2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicsAdS/CFT, the radial direction and scale

Holography — reading a four-dimensional field theory through the geometry of gravity in one more dimension

The claim of holography is this. A field theory without gravity in $d$ dimensions describes, in other words, the same physics as a theory with gravity in $d+1$ dimensions. The extra dimension is "depth" from the boundary; the former is called the boundary theory and the latter the interior (bulk) theory. The representative example is Maldacena (1997, ATMP 1998; the first year is when it appeared on arXiv, the second the year of the journal).

On the left is four-dimensional $\mathcal N=4$ supersymmetric Yang–Mills theory (SYM below), on the right ten-dimensional string theory. Type IIB is one of the ten-dimensional superstring theories, differing from the type IIA of 7-05 in the combination of handedness (chirality) of its supersymmetries. The dimensions are 10 = 4 (boundary) + 1 (depth) + 5 ($S^5$) (7-05 §09). The original paper writes 'We are led to the conjecture', and (1) is still a conjecture Physics. This chapter deals with the entrance; models in which the mass gap turns into geometry are left to 7-09.

Detour: why "hologram" — counting by area

Open strings ending on $N$ stacked D3-branes (membranes with three spatial dimensions on which the ends of open strings sit) become $U(N)$ gauge fields at low energy (7-07). Seen from gravity, the same branes curve spacetime. Maldacena read that in the low-energy limit the theory on the branes and the spacetime just near the branes describe the same thing Physics. The metric (line element) $ds^2$ is the formula giving the squared distance when the coordinates shift slightly, and the length of a curve is $\int ds$ (for flat space $ds^2=dx_1^2+dx_2^2+\cdots$). The metric made by the branes is

Here $x_\parallel$ are the four coordinates along the branes (in $dx_\parallel^2$ only the time direction has the opposite sign; 2-04), $r$ is the distance from the branes, $\alpha'=\ell_s^2$ is the square of the string length, and $g_s$ is the string coupling. As $r\to0$, $f\approx R^4/r^4$ ($R^4=4\pi g_sN\alpha'^2$), and setting $z=R^2/r$ gives

the product of $AdS_5$ of radius $R$ and $S^5$ Known. There are two ways of writing the correspondence of couplings. In the convention widely used today $g_{\rm YM}^2=4\pi g_s$ and $R^4/\alpha'^2=4\pi g_sN=\lambda$ ($\lambda=g_{\rm YM}^2N$); in the convention of the original paper, $g_{\rm YM}^2=2\pi g_s$, $R^4/\alpha'^2=2\lambda$ Known. The difference is only in the normalisation of the action; today's convention is the normalisation in which the instanton action is $8\pi^2/g_{\rm YM}^2$ (the review of Aharony–Gubser–Maldacena–Ooguri–Oz (3.8), §4.2). (2) of 7-04 ($\frac{1}{2g^2}\int\operatorname{tr}F^2$ with matrix traces, $\lambda=g^2N$) has the same normalisation, so with the $\lambda$ of 7-04, $R^4/\alpha'^2=\lambda$. In this chapter we do not use $\lambda$ directly and write $R^2/\alpha'=\sqrt{4\pi g_sN}$.

Supergravity (the field theory with gravity that remains at low energy in string theory) is reliable when $R\gg\ell_s$, that is $g_sN\gg1$, and quantum effects in the interior correspond to the $1/N$ corrections of the gauge theory (original paper) Physics. Making this compatible with $g_s\ll1$ requires $N\gg g_sN\gg1$.

Example: taking the curvature radius to be 5 times the string length, $N=3$ gives $g_s=16.6$, a strong string coupling, and only at $N=100$ does it come down to 0.5 Computed.

Anti-de Sitter space $AdS_{d+1}$ is, in a flat space whose metric (§02) has signature $(-,-,+,\dots,+)$, the hyperboloid

and the group $SO(2,d)$ preserving (5) is its symmetry as it stands (appendix of Maldacena). Choosing coordinates again gives the form of the first half of (4) (Poincaré coordinates). Below we consider the Euclidean version, moved to imaginary time (2-04). Writing the boundary coordinates as $x=(x_1,\dots,x_d)$,

For $d=1$ it is the upper half-plane, and Figure 1 draws this case. Example: the length going up along $x=0$ from $z=1$ to $z=\varepsilon$ is $\int_\varepsilon^1R\,dz/z=R\ln(1/\varepsilon)$, infinite as $\varepsilon\to0$. The boundary $z=0$ is infinitely far away Known. The dimension of the symmetry is $6\cdot5/2=15$ for $SO(2,4)$ (in the Euclidean version $SO(1,5)$, of the same dimension 15; 1-03), and the four-dimensional conformal transformations number 15 (translations 4, Lorentz 6, dilatation 1, special conformal 4); in the words of Witten (1998a), 'SO(2, d) ... acts on M_d as the conformal group' Known.

In (6), under $(x,z)\mapsto(sx,sz)$ both numerator and denominator are multiplied by $s^2$, and $ds^2$ does not change. This isometry of the interior (a transformation preserving length) is, on the boundary, the dilatation $x\mapsto sx$: "looking at something $s$ times larger on the boundary" and "looking $s$ times deeper" become the same operation. In the words of Susskind–Witten (1998), 'scale size gets transmuted into a spatial dimension' Known.

The geodesics of (6) (the shortest curves joining two points) are semicircles orthogonal to the boundary, and the semicircle hanging from the boundary interval $[-\ell/2,\ell/2]$ goes down to depth $\ell/2$. The length cut off at depth $\varepsilon$ is logarithmic in $\ell$.

The amplitude for a particle of mass $m$ to propagate through the interior is roughly $e^{-mL}$, so exponential decay turns into a power law on the boundary. Susskind–Witten (2.5) follow the same idea, writing it on a model of the sphere in a form without coefficients, $\delta^m/|X_1-X_2|^m$. The exponent $2mR$ is an approximation for large $mR$; exactly it is (9) Physics.

ComputedThe numbers in this figure were computed on this machine.

Cutting near the boundary at $z=\varepsilon$ is an ultraviolet cutoff of the boundary theory. Susskind–Witten write 'the infrared regulator δ in the bulk theory also plays the role of an ultraviolet regulator in the boundary super Yang Mills theory' ($\delta$ corresponds to the $\varepsilon$ of this chapter). They then divided the cut sphere into a lattice of size $\delta$, assumed one degree of freedom per field per lattice cell, and counted the degrees of freedom as $N^2/\delta^3$ (ibid. (3.1)) Physics. In Figure 1 the width of the tiles doubles at each level down, which recalls one step of the renormalisation group, grouping blocks in pairs (6-05). More precisely, it stands at the level where de Boer–Verlinde–Verlinde (1999, JHEP 2000) derived the form of the Callan–Symanzik equation (the formula describing how correlation functions change when the scale is changed; 4-04) from the Hamilton–Jacobi equation describing radial evolution in classical supergravity (an equation set up by regarding the action as a function of the values at the end), and proposed 'We propose a direct correspondence' Physics.

In the prescription of Gubser–Klebanov–Polyakov (1998) and Witten (1998a), the boundary value $\phi_0$ of an interior field $\phi$ is read as the source of a boundary operator $\mathcal O$ (an external field added to the action in the form $\int\phi_0\mathcal O\,d^dx$), and the following are set equal. The right end is the action $I$ of the classical solution in the limit where supergravity can be used ($N,\lambda\to\infty$). The abstract of GKP begins 'We suggest'; it is a proposal Physics.

One consequence is the relation between mass and dimension. For solutions independent of $x$, the equation of a field of mass $m$ is $z^{d+1}(z^{1-d}\phi')'=m^2R^2\phi$ (Witten 1998a (2.37)), and putting in $\phi=z^\alpha$ gives $\alpha(\alpha-d)=m^2R^2$. We write the larger root as $\Delta$ (the letter customary in the literature, distinct from the $\Delta$ of the mass gap).

A solution with $\phi\approx\phi_0z^{d-\Delta}$ near the boundary extends smoothly into the interior. Under the dilatation $(x,z)\mapsto(sx,sz)$, $\phi_0$ is multiplied by $s^{\Delta-d}$, so for $\int\phi_0\mathcal O$ to be invariant the scaling dimension of $\mathcal O$ (the $\Delta$ such that the operator is multiplied by $s^{-\Delta}$ under the dilatation $x\mapsto sx$) must be $\Delta$, and the two-point function is proportional to $|x|^{-2\Delta}$ Physics. The condition for real roots, $m^2R^2\ge-d^2/4$, is the Breitenlohner–Freedman bound, which allows negative squared masses down to this point Known.

Example: if massless, $\Delta=4=d$, and the $\phi_0$ in $\int\phi_0\mathcal O\,d^4x$ is a dimensionless number, that is, it plays the role of a coupling. The dilaton (the scalar whose expectation value fixes the string coupling) corresponds to $\operatorname{tr}F^2$. For the $\mathcal N=4$ theory, Witten (1998a §2.6) states that the operator obtained by differentiating the Lagrangian density with respect to the coupling (which includes $\operatorname{tr}F^2$) corresponds to the dilaton with $m^2=0$, and Csáki–Ooguri–Oz–Terning, citing earlier literature, write 'the operator tr F² in four dimensions corresponds to the dilaton field …' Physics. $\operatorname{tr}F^2$ creates the $0^{++}$ glueball (7-01), so in 7-09 we read glueball masses from the dilaton equation.

The energy $E(L)$ of a heavy quark pair at distance $L$ is read from the Wilson loop (5-01). In the prescription put forward at about the same time by Maldacena (1998) and Rey–Yee (1998, EPJC 2001), the quarks are the endpoints of a string ending on the boundary, and the area (Nambu–Goto) of the string hanging into the interior gives $E$ Physics. In (4) the tension $1/(2\pi\alpha')$ is multiplied by $R^2/z^2$.

The integrand does not contain $x$, so $z^2\sqrt{1+z'^2}=z_0^2$ ($z_0$ is the depth of the bottom) is conserved, giving the right-hand formula. The string goes down to a depth $z_0=0.8346L$ proportional to $L$. Subtracting the divergence at the boundary (the self-energy of point charges) by the amount for two straight strings gives

where the constant agrees between numerical integration and the closed form to 8 digits, and is the same as (4.6)(4.8) of the original paper Computed. The original paper writes 'the energy goes as 1/L, a fact which is determined by conformal invariance. Notice that the energy goes as (gN)^1/2 as opposed to gN which is the perturbative result' Physics.

ComputedThe numbers in this figure were computed on this machine (the shape of the string from a table of the integral (10), $E$ from (11)(12)).

The $1/L$ of (11) is a consequence of conformality, so confinement ($E\approx\sigma L$, 7-01) needs a scale. The crudest model is to cut the interior at depth $z_w$. Polchinski–Strassler (2001, PRL 2002) stated that if there is a gap 'the gravitational redshift has a nonzero lower bound' and used the approximation of cutting the geometry Physics. Minimising (10) in this model, for $L\le L_c=1.19814z_w$ the conformal solution (bottom depth $0.8346L\le z_w$) can be used as it is. For $L>L_c$ the bottom of the string reaches the wall. Where it touches the wall the string is tangent to it (slope 0), so the conserved quantity $z^2\sqrt{1+z'^2}$ becomes $z_w^2$, and the descending part is half of the conformal solution with $z_0=z_w$. The halves on the two sides use up a width $L_c$, and the remaining $L-L_c$ is joined by a straight line along the wall.

A part proportional to $L$ (an area law) appears, and at $L=L_c$ it joins (11) in both value and slope Computed. Example: at $L=8z_w$ in Table 4 it is $+0.892$ against the conformal $-0.029$. As in (12) it has the form $\sigma(L-L_c)$ plus a constant, and for large $L$ it lies on a straight line of slope $\sigma$. The string tension carries a factor $\sqrt{4\pi g_sN}$, while the glueball mass fixed by the box up to the wall ($\sim1/z_w$) does not, so the larger the coupling, the further apart the two scales (the $\lambda$-dependence of the ratio and its sources are in 7-09 §08). This is arithmetic within the model; the actual geometry is treated in 7-09 and 7-10.

"Rigorous" has two meanings: being correct as mathematics closed on one side alone (hyperbolic geometry, Lie groups, ordinary differential equations), and the definitions of both sides agreeing within a model. Neither is a theorem of four-dimensional Yang–Mills theory, and (1) is not proved either.

The last row is a theorem of Witten–Yau (1999). We prepare three terms. An Einstein manifold is a manifold whose Ricci curvature (curvature averaged over directions) is a constant multiple of the metric; here the constant is negative (the space (6) is an example). A conformal boundary (in Penrose's sense) is a boundary at infinity, attached when multiplying the metric by the square of a function vanishing on the boundary makes it extend smoothly to the boundary (for (6), $z^2ds^2$ extends to $z=0$). $H_n(M;\mathbb Z)=0$ means, roughly, that $M$ has no $n$-dimensional "holes". If $M$ is $n+1$-dimensional, $N$ is its $n$-dimensional conformal boundary, $M$ is complete (not cut off partway in any direction), and the scalar curvature of $N$ (curvature further condensed into a single number) is positive, then

(abstract: 'Let M be a complete Einstein manifold of negative curvature ... N must be connected') Known. What this chapter uses is only the conclusion that "the boundary $N$ is in one piece (connected)". AdS/CFT is the motivation, but it is not a theorem about the duality itself. Example: the Physics rows of Table 5 all use (1) as a premise.

The boundary theory of (1) is not pure Yang–Mills theory. It has six scalar fields and fermions, is conformal so its coupling does not run (in contrast with 4-06), and no scale $\Lambda$ is born (4-07). The $N,\lambda\to\infty$ where the gravity computation works is the opposite side from what the problem statement (0-03) asks for: any compact simple group, and a theory that agrees with asymptotic freedom ($\lambda\to0$) at short distances. A conformal theory has no gap: if there is a state of mass $m>0$, dilatation gives states of mass $sm$ (for any $s>0$) too.

$P^2$ is the square of the four-momentum (2-04), and the $1/L$ of (11) is another face of this circumstance Known. Example: the Clay page on the problem still writes of the mass gap 'no proof of this property is known'. There is no example in which holography has shown the mass gap as a mathematical theorem (models with a gap at the level of physics are in 7-09); string theory has no non-perturbative definition, (1) is a conjecture, and the limits lie outside the problem statement, all at once (9-02). The road back to four dimensions is in 7-10, and an example where a gap is visible from a duality under supersymmetry in 7-11.

What this chapter can say

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Next chapter to read: 7-09 The mass gap becomes geometry — reading glueball masses as eigenvalues in the geometry of a shrinking circle.

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