Polyakov action with Kalb-Ramond field defined globally?

In string theory, with the addition of the anti-symmetric $$B$$-field, the Polyakov action takes the form: $$S=\frac{1}{4 \pi \alpha^{'}} \int_{\sum}d \sigma d \tau (\cdots + \epsilon^{\alpha \beta} B_{\mu \nu} \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} + \cdots).$$ Now the $$B$$-field is a locally defined 2-form, that is not necessarily globally defined, however the associated $$H$$-flux given by $$H=dB$$ is, and the $$B$$-field transforms as higher dimensional connection form, known as a gerbe.

I don't know how to make sense of the above integral given the $$B$$-field is only locally defined, but the integral goes over all of the worldsheet. How can one make sense of a globally defined integral over locally defined objects, where these objects don't transform in the required way (they transform as gerbes not tensors) for the integral to make global sense.?

• What do you mean say that B-field transform as gerbes? B-field is Lorentz tensor in indeces $\mu, \nu$ and worldsheet scalar . Dec 28 '19 at 11:27
• this is a good question. see Alvarez, O., inspirehep.net/record/206164?ln=en Jan 6 '20 at 14:45

This $$B$$-term is equivalent to the $$n\Gamma$$ term in WZW sigma models. You can think of this integral as a 3-dimensional integral of $$H$$ in $$V$$, where $$\partial V=\Sigma$$, being $$\Sigma$$ the worldsheet
$$\oint_{\Sigma} B = \int_{V} H$$
since $$H$$ is globally defined all your worries should disappear. Dirac quantization for the NS-NS fluxes guarantee that physics will not be sensitive to how you extend the worldsheet fields from $$\Sigma$$ to $$V$$.