History: The Ever-Present Past

We are interested in finding a dual to Yang–Mills. As argued in Section 2.6.2 the closest known approximation to Yang-Mills, at least in the low energy limit, is given the solution of a consistent truncation of Type IIA Supergravity (where only the metric, the dilaton and the F(4) form are switched on)

S_IIA= 1 2κ²₁₀

Z

d¹⁰X√

−g



e^−2φ R+ 4∂Mφ∂^Mφ
− 1 2|F₍₄₎|²



, (4.1.1) corresponding to Nc D4–branes wrapped on S¹. We rewrite the metric:

ds² = U R

3/2

η_µνdx^µdx^ν + f (U )dτ²
+ R U

3/2

 dU²

f(U ) + U²dΩ42



, (4.1.2)

e^φ = gs

 U R

3/4

, F₍₄₎ = dC(3) = 2πNc

Vol(S⁴)ω_S⁴ , (4.1.3) where f (U ) is given by

f(U ) = 1 −U_KK³

U³ . (4.1.4)

With the AdS /CFT dictionary R³ = 9

4 , U_KK = MKK = 1 , g_s = 1 2π

g²_YM MKKls

, 2

9M_KK² l²_s = λ⁻¹ . (4.1.5) This metric, in particular the “cigar” described in Figure 2.8, not only provides an explicit energy scale, but it is also able to describe qualitatively most of the properties of low energy Yang–Mills. In particular we have

 Mass gap: the mass spectrum of the theory is discrete hence there are no arbitrarily light glueballs (i.e. colourless bound states of gluons).

 Confinement: the potential of an external heavy quark–antiquark couple grows linearly with the distance. There is also a deconfined solution that we won’t discuss. There is a phase transition between these two called Hawking–Page phase transition [42, 79].

 As we will see when D8 branes are introduced, it also describes the spon-taneous chiral symmetry breaking.

We should point out that this model is not precisely the dual of YM essentially for two reasons. The first one is that the compactification along τ creates a whole towers of massive (∼ MKK) fields in the adjoint representation of SU (Nc), which we are unable to decouple. The reason for this is that the parameter weighting the decoupling is Ts/M_KK² ∼ λ (Ts being the string tension from the rectangular Wilson loop); sadly however, in order to have a reliable supergravity description, we must take λ  1, which is the opposite of the decoupling limit. Secondly the S⁴ factor has an holographic interpretation as a global SO(5) symmetry, of which there is no trace in YM theory. Implicitly we will take a multipole expansion in the SO(5) spherical harmonics retaining only the ` = 0 term, so that these extra degrees of freedom do not appear in the holographic description.

Let us now show explicitly some of the properties claimed above, namely confinement and mass gap.

4.1.1 Confinement in the Witten model

The potential between an external heavy quark–antiquark couple can give us information abound the confinement of the theory: if asymptotically it grows linearly we say that the theory is confining. In order to compute the potential one has to take the expectation value of the Wilson loop along a rectangle with one (Euclidean) time direction.

hW_Ci = e^{−T V (l)} , (4.1.6)

T and l being, respectively, the time and space sides of the rectangle C. As we said in Chapter 2 this expectation value can be computed holographically as the on shell Nambu–Goto action of a string worldsheet whose boundary is C. The key element in this computation is that the U coordinate has a minimum value UKK. Hence, since the metric diverges at the boundary U → ∞ where C stands, the worldsheet will try to minimize its area by laying on UKK as much as possible.

This results in a “bathtub” shaped worldsheet (see Figure 4.1). The Wilson loop in only the integral in the bottom of the bathtub, the two walls being the (infinite) contributions related to the heavy quark masses.

Let us embed the wolrdsheet in the space (x⁰_E, x¹), x⁰_E being the Euclidean time, with U ≡ U (x¹). The Nambu–Goto action reads

SNG = 1 2πα⁰

Z T 0

dx⁰_E Z l/2

−l/2

dx¹ q

g00 g11+ U⁰²(x¹)gU U
. (4.1.7) In the limit l → ∞ the worldsheet is laying on U = UKK, a part from the divergent contribution of the walls that we ignore. The integral becomes simply

S_NG ∼

l1

T l 2πα⁰

 U_KK R

3/2

+ (divergent part) . (4.1.8) We have obtained that V (l) grows linearly, with a string tension

T_s = 1 2πα⁰

 U_KK R

3/2

= 2λ

27πM_KK² . (4.1.9)

This indicates that the theory is confining, as announced.

U = ∞

U = UKK

−l/2 l/2 ← −l/2 l/2 →

Figure 4.1: Section x⁰_E = const. of the worldsheet. The vertical direction is U and the horizontal direction is x¹. When l is not too large the surface does not lie completely of U = U_KK; as l grows it forms the “bathtub” shape.

4.1.2 Mass gap in the Witten model

Yang–Mills is expected to exhibit a mass gap, i.e. a lower bound on the mass spectrum of the theory above zero. The bound states (which are colourless due to

confinement) are called glueballs. In this simple argument let us consider a scalar parity even glueball, which will be related to the operator Tr F². The holographic correspondence tells us that this operator is dual to a scalar field in the gravity side. Let us then consider a scalar field φ in the Witten’s background. We look for solutions to the Klein–Gordon equation

 φ − m²φ = 0 , (4.1.10)

with k² = −M_g² < 0. Mg is the mass of the glueball while m is related to the dimension of the operator Tr F² by means of the holographic correspondence (2.5.24).

Let us suppose that φ does not depend on the S⁴ angles, we choose the ansatz φ(x, U ) ≡ e^ik·xϕ(U ) . (4.1.11) Expanding the Klein–Gordon equation one finds an equation for ϕ



− 3

2U^3/2k²− m²



ϕ+ 1

3U^5/2 7U³− 1
ϕ⁰+ 2

3U^3/2(U³− 1)ϕ⁰⁰ = 0 , (4.1.12) where the primes are derivatives w.r.t. U and we have used the dictionary in (2.6.33). Now we must impose some suitable boundary conditions:

i) Normalizability of ϕ at U → ∞.

ii) At the tip of the cigar we must have ϕ⁰(UKK) = 0 otherwise there would be a cusp in the plane (U, τ ).

Now we have reduced our problem to a one dimensional Schr¨odinger problem with a potential that grows both at U → ∞ and U → UKK. It is known that the eigenvalues M_g² are discrete and with a finite gap. We can compute some of them numerically with the shooting method:

M_g² = {4.33, 10.11, 17.78, . . .} , (4.1.13) all expressed in units of M_KK² . The theory exhibits a mass gap as announced.