Simple weak modules for the fixed point subalgebra of the Heisenberg vertex operator algebra of rank $1$ by an automorphism of order $2$ and Whittaker vectors

Abstract

Let $M(1)$ be the vertex operator algebra with the Virasoro element $\omega$ associated to the Heisenberg algebra of rank $1$ and let $M(1{)}^{+}$ be the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of order $2$. We classify the simple weak $M(1{)}^{+}$-modules with a non-zero element $w$ such that for some integer $s\ge 2$, ${\omega }_{i}w\in \mathbb{C}w$ ($i=\lfloor s/2\rfloor +1,\lfloor s/2\rfloor +2,\dots ,s-1$), ${\omega }_{s}w\in {\mathbb{C}}^{\times }w$, and ${\omega }_{i}w=0$ for all $i>s$. The result says that any such simple weak $M(1{)}^{+}$-module is isomorphic to some simple weak $M(1)$-module or to some $\theta$-twisted simple weak $M(1)$-module.

20 pages; typos corrected

References in corpus (2)

• Asymptotically free N=2 theories and irregular conformal blocks
• Whittaker modules for the affine Lie algebra $A_1^{(1)}$

Cited by in corpus (10)

• Simple Whittaker modules over free bosonic orbifold vertex operator algebras
• Whittaker modules for a subalgebra of N=2 superconformal algebra
• On irreducibility of modules of Whittaker type for cyclic orbifold vertex algebra
• The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order $2$ (Part $1$)
• The Heisenberg-Virasoro Lie conformal superalgebra
• Simple weak modules for some subalgebras of the Heisenberg vertex algebra and Whittaker vectors
• Representations of the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order $2$
• Simple restricted modules over the Heisenberg-Virasoro algebra as VOA modules
• Simple restricted modules for Neveu-Schwarz algebra
• The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order $2$