Expectation of 2-form field $B_{MN}$ in string theory

Question

In the context of string theory, in particular when we are dealing with a low energy effective action, if we have an effective action of the form,

$$S_{\mathrm{eff}} \sim S^{(0)} + \alpha S^{(1)} + \alpha^2 S^{(2)} + \ldots$$

where $\alpha$ is the Regge slope,

$$S^{(0)} \sim \int\limits \! d^Dx \, \, H_{MNP}H^{MNP},$$

and $H_{MNP} = \partial_M B_{NP} + \partial_N B_{PM} +\partial_P B_{MN}$. When we are asked to put $H_{MNP}=0$ to compute the list of terms corresponding to the tensor expectation relative to the field, what are we supposed to do exactly?

One more question: How can we find the field equations of the tensor to any order?

Answer 1

The effective action you are describing can be obtained in a given string theory by asking weyl invariance to hold when you put all the quantum corrections (trying to make it hold order by order of perturbation theory, in this case). Every quantum correction to the beta function of this field makes you go deeper in $\mathcal{O}((\alpha')^n)$. The worlsheet action that you use to this is $S_{polyakov}=\frac{1}{2\pi\alpha'}\int d^2 \sigma [\eta_{\mu\nu}\partial x^\mu \tilde \partial x^\nu+B_{\mu\nu}(\partial x^\mu \partial x^\nu+\tilde \partial x^\mu \tilde \partial x^\nu)]$, if I remember well.

Also, you can get it by studying what effective action leads you to the amplitudes given by the vaccum expectation values of a product of the allowed vertex operators, that are given in good part by conformal invariance.

What book or paper are you following?