The resolution of Euclidean massless field operators of higher spins on $\Bbb R^6$ 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 $\Bbb R^{5,1}$. Its Euclidean version $\mathscr{D}_0$ 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 $\mathscr{D}_0$: $$L^2_\varphi(\Bbb R^6, \mathscr{V}_0)\overset{\mathscr{D}_0}\longrightarrow L^2_\varphi(\Bbb R^6,\mathscr{V}_1)\overset{\mathscr{D}_1}\longrightarrow L^2_\varphi(\Bbb R^6, \mathscr{V}_2)\overset{\mathscr{D}_2}\longrightarrow L^2_\varphi(\Bbb R^6, \mathscr{V}_3)\longrightarrow 0,$$ with suitable operators $\mathscr{D}_l$ and vector spaces $\mathscr{V}_l$. Namely, we can solve $\mathscr{D}_{l}u=f$ in $L^2_\varphi(\Bbb R^6, \mathscr{V}_{l})$ when $\mathscr{D}_{l+1} f=0$ for $f\in L^2_{\varphi}(\Bbb R^6, \mathscr{V}_{l+1})$. 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 $L^2_{\varphi}$ integrable. As a corollary, we prove that $$ P(\Bbb R^6, \mathscr{V}_0)\overset{\mathscr{D}_0}\longrightarrow P(\Bbb R^6,\mathscr{V}_1)\overset{\mathscr{D}_1}\longrightarrow P(\Bbb R^6, \mathscr{V}_2)\overset{\mathscr{D}_2}\longrightarrow P(\Bbb R^6, \mathscr{V}_3)\longrightarrow 0$$ is a resolution, where $P(\Bbb R^6, \mathscr{V}_l)$ is the space of all $\mathscr{V}_l$-valued polynomials. This provides an analytic way to construct a resolution of a differential operator acting on vector valued polynomials.