According to the gauge/gravity duality, string theory and a gauge theory can describe the same physical system. It is then logical that both descriptions should respond to perturbations in a similar way [38]. More precisely, studying the perturbations in each of the two theories could be very useful in creating a
dictionary that links the two different descriptions [47, 48], based on the responses to the perturbations.
Let us study a known example that can provide some insight on how to relate the two theories. The SYM coupling gY M is related to the string coupling through
(3.29) and also the string coupling gs is given by the expectation value of the
dilaton field at the AdS5 boundary (r → ∞). If a perturbation (sometimes called
constant, then the expectation value of the dilaton field at the boundary should change as well so that the relation (3.29) will still hold. This example can be generalised by considering a general deformation of the gauge theory of the form
S → S +
Z
d4x φ0(x)O(x), (3.33)
where O(x) is a local, gauge invariant operator and φ0(x) is the source of the
operator. The gY M in our example is a source to some operator. Therefore, in the general case there should be a field in the gravity side whose asymptotic value matches with the source φ0(x), as the dilaton in the above example, for each
operator in the field theory [37]. This one to one matching between the fields in the bulk and the field theory operators is called field/operator matching. In [47, 48] the field/operator matching was stated in a path integral language as
Zstring h φ(~x, r)¯¯r=∞= φ0(x) i = D eRd4x φ0(x)O(x) E F T, (3.34)
where Zstring is the full partition function in string theory and the r.h.s is the
generating functional in the field theory. This is the most general operator/field matching recipe arising from the gauge/gravity duality. It is possible to
approximate the string theory partition function with the supergravity partition function, when using the weak form of the correspondence (see 3.3.4), and therefore (3.34) becomes e−ISU GRA ∼ D eRd4x φ0(x)O(x) E F T, (3.35)
where ISU GRA is the on-shell supergravity action19.
The next obvious question is how to determine the dual field for a given field theory operator? This is not always possible but a guide for the identification should be provided by the quantum numbers and symmetries. Specifically, the field and operator should share the same quantum numbers under the global symmetries of the theory [37].
19It is possible to have more than one on-shell supergravity actions, if there are more than one solutions to the equation of motion of the bulk field of interest. In this case, it is necessary to decide which solution is the real vacuum of the theory by comparing their free energies [32].
To explore further the operator/field matching it is very instructive to consider the case of a five dimensional massive scalar field Φ(~x, r) living in AdSd+1× S5. This
field is dual to some operator O(x), which lives on the d dimensional Minkowski boundary of AdSd+1× S5, where the gauge theory lives. The bulk action is the following
S = −1
2 Z
drddx√−g¡gM N∂MΦ∂NΦ + m2Φ2¢, (3.36) where M, N = 0, · · · , d + 1. It is convenient to Fourier decompose the field Φ in the Minkowski directions xµ and then calculate its equation of motion which is the
following
r1−d∂r(rd+1∂rΦ) − (k2R2r−2+ m2R2)Φ = 0, (3.37) where kµ= (E, ~p) and therefore k2 = −E2+ ~p2. The (3.37) describes the scalar
field Φ in the bulk. Since a connection between the field Φ in the bulk and the operator on the boundary is investigated, the study is limited to the r → ∞ case. The solution to the asymptotic equation of (3.37) is20
Φ(~x, r) ∼ A(x)r∆−d+ B(x)r−∆, (3.38) where ∆ = d 2+ ν and ν = r m2R2+d2 4 . (3.39)
The next step is the interpretation of the asymptotic solution. There are two linear contributions of which the first term ∼ A(x) is non-normalisable and the second term ∼ B(x) is normalisable21. The non-normalisable term is dominant at the
boundary and therefore it is natural to match the non-normalisable asymptotic value of the field to the source of the operator O(x) at the boundary. The solution is singular at the boundary and needs to be regularised. The exact matching is
φ0(x) = lim
r→∞r
d−∆Φ(~x, r). (3.40)
20The solution is Fourier transformed back to position coordinate space.
21It is possible that both terms are normalisable when the mass is in the range −d2
4 ≤ m
2R2 ≤
−d2
This last expression implies that the source acquires a mass dimension d − ∆ and therefore the operator O(x) has mass dimension ∆.
The normalisable terms should be identified with the states of the boundary theory and more specifically the coefficient B(x) should be identified with the expectation value of the operator O(x) [37].
Finally, let us make some observations regarding (3.39). This relation indicates a correspondence between the mass dimension of the gauge theory operator and the mass of the field in the bulk. For example, in the case of a massless scalar field the mass dimension of the corresponding operator is d. The relation between mass and dimension can be generalised for any p-form field in the bulk and the expression that relates the mass of the bulk field with the dimension of the field theory operator is
m2R2 = (∆ − p)(∆ + p − d). (3.41) Moreover, the definition of ν imposes a limit on the values of mass as a result of the requirement that the square root should be real. The specific limit is
m2R2 ≥ −d
2
4 . (3.42)
The conclusion from the above relation is that the mass of the scalar field in the bulk can have negative values as long as it doesn’t become too negative. Generally, all the fields in the bulk have a minimum allowed value for the mass which is called Breitenlohner-Freedman(BF) bound [49, 50] 22.