Worldsheet moduli (${\mathcal{M}}_{g,n}$)

Worldsheet moduli parameterize inequivalent conformal structures and marked insertion points after worldsheet diffeomorphisms and Weyl invariance are fixed. At each genus, string perturbation theory integrates over the corresponding moduli space.

The worldsheet conformal Killing group is the connected residual group of conformal automorphisms of the chosen compact Riemann surface. Its infinitesimal vectors satisfy the Conformal Killing equation, or $P_{1}v=0$ for the worldsheet diffeomorphism ghost operator. Its complex dimensions are three on a sphere, one on a torus, and zero for genus at least two. The sphere group consists of Möbius transformations, while the torus's connected group is translation. Additional discrete automorphisms are handled by the moduli quotient rather than counted as continuous zero modes.

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.

The torus worldsheet path integral is invariant under the modular group acting on its complex parameter $\tau$. Integrating over the standard fundamental domain of the modular group counts each inequivalent torus once and excludes the short-proper-time ultraviolet region.