Abstract

Abstract We construct $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over the non-compact exceptional Hermitian symmetric spaces $$ \mathrm{\mathcal{M}}={E}_{6\left(-14\right)}/\mathrm{SO}(10)\times \mathrm{U}(1) $$ ℳ = E 6 − 14 / SO 10 × U 1 and E 7(−25) /E 6 × U(1). In order to construct them we use the projective superspace formalism which is an $$ \mathcal{N}=2 $$ N = 2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the $$ \mathcal{N}=1 $$ N = 1 superfields, once the Kähler potentials of the base manifolds are obtained. We derive the $$ \mathcal{N}=1 $$ N = 1 supersymmetric nonlinear sigma models on the Kähler manifolds $$ \mathrm{\mathcal{M}} $$ ℳ . Then we extend them into the $$ \mathcal{N}=2 $$ N = 2 supersymmetric models with the use of the result in arXiv:1211.1537 developed in the projective superspace formalism. The resultant models are the $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over the Hermitian symmetric spaces $$ \mathrm{\mathcal{M}} $$ ℳ . In this work we complete constructing the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces.

Highlights

  • Theories in the Higgs branch are the NLSMs on the hyperkahler manifolds [7, 8]

  • In order to construct them we use the projective superspace formalism which is an N = 2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the N = 1 superfields, once the Kahler potentials of the base manifolds are obtained

  • We extend them into the N = 2 supersymmetric models with the use of the result in arXiv:1211.1537 developed in the projective superspace formalism

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Summary

General case

The action is written by the function of the superfields representing the polar multiplets Υ and Υ , which are called an arctic superfield and an antarctic superfield respectively. The action (2.12) is written by the base manifold coordinate Φ and the tangent vector Σ This action represents the N = 2 SUSY model on the tangent bundle over the Kahler manifold. The rest of the work is to derive the Kahler potential of the cotangent bundle over the Kahler manifold It is carried out by changing the tangent vectors Σ’s in (2.12) into chiral one-forms, cotangent vectors Ψ’s. Where U is a complex unconstrained superfield and Ψ is a chiral superfield This action goes back to the tangent bundle action (2.12) after eliminating the chiral superfields Ψ and Ψby their equations of motion. The variables (ΦI , ΨJ ) parameterize the cotangent bundle over the Kahler manifold and the action gives the Kahler potential of the cotangent bundle over the Kahler manifold

HSS case
Kahler potential
Cotangent bundle
Conclusion
A Clifford algebra
Full Text
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