The MFF Singular Vectors in Topological Conformal Theories

TL;DR

This paper demonstrates that singular vectors in topological conformal theories are equivalent to those in $sl(2)$ Kac--Moody algebra, providing a new formula for topological singular vectors and analyzing their behavior in free-field models.

Contribution

It establishes a direct correspondence between topological and $sl(2)$ singular vectors and derives a general expression for topological singular vectors based on the MFF formula.

Findings

Singular vectors in topological theories are identical to those in $sl(2)$ algebra.

Derived a general formula for topological singular vectors from MFF.

Topological singular states vanish in Witten's free-field construction.

Abstract

It is argued that singular vectors of the topological conformal (twisted $N=2$) algebra are identical with singular vectors of the $sl(2)$ Kac--Moody algebra. An arbitrary matter theory can be dressed by additional fields to make up a representation of either the $sl(2)$ current algebra or the topological conformal algebra. The relation between the two constructions is equivalent to the Kazama--Suzuki realisation of a topological conformal theory as $sl(2)\oplus u(1)/u(1)$. The Malikov--Feigin--Fuchs (MFF) formula for the $sl(2)$ singular vectors translates into a general expression for topological singular vectors. The MFF/topological singular states are observed to vanish in Witten's free-field construction of the (twisted) $N=2$ algebra, derived from the Landau--Ginzburg formalism.