Two-form gauge field ($B\mapsto B+d\alpha$)

An antisymmetric tensor potential $B$ has gauge transformation $B\mapsto B+d\alpha$, where $\alpha$ is a one-form. Its field strength is the three-form $H=dB$, invariant because $d^2=0$. The Kalb–Ramond field is the string-theory example.

The coupling $q{\int }_{\Sigma }{X}^{\ast }B$ integrates the pullback of the two-form gauge field over the string worldsheet. Its gauge variation is $q{\int }_{\partial \Sigma }{X}^{\ast }\alpha$, by Stokes theorem. It therefore vanishes for a closed worldsheet, while an open surface requires suitable boundary fields or boundary-state transformations.