# 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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Web of Science

Works referenced in this record:

##
Phases of N = 2 theories in two dimensions

journal, August 1993

- Witten, Edward
- Nuclear Physics B, Vol. 403, Issue 1-2

##
Vortex Strings and Four-Dimensional Gauge Dynamics

journal, April 2004

- Hanany, Amihay; Tong, David
- Journal of High Energy Physics, Vol. 2004, Issue 04

##
Effective world-sheet theory for non-Abelian semilocal strings in $\mathcal{N}\mathbf{=}2$ supersymmetric QCD

journal, June 2011

- Shifman, M.; Vinci, W.; Yung, A.
- Physical Review D, Vol. 83, Issue 12

##
Three-loop β-function for the bosonic non-linear sigma model

journal, November 1987

- Graham, S. J.
- Physics Letters B, Vol. 197, Issue 4

##
Heterotic flux tubes in $\mathcal{N}=2$ supersymmetric QCD with $\mathcal{N}=1$ preserving deformations

journal, June 2008

- Shifman, M.; Yung, A.
- Physical Review D, Vol. 77, Issue 12

##
Instatons, the quark model, and the 1/N expansion

journal, March 1979

- Witten, Edward
- Nuclear Physics B, Vol. 149, Issue 2

##
Composite non-Abelian strings with Grassmannian models on the worldsheet

journal, September 2019

- Ireson, Edwin; Shifman, Mikhail; Yung, Alexei
- Physical Review Research, Vol. 1, Issue 2

##
Remarks on the Novikov-Shifman-Vainshtein-Zakahrov β functions in two-dimensional N = ( 0 , 2 ) supersymmetric models

journal, March 2019

- Chen, Jin; Shifman, Mikhail
- Physical Review D, Vol. 99, Issue 6

##
Exact Gell-Mann-Low function of supersymmetric Kähler sigma models

journal, December 1984

- Morozov, Alexei Y.; Perelomov, Askold M.; Shifman, Michael A.
- Nuclear Physics B, Vol. 248, Issue 2

##
Heterotic vortex strings

journal, May 2007

- Edalati, Mohammad; Tong, David
- Journal of High Energy Physics, Vol. 2007, Issue 05

##
On isometry anomalies in minimal 𝒩 = (0,1) and 𝒩 = (0,2) sigma models

journal, September 2016

- Chen, Jin; Cui, Xiaoyi; Shifman, Mikhail
- International Journal of Modern Physics A, Vol. 31, Issue 27

##
Quantum dynamics of low-energy theory on semilocal non-Abelian strings

journal, September 2011

- Koroteev, P.; Shifman, M.; Vinci, W.
- Physical Review D, Vol. 84, Issue 6

##
$\mathcal{N}\mathbf{=}(0,2)$ deformation of the $CP(1)$ model: Two-dimensional analog of $\mathcal{N}\mathbf{=}1$ Yang-Mills theory in four dimensions

journal, February 2012

- Cui, Xiaoyi; Shifman, M.
- Physical Review D, Vol. 85, Issue 4