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

Abstract

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 $6$-dimensional physics including the massless field operators of higher spins on Lorentzian space ${\mathbb{R}}^{5,1}$. Its Euclidean version ${\mathcal{D}}_0$ and their function theory are discussed in \cite{wangkang3}. In this paper, we construct an exact sequence of Hilbert spaces as weighted $L^2$ spaces resolving ${\mathcal{D}}_0$: $$L_{\phi }^2({\mathbb{R}}^6,{\mathcal{V}}_0)\stackrel{{\mathcal{D}}_0}{⟶}{L}_{\phi }^2({\mathbb{R}}^6,{\mathcal{V}}_1)\stackrel{{\mathcal{D}}_1}{⟶}{L}_{\phi }^2({\mathbb{R}}^6,{\mathcal{V}}_2)\stackrel{{\mathcal{D}}_2}{⟶}{L}_{\phi }^2({\mathbb{R}}^6,{\mathcal{V}}_3)⟶0,$$ with suitable operators ${\mathcal{D}}_l$ and vector spaces ${\mathcal{V}}_l$. Namely, we can solve ${\mathcal{D}}_{l}u=f$ in $L_{\phi }^2({\mathbb{R}}^6,{\mathcal{V}}_l)$ when ${\mathcal{D}}_{l+1}f=0$ for $f\in L_{\phi }^2({\mathbb{R}}^6,{\mathcal{V}}_{l+1})$. 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 $L^2$ spaces, an advantage of which is that any polynomial is $L_{\phi }^2$ integrable. As a corollary, we prove that $$P({\mathbb{R}}^6,{\mathcal{V}}_0)\stackrel{{\mathcal{D}}_0}{⟶}P({\mathbb{R}}^6,{\mathcal{V}}_1)\stackrel{{\mathcal{D}}_1}{⟶}P({\mathbb{R}}^6,{\mathcal{V}}_2)\stackrel{{\mathcal{D}}_2}{⟶}P({\mathbb{R}}^6,{\mathcal{V}}_3)⟶0$$ is a resolution, where $P({\mathbb{R}}^6,{\mathcal{V}}_l)$ is the space of all ${\mathcal{V}}_l$-valued polynomials. This provides an analytic way to construct a resolution of a differential operator acting on vector valued polynomials.

23 pages