I want to look at the following theory in $1+1$ dimensions with $\Phi$ being the chiral superfield,

$L = \int d^2x d^4\theta \bar{\Phi}\Phi - \int d^2x d^2\theta \frac{\Phi^{k+2}}{k+2} - \int d^2x d^2\bar{\theta} \frac{\bar{\Phi}^{k+2}}{k+2} $

  • How does one show that the above theory has the $\cal{N}=2$ superconformal symmetry? (..I guess that is a claim that I see in various literature..)

  • How does one calculate the charge of the chiral primary states in this theory and which is claimed to be $\frac{n}{k+2}$ for $n=0,1,2,..,k$? And can one explicitly enumerate those states?

  • How does one show that the index $Tr(-1)^F$ for the potential $\frac{\Phi^{k+2}}{k+2}$ is $k+1$?

  • 1
    $\begingroup$ You should specify that you are using the 4d chiral superfield (although it is obvious from context). This is the 2d dimensional reduction of the 4d Wess-Zumino model, and k>=0 for superconformality (no mass term and homogenous superpotential). The N=2 SUSY is 2d N=2 susy, really (2,2) SUSY (left and right SUSY are independent in 1+1 d). and it corresponds to the N=1 SUSY in 4d of the ordinary Wess Zumino model with a superpotential. This model is solvable in a certain sense for any k, in that it has an exact local Nicolai map. This should allow you to calculate T's and L's (didn't do it). $\endgroup$ – Ron Maimon May 7 '12 at 7:09
  • $\begingroup$ @Ronn Maimon May be you can give some reference to where such a calculation might be done? From where one can learn to prove such stuff? $\endgroup$ – user6818 May 8 '12 at 22:32
  • $\begingroup$ I gave an "answer" which doesn't answer anything--- it just shows you a better way to think about the model. For your problem, you work out the SUSY transformations explicitly, compute the T and show T is traceless (Tzzˉ=0), then find the OPE's with the field using their transformation law, and actually answer the questions here. I didn't do this below, I didn't work it out, neither is the conformality obvious in the stochastic formulation as I thought it might be, if it exists, it mixes up the field nontrivially. You need to explicity work out the content by OPE and constructing the SUSY's. $\endgroup$ – Ron Maimon May 9 '12 at 16:47

I don't know whether this is an exercise, but there is a famous Nicolai map for this system, which is as follows: write the complex field $\phi = \phi_1 + i\phi_2 $ which is the scalar part of the chiral field in terms of the real and imaginary components. Then consider the stochastic equation (Euclidean space):

$$ \partial_\bar{z} \phi + W(\phi) = \eta $$

Where $\eta$ is complex white noise, meaning a random variable which is random from point to point, and W is a complex function of the field values. In terms of real and imaginary parts, calling the two space coodinates x,y:

$$ \partial_x \phi_1 + \partial_y \phi_2 + W_1(\phi_1,\phi_2)= \eta_1 $$ $$ \partial_y \phi_1 - \partial_x \phi_2 + W_2(\phi_1,\phi_2)=\eta_2 $$

The probability of $\eta$ having a given value is the product of a gaussian at each point in space, this is what it means to have a stochastic equation:

$$ P(\eta) = e^{-\int {1\over 2} (\eta_1^2 + \eta_2^2)} $$

In this form, you do an unweighted path integral over $\eta$ to find the probability of a configuration. This means, generate $\eta$ according to this trivial distribution (you can make a lattice and generate the $\eta$s as independent Gaussians at each point). Then use the nonlinear equations above to find the field $\phi$, and this gives you a configuration of the stochastic equation.

Then the correlation functions of the scalar are given by the path integral

$$ \langle \phi_i(x)\phi_(x') \rangle = \int \phi_i(x)\phi_j(x') P(\eta) D\eta $$

So long as P is normalized properly, meaning you divide by the path integral without insertions.

The magic of the Nicolai map (or the Parisi Sourlas supersymmetry of any stochastic equation) is when you change variables to do the path integral over $\phi$. You substitute in for $\eta$ in terms of $\phi$, and you need a determinant to change from the $\eta$ variable (where the path integral measure is uniform) to the $\phi$ variables (where in the normal Stratonovich convention for products in the path integeral, it isn't).

You get

$$ S= -{1\over 2} \int (\partial_x \phi_1 + \partial_y \phi_2 +W_1 )^2 + (\partial_y\phi_1 - \partial_x\phi_2+ W_2)^2 $$

and the path integral

$$ \int e^{-S} \mathrm{det}(\partial_x +\partial_1 W_1, \partial_y + \partial_2 W_1 ; \partial_y + \partial_2 W_2,-\partial_x + \partial_2 W_2)$$

Where the semicolon separates lines of a matrix (I don't know how to tex that). The fermionic action will give the 2d Fermionic part of the full SUSY action in 2d.

First note that the bosonic action reproduces the desired free field action in the derivative parts

$$ S_f = -{1\over 2} \int |\nabla\phi_1|^2 + |\nabla\phi_2|^2 $$

In the interacting parts, you get

$$ S_i = -{1\over 2} \int W_1^2 + W_2^2 $$

Which is going to be the superpotential interaction for the bosonic field at the end of the day. But you also get cross terms which wreck rotational invariance in general

$$ S_c = - \int \partial_x \phi_1 W_1 + \partial_y\phi_2 W_1 +\partial_y\phi_1 W_1 - \partial_y\phi_2 W_2 $$

These cross terms have to cancel to get a rotationally invariant system. From this, you learn that (W_1 + iW_2) has to be a holomorphic function of $\phi_1 + i\phi_2$, which is a completely different demonstration of the holomorphy of the superpotential, not going through superspace or diagrams, but from requiring rotational invariance of the stochastic form of the Euclidean theory.

The quickest way to see that holomorphy is required (you can figure it out for yourself by trying examples) is to write the cross terms in holomorphic form: they are the real part of the expansion in components of

$$ \partial_{z}\phi W(\phi) = \partial_z V $$

Where V is the anti-derivative of the holomorphic function W. V is also holomorphic, and differentiating it with respect to $\bar\phi$ gives zero. The cross terms are now perfect derivatives (but you need a chain-rule respecting interpretation of the noncommuting products, this is the one that automatically gives the determinant as written).

The determinant now adds the Fermionic action

$$ S_f = \bar\psi (\sigma\cdot\partial + V'') \psi $$

For two component fermions with the choice of 2d Euclidean i-free gamma matrices $\sigma_x, \sigma_z$ (which are real, anticommute with each other, and square to 1). The free action can be rewritten in terms of left movers and right movers to see that there are two of each, and the resulting action is

$$ S = \int |\nabla\phi|^2 + |V'(\phi)|^2 + \bar{\psi}(\sigma\cdot\nabla + V'')\psi $$

And this is your dimensionally reduced Euclidean Wess-Zumino action, using $V(\phi) = {\phi^{k+3}\over (k+2)(k+3)}$ (V is just the antiderivative of W, which is just the polynomial superpotential you were given). The (2,2) SUSY (two left-moving and two right-moving fermions each SUSic with the scalar) is automatic, because it is the Parisi Sourlas SUSY of any stochastic system.

The Nicolai map immediately gives you a ton of things: it normally gives you the exact ground state wavefunction for the bosonic fields, as this is the statistical distribution of the associated stochastic equation, which is the exponential of V (and some derivative parts). Unfortunately, in this case, there are severe infrared problems with the distribution due to the derivative parts, so I was never able to write the ground state wavefunction down analytically in a way that made it sensible.

The Nicolai map automatically gives you a way to simulate the theory on a lattice--- just generate $\eta's$ and do the nonlinear transformations. This has become an industry in the last decade, with one of the leading figure Simon Catterall at Syracuse. SUSY systems are notoriously hard to simulate keeping the SUSY exact. The handful of Nicolai map systems (this one, and SUSY QM) are the only exceptions, where simulating the SUSY system is easier than the non-SUSY system. Thankfully, this includes matrix theory, and perhaps it includes N=4 SYM (Catterall wants to do this theory too), although there I don't know how to do it (but I always have the nagging feeling that it can be done for many more systems, we just are missing a crucial simple idea--- Catterall does it without mentioning the explicit Nicolai map (although this was his starting point), but with a lattice subset SUSY which is a more technically annoying way to say a similar thing, for the reason, I suppose, that he wants a generalization of the method that allows lattice simulation for cases where the Nicolai map isn't explicit like this).

This has very little relation to the questions you pose directly, but you asked for it. You need the SUSY generators, from this you find the stress energy tensor, and from this you find the primary fields, and so on. All this you can do directly without knowing anything about this stuff, but this stuff makes the SUSY in the model completely intuitive.


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