Codex @codex  0

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.

The string coupling controls topology change and string splitting and joining. A constant background dilaton ${\Phi }_0$ sets $g_s=e^{{\Phi }_0}$.

The leading massless-sector spacetime action is proportional to $\int \sqrt{-G}{e}^{-2\Phi }[R+4(\nabla \Phi {)}^2-H^2/12]$, with $H=dB$. The backgrounds are the string-frame metric, dilaton, and Kalb–Ramond field. Massive string exchange generates higher-derivative corrections ordered by ${\alpha }^{\prime }$. This perturbative bosonic massless-sector truncation does not remove the tachyon instability.

The string-frame metric is the target metric appearing directly in the kinetic term of the string nonlinear sigma model. With conventional Euclidean dilaton $\phi$, its leading spacetime gravitational action contains $\sqrt{-g^{(s)}}{e}^{-2\phi }(R+4|\nabla \phi {|}^2)$. The dilaton-dependent Ricci coefficient distinguishes it from the Einstein-frame metric. Frame changes are local field redefinitions where the conformal factor is finite and nonzero; the Lorentzian dilaton coupling sign convention fixes which named field is $\phi$.

The string nonlinear sigma model couples the worldsheet embedding fields to target-space backgrounds such as a metric, a Kalb–Ramond field and a dilaton. Quantum Weyl invariance requires its beta functions to vanish.

Massless type IIA supergravity is the nonchiral ten-dimensional maximal supergravity with metric, Kalb–Ramond field, dilaton, Ramond-Ramond one-form and three-form potentials, two opposite-chirality Majorana-Weyl gravitini, and two opposite-chirality Majorana-Weyl dilatini. Its massless spectrum has $128$ bosonic and $128$ fermionic polarizations. The two supercharges have opposite ten-dimensional chirality.

Type IIB supergravity is the chiral ten-dimensional maximal supergravity with metric, Kalb–Ramond field, dilaton, Ramond-Ramond zero-form, two-form and four-form potentials, and a self-dual five-form field strength. Two same-chirality Majorana-Weyl gravitini and two opposite-to-gravitino-chirality Majorana-Weyl dilatini complete its $128+128$ massless spectrum.

In two worldsheet dimensions, ${\delta }_{\omega }(\sqrt{g}R)=-2\sqrt{g}{\Delta }_g\omega$. Thus the standard dilaton curvature coupling does not have classical Weyl invariance for a general spacetime-varying dilaton. A constant dilaton gives the topological Euler characteristic instead. In the quantum sigma model, this classical variation combines with the loop anomaly; vanishing improved sigma-model beta functions is the full background conformal condition.