Moduli matrices of the vacua and walls on SO (2 N )/ U ( N )
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oduli Matrices of the Vacua and Walls on SO(2N)/U(N)1 M. Araia and Su. Shinb a
Institute of Experimental and Applied Physics, Czech Technical University in Prague, Horska 3a/22, 128 00 Prague 2, Czech Republic bBogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow region, 141980 Russia email: [email protected] Abstract—We construct parallel domain walls on the SO(2N)/U(N) manifold by using the moduli matrices, which were originally formulated on the Grassmann manifold. We propose a method to impose a quadratic constraint to the moduli matrices. This talk is based on [3]PRD83,125003(2011). This article is prepared for the proceedings of International Workshop on Supersymmetries and Quantum SymmetriesSQS 2011. DOI: 10.1134/S1063779612050024 1
We study parallel domain walls of the massive non linear sigma model (NLSM) on the SO(2N)/U(N) manifold by using the moduli matrices [1]. Discrete vacua can be induced by a mass term. The Bogo mol’nyi–Parasad–Sommerfield (BPS) solutions describe walls interpolating the vacua. We use the moduli matrices, which are the coefficients of the vacua and the BPS solutions. We show that the moduli matrices are on SO(2N)/U(N). We discuss the moduli matrices of domain walls.
We consider the case that fields are static and all the fields depend only on the x1 coordinate. We also take account of the Poincare invariance on the twodimen sional worldvolume of walls. The BPS equation can be derived from the Bogomol’nyi completion of the Hamiltonian as
The massive Lagrangian with four supersymmetries can be obtained in three dimensions by dimensional reduction of the ᏺ =1 massless NLSM [2] in four dimensions, which is Kahlerian. The Lagrangian is obtained by imposing an Fterm constraint to the Grassmann manifold G2N, N. We only consider the bosonic part of the Lagrangian:
Σ ba − iv ab = (S −1∂S )ab , φia = (S −1)ab f bi . The BPS solutions to (3) are
2
2
ᏸ bos 3D = − Dmφia − iφaj M ij − iΣ ab φib + Fai
2
+ 1 (Dabφb i φaa − Daa ) + ((F0 )ab φib J ij φaTj 2 + (φ0 )ab Fbi J ij φaTj + (φ0 )ab φib J ij FaTj + c.c.),
(1)
(6) S ' ba = VacS cb, H ' i0a = Vac H 0i c , where V ∈ GL(N, C). The V defines an equivalent class of the sets of the matrix functions and the moduli matrices (S, H0). This is called the worldvolume sym metry. The Dterm and Fterm constraints of the Lagrangian (1) become
The index m denotes threedimensional spacetime coordinates. The indices i, j are for flavor numbers (i, j = 1, …, 2N) and the indices a, b are for color num bers (a, b =1, …, N). J is an invariant tensor of O(2N) and M i j is the Cartan matrix of SO(2N). The compo nents mi (i = 1, …, N) are real and positive parameters with a condition mi > mi + 1.
(7)
(8) = 0. Equations (6) and (8) are the definition of SO(2N)/U(N). Thus the moduli matrices H0, which parameterize the moduli space of the vacua and domain walls are on SO(2N)/U(N). For N = 2 case, the moduli matrices for the vacua of (1) are obtained by the relation (5) as ⎛1 0 0 0⎞ ⎛0 1 0 0⎞ (9) H0 1 = ⎜ , H0 2 = ⎜ ⎟ ⎟. ⎝0 0 1 0⎠
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