• Corpus ID: 234357709

The resolution of Euclidean massless field operators of higher spins on $\Bbb R^6$ and the $L^2$ method

@inproceedings{Kang2021TheRO,
  title={The resolution of Euclidean massless field operators of higher spins on \$\Bbb R^6\$ and the \$L^2\$ method},
  author={Qianqian Kang and Wei Wang and Yuchen Zhang},
  year={2021}
}

• Qianqian Kang, Wei Wang, Yuchen Zhang
• Published 11 May 2021
• Mathematics

The resolution of 4-dimensional massless field operators of higher spins was constructed by Eastwood-Penrose-Wells by using the twistor method. Recently physicists are interested in 6dimensional physics including the massless field operators of higher spins on Lorentzian space R. Its Euclidean version D0 and their function theory are discussed in [14]. In this paper, we construct an exact sequence of Hilbert spaces as weighted L spaces resolving D0: L 2 φ(R 6 ,V0) D0 −→ L 2 φ(R 6 ,V1) D1 −→ L 2 φ(R 6 ,V2) D2 −→ L 2 φ(R 6 ,V3) −→ 0, with suitable operators Dl and vector spaces Vl. Namely, we can solve Dlu = f in L 2 φ(R ,Vl) when Dl+1f = 0 for f ∈ L 2 φ(R ,Vl+1). This is proved by using the L 2 method in the theory of several complex variables, which is a general framework to solve overdetermined PDEs under the compatibility condition. To apply this method here, it is necessary to consider weighted L spaces, an advantage of which is that any polynomial is Lφ integrable. As a corollary, we prove that P (R,V0) D0 −→ P (R,V1) D1 −→ P (R,V2) D2 −→ P (R,V3) −→ 0 is a resolution, where P (R,Vl) is the space of all Vl-valued polynomials. This provides an analytic way to construct a resolution of a differential operator acting on vector valued polynomials.