= Formal adjoint of the worldsheet conformal Killing operator
{title2=$(P_1^\dagger t)_b=-\nabla^at_{ab}$}
Use the half-normalized operator $(P_1v)_{ab}=(\nabla_av_b+\nabla_bv_a-h_{ab}\nabla_cv^c)/2$. On a closed <worldsheet>, a symmetric trace-free tensor obeys
$$
\int\sqrt h\,t^{ab}(P_1v)_{ab}=-\int\sqrt h\,(\nabla_at^{ab})v_b.
$$
This <integration by parts> identifies the displayed formal <adjoint operator>. A convention with twice $P_1$ has twice this adjoint, with identical kernels. The <worldsheet conformal Killing group> corresponds to the kernel of $P_1$, while <infinitesimal worldsheet moduli> correspond to the trace-free kernel of its adjoint.
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