# N=4 SYM from Klebanov Witten field theory

+ 2 like - 0 dislike
3413 views

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

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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysi$\varnothing$sOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.