Counting Ghost Numbers in String Theory | Beltrami Parametrization

[christodouloum]

Hello I am new in string theory and I am wrestling with a paper in Beltrami parametrization. Although I have spend time studying ghosts, brs symmetry etc my knowledge in differential geometry is very limited (for string theory). My question is how do I count practically the grading of an object (it is defined as it's ghost number plus form degree).

Do I read this from the Lagrangian?

Second question: Where can I find the full Lagrangian with terms for antighosts?

 

[mitchell porter]

What is the paper? This will be easier to answer if I can see what you're reading.

 

[christodouloum]

Hi, thanks for the interest! The paper is Physics Letters 196B, 142 (1987) by L.Baulieu and M.Bellon.

In trying to reproduce the calculations I see that I am definitely missing something. I believe I do not understand how to use the graded commutation and derivation correctly.

Specifically I do not understand:

1. how in equation (11) the plus and minus components of d from (8a) mix with those of lambda. The calculation should be straightforward from (8a) , (10) and the zero torsion condition after (11).

2. what exactly is meant by symmetrical product of forms which is mentioned in (7). I can see that in (7) it means to change the index signs in the right parenthesis before doing the multiplication.

3. if the spin connection omega is even or odd graded. From calculations I did I see that it must probably be graded even but I do not understand how I can tell from my equations.

Any help much appreciated since very few people can help me with this

 

[mitchell porter]

Now we can struggle together. This could take a while... I want to think about the geometric meaning of the paper's thesis first, so if anyone else out there wants to tackle the algebra right away, please, go ahead.

 

[christodouloum]

I am happy to see that you are interested! My main problem too is indeed intuition as you have well identified. In the meantime I have made progress with the algebra and solved some of my questions. I have to harry up with the calculations since I am doing this as part of a project. When you are ready we can have a discussion on what is going on and attack the geometric interpretation and arguments, although I am not sure of how much help a mere grad student like me could be.

 

[christodouloum]

Do you think we can start from identifying what mathematical objects are the things that are here?

For starters

e^{\pm} are the two components of a one-form
\omega is a one form
\xi^{\alpha} are the components of a fermionic vector field, so each one is a fermionic variable

what confuses me is the zweibein. From what I knew so far I thought it should be a basis of vector fields and being the components of a one form here confuses me.

 

[mitchell porter]

I think one should begin by distinguishing between physical objects and ghosts. Physically, there's a metric and a frame field (and from these you can construct a spin connection, torsion, curvature...). Then you have ghosts for diffeomorphisms, scale transformations, and local rotations.

But I don't understand the complex coordinates. I have the usual beginner's trouble with thinking of z and \bar{z} as independent variables, but OK, maybe if I think about it for long enough, I'll get used to it. However, what I'm really not getting is the association between the frame vectors and the complex coordinates. Is e^+ supposed to depend only on z, and e^- only on \bar{z}?