” structures. (In those special cases when it does, the third and higher loops do not vanish.) In this paper we address the issue of renormalization group (RG) running in the parentdaughter pairs gauged linear sigma models (GLSM)/NLSM for such target spaces
The question we address now is the relation between two results: Eq (5) in GLSM and Eq (15) in NLSM
Summary
In 1979 Witten suggested [1] an ultraviolet (UV) completion for CPðN − 1Þ, one of the most popular nonlinear sigma models (NLSM), with the aim of large-N solution of the latter He considered both nonsupersymmetric and N 1⁄4 ð2; 2Þ versions. Of special importance is the case in which the number of positive charges N is equal to that of the negative charges N .4 In such GLSMs the FayetIliopoulos parameter is not renormalized [assuming N 1⁄4 ð2; 2Þ]. (In those special cases when it does, the third and higher loops do not vanish.) In this paper we address the issue of RG running in the parentdaughter pairs GLSM/NLSM for such target spaces. The number of emergent structures grows in higher loops in the nonsupersymmetric case [see (66)], so that these NLSMs are not renormalizable in the conventional sense of this word. VII we work out the N 1⁄4 ð0; 2Þ versions of the WCPðN; N Þ models
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