Gauge-invariant string coupling to a two-form ($q{\int }_{\Sigma }{X}^{\ast }B$)

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.