2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicsthe cigar and glueballs

The mass gap becomes geometry — when a circle shrinks and spacetime ends, a gap opens in the spectrum

The boundary theory of AdS is conformal and has no gap (7-08 (14)), and in 7-08 §07 a scale was put in with a wall placed by hand. In this chapter we look at models in which the geometry itself ends at a finite depth. Witten (1998) read two things from this Physics. The area law, from the coefficient of the boundary directions $dx_i^2$ being at least a positive constant in the interior. The gap, from the wave equation in the interior having only positive, discrete eigenvalues $m_1^2\lt m_2^2\lt\cdots$. Said on the boundary, the latter means that two-point functions such as that of $\operatorname{tr}F^2$ become sums of free propagators.

The fraction on the right is the propagator of mass $m$ in three dimensions Known; at long distance the first term dominates, and the correlation length is $1/m_1$ (2-05 §05, 4-11). Example: looking at the two-point function of $\operatorname{tr}F^2$ in the model of §05 (the lowest dilaton mode), in the normalisation of §03 where the temperature is $T=1/\pi$, $m_1=3.404$ and the correlation length is 0.294 Computed. But among all the supergravity fields, another scalar coming from fluctuations of the metric is lighter, $m^2=5.457$ (the dilaton is 11.588) (Brower–Mathur–Tan 2000; §07) Physics. (1) is the leading term in $1/N$, and Witten writes that interactions destroy the discreteness but the gap remains.

Thermal equilibrium at temperature $T$ is the path integral with imaginary time wound on a circle of period $\beta=1/T$ (the inverse temperature; the $\beta$ of this chapter is not the lattice inverse coupling) (2-07 §06). Bosons return to themselves after one turn, and fermions change sign (antiperiodic), so the momenta in the direction of the circle are

Known. Fermions have no $\omega=0$ and drop out at long distances. Witten put the $\mathcal N=4$ theory on $\mathbb R^3\times S^1$ to break supersymmetry, and since the scalars also become heavy at one loop, held that a three-dimensional gauge field remains at long distances Physics. Csáki–Ooguri–Oz–Terning (1998; COOT below) call this $\mathrm{QCD}_3$. As a metaphor, the circle is a mould. More precisely, the radius $R$ of the circle is an ultraviolet cutoff of the three-dimensional theory (7-06 §08). Example: the lowest fermion momentum is $\pi T$, and the lowest non-zero boson momentum $2\pi T$.

If there is a circle on the boundary, the interior becomes an AdS black hole. Witten scaled that solution to obtain the following metric (the overall $R^2$ omitted) Physics.

$b$ is the curvature radius, and $\rho\to\infty$ is the boundary, where it returns to AdS (7-08 §03). At $\rho=b$, $g=0$ and the circle shrinks. Written with the distance $r$ from the tip ($\rho\approx b+r^2/b$),

these are polar coordinates in the plane. But if the period of $\tau$ deviates from $\pi b$, the tip is the vertex of a cone (a singularity). Smoothness fixes the period to $\pi b$. Near the boundary, dividing the metric by $\rho^2$ gives $d\tau^2/b^2+\sum dx_i^2$, so the circumference measured on the boundary is $\pi b/b=\pi$, and the temperature is fixed to $T=1/\pi$ independently of $b$ Known. The numbers of this chapter are written in this normalisation (circumference $\pi$, $T=1/\pi$). "$b=1$" in the tables means that $b$ was set to 1; the eigenvalues of (7) do not depend on $b$, since $b$ disappears on setting $\rho=bu$.

$(\rho,\tau)$ is a cigar-shaped disc. As a metaphor, spacetime "ends there". More precisely, the tip is the centre of the disc. For $b=1$ it can be written $\rho^2=\cosh 2r$, agreeing with numerical integration to 8 digits Computed.

ComputedThe numbers in this figure were computed on this machine (the curves are $\rho^2=\cosh2s$; the table gives values from numerical integration, and closer to the tip $C/(2\pi s)$ comes closer to 1).

Detour: the temperature of a black hole is the same computation

The Wilson loop is fixed as $e^{-\alpha(D)}$ by the regularised area $\alpha(D)$ of a surface in the interior (7-08 §06) Physics. In (3) the surface can only enter $\rho\ge b$, and the coefficient $\rho^2$ of $dx_i^2$ is at least $b^2$, so with $A$ the enclosed area,

and the area law (7-01 §04) comes out (Witten). We computed the area per unit length $E(L)$ of a string joining two points at distance $L$ by the same method as 7-08 (10).

At short distances it is the conformal $1/L$, at long distances $E\approx L-2.000$, and the slope 1 is $\sigma=b^2$ Computed. The string goes down to the bottom $\rho=b$ and lies flat. No wall (7-08 §07) is needed. Example: rescaling the boundary scale by $s$ makes the temperature $T/s$ and multiplies the tension by $1/s^2$, so $\sigma\propto T^2$ (Witten).

$\operatorname{tr}F^2$ corresponds to the dilaton (7-08 §05). Witten studied waves $\phi=f(\rho)e^{ik\cdot x}$ on (3) that do not depend on $\tau$ (waves with momentum in the $\tau$ direction are heavier; a comparison within the same field; §08). The equation is

(for $b=1$, $\rho^3g=\rho^5-\rho$) Physics, where $m$ is the mass of a three-dimensional particle. There are two conditions. Smooth at the tip: one solution diverges as $\log(\rho-b)$, and the smooth solution is unique up to a constant multiple. Normalisable at infinity: of the solutions $f\sim1$ and $f\sim\rho^{-4}$, the latter. The constant component is the source on the boundary (7-08 (8)).

Smoothness at the tip is the condition that the slope measured with the radial coordinate $r$ of polar coordinates in the plane ((4)) is 0. Since $\rho-b\propto r^2$, the slope measured with $\rho$ is not 0; from (7), $f'(1)=-m^2f(1)/4$ ($b=1$), which for the first eigenfunction is $-2.90f(1)$ Computed. The abstract of de Mello Koch et al. (1998) also states this condition: there is no smooth solution whose slope in $\rho$ vanishes at the horizon. The method is shooting: the zeros of the constant component $A(m^2)$ that the smooth solution from the tip leaves at infinity are the eigenvalues.

ComputedThe numbers in this figure were computed on this machine (the bottom is a table of a precise integration; the top is a Runge–Kutta integration of the same formula in the page, agreeing with the table to 4 digits).

Shooting and the Chebyshev collocation method (moved to $u=1/\rho^2$) agree to 7 digits and reproduce the 6 rows of Table 1 of COOT Computed. The ratio to the temperature is $m_1/T=10.69$.

Positive. Multiplying (7) by $\rho f$ and integrating by parts, the boundary terms at the tip ($\rho^3g=0$) and at infinity vanish and

If the numerator is 0 then $f$ is constant, but a constant is not normalisable ($\int^\infty\rho\,d\rho=\infty$), so $m^2\gt0$ Known. The minimum of the right-hand side is $m_1^2$, and the gap is the best constant of a Poincaré inequality (1-10, 2-05 §08).

Discrete. Changing the variable to $t=\int_1^\rho d\rho'/\sqrt{\rho'^4-1}$ and the unknown function to $\psi=(\rho^6-\rho^2)^{1/4}f$ (a Liouville transformation; $b=1$), (7) becomes the Schrödinger equation on the interval $[0,T_L]$

Known. $V=(\rho^6-\rho^2)^{-1/4}\,\frac{d^2}{dt^2}(\rho^6-\rho^2)^{1/4}$ diverges at both ends: $V\approx-1/(4t^2)$ at the tip, $V\approx(15/4)/(T_L-t)^2$ at the end at infinity Computed. Even so, the length of the box $T_L$ is finite, so the eigenvalues are discrete, just as for a particle in a box.

The type of each end also fixes the phase of the approximation. At an end where the potential is $(\nu^2-\tfrac14)/u^2$ ($u$ the distance from the end), the allowed solution is $\sqrt u\,J_\nu(mu)$ ($J_\nu$ the Bessel function of order $\nu$), which away from the end oscillates like $\cos(mu-\nu\pi/2-\pi/4)$. The tip has $\nu=0$ and brings in a phase $\pi/4$, the end at infinity has $\nu=2$ and brings in $5\pi/4$, so the $n$-th eigenvalue is $m_nT_L\approx\pi(n-1)+\pi/4+5\pi/4=\pi(n+\tfrac12)$, that is $m_n\approx(\pi/T_L)(n+\tfrac12)$. At $n=6$ the difference shrinks to 0.6% in $m^2$ (0.3% in $m$) Computed. The $6n(n+1)$ of Table 2 is an approximate formula from exact WKB analysis, which COOT cite as a private communication from Takei–Koike. Exact WKB analysis is a theory that handles rigorously divergent series expanded in a large parameter by summing them with Borel's method (4-12), and this formula assumes their summability (COOT note that there is no proof in this case).

In AdS the length $\int_0d\rho/\rho^2$ down to the lower end $\rho=0$ is infinite, solutions oscillate deep inside, and the spectrum is continuous from 0 (Witten). "Spacetime ends" here means "the Liouville length is finite". In the five-dimensional models of Gürsoy–Kiritsis–Nitti (2007) too, it is stated, by analysis within the model, that for the exponent $\alpha$ of the growth of the potential deep inside, confinement and a gap hold for $\alpha\ge1$, and a purely discrete spectrum for $\alpha\gt1$ (at $\alpha=1$, a continuous spectrum above the gap). The same paper also has the sentence 'confinement and discrete spectra imply each other' Physics.

What supergravity produces are numbers in units of a single scale, so COOT matched the lowest dilaton state to the $0^{++}$ of the lattice (5-08) (in three dimensions, the lattice $N\to\infty$ value 4.065, printed as 4.07 in the table of COOT) and compared the rest. $0^{--}$ comes from the equation of a two-form field.

Taking the dilaton as the reference, the ratio $0^{--}/0^{++}=1.50$ agrees with the lattice $1.45\pm0.03$ to 3%, and the excited states come out 14% and 24% higher than the $N\to\infty$ lattice Computed. Which scalar to take as the reference is treated in "Later analysis" below.

Four dimensions is the D4-brane background of 7-07 §09 (radius $R_2$ of the antiperiodic circle). The equation of the dilaton-type scalar of COOT, with $y^2=\rho$ (the $\lambda$ of the original paper) and in units placing the tip at $y=1$ (then $M_{\rm KK}=1/R_2=3$), is the following (the solutions at infinity are $1$ and $y^{-9}$) Physics.

$0^{-+}/0^{++}$ is 1.14 in COOT and $1.36\pm0.32$ on the lattice of the time. COOT obtain $0^{-+}$ from a component of the 11-dimensional three-form field. Solving that equation on this machine, the lowest value is $m^2=53.376$ (1.93 GeV if the $0^{++}$ of (10) is 1.61 GeV, ratio 1.20), agreeing with the value for the same equation (N4) in Table 2 of Brower–Mathur–Tan (2000; BMT below) Computed. The 1.83 GeV in the table of COOT cannot be reproduced from this equation. For the lattice $0^{++*}$, a note says it may be a bound state of two glueballs.

Later analysis — which scalar is lightest

BMT turned all the supergravity fields (metric, three-form, size of the sphere) into radial eigenvalue problems. We list the four-dimensional scalar-type modes. "Assignment" is the quantum numbers $J^{PC}$ read from the operator to which the mode couples.

The T4 equation differs from (10) only in the first-order term.

In the later analysis, the dilaton-type $0^{++}$ coupling to $\operatorname{tr}F^2$ is degenerate with $2^{++}$ at $1.567\,M_{\rm KK}$, and the lightest $0^{++}$ is another mode, S4, at $0.901\,M_{\rm KK}$ Physics. The 2.03 in Table 4 is the value from formula (10) of COOT. In three dimensions too there is a scalar (S3 of BMT, 5.457) lighter than the dilaton (11.588) Computed. BMT write 'the lowest mass scalar comes from the gravitational multiplet, not the dilaton', and call the agreement of ratios to a few per cent 'a misconception'. At the same time they write that the overall pattern of spins and masses corresponds well with the lattice results Physics.

So the ratios move depending on which modes are taken as numerator and denominator. The three-dimensional $0^{--}/0^{++}$ is 2.19 with S3 as the reference. In four dimensions, the three-form value of COOT divided by T4 is 1.55, and with the assignment of BMT ($0^{-+}$ is V4) V4/T4 is 1.20 and V4/S4 is 2.09 Computed. For $SU(3)$ on today's lattice, $0^{-+}/0^{++}=1.549(16)$ and $0^{++*}/0^{++}=1.720(16)$ (the table of 7-01). The holographic ratios with the reference of COOT (1.14, 1.20, 1.48) come out 26%, 23% and 14% smaller respectively Computed.

Glueball masses are fixed by an eigenvalue problem and do not depend on the coupling $\lambda=g_{\rm YM}^2N$, but the string tension is proportional to $\lambda$. In the D4 model, from formula (2.7) of Kruczenski–Mateos–Myers–Winters (2003) Physics

(it may be off by a factor of 2 depending on the convention for $\lambda$). The value changes depending on which is put into $m_{0^{++}}/M_{\rm KK}$ on the right: 2.03 from formula (10) of COOT, 1.567 from T4 of BMT (dilaton type), or 0.901 from S4 (the lightest).

At $\lambda\gg1$, where supergravity is reliable, the ratio goes to 0 as $\lambda^{-1/2}$ Computed. The $\lambda$ matching 3.072 is 3.7 to 19, moving by a factor of about 5 depending on which mode is taken, and in every case it is a fit, not a prediction. Klebanov–Strassler too acknowledge a 'large separation of scales' between the tension and the glueballs. On the lattice it is 3 to 4 (7-03).

In models cut off by a wall, the $\lambda$-dependence of the ratio is $\lambda^{-1/4}$. With a hard wall ((12) of 7-08) the tension is $\sigma=\sqrt\lambda/(2\pi z_w^2)$ and the glueball mass $\sim1/z_w$ (Boschi-Filho–Braga 2003, imposing a Dirichlet condition at the wall). With a soft wall the tension is $\frac{R^2}{\alpha'}\frac{e}{4\pi}c$, from (19) of Andreev–Zakharov (2006) and $g=R^2/\alpha'$, and the squared mass is proportional to $c$ ((15) of Karch–Katz–Son–Stephanov 2006; the case of mesons). In both the tension is proportional to $\sqrt\lambda$, so the ratio goes as $\lambda^{-1/4}$; the difference from the $\lambda^{-1/2}$ of the D4 model comes from the tension in (11) being proportional to $\lambda$ Physics.

The other is the scale of the circle. In three dimensions, in the normalisation of §03 the circumference on the boundary is $\pi$, so $R=1/2$ and $m_1=3.404=1.70/R$ (COOT use another normalisation, placing the horizon at $b=1/(2R)$). In four dimensions $0^{++}$ is $0.90/R_2$ to $2.03/R_2$, and whichever mode is taken, the glueballs are of the same order as the circle Computed. The KK modes on the sphere (7-06 §02) are also 'of the same order' (COOT), and the states regarded as glueballs are chosen by hand. Removing them needs $\lambda\to0$, which is the 'opposite' of the direction in which supergravity works. Witten himself called the computation at small $\eta=g_4^2N$ 'a tall order' Physics. On the order of limits, see 7-10.

Klebanov–Strassler (2000) placed D3-branes (7-08 §02) on a six-dimensional cone (the conifold) and built a background in which the singularity at the tip is deformed in the IR, leaving a finite sphere.

Example: restricting $z_i$ to real numbers, the right is an $S^3$ of radius $\epsilon$, the sphere left at the tip Known. The flux of the five-form field through a cross-section of the cone (a quantity corresponding to the number of D3-branes) changes logarithmically with the radius $r$: $\mathcal K(r)=N+a\,g_sM^2\ln(r/r_0)$. $g_s$ is the string coupling (7-08 (3)), $M$ the number of "fractional branes" wrapped on the sphere in the cone, and $a$ a constant of order 1 (original paper). The authors read this decrease as a sequence (a cascade) in which the gauge group $SU(N+M)\times SU(N)$ shrinks step by step as $N\to N-M$ Physics. The deformation of the tip is held to be the breaking $\mathbb Z_{2M}\to\mathbb Z_2$ of the R-symmetry (the chiral symmetry of this theory) of the $\mathcal N=1$ supersymmetric Yang–Mills remaining in the IR, made visible as geometry Physics. At the tip the coefficient of $dx^2$ is finite, so an area law comes out as in §04. But the theory is $\mathcal N=1$ supersymmetric, the UV cascade 'goes on forever', and the KK modes of $S^3$ and the glueballs 'both scale as $m/(g_sM)$'.

In every model the gap is shown as "a linear eigenvalue problem on a fixed background", and the part saying that the background describes pure Yang–Mills theory on $\mathbb R^4$ remains a conjecture. The Clay page too writes 'no proof of this property is known'. The distance from the problem statement (0-03) is in 7-10, the seams in 9-02.

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