String Theory and M-Theory

12.4 Gauge/string duality for the conifold and generalizations 671
N = 1 examples
The five-dimensional space X5 is called a Sasaki–Einstein space, if it is an
Einstein space and if the six-dimensional cone over X5 is a noncompact
Calabi–Yau space (with a conical singularity). Then 3/4 of the supersymmetry
is broken. Therefore, the dual gauge theory on the world-volume
of the D3-branes should be an N = 1 superconformal gauge theory. The
formula for the AdS radius is modified to
R 4 = 4πλℓ 4 s
Vol(S5 )
. (12.143)
Vol(X5)
For this ratio to be meaningful, it is important to choose coordinates in
which Rmn = gmn for the Sasaki–Einstein space.
T 1,1 and the conifold
Chapter 10 introduced a noncompact Calabi–Yau space, called the conifold,
with this structure. Recall that it was defined as a hypersurface in £ 4 by the
simple equation (w A ) 2 = 0. In this case, the five-dimensional space X5
is T 1,1 = SU(2) × SU(2)/U(1), which has SU(2) × SU(2) × U(1) isometry.
T 1,1 is the simplest nontrivial case of a Sasaki–Einstein space. Its metric was
given in Chapter 10. As was explained there, it has the topology S 3 × S 2 .
This example is the simplest case of an infinite family of possible choices.
This section explores this example in some detail, and then comments very
briefly on the other ones. The bulk theory contains vector superfields that
realize the SU(2) × SU(2) symmetry. There is also a U(1) gauge field in the
AdS5 supergravity multiplet. As is always the case, these local symmetries
of the bulk theory correspond to global symmetries of the dual gauge theory.
In particular, the part coming from the supergravity multiplet, which is U(1)
in this case, is dual to the global R symmetry of the gauge theory. This R
symmetry is contained in the superconformal algebra SU(2, 2|1).
The dual gauge theory
Let us now describe the gauge theory in more detail. The T 1,1 space can be
obtained by smoothing out the 2 orbifold theory described above. This fact
allows us to deduce that this is also an SU(N) × SU(N) gauge theory. Each
N = 2 hypermultiplet decomposes into two N = 1 chiral supermultiplets.
Thus, the gauge theory has two chiral superfields, denoted Ai, transforming
under the gauge group as (N, N) and two chiral superfields, denoted Bi,
transforming as (N, N). The Ai fields form a doublet of one SU(2) symmetry
and the Bi fields form a doublet of the other SU(2). All four fields Ai and