The Quantum Structure of Space and Time (293 pages)

Emergent Spacetime 191

5.3.2 Igor Klebanov: Confinement, Chiral Symmetry Breaking

and String Theory

The AdS/CFT duality [l-31 provides well-tested examples of emergent spacetimes.

The best studied example is the emergence of an Ads5 x S5 background of type

IIB string theory, supported by N units of quantized Ramond-Ramond flux, from

the JV = 4 supersymmetric SU(N) gauge theory. The emergent spacetime has radii

of curvature L = (g$MN)1/4&‘. In the ’t Hooft large N limit, g$MN is held

fixed [4]. This corresponds to the classical limit of the string theory on Ads5 x S5

(the string loop corrections proceed in powers of 1/N2). The traditional Feynman

graph perturbative expansion is in powers of the ‘t Hooft coupling g$,N. The

AdS/CFT duality allows us to develop a completely different perturbation theory

that works for large ’t Hooft coupling where the emergent spacetime is weakly

curved. The string scale corrections to the supergravity limit proceed in powers of

2 N

& = (SYM ).

The metric of the Poincark wedge of Adsd is

i=l

Here z E [0, m) is an emergent dimension related to the energy scale in the gauge

theory. In the AdS/CFT duality, fields in Ads space are dual to the gauge invariant

local operators [2, 31. The fundamental strings are dual to the chromo-electric flux

lines in the gauge theory, providing a string theoretic set-up for calculating the quark

anti-quark potential [5]. The quark and anti-quark are placed near the boundary of

Anti-de Sitter space (z = 0), and the fundamental string connecting them is required

to obey the equations of motion following from the Nambu action. The string bends

into the interior (z > 0), and the maximum value of the z-coordinate is proportional

to the separation r between quarks. An explicit calculation of the string action gives

an attractive Coulombic gQ potential [5]. Historically, a dual string description was

expected mainly for confining gauge theories where long confining flux tubes have

string-like properties. In a pleasant surprise, we have seen that a string description

can apply to non-confining theories as well, due to the presence of extra dimensions

in the string theory.

It is also possible to generalize the AdS/CFT correspondence in such a way that

the QQ potential is linear at large distances. In an effective 5-dimensional approach

[6] the necessary metric is

dz2

ds = - 22 +a2(z)( -

and the space must end at a maximum value of z where the “warp factor” u2(z,,)

is finite. Placing widely separated probe quark and anti-quark near z = 0, we find
that the string connecting them bends toward larger z until it stabilizes at z,,
where its tension is minimized at the value w. Thus, the confining flux tube