The last week of the semester was dedicated to Feynman rules in superspace. Before we go into that, it is good to review and set our conventions in superspace.

A remark: as in any quantum field theory analysis, here we will see the need for regularization. Now we need to to preserve the supersymmetry of the theory and the task of finding a regulator that obeys this criteria is non-trivial. Nevertheless, there exist such regularization schemes, namely dimensional reduction.

In any case, as action we will take the Wess-Zumino model coupled to super Yang-Mills. Of course this theory only has simple supersymmetry, but actually this is one of the only cases where superspace methods are useful. I do not want to rule out superspace methods for extended supersymmetry since one never knows what one might end up doing for his/her Ph.D. thesis. The action has the form:

.

The prepotential is

.

In four-dimensional Minkowski spacetime the Lorentz group is doubly covered by . We will label spinors with a Weyl index . The supercovariant derivatives, (giving objects that are covariant under supersymmetric transformations) are defined as follows:

Now we set the convention for raising and lowering Weyl indices (which actually we do not follow in the definition of the supercovariant derivatives): Weyl indices are raised with the two-dimensional antisymmetric symbol in the ¨from north-west to south-east¨ fashion. Namely,

The bars on spinors with dotted indices will be omited in what follows (they are redundant anyway). For example, we have the anticommutators for the supercovariant derivatives:

Now we move on to integration. We start with

Similarly for the dotted coordinates:

This expression follows from the single integration:

The integral of anticommuting variables can be written in terms of the derivative.

This will be useful later. Total supercovariant derivatives integrate to zero:

Here we have ignore boundary terms. Integration over all superspace can be expressed as

Now we turn to chiral superfields. A superfield is a function that is defined on superspace (i.e. it has dependence on the spacetime coordinates and the supercoordinates also). A chiral superfield satisfies the constraint:

Either      or      but not both.

With chiral superfields one can do wonders. For example, sticking to the first choice for constraint defining a chiral superfield,  we have the property

This is useful when re-writing chiral integrals as integrals over the whole supercoordinates:

We have forgotten about Dirac delta functions for supercoordinates! For a single supercoordinate we define:

In particular, for the case of the identity function we get

This result can be generalized to the full superspace integral:

(Writing the integral as derivatives as discussed above.) The solution to this equation is

We can mention some properties related to this delta function:

I would like to consider the gravity background:

with the case that We saw previously that the equations of motion were given by:

     and    

Now we take the warp factor to have the form . Then we have

We will introduce a new symbol,

     with a function of the worldsheet coordinates.

The differential equation now looks like:

We now assume that can be factored into functions of each coordinate, . Then we can solve this PDE when

     with a constant.

Separation of variables gives the following differential equation:

with

     and    

Inserting this into Mathematica gives:

     which means    

Notice that the case is interesting: for the constraints we have used the solution is an exponential function of a quadratic polynomial.

Let us consider the following gravitational background:

and the Polyakov lagrangian in conformal gauge with Euclidean Lorentzian signature:

The Euler-Lagrange equations can be found to be:

     and    

We look at this and think how can we make these equations simpler. The first equation can be solved with:

This is a “T-duality” transformation. For the second equation we see that if then we get

with the flat-space Polyakov lagrangian in terms of the new coordinates :

We can integrate this equation to find an expression for in terms of the solutions of the equations of motion and the flat-space lagrangian:

Then, pluging this back into the classical action we get

Finally we can write

so we can write the expression in the exponential as a sum of integrals over the worldsheet coordinates.

On the other hand, if instead we have then we can write:

with

The lagrangian can be writen as

We have found an expression for the classical lagragian in some (not-so) arbitrary gravitational background in terms of the coordinates and the flat-space lagrangian. The problem is we do not get any information about the solution for the equations of motion.

]]> http://indexguy.wordpress.com/2008/08/22/a-curved-lagrangian-in-terms-of-a-flat-one/feed/ 2 IG http://indexguy.wordpress.com/2008/08/12/ego-breaking/ http://indexguy.wordpress.com/2008/08/12/ego-breaking/#comments Wed, 13 Aug 2008 01:33:01 +0000 Index Guy http://indexguy.wordpress.com/?p=182 ]]>

The hardest part of research is to overcome one’s ego and know when one’s ideas are leading nowhere. Also, to accept advice from others.

Let us start with the following background metric:

where the functions and are functions of the extra coordinate . The case of anti-de Sitter space corresponds to

Under the “T-duality” transformation

   and  

the metric in the dual space takes the form

but now

   and  

Since we can always bring the metric in this form, we will just consider the initial case and see what can we do with it. Note that it could be the case that the change of variables between could be of the general form

   which means  

This case could be more complicated… For now we will just stick with the first change of variables introduced.

I will consider a classical string, and I will set one of the (spatial) coordinates equal to zero and work with a static gauge: the other two spatial coordinates describe the worldsheet of the strings. I am interested in a configuration very similar to that on the Alday-Maldacena paper. The one thing I could change is to allow the polygon where the worldsheet end to be time-like instead of light-like. This is achieved by setting

with being a positive real number that is greater than unity. Our metric has signature .

In anticipation we will define the following symbols:

with the index taking values of 1 and 2. Our coordinate vector in the static gauge looks like:

The Nambu-Goto action involves the determinant of the induced metric,

Evaluating we get:

        and    

Then, after plugging all these into the expression for the determinant we get:

We see a factor of popping out of the action. This tells us that the worldsheet has Euclidean signature.

For convenience we will define

The lagrangian then looks like:

Let us now look at the equations of motion for the fields and . The first field is a bit more simpler:

     which implies    

This is the expression for a two-dimensional vector that has vanishing divergence. Then it follows that the components of this vector must be of the form:

     and    

for some other field that is a function of both worldsheet coordinates. Note that adding a constant to does not affect the equations of motion, so it may correspond to some coordinate with a shift symmetry. Let us look at the conjugate momenta:

     and    

We have denoted

One can see that the ratio of the conjugate momenta has a nice form:

One can also use this expression to write the ratio of the warp functions in terms of derivatives of the fields:

We see that the metric would blow up when the fields have orthogonal gradients.

On Thursday night I was reading Polyakov’s contribution to the book 50 years of Yang-Mills theory. This is my version of a bed time story.

Polyakov mentions some of his earlier work and how he solved different problems. I actually got tired after a while, and decided to work and solve my own problem. I had been sort of running away from it with feelings of overwhelming difficulty. There is also a chance that reading this post on the (now dead) string coffee table provided some motivation. In any case, here we go…

Let us consider the following metric in 5-dimensional spacetime:

We have coordinates and we will work in a static gauge where a worldsheet will be parametrized by the two spacetime coordinates . We will set .

Then taking derivatives of the coordinate vector we have:

   and  

Then, the ingredients that go into the Nambu-Goto lagrangian are:

   and  

The cross term goes as:

The determinant of the induced metric will be denoted by:

We see that since we have:

and

we get some cancellations in . In the end we can write

We can write the lagrangian as:

with

It turns out that all that I have been doing for the past few weeks has been slightly off-track. Back to the main source: Alday and Maldacena.

Today I found this:

Luis Fernando Alday and Radu Roiban, Scattering amplitudes, Wilson loops and the Strings/Gauge Theory correspondence, arXiv:0807.1889.

At this moment it looks really helpful.

I have become more comfortable with the “T-dual” transformation and now I am confident on what to consider.

Exact adS can be described by the metric

After the T-transformation the metric returns to the same form but now ,

Both of these spaces can be described by embedding a hyper-surface in . They both have scale invariance.

There are two extra-dimensional spaces that one can obtain by embedding a hyper-surface in and reduce to adS in some limit.

Case I:

For this case anti-de Sitter space is obtained by setting the extra coordinate equal to . The metric in the original Poincaré coordinates looks like:

After the T-transformation (leaving the coordinate intact) one obtains

Case II:

Now adS is reached when the extra coordinate vanishes. In Poincaré coordinates,

While in the T-coordinates we obtain

Note that both of these cases have scale-invariance in the coordinates and . These two cases are atractive because they can be constructed straight from a higher-dimensional reduction. On the other hand one can just modify the adS metric and see what happens. Here are some of the modifications.

• Soft wall:
• Hard wall:
• Cutoff:

I might not want to keep the scale invariance so obvious for the coordinates.

Today I found yet another space that looks promising. This one looks even more nicer than the previous cases. To obtain the metric one can start with the product of 3-d Minkowski and d-dimensional Minkowski space. The coordinates can be taken as

with a two-dimensional Minkowski index and a d-dimensional Minkowski index. Let us write the metric as:

Now one imposes the condition:

and introduces “Poincaré coordinates” that satisfy this constrain. These can be defined as:

Then writing the metric in terms of these coordinates one obtains:

One can see that when we set we obtain exact d-dimensional anti-de Sitter space. This metric is very atractive. It contains more symmetry than the other cases that I have consider previously. Besides the first term being invariant under Poincaré transformations, one also has an invariance under the rescalling of the coordinates. I shall explore this space further.