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

We study ${\mathbb Z}$-graded thread $W^+$-modules $$V=\oplus_i V_i, \; \dim{V_i}=1, -\infty \le k< i < N\le +\infty, \; \dim{V_i}=0, \; {\rm \; otherwise},$$ over the positive part $W^+$ of the Witt (Virasoro) algebra $W$. There is well-known example of infinite-dimensional ($k=-\infty, N=\infty$) two-parametric family $V_{\lambda, \mu}$ of $W^+$-modules induced by the twisted $W$-action on tensor densities $P(x)x^{\mu}(dx)^{-\lambda}, \mu, \lambda \in {\mathbb K}, P(x) \in {\mathbb K}[t]$. Another family $C_{\alpha, \beta}$ of $W^+$-modules is defined by the action of two multiplicative generators $e_1, e_2$ of $W^+$ as $e_1f_i=\alpha f_{i+1}$ and $e_2f_j=\beta f_{j+2}$ for $i,j \in {\mathbb Z}$ and $\alpha, \beta$ are two arbitrary constants ($e_if_j=0, i \ge 3$). 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.