N=4 SYM from Klebanov Witten field theory

Question

This is with reference to M. J. Strassler's lectures on "The Duality Cascade" pg. 46. I want to see how $\mathcal{N}=4$ SYM emerges when D3 branes, in the KW setup, are placed at smooth point of the conifold, purely from field theory point of view.

That is: Given a $U(N) \times U(N)$ quiver with $A_i$ and $B_i$ in the bi-fundamental and the anti-bi-fundamental respectively and an $SU(2) \times SU(2)$ invariant quartic superpotential (which vanishes for $N=1$), in what manner do we higgs the theory so as to get $\mathcal{N}=4$ SYM?

For the case of $N=1$ there is a comment in the lectures. It says,

... suppose we just have one D3 brane and we allow $A_1B_1$ to have an expectation value, so that the D-brane sits at some point away from the singular point of the conifold. Then the gauge group is broken to $U (1)$, and six scalars remain massless — the six possible translations of the D3-brane away from its initial point — exactly the number needed to fill out an $\mathcal{N}= 4$ $U(1)$ vector multiplet.

I want to understand this statement. Please help me understand how to work out for case of $N=1$. I can try for the generic value of $N$.

This post imported from StackExchange Physics at 2014-08-31 18:06 (UCT), posted by SE-user Orbifold

Answer 1

The conifold is the space of vacua of the Klebanov-Witten theory. It is obtained as the quotient of the complex four dimensional space of fields $A_1$,$A_2$,$B_1$,$B_2$ by the action of the gauge group $U(1) \times U(1)$. In this action, the diagonal $U(1)$ fixes everyone and only the point $(0,0,0,0)$ is fixed by the full $U(1) \times U(1)$. This implies that the gauge group of the theory is $U(1) \times U(1)$ at the singular point of the conifold and is Higgsed to the diagonal $U(1)$ for a point away from the singularity. At the singular point, the four fields $A_1$,$A_2$,$B_1$, $B_2$ can freely fluctuate despite the fact that the conifold is complex three dimensional: it is the definition of a singular point to have a tangent space of dimension, here complex four, greater than the dimension of the manifold. At a point away from the singularity, the four fields $A_1$,$A_2$,$B_1$, $B_2$ are in the adjoint of the diagonal $U(1)$ (of course, adjoint of $U(1)$ is trivial but I keep a terminology which works for the general $U(N)$ case) but one linear combination is frozen because the tangent space at such a point is complex three dimensional (for example, if $A_1 B_1$ is the only combination with a non-trivial expectation value, computation of the tangent space shows that the $A_2 B_2$ direction is frozen). This implies that the theory at this point is a $U(1)$ gauge theory with three massless chiral multiplets in the adjoint representation. This is exactly the content of a $U(1)$ $N=4$ vector multiplet (indeed, a $N=4$ vector multiplet is the same thing as a $N=2$ vector multiplet with a massless $N=2$ hypermultiplet in the adjoint, which is the same thing as a $N=1$ vector multiplet with three massless $N=1$ chiral multiplets in the adjoint). 

The $U(N)$ case is similar. The only new thing is to check that the superpotential of the Klebanov-Witten theory  reduces to the superpotential of the $N=4$ super Yang-Mills at a Higgsed point of the moduli space.