${\mathrm{\Phi }}_r$ are therefore half-integer moded for $w$ even and integer moded for $w$ odd. The surviving wedge modes behave as generalized zero modes (see the “Comments” paragraph at the end of the Letter) and thus only one copy of them survives in the gauge fixed theory. (Before gauge fixing, we have independent left- and right-moving oscillators.)

We also have ${\mathcal{C}}_n|0{⟩}_w=L_0|0{⟩}_w=0$ for $n>0$. As a consequence, $|0{⟩}_w$ is a physical state.

Note that the $†$ in ${\mu }^{†}$, ${\lambda }^{†}$, etc., does not refer to the complex or Hermitian conjugate; these are independent oscillators.

Our convention differs relative to [7, 8], where we set $|0{⟩}_w=[|0⟩{]}^{(w)}$ with action ${\lambda }_r^{\alpha }[|0⟩{]}^{(w)}=[{\sigma }^w({\lambda }_r^{\alpha })|0⟩{]}^{(w)}$. Thus, the convention of spectral flow below corresponds to ${\sigma }^w({\lambda }_r^{\alpha })={\lambda }_{r+(w/2)}^{\alpha }$ and similarly for the other modes.

In terms of the scalar field ${\varphi }^i$ of supersymmetric Yang-Mills theory, this is the multiplet built from the completely symmetric traceless field $\mathrm{Tr}[{\varphi }^{(i_1}\cdots {\varphi }^{i_w)}]$ and its $\mathfrak{psu}(2,2|4)$ descendants.

A helpful parallel might be with the holomorphic A-model topological string, which nevertheless captures the (nonchiral) de Rham cohomology of the target space.