Graded Thread Modules over the Positive Part of the Witt (Virasoro) Algebra

We study $\mathbb{Z}$-graded thread $W^{+}$-modules $$V={\oplus }_i{V}_i,\dim V_i=1,-\mathrm{\infty }\le k<i<N\le +\mathrm{\infty },\dim V_i=0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},$$ over the positive part $W^{+}$ of the Witt (Virasoro) algebra $W$. There is well-known example of infinite-dimensional ($k=-\mathrm{\infty },N=\mathrm{\infty }$) two-parametric family $V_{\lambda ,\mu }$ of $W^{+}$-modules induced by the twisted $W$-action on tensor densities $P(x)x^{\mu}(dx)^{-\lambda}, \mu, \lambda \in {\mathbb K}, P(x) \in {\mathbb K}[t]$\mathrm{.}Anotherfamily$C_{\alpha, \beta}$of$W^+$-modules is defined by the action of two multiplicative generators $e_1,e_2$ of $W^{+}$ as $e_1{f}_i=\alpha f_{i+1}$ and $e_2f_j=\beta f_{j+2}$for$i,j \in {\mathbb Z}$and$\alpha, \beta$ are two arbitrary constants ($e_i{f}_j=0,i\ge 3$). We classify $(n+1)$-dimensional graded thread $W^{+}$-modules for $n$ sufficiently large $n$ of three important types. New examples of graded thread $W^{+}$-modules different from finite-dimensional quotients of $V_{\lambda ,\mu }$ and $C_{\alpha ,\beta }$ were found.