(^192) The Quantum Structure of Space and Time
is described by a fundamental string placed at z = zmax parallel to one of the
xi-directions. This establishes a duality between “emergent” chromo-electric flux
tubes and fundamental strings in certain curved string theory backgrounds.
Several 10-dimensional supergravity backgrounds dual to confining gauge theo-
ries are now known, but they are somewhat more complicated than (2) in that the
compact directions are “mixed” with the 5-d (xp, z) space. Witten [7] constructed
a background in the universality class of non-supersymmetric pure glue gauge the-
ory. While in this background there is no asymptotic freedom in the UV, hence
no dimensional transmutation, the background has served as a simple model of
confinement where many infrared observables have been calculated using the clas-
sical supergravity. For example, the lightest glueballs correspond to normalizable
fluctuations around the supergravity solution.
Introduction of N = 1 supersymmetry facilitates construction of confining
gauge/string dualities. A useful method to generate N = 1 dualities (for reviews,
see [8, 91) is to place a stack of N D3-branes at the tip of a Calabi-Yau cone, whose
base is Ys. In the near-horizon limit, one finds the background Ads5 x Y5, which
is conjectured to be dual to the superconformal gauge theory on the D3-branes.
Furthermore, for spaces Ys whose topology is S2 x S3, the conformal invariance
may be broken by adding M D5-branes wrapped over the S2 at the tip of the cone.
The gauge theory on such a combined stack is no longer conformal; it exhibits a
novel pattern of quasi-periodic renormalization group flow, called a duality cascade
To date, the most extensive study of a cascading gauge theory has been carried
out for a 6-d cone called the conifold. Here one finds a N = 1 supersymmetric
SU(N) x SU(N+M) theory coupled to chiral superfields All A2 in the (N, m)
representation, B1, B2 in (N, N + M), with a quartic superpotential [ll]. The M
wrapped D5-branes create M units of R-R flux through the 3-cycle in the conifold.
This flux creates a “geometric transition” to the deformed conifold Ct=, w: = c2 ,
where the 3-cycle is blown up. An exact non-singular supergravity solution dual
to the cascading gauge theory, incorporating the 3-form and the 5-form R-R field
strengths and their back-reaction on the geometry, is the warped deformed conifold
PO, 91.[lo1
ds2 = h-1/2(~) (-(dzO)’ + (dzz)’) + h1/2(~)difi , (3)where d.5; is the Calabi-Yau metric of the deformed conifold with radial coordinate
r. The 5-from R-R field, which is dual to N, decreases as the theory flows to the
infrared (towards smaller T).
What is the field theoretic interpretation of this effect? After a finite amount of
RG flow, the SU(N + M) group undergoes a Seiberg duality transformation [12].
After this transformation, and an interchange of the two gauge groups, the new
gauge theory is SU(N) x SU(N + M) with the same matter and superpotential,
and with fi = N - M. The self-similar structure of the gauge theory under the
Seiberg duality is the crucial fact that allows this pattern to repeat many times.