Hey guys,(adsbygoogle = window.adsbygoogle || []).push({});

I haven't posted on here for quite a while, so hello to everybody.

I've been trying to derive the stress-energy tensor for the ghost LaGrangian:

[tex] \int d^2 \sigma \sqrt{g} \left( b_{\alpha\beta} \nabla^\alpha c^{\beta} + \omega b^{\alpha\beta} g_{\alpha\beta} \right)[/tex]

for quite a while now. It is a conformal field theory and to word out the central charge the most efficient way is through the T(z)T(w) OPE. Both fields, b and c, are anticommuting. b is a symmetric tensor. [tex]\omega[/tex] is a Lagrange multiplier to enforce the tracelessness of b. I've got tantalisingly close, but no cigar. So really need some help here ;).

Firstly, I gauge fixed the WS metric to be a flat one, so that [tex]\nabla^\alpha c^{\beta} = \partial^{\alpha} c^{\beta} [/tex].

Now you can proceed two ways.

1) Couple the theory to gravity (which involves reinstating the covariant derivative), and vary the metric. This gives a symmetric SE tensor. This I have done, and it works out fine.

2) Peform an infinitesimal change in the coordinates [tex]x^\mu \rightarrow x^\mu + \epsilon^\mu (x)[/tex]. The change in the action will be of the form

[tex]\int d^2 \sigma T_{\alpha\beta} \partial^\alpha \epsilon^\beta[/tex],

so you can read off the SE tensor.

The method I want to make work is the second (because I'm stubborn and want to do it both ways).

In complex coordinates, I get:

[tex] T_{zz} \equiv T = b\partial c + c \partial b [/tex] .

Compare this with the correct result, as quoted in Polchinski, BBS, Steve Tong's DAMTP notes:

[tex] T_{zz} \equiv T = 2 \partial c b + c \partial b [/tex] .

Now the tensor obtained this method is not symmetric, so there are things you can add to it to make it so. But I don't think I've found the 'right thing' to bring it into the standard symmetric form. Nor do I know how much this matters the context of the CFT. By adding various things conserved quantities to the the SE tensor I've found quite a few different forms. Can they all be used in the CFT OPEs, to find the central charge, etc? I'm a bit worried that I can find quite so many different forms for a single object.

Thanks for listening to my rambling. I know it's reasonably technical, but I've been stuck on it for weeks, so any help is appreciated. I've check my calculation as best I can, although there are quite a few terms (e.g. from the tensor transformations of b and c),so it's open to error.

Ideas anybody? Some of these mentor bods floating around? ;)

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# Worldsheet ghost stress-energy tensor

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