In the last posting, we derive some equations. Using these results as an exercise let's compute the equation of motion of low-energy effective action of string theory.
The equations of motion are following
Note H=dB which is totally anti-symmetric (note B is two form, anti-symmetric which we called Kalb-Ramond two form)
Variation of B
Let's do variation about B
Before that recall the definition of H
Simply one can write H=dB, and since B is two-form, anti-symmetric, we can write H explicitly as follows
Now do the variation
First parts comes from variation of H^2 and the second part we used
and the last line we used
Here we used the symmetric properteis of Connection and totally anti-symmetric properties of H
Thus
via
up to total derivatives
Thus we have
Variation about scalar Phi
Variation about scalar Phi we have
Here we used
Checking in terms of action
Recall the definition of beta function
which is nothing but
Thus
Variation about metric
First do variation about metric G
To obtain this equation we used
and
To proceed the computation using following equations
Specially
In the process we used following equation with vanishing total-derivatives terms
Now combing results we have
Thus
The first equation is nothing but equation of motion for Phi. Thus recall the definition of beta function we have
summary
By solving linear equations we have
