The conjectured correspondence between N = 4 SU(N) super Yang-Mills theory in four-dimensional flat spacetime and type IIB string theory on AdS5×S5 pre-
sented in the preceding section is generally imposed and only of qualitative nature. Therefore, in this section, we are interested in more precise relations between the two theories to be able to extract quantitative results from the AdS/CFT correspondence. These explicit relations are often referred to as the AdS/CFT dictionary.
First, the various coupling constants of both theories are related as in (2.28). The parameterN was defined to be the flux of the type IIB five-form field strength. In the gauge theory N corresponds to the parameter of the SU(N) gauge group. Second, the symmetry groups of the two theories agree and are given by the supergroup P SU(2,2|4) with the bosonic subgroup SU(2,2)×SU(4). As already mentioned in section 2.3.1, on the string theory side AdS5 has the spacetime
symmetry group SO(2,4) ∼= SU(2,2) and the S5 has isometry SO(6) ∼= SU(4).
Moreover,AdS5×S5 realizes all 32 supercharges of the type IIB string theory. On
the gauge theory side, we have a conformally invariant quantum field theory with conformal group SO(2,4). The SU(4) symmetry of the S5 arises on the gauge
theory side as a global SU(4) R-symmetry of the 32 supercharges.
Third, the field content of both theories should be explicitly related in order to perform calculations on the supergravity side and to transfer the results to the strongly-coupled field theory side where calculations are difficult. Such explicit mappings between supergravity fields φ and gauge-invariant operators O of the CFT were provided by Witten [67] and also by Gubser, Klebanov and Polyakov [68]. According to these works, the boundary values of supergravity fields are considered to be sources which couple to operators in the dual gauge theory, and the supergravity partition function is identified with the generating functional of gauge theory correlation functions,
D
eRd4xφ0OE
CFT=ZS[φ0], (2.29)
where the left hand side is the generating functional containing the coupling of a gauge theory operator to the boundary value φ0 of a supergravity field φ. The
2.5 A more precise correspondence 29
right hand side is the supergravity partition function evaluated at the boundary which can be computed via the classical supergravity action IS,
ZS(φ0) = e−IS(φ) φ=φ0 . (2.30)
With relation (2.29) at hand, we can compute correlation functions ofO simply by taking functional derivatives of the classical supergravity action with respect toφ0.
The formula (2.29) applies in general such that each field propagating in the bulk is in a one-to-one correspondence with an operator in the gauge theory. However, the supergravity action diverges, because of the infinite volume of theAdS5 spacetime
and thus needs to be appropriately renormalized. A renormalization procedure was developed in [69, 70] which consists in adding covariant counterterms to the divergent action yielding a finite action. Furthermore, as in any background with boundary, the action has to be supplemented by a Gibbons-Hawking term [71] which is a boundary term rendering the variational problem well-defined. Finally, the resulting five-dimensional renormalized action Sren consists of three terms,
Sren =Sbulk+SGH+Scounter, (2.31)
whereSGH denotes the Gibbons-Hawking term andScounter includes the necessary
counterterms. Evaluating this action on the solution then yields the renormalized on-shell action.
The general solution of the bulk field equations has to satisfy Dirichlet bound- ary conditions. An ansatz for the supergravity fields is given by the decomposition with respect to a basis of spherical harmonics Y∆ on the S5 [8],
φ(r, x, y) =
∞
X
∆=0
φ∆(r, x)Y∆(y), (2.32)
where r, x and y denote the coordinates on AdS5 and on S5, respectively. Now,
the fields φ∆(r) are effectively defined on AdS51. Asymptotically, the supergrav-
ity fields φ∆(r) are free and satisfy the free field equations of motion. The S5
compactification contributes to the masses of the fields. Thus, for instance, for a scalar field this yields the mass relation [67]
m2 = ∆(∆−4), (2.33)
1It is important to note, that the truncation of ten-dimensional supergravity on AdS 5×S5
to supergravity onAdS5 is not evident, since the usual Kaluza-Klein method is not applicable.
Kaluza-Klein compactification consists in dimensional reduction on a space of very small radius which leads to large masses of the Kaluza-Klein modes. In a low-energy approximation, these Kaluza-Klein modes decouple from the low-energy states. However, in the case of AdS5×S5
both spaces have equal radius L which spoils the usual Kaluza-Klein argument. Nevertheless, there is a consistent truncation toN = 2 supergravity onAdS5. Examples are given in [72–74].
and the asymptotic solution for r→ ∞ takes the form
φ∆(r∞, x) =r∆−4φ0(x) +r−∆hO(x)i, (2.34)
where x denotes coordinates along the boundary of AdS5. Since the supergrav-
ity field φ∆ is dimensionless, the operator O has dimension ∆. The boundary
condition on the supergravity field becomes φ0 = limr→∞r∆−4φ0(x) [67]. Similar
relations exist for non-scalar fields such as fermions and tensor fields on AdS. Thus, the boundary condition is determined such that the mass of the supergrav- ity and the conformal dimension of the gauge theory operator match in a certain way.
In this thesis, we are only concerned with one-point functions of operators O
in the presence of sources. Thus we ask questions as given a supergravity field to which operator does it correspond on the gauge theory side. The answer is given by the computation rule [69, 70],
hO(x)i= p 1
G0(x)
δSren
δφ0(x)
, (2.35)
where Sren is now the renormalized on-shell action.
Examples of the field-operator mappings (2.35) are, for instance, the relation between the metric fieldGM N on the supergravity side and the energy-momentum
tensor Tµν on the gauge theory side, as well as the relation between a gauge field
AM on the supergravity side and a currentJµ on the gauge theory side which are
given by AM(r, x)→hJµ(x)i= 1 p G0(x) δSren δA0µ(x) , (2.36) GM N(r, x)→hTµν(x)i= 2 p G0(x) δSren δG0µν(x) . (2.37)
So far, we have seen how the correspondence is implemented. In the following section, we explain how the correspondence is extended to describe CFTs at finite temperature as well as in the presence of a finite charge density.