Three-Pronged Junctions on SO (2 N )/ U ( N ) and Sp ( N )/ U ( N )
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HYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
Three-Pronged Junctions on SO(2N)/U(N) and Sp(N)/U(N) T. Kima, * and Su. Shina, ** a
Department of Physics and Institute of Basic Science, Sungkyunkwan University, Suwon, 16419 Republic of Korea *e-mail: [email protected] **e-mail: [email protected] Received November 15, 2019; revised January 15, 2020; accepted February 28, 2020
Abstract—We report on the results of our work [1], which discusses 1 = 2 nonlinear sigma models on the quadrics of the Grassmann manifold and three-pronged junctions of the mass-deformed nonlinear sigma models on SO(8)/U (4) and Sp(3)/U (3) . This article is prepared for the Proceedings of International Workshop “Supersymmetries and Quantum Symmetries—SQS’2019”, which was held in Yerevan from 26 to 31 August, 2019. The talk was based on [1]. DOI: 10.1134/S1547477120050222
It is shown in [2] that we can construct threepronged junctions of the mass-deformed nonlinear sigma model on the cotangent bundle of the Grassmann manifold in the moduli matrix formalism without using the Plücker embedding. We can apply the method to mass-deformed nonlinear sigma models on the other Hermitian symmetric spaces. The work [1] discussed 1 = 2 nonlinear sigma models on the quadrics of the Grassmann manifold and three-pronged junctions of the mass-deformed nonlinear sigma models. The hyper-Kähler nonlinear sigma model on the cotangent bundle of the Grassmann manifold N +M ) , [3–5] is T * GN + M ,M , GN + M ,M = SU (NSU)×(SU (M )×U (1)
S = d4x
{ d θ [Tr (ΦΦe 4
V
Φ J ΦT = 0, 0 IN J = , eI N 0
2
(3)
1, for SO(2N )/U (N ) (4) e= −1, for Sp(N )/U (N ).
The parametrisation (2) shows that the constraint (3), which is f T + ef = 0 , ensures that Ψ obeys
+ ΨΨ e −V − cV )]
+ d θ [Tr ( Ξ ( ΦΨ − bI M ))
}
manifold G2N,N. The Kähler nonlinear sigma models on SO(2N )/U (N ) and Sp(N )/U (N ) are embedded to the Kähler nonlinear sigma model on G2N ,N [6], which corresponds to (1) with Ψ = 0, by
ΨT J Ψ = 0. (1)
+ (conjugate transpose)] , (c ∈ R ≥0, b ∈ C). Chiral fields Φ and ΨT are M × (N + M ) matrices. Vector field V and complex field Ξ are M × M matrices. There are two cases b = 0 and b ≠ 0 , which are related by the SU (2)R symmetry. We study the b = 0 case in this paper. The superfields in this case can be parametrised [5] by
− fg (2) Φ = (I M f ) , Ψ = . g We can show that f lives on the base Grassmann manifold whereas g lives on the cotangent space [3, 5]. The Hermitian symmetric spaces SO(2N )/U (N ) and Sp(N )/U (N ) are submanifolds of the Grassmann
(5)
Therefore the only consistent 1 = 2 nonlinear sigma models on SO(2N )/U (N ) and Sp(N )/U (N ) for b = 0 is obtained by imposing constraints (3) and (5) on the model (1) with b = 0 [1]. The mass-deformed nonlinear sigma models [1] are obtained by adding the potential terms that are discussed in [5]:
S = d x 4
{ d θ [Tr (ΦΦe 4
V
+ ΨΨ e
−V
+ d θ Tr ( ΞΦΨ + Φ M Ψ
− cV )]
2
}
(6)
+ Φ 0Φ J ΦT + Ψ 0 ΨT J Ψ ) + (c.t.) . The models (6) have discret
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