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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.?

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

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