Constant Curvature Holomorphic
Solutions of the Supersymmetric G(2, 4)
Sigma Model
Véronique Hussin, Marie Lafrance, and ˙
Ismet Yurdu¸ sen
Abstract We explore the constant curvature holomorphic solutions of the supersymmetric Grassmannian sigma model G(M, N ) using in particular the gauge
invariance of the model. Supersymmetric invariant solutions are constructed by
generalizing a known result for CP N −1 . We show that other such solutions also
exist. Considering the simplest case of G(2, N) model, we give necessary and
sufficient conditions for getting the constant curvature holomorphic solutions. Since,
all the constant curvature holomorphic solutions of the non-supersymmetric G(2, 4)
sigma model are known, we treat this example in detail.
Keywords Supersymmetric · Grassmannian sigma model · Gauge invariance
PACS numbers: 12.60.Jv, 02.10.Ud, 02.10.Yn
1 Introduction
For a long time searching for exact solutions of integrable models has been a lively
subject of great interest to the mathematics and physics communities. In particular,
the integrable non-supersymmetric (non-susy) CP N −1 sigma model has found
many applications in physics [1–4]. The solutions of this model have also been used
to construct solutions of more general Grassmannian sigma models [5]. Another
extension of these models is to consider the supersymmetric (susy) generalizations.
The main motivation to study susy models is to include fermions into the theory
[6]. Although, there exist many ways of including fermions into the Grassmannian
models, the most interesting is the one which renders supersymmetry.
V. Hussin () · M. Lafrance
Centre de Recherches Mathématiques, Montréal, QC, Canada
e-mail: veronique.hussin@umontreal.ca; marie.lafrance@umontreal.ca
˙
I. Yurdu¸ sen
Hacettepe University, Ankara, Turkey
e-mail: yurdusen@hacettepe.edu.tr
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_8
91
Solutions of the Supersymmetric G(2, 4)
Sigma Model
Véronique Hussin, Marie Lafrance, and ˙
Ismet Yurdu¸ sen
Abstract We explore the constant curvature holomorphic solutions of the supersymmetric Grassmannian sigma model G(M, N ) using in particular the gauge
invariance of the model. Supersymmetric invariant solutions are constructed by
generalizing a known result for CP N −1 . We show that other such solutions also
exist. Considering the simplest case of G(2, N) model, we give necessary and
sufficient conditions for getting the constant curvature holomorphic solutions. Since,
all the constant curvature holomorphic solutions of the non-supersymmetric G(2, 4)
sigma model are known, we treat this example in detail.
Keywords Supersymmetric · Grassmannian sigma model · Gauge invariance
PACS numbers: 12.60.Jv, 02.10.Ud, 02.10.Yn
1 Introduction
For a long time searching for exact solutions of integrable models has been a lively
subject of great interest to the mathematics and physics communities. In particular,
the integrable non-supersymmetric (non-susy) CP N −1 sigma model has found
many applications in physics [1–4]. The solutions of this model have also been used
to construct solutions of more general Grassmannian sigma models [5]. Another
extension of these models is to consider the supersymmetric (susy) generalizations.
The main motivation to study susy models is to include fermions into the theory
[6]. Although, there exist many ways of including fermions into the Grassmannian
models, the most interesting is the one which renders supersymmetry.
V. Hussin () · M. Lafrance
Centre de Recherches Mathématiques, Montréal, QC, Canada
e-mail: veronique.hussin@umontreal.ca; marie.lafrance@umontreal.ca
˙
I. Yurdu¸ sen
Hacettepe University, Ankara, Turkey
e-mail: yurdusen@hacettepe.edu.tr
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_8
91