# IIB Supergravity from worldsheet (super)conformal invariance of Green-Schwarz string

How are low energy effective actions derived in string theory?

I began to wonder what is the coupling of the string to the other sugra fields. In almost all textbooks there is information how the string can be described in an arbitrary background with fields $$g_{\mu\nu}$$, $$b_{\mu\nu}$$ and $$\phi$$. Then, conditions for worldsheet conformal invariance give us beta functions for each field and the corresponding equations of motion for the metric, Kalb-Ramond field and dilaton.

So, if we want to get the equations of motion of IIB sugra, I guess we first need to have an action for the Green-Schwarz string in background fields: $$g_{\mu\nu}$$, $$b_{\mu\nu}$$ and $$\phi$$, RR forms $$F_1$$, $$F_3$$, $$F_5$$ and gravitini and dilatini.

But, as far as I know, the string cannot couple to RR-fields for instance.

The idea is to have that string action (in arbitrary IIB background) and then, form beta functions, obtain the equations of motion of all the background fields (bosonic and fermionic).

Has this route to the supergravity effective theory been taken in some paper? (if YES, could you sketch the procedure and give some reference)

Is this route imposible? perhaps just because the string doesn't couple to all sugra fields.

• I was reading your reference "$\beta$-FUNCTIONS FOR THE GREEN-SCHWARZ SUPERSTRING", but I cannot see the coupling to any gravitino $\psi_\mu$ nor dilatino $\lambda$. Could you clarify if there is any problem with the coupling of the worldsheet to the gravitino or dilatino? I'm interested in particular to the fermionic equations of motion. Apr 19, 2019 at 13:09
• The fermionic fields will always have zero expectation value since they are fermionic numbers, so they will not have a background value. But they are contained in the worldsheet action. You can see that if you pay attention to the fact that these formulas in the paper are in terms of superfields, so the superpartners of the bosonic fields could be obtained in the theta expansion. The first component of $E_m^{\alpha}$ for instance is the gravitino. Apr 19, 2019 at 14:39