String nonlinear sigma model

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.

The Kalb–Ramond field is an antisymmetric two-form background coupled to the oriented worldsheet area element. Its gauge-invariant field strength is the three-form $H=dB$.

The H-field is the three-form field strength of a Kalb–Ramond field. It obeys the Bianchi identity $dH=0$ and contributes quadratically to the leading string background equations.

A sigma-model beta function measures the failure of a worldsheet background coupling to remain scale invariant. Vanishing metric, B-field and dilaton beta functions gives the spacetime field equations order by order in ${\alpha }^{\prime }$.