# String Theory, Ghost SCFTs

In the section on free SCFTs in chapter 10, in equation (10.1.19) of Polchinski's volume 2, it isn't clear to me how he writes down the fermionic stress tensor. Shouldn't that correspond to a world sheet supersymmetry transformation? If so, how is he able to write that expression without mentioning what the corresponding supersymmetry transformations are?

• I am sure that you would have a better chance of an answer if you could include the equation and surrounding text in your post. Even a photo.
– user108787
Commented Nov 11, 2016 at 4:20
• I think the pedagogy here is that $T_F$ is chosen to satisfy the general N=1 superconformal algebra in OPE form, given by (10.1.15a-c). From the OPEs of $T_F$ with the fields $b,c, \beta$ and $\gamma$, the superconformal transformations can be found. They can also be found in the literature. Commented Jan 13, 2017 at 13:41

I think the pedagogy here is that $T_F$ is chosen to satisfy the general $N$=1 superconformal algebra in OPE form, given by (10.1.15a-c). From the OPEs of $T_F$ with the fields $b,c, \beta$ and $\gamma$, the superconformal transformations can be found. The explicit forms of these transformations should be available in the literature.

To calculate the energy-momentum tensor, you need to look at the variation of the action with respect to world-sheet transformations, and use Noether's theorem.

An infinitesimal local world-sheet translation is given by $$z\rightarrow z'=z+\epsilon v(z)$$ Since $$b$$ and $$c$$ are tensor fields with weights $$\lambda$$ and $$1-\lambda$$, they transform as $$b'(z')=\left(\frac{\partial z'}{\partial z}\right)^{-\lambda} b(z)$$ and so on.

From here you can find the infinitesimal variations of the fields and calculate the infinitesimal variations of the action, and easily read off the energy-momentum tensor using Noether's trick.

To calculate the supercurrents, you have to calculate the same analysis for world-sheet superconformal transformations.

I see that you are reading volume 2, and the basic $$bc$$ and $$\beta\gamma$$ CFT's are introduced in volume 1. I suggest you go back and study volume 1 chapter 2.

Best of luck.