# From gauged linear sigma models to geometric representation of $\mathbb{W}\mathbb{C}\mathbb{P}(N,\tilde{N})$ in 2D

## 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 $$\mathbb{WCP}(N,\tilde{N})$$ 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.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Minnesota, Minneapolis, MN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1582745

- Alternate Identifier(s):
- OSTI ID: 1802274

- Grant/Contract Number:
- SC0011842

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 101 Journal Issue: 2; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Astronomy & Astrophysics; Physics

### Citation Formats

```
Sheu, Chao-Hsiang, and Shifman, Mikhail. From gauged linear sigma models to geometric representation of W C P ( N , N ˜ ) in 2D. United States: N. p., 2020.
Web. doi:10.1103/PhysRevD.101.025007.
```

```
Sheu, Chao-Hsiang, & Shifman, Mikhail. From gauged linear sigma models to geometric representation of W C P ( N , N ˜ ) in 2D. United States. https://doi.org/10.1103/PhysRevD.101.025007
```

```
Sheu, Chao-Hsiang, and Shifman, Mikhail. Tue .
"From gauged linear sigma models to geometric representation of W C P ( N , N ˜ ) in 2D". United States. https://doi.org/10.1103/PhysRevD.101.025007.
```

```
@article{osti_1582745,
```

title = {From gauged linear sigma models to geometric representation of W C P ( N , N ˜ ) in 2D},

author = {Sheu, Chao-Hsiang and Shifman, Mikhail},

abstractNote = {In this paper two issues are addressed. First, we discuss renormalization properties of a class of gauged linear sigma models (GLSM), which reduce to $\mathbb{WCP}(N,\tilde{N})$ 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.},

doi = {10.1103/PhysRevD.101.025007},

journal = {Physical Review D},

number = 2,

volume = 101,

place = {United States},

year = {2020},

month = {1}

}

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

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