Polyakov path integral

The Polyakov path integral integrates over string embeddings and worldsheet metrics, divided by worldsheet diffeomorphisms and Weyl transformations. Gauge fixing introduces worldsheet ghosts and moduli integrals.

The Polyakov path integral integrates over string worldsheet metrics as well as embedding fields, modulo worldsheet diffeomorphisms and Weyl transformations. Summing surface topologies gives perturbative two-dimensional quantum gravity coupled to the coordinate matter fields. Gauge fixing retains worldsheet moduli and introduces ghosts; a conformal gauge is locally flat, not a global flat metric on every genus. Cancellation of the worldsheet Weyl anomaly makes the critical bosonic version consistent.

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.

A closed-string scattering amplitude is a worldsheet correlation function of external string vertex operators, integrated over their unfixed positions and worldsheet moduli, with the necessary worldsheet ghost fields.

For a stable $n$-point closed-string scattering amplitude on a genus-$g$ surface, let $d_g$ count complex unmarked worldsheet moduli and $k_g$ count complex dimensions of the worldsheet conformal Killing group. Integrating the $n$ vertex positions and moduli, then fixing $k_g$ positions, leaves $d_g+n-k_g$ complex integrations. The punctured Riemann surface moduli dimension identifies this as $3g-3+n$. Faddeev-Popov ghost fields and antighost insertions implement the removed zero modes and the surviving modulus measure; they do not add independently integrated worldsheet coordinates.

The tree-level Polyakov path integral for closed-string tachyons reduces in conformal gauge to their vertex-position integral on the Riemann sphere. Three positions are fixed by the residual Möbius group; the other $N-3$ complex positions are integrated. The embedding-field Gaussian integral produces momentum conservation and the Koba-Nielsen factor.

At genus zero the residual Möbius transformation group fixes three marked insertion points, leaving one complex position to integrate. The corresponding worldsheet ghost fields give $|z_{ij}{z}_{jk}{z}_{ki}{|}^2$ for the three fixed points. For on-shell weight-$(1,1)$ string vertex operators, these ghost factors cancel the coordinate dependence of the fixed-point normalization, including the limit at infinity. The remaining Koba-Nielsen factor for four tachyons is the integrand of the complex beta integral, producing the Virasoro–Shapiro amplitude.

A single crossing-symmetric scattering amplitude has equivalent pole expansions in different exchange channels. Its scattering-amplitude factorization reproduces the string spectrum and couplings in each channel without adding the channel expansions again.

A worldsheet Green function inverts the scalar kinetic operator after its zero modes have been removed. In a locally flat complex coordinate it satisfies $-(\pi {\alpha }^{\prime }{)}^{-1}\partial \bar{\partial }G(z,w)={\delta }^{(2)}(z-w)$ and is $G(z,w)=-({\alpha }^{\prime }/2)\log |z-w{|}^2$ up to harmonic and constant terms.

A string vertex operator inserts an asymptotic string state at a point of the worldsheet. Its conformal weights encode the physical-state and mass-shell conditions.

These insertions are described by the state-field correspondence of a vertex operator algebra; an individual insertion is not the whole algebra.

An integrated closed-string vertex is inserted as $\int d^{2}zV(z,\bar{z})$. In conformal gauge its matter operator must have conformal weights $(1,1)$ so that the insertion is invariant under worldsheet coordinate changes.

At the first massless closed-string level, the matter vertex has the form ${ϵ}_{\mu \nu }:\partial X^{\mu }\bar{\partial }{X}^{\nu }{e}^{ik\cdot X}:$, with null momentum and a polarization tensor transverse in both indices. Its symmetric trace-free, antisymmetric and trace sectors describe the graviton, Kalb–Ramond field and dilaton.

A closed-string polarization tensor specifies the target-spacetime tensor state multiplying $\partial X^{\mu }\bar{\partial }{X}^{\nu }{e}^{ik\cdot X}$. Its decomposition into symmetric trace-free, antisymmetric and trace parts separates the graviton, Kalb–Ramond field and dilaton polarizations.

Changing a massless closed-string polarization by a momentum-longitudinal tensor changes the integrated vertex by a worldsheet total derivative, equivalently by a BRST-exact state. The corresponding target-space transformations are linearized diffeomorphisms and two-form gauge transformations.

A string mass-shell condition relates the target momentum of a vertex operator to its oscillator level. Together with polarization constraints, it makes the matter vertex have the conformal weights required for an integrated insertion.

The bosonic closed-string tachyon vertex is the normal-ordered exponential $:e^{ip\cdot X}:$ integrated over its worldsheet position. Pairwise contractions of such exponentials generate the Koba-Nielsen factor.