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Register a new userPolar supermultiplets, Hermitian symmetric spaces and hyperkahler metrics
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We address the construction of four-dimensional N=2 supersymmetric nonlinear sigma models on tangent bundles of arbitrary Hermitian symmetric spaces starting from projective superspace. Using a systematic way of solving the (infinite number of) auxiliary field equations along with the requirement of supersymmetry, we are able to derive a closed form for the Lagrangian on the tangent bundle and to dualize it to give the hyperkahler potential on the cotangent bundle. As an application, the case of the exceptional symmetric space E_6/SO(10) times U(1) is explicitly worked out for the first time.
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We review the projective-superspace construction of four-dimensional N=2 supersymmetric sigma models on (co)tangent bundles of the classical Hermitian symmetric spaces.
We describe the relation between supersymmetric sigma-models on hyperkahler manifolds, projective superspace, and twistor space. We review the essential aspects and present a coherent picture with a number of new results.
In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a natural piecewise $ell^infty$ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural piecewise $ell^infty$ metric which is coarsely Helly. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. The only exception is the special linear group: if $n geq 3$ and $mathbb{K}$ is a local field, we show that $operatorname{SL}(n,mathbb{K})$ does not act properly and coboundedly on an injective metric space.
We formulate four-dimensional $mathcal{N} = 1$ supersymmetric nonlinear sigma models on Hermitian symmetric spaces with higher derivative terms, free from the auxiliary field problem and the Ostrogradskis ghosts, as gauged linear sigma models. We then study Bogomolnyi-Prasad-Sommerfield equations preserving 1/2 or 1/4 supersymmetries. We find that there are distinct branches, that we call canonical ($F=0$) and non-canonical ($F eq 0$) branches, associated with solutions to auxiliary fields $F$ in chiral multiplets. For the ${mathbb C}P^N$ model, we obtain a supersymmetric ${mathbb C}P^N$ Skyrme-Faddeev model in the canonical branch while in the non-canonical branch the Lagrangian consists of solely the ${mathbb C}P^N$ Skyrme-Faddeev term without a canonical kinetic term. These structures can be extended to the Grassmann manifold $G_{M,N} = SU(M)/[SU(M-N)times SU(N) times U(1)]$. For other Hermitian symmetric spaces such as the quadric surface $Q^{N-2}=SO(N)/[SO(N-2) times U(1)])$, we impose F-term (holomorphic) constraints for embedding them into ${mathbb C}P^{N-1}$ or Grassmann manifold. We find that these constraints are consistent in the canonical branch but yield additional constraints on the dynamical fields thus reducing the target spaces in the non-canonical branch.
We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.
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