Infinitesimal worldsheet modulus ($h^{ab}{t}_{ab}=0,{\nabla }^a{t}_{ab}=0$)

After quotienting infinitesimal worldsheet diffeomorphisms and Weyl transformations, choose a metric variation orthogonal to the gauge directions. Orthogonality to Weyl scaling makes its trace zero; the formal adjoint of the worldsheet conformal Killing operator makes its divergence zero. These are transverse-traceless variations of the two-dimensional metric. In a local conformal complex coordinate, their nonzero components satisfy ${\partial }_{\bar{z}}{t}_{zz}=0$ and its conjugate, linking them to holomorphic quadratic differentials.