2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicsstring theory and extra dimensions

The whole book　Contents and reading orders

String theory and extra dimensions — 26, 10 and 11 are numbers that come from conditions of consistency

What this chapter shows — That quantising a string as a fundamental object takes away the free choice of the dimension of spacetime. How 26 for the bosonic string is derived in two ways (the first excited state / the central charge), and 10 for the superstring by the same balance. That 11 is a number that comes from supergravity.

Prerequisite chapters — 7-04 Large N. We use the zero-point energy and $-1/12$ of 7-03. Group representations (1-02), Lorentz transformations (2-04) and the harmonic oscillator (2-06) are written on the spot as far as needed.

Said first — The derivations are at the level of physics. The two-dimensional computations on the world sheet are exact, but string theory has no non-perturbative definition, and the existence of M-theory is a hypothesis. The story of extra dimensions shows nothing about the construction of four-dimensional Yang–Mills or about the mass gap. This site shows neither the mass gap nor the construction of the continuum limit.

The order of this chapter

Making the string a fundamental object — how it differs from the string of the tube
Counting the transverse oscillations — the mass formula
Why 26 (I) — the first excited state is massless
Counting the levels — the tachyon and the growth (Detour: partition numbers and Hardy–Ramanujan, Detour: the growth of states and the Hagedorn temperature)
Why 26 (II) — the balance of the central charge
Why 10 — world-sheet supersymmetry (Detour: 26 − 10 = 16 and the heterotic string)
Why 11 — the number of supergravity
The "critical dimension" is a critical central charge — it need not be geometry
Back to Yang–Mills — the string of the tube, and holography

Making the string a fundamental object

7-03 treated the tube of confinement as a single string, and in 7-04 §05 $1/N$ played the role of the string coupling. String theory too is said to have begun from models of mesons (dual resonance models). The scattering amplitude of Veneziano (1968) is often cited as the starting point; for example, the abstract of Neveu–Schwarz (1971) opens with 'A new dual-resonance model of mesons is constructed …' Known. For mesons, spin $J$ and the square of the mass line up on a straight line (Regge trajectories). If a string of tension $T$ and length $2R$ rotates with angular velocity $1/R$ , its ends move at the speed of light, and integrating the energy and the angular momentum (2-04) gives $E=\pi TR$ and $J=\pi TR^2/2$ , that is

J=\alpha' E^2,\qquad \alpha'=\frac{1}{2\pi T}. \tag{1}

For the quantised open string, the maximum spin at level $N$ is $J_{\max}=\alpha'M^2+1$ (Tong §3.1.3). Putting the $\sqrt\sigma=485$ MeV of 7-01 into $T$ gives $\alpha'=0.677\ \mathrm{GeV}^{-2}$ , $\sqrt{\alpha'}\approx0.16$ fm Computed. The fundamental string of this chapter is not the effective theory of a tube; it is required to be quantisable without contradiction at however short a distance. Then the dimension $D$ of spacetime stops being a free input. In this chapter, following the string literature, we write $D$ for the dimension of spacetime. It is the same as the $d$ of 7-03 and of the table of notation in the appendix, not the dimension of space alone.

Counting the transverse oscillations: the mass formula

A string sweeps out a surface (the world sheet) in spacetime $\mathbb{R}^{1,D-1}$ . In light-cone gauge we use the light-cone coordinates $X^\pm=(X^0\pm X^{D-1})/\sqrt2$ , recombining time $X^0$ and one longitudinal direction $X^{D-1}$ , and make $X^+$ proportional to the world-sheet time $\tau$ ( $X^+=x^++\alpha'p^+\tau$ ; $x^+$ and $p^+$ are the $+$ components of the position and momentum of the centre of mass of the string; Tong (2.13)). The oscillations of $X^-$ do not vanish, but they are fixed by the transverse oscillations through the constraint equations. Only the $D-2$ transverse directions oscillate independently (Goddard–Goldstone–Rebbi–Thorn 1973) Known. This is the same counting as the transverse displacement $h\in\mathbb{R}^{d-2}$ of 7-03 ( $d=D$ ). The $n$ -th oscillation in direction $i$ is a harmonic oscillator (2-06), and each application of the creation operator $\alpha^i_{-n}$ raises the level $N$ by $n$ . The mass of the open string is

\alpha' M^2 = N - a \tag{2}

where $-a$ is the zero-point energy (the sum of $\frac12 n$ over all oscillators). Its finite part is the $-1/12$ of 7-03 (7-03 §04), so

a = -\frac{D-2}{2}\sum_{n\ge1}n\ \longrightarrow\ -\frac{D-2}{2}\cdot\Big(-\frac{1}{12}\Big)=\frac{D-2}{24}. \tag{3}

The Lüscher term $-\pi(D-2)/(24R)$ of 7-03 and the $(D-2)/24$ of the Nambu–Goto levels (7-03 §06) are this same number Known. Tong admits that this treatment of the sum is 'famously unconvincing', and derives it again by the central-charge method, the same as §05 of this chapter (Tong §4.4.1). The closed string has right-moving oscillations $\alpha$ and left-moving oscillations $\tilde\alpha$ , and its levels satisfy $N=\tilde N$ (Tong (2.26)).

Why 26 (I): the first excited state must be massless

The internal states of a particle form a representation of the rotations that leave its momentum unchanged (the little group) (1-02; Wigner 1939). For mass $M\gt0$ we can go to the rest frame and the little group is $SO(D-1)$ ; for zero mass it is the $SO(D-2)$ that preserves the momentum $(p,0,\dots,0,p)$ (precisely, it is the group $ISO(D-2)$ of rotations and translations in $D-2$ dimensions, but what acts on finitely many states is the rotation part $SO(D-2)$ ). A photon in four dimensions has 2 polarisations, a massive spin-1 particle 3 Known. The level-1 states $\alpha^i_{-1}|0\rangle$ of the open string are a transverse vector with $D-2$ states, one short of the $D-1$ components a massive vector needs. To keep Lorentz symmetry it must be massless, and

\alpha'M^2 = 1-a = 0 \ \Longrightarrow\ a=1 \ \overset{(3)}{\Longrightarrow}\ \frac{D-2}{24}=1 \ \Longrightarrow\ D=26. \tag{4}

Tong makes the same argument for the first excited state of the closed string ( $(D-2)^2$ states) and writes 'these states will have to be massless. And this is only the case if the dimension of spacetime is D = 26' Physics.

Checking with the Lorentz algebra

More carefully, we check whether the conserved quantities $M^{\mu\nu}$ that generate Lorentz transformations (rotations and boosts) satisfy the relations of the Lorentz algebra in light-cone gauge too. $M^{\mu\nu}$ includes the angular momentum carried by the oscillations of the string. The crux is $[M^{i-},M^{j-}]=0$ , where $i,j$ are transverse directions and $-$ is the light-cone direction $X^-$ (Tong §2.4). Since the oscillations of $X^-$ are fixed by the transverse oscillations, this commutator can be written with the transverse oscillators.

[M^{i-},M^{j-}]\ \propto\ \sum_{n>0}\Big[\Big(\frac{D-2}{24}-1\Big)n+\frac{1}{n}\Big(a-\frac{D-2}{24}\Big)\Big]\big(\alpha^i_{-n}\alpha^j_n-\alpha^j_{-n}\alpha^i_n\big)+(\alpha\leftrightarrow\tilde\alpha) \tag{5}

The proportionality factor $2/(p^+)^2$ is omitted. This is the formula for the closed string, and $(\alpha\leftrightarrow\tilde\alpha)$ is a term of the same form for the left-moving oscillations. It vanishes for every $n$ only when $D=26$ and $a=1$ Known. The original source is Goddard–Goldstone–Rebbi–Thorn (1973), whose abstract also states that the angular momentum operators generate the Lorentz group $O(D-1,1)$ 'only when the first excited state is a photon, i.e., a spin-one massless state, and when D = 26'. The first excited state of the closed string has $24^2=576=299+276+1$ states Computed, and Tong assigns these to the symmetric traceless, antisymmetric and trace fields $G_{\mu\nu},B_{\mu\nu},\Phi$ of $SO(24)$ , reading the first as a massless spin 2, that is, the graviton Physics.

Figure 1 Varying D moves the levels

\alpha'M^2=N-a

up and down (the length of a line is the logarithm of the number of states). The thick level 1 (1/2 for the superstring) sits on

M^2=0

only for D=26 for the bosonic string and D=10 for the superstring

D Superstring (NS)

D = 26, a = 1, level 1: 24 states, α′M² = 0 → consistent as a massless vector (SO(24))

ComputedThe numbers in this figure were computed on this machine. If JavaScript is disabled, see (4) and Table 2.

Counting the levels: the tachyon and the growth of the number of states

$N=0$ of the open string at $D=26$ has $\alpha'M^2=-1$ : a tachyon, with negative squared mass (Tong §3.1.1). It is a field whose mass term has the opposite sign (2-02), meaning that we are expanding around a maximum of the potential Physics. The number of states $d_N$ at level $N$ is the number of ways of splitting $N$ into a sum of "positive integers in 24 colours".

\sum_{N\ge0} d_N\,q^N=\prod_{n\ge1}(1-q^n)^{-24} \tag{6}

The 324 at level 2 is the dimension of the symmetric traceless rank-2 tensor of $SO(25)$ , and the 3200 at level 3 is $2900+300$ (symmetric traceless rank 3 and antisymmetric rank 2) Computed. According to Tong all excitations fit into representations of $SO(D-1)$ , and the condition is on level 1 alone. That the numbers at levels 2 and 3 match sums of dimensions of representations of $SO(D-1)$ we checked for $D=4$ – $39$ Computed. Both are polynomials in $D$ of degree at most three, so this makes them identities valid for any $D$ .

Table 2 Levels of the open bosonic string ( $D=26$ ). The ratio is to the Bessel form of the detour ( $n=N-1$ )
N	$\alpha'M^2$	$d_N$	Ratio
0	−1	1
1	0	24
2	1	324	1.000009
3	2	3200	0.999998
10	9	6.392×10⁸	1.00000000
400	399	1.446×10⁹¹	1.00000000

ComputedThe numbers in this table were computed on this machine.

Detour: partition numbers and Hardy–Ramanujan

Detour

A subsection you may skip.

With one colour the coefficients of (6) are the partition numbers $p(N)$ ( $p(4)=5$ from $4=3+1=2+2=2+1+1=1+1+1+1$ ), and Hardy and Ramanujan (1918) gave $p(N)\sim\frac{1}{4\sqrt3\,N}e^{\pi\sqrt{2N/3}}$ Known. For (6) with 24 colours, setting $n=N-1$ gives $d_N\approx\frac{1}{\sqrt2}n^{-27/4}e^{4\pi\sqrt n}$ , but the approach is slow, and the ratio is still 0.71 at $N=400$ . What fits well is the form $2\pi n^{-13/2}I_{13}(4\pi\sqrt n)$ using the modified Bessel function $I_{13}$ , which already agrees to 5 digits at $N=2$ (Table 2) Computed. $I_\nu(x)$ is a function that behaves like $e^x/\sqrt{2\pi x}$ when $x$ is large. Putting this in gives $2\pi n^{-13/2}\,e^{4\pi\sqrt n}/\sqrt{8\pi^2\sqrt n}=\frac{1}{\sqrt2}n^{-27/4}e^{4\pi\sqrt n}$ , and the two formulas agree at large $n$ . The ratio 0.71 is the error from replacing $I_{13}$ by this approximation.

Detour: the growth of states and the Hagedorn temperature

Detour

A subsection you may skip.

An open string at level $N$ has $\alpha'M^2=N-1$ , so setting $n=N-1$ as in the detour above gives $\alpha'M^2=n$ , that is $M=\sqrt{n/\alpha'}$ . From the asymptotic form of the detour above, the number of states grows at the rate $e^{4\pi\sqrt n}=e^{4\pi\sqrt{\alpha'}M}$ (the power of $n$ in front changes slowly compared with the exponential). Write $\Theta$ for the temperature (to distinguish it from the tension $T$ ; in units with Boltzmann's constant 1). The weight of a state of mass $M$ is the Boltzmann factor $e^{-M/\Theta}$ (2-08), so the product of number and weight has the form $e^{(4\pi\sqrt{\alpha'}-1/\Theta)M}$ , and for $\Theta\gt\Theta_H=1/(4\pi\sqrt{\alpha'})$ it grows with $M$ and the sum diverges. This $\Theta_H$ is the Hagedorn temperature Physics. Putting $d=D=26$ and $\sigma=1/(2\pi\alpha')$ into the Hagedorn length $L_H$ of 7-03 §06 ( $\sigma L_H^2=\pi(d-2)/3$ , where $\sigma$ is the tension, the $T$ of this chapter) gives $L_H=4\pi\sqrt{\alpha'}$ , and its reciprocal $1/L_H$ coincides with $\Theta_H$ Computed. It is the same reading as in 7-03, where $1/L$ became the temperature when the periodic direction was regarded as time.

Why 26 (II): the balance of the central charge

The second derivation looks at the $D$ fields $X^\mu$ on the world sheet as two-dimensional free scalar fields (2-02). A two-dimensional theory whose form does not change under a change of scale (a conformal field theory) has a central charge $c$ , and one free scalar has $c=1$ . More precisely, a conformal field theory is a theory invariant under angle-preserving maps (conformal transformations), of which changes of scale are a part. In two dimensions there are infinitely many such maps, and this constrains the theory strongly. $c$ appears in the ground-state energy on a cylinder of circumference $L$ (when the left and right $c$ are equal).

E_0(L)=-\frac{\pi c}{6L} \tag{7}

(Blöte–Cardy–Nightingale 1986, Affleck 1986; Tong §4.4.1) Known. The $-\pi(D-2)/(6L)$ of the closed tube of 7-03 is the case $c=D-2$ , and about the $-\pi c/(24R)$ of a tube with ends Tong writes 'c counts the number of degrees of freedom of the QCD flux tube'. The Lüscher coefficient measures the central charge of the tube.

In Polyakov's formulation the metric of the world sheet is also a field to be integrated, and changes of coordinates and changes of scale of the metric (Weyl transformations) are gauge symmetries. Fixing the gauge brings in auxiliary fields (ghosts; the same mechanism as Faddeev–Popov in 4-08), with central charge $-26$ . A quantum breaking of a gauge symmetry (an anomaly) is not allowed, so the total must be 0 (Tong §5.3) Physics.

c_{\text{matter}}+c_{\text{ghost}} = D\cdot1+(-26)=0\ \Longrightarrow\ D=26 \tag{8}

The $D-2$ fields and $-1/12$ in light cone, and the $D$ fields of the covariant form with the ghosts' $-26$ , select the same 26. More precisely, (8) is a condition only as long as the scale of the metric (Weyl) is kept as a gauge symmetry. For $D\ne26$ the scale degree of freedom does not disappear but remains, and behaves as a new field called the Liouville field (the non-critical string of Tong §5.3.2). The abstract of Polyakov (1981) too writes 'At D = 26 the string dynamics is that of harmonic oscillators …, otherwise it is described by the nonlinear integrable theory', treating the $D\ne26$ side as a different theory.

Why 10: world-sheet supersymmetry

The bosonic string has no fermions (relatives of the electron and the quarks), and has a tachyon. Add to the world sheet fermions $\psi^\mu$ as partners of $X^\mu$ ( $c=1/2$ each), make the world-sheet supersymmetry that exchanges the two a gauge symmetry, and the ghosts add central charge $+11$ .

D\Big(1+\frac12\Big)+(-26+11)=0\ \Longrightarrow\ D=10 \tag{9}

In Tong §5.3.1, 'c = 26 − 11 = 15' and 'D(1 + 1/2) = 15' Physics. We check it in light cone too. Depending on the boundary condition, the fermion oscillations take half-integer labels $r=\frac12,\frac32,\dots$ (Neveu–Schwarz, NS) or integer ones (Ramond, R), and their zero-point energy has the opposite sign, $-\frac12\sum r$ . The finite parts are in Table 1.

Table 1 The sum minus $1/\varepsilon^2$ (cutoff $e^{-\varepsilon r}$ )
ε	Integer $\sum n\,e^{-\varepsilon n}$	Half-integer $\sum r\,e^{-\varepsilon r}$
0.1	−0.0832917	0.0416302
0.001	−0.0833333	0.0416667
→ 0	−1/12	+1/24

ComputedThe numbers in this table were computed on this machine (closed forms checked against direct sums).

The zero-point energy per transverse direction is $-\frac1{24}-\frac1{48}=-\frac1{16}$ for NS and $-\frac1{24}+\frac1{24}=0$ for R Computed. The NS levels are $\alpha'M^2=N-\frac{D-2}{16}$ ( $N$ also takes half-integers), and the first excited state $\psi^i_{-1/2}|0\rangle$ is again a vector with $D-2$ components. The massless condition $\frac12-\frac{D-2}{16}=0$ gives $D=10$ , agreeing with (9) ("Superstring" in Figure 1). The tachyon of the NS ground state, $\alpha'M^2=-\frac12$ , is said to be removed by a rule that discards part of the states (attributed to Gliozzi–Scherk–Olive 1977) Physics.

Detour: 26 − 10 = 16 and the heterotic string

Detour

A subsection you may skip.

Green–Schwarz (1984) named $SO(32)$ and $E_8\times E_8$ as the groups for which the gauge and gravitational anomalies in 10 dimensions cancel, and Gross–Harvey–Martinec–Rohm (1985) built the heterotic string as 'a chiral combination of the closed D=26 bosonic and D=10 fermionic strings' Known. The difference between 26 and 10 is 16, and $SO(32)$ and $E_8\times E_8$ both have rank 16 and dimension 496 ( $E_8$ has 240 roots and rank 8, giving 248; 1-04) Computed. How this 16 becomes a gauge group is in 7-07.

Why 11: a number of supergravity, not of strings

11 is not a critical dimension of strings. The abstract of Cremmer–Julia–Scherk (1978) reads 'We present the action and transformation laws of supergravity in 11 dimensions' Known. Supergravity is a field theory containing the graviton and its supersymmetric partner (the gravitino), and that 11 is its upper limit is said to follow from the classification of Nahm (1978).

The skeleton is this Physics. Supercharges are the generators of the symmetry that exchanges bosons and fermions (supersymmetry), and $Q$ is the number of their real components. With $Q$ supercharges, the range of helicities (the amount of rotation about the direction of motion) of a massless multiplet is $Q/8$ , and for spin at most 2 (range at most 4) $Q\le32$ . 11 dimensions is said to be the last dimension that fits within this.

The fields of 11-dimensional supergravity are the graviton, a 3-form and the gravitino. A 3-form is an antisymmetric field with three indices, the electromagnetic potential $A_\mu$ (2-03) with three indices instead of one. Counting the degrees of freedom with the little group $SO(9)$ (the $SO(D-2)$ of $D=11$ ), the 128 bosons, graviton $45-1=44$ and 3-form $\binom93=84$ , balance the gravitino's $9\cdot16-16=128$ Computed. In counting the gravitino we used, only as a number, that a spinor of $SO(9)$ (the quantity carrying the internal degrees of freedom of a fermion) has 16 components.

Witten (1995) argued in the abstract 'It is argued … that eleven-dimensional supergravity arises as a low energy limit of the ten-dimensional Type IIA superstring'. The IIA string is a kind of closed superstring in 10 dimensions in which the supersymmetries carried by the right- and left-moving oscillations have opposite handedness (chirality). With $\lambda$ the coupling of the IIA string, the radius of the 11th circle in 11-dimensional units is

r(\lambda)=\lambda^{2/3} \tag{10}

(formula (2.13) of that paper) Physics. At weak coupling the circle shrinks, and states with momentum $n$ around the circle have mass $\sim|n|/\lambda$ in the string metric (formula (2.14)): heavy and not seen. As a metaphor, the 10 dimensions of the string are how a thin 11-dimensional tube looks from afar. More precisely, this is a conjecture from low-energy arguments, and the IIA string at strong coupling has no independent definition. Hořava–Witten (manuscript 1995, published 1996) proposed a link between the heterotic string and 11 dimensions in the form 'We propose'. The following paper (1996) argued that the gauge group is fixed to one $E_8$ for each boundary, wrote that this gives new material testing 'the hypothesis that there is a consistent quantum M-theory', and treated the existence of M-theory as a hypothesis. This 11-dimensional theory (M-theory) has no complete definition.

The "critical dimension" is a critical central charge: extra dimensions need not be geometry

What (8)(9) require is not a dimension but a central charge. Tong writes 'the "critical dimension" of string theory is something of a misnomer: it is really a "critical central charge"' Known. For flat four-dimensional spacetime, for the bosonic string it suffices to add any conformal field theory with $c=22$ to the $c=4$ of $X^0,\dots,X^3$ , and Tong continues 'It is what we really mean when we talk about the "extra hidden dimensions of string theory"'. For the superstring flat four dimensions give $c=6$ and the rest is $c=9$ Computed. As a metaphor, only the total of the ledger is fixed and the items are free. More precisely, the internal theory must also satisfy other conditions (world-sheet supersymmetry and so on), and it can be read as geometry, as a manifold, only in special cases. How the extra dimensions then become invisible is in 7-06, and how gauge fields come out in 7-07.

Figure 2 Varying the number of flat dimensions

d_{\text{flat}}

changes the breakdown of the central-charge ledger. The internal theory (hatched) takes up what is removed, and the total stays 0

Number of flat dimensions Superstring

Bosonic string, flat 4 dims c = 4, internal c = 22, ghosts c = −26, total 0

ComputedThe arithmetic of this figure was checked on this machine. If JavaScript is disabled, read it as $d_{\text{flat}}+c_{\text{int}}=26$ (for the superstring $\frac32d_{\text{flat}}+c_{\text{int}}=15$ ).

Back to Yang–Mills: the four-dimensional tube, and the dimensions of holography

Then is the string of the four-dimensional tube of 7-03, not being 26-dimensional, inconsistent? The abstract of Polchinski–Strominger (1991) states that the long-distance effective theory of four-dimensional strings such as 'QCD strings' is 'an interacting Poincaré-invariant conformal field theory with four bosons and c=26', and showed the compatibility to second order in perturbation Physics. The 4 bosons, which would have $c=4$ if free, carry $c=26$ through their interaction. What the critical dimension constrains is the fundamental string; the effective theory of a long tube is consistent as it is, in four dimensions. But the effective theory breaks down at short distances, and there is no rigorous derivation (7-03 §06).

The extra dimensions of the fundamental string return to Yang–Mills through holography (7-08), and the breakdown differs from model to model. Maldacena's $AdS_5\times S^5$ ( $AdS_5$ is a five-dimensional spacetime of constant negative curvature, with the four-dimensional directions and a radial direction; $S^5$ is the five-sphere) is 10 = 4 + radial + $S^5$ ; Witten 1998 §4 is 11 = 4 + two circles + radial + $S^4$ (one circle fewer, 10, on reducing to IIA); the improved holographic QCD of Gürsoy–Kiritsis is 5 = 4 + radial. The dimensions that make the gap also differ by model: in Witten's type, a two-dimensional cigar formed by the radial direction and a shrinking circle; in the bottom-up type, a single radial direction and an infrared singularity (7-09) Physics. These extra dimensions are tools of a dual description, not claimed to be real dimensions, and are a separate matter from the experimental lower bounds (7-06).

Seam

(1) String theory has no non-perturbative definition, and the very existence of M-theory is a hypothesis. (2) Holographic computations work on the side of $N\to\infty$ and 't Hooft coupling $\to\infty$ , the opposite direction to the limit that returns to four-dimensional pure Yang–Mills (7-10). (3) The problem statement (Jaffe–Witten) writes 'at present we do not know' of a non-trivial relativistic field theory in four dimensions satisfying axioms such as Wightman's (4-09). The road of extra dimensions does not construct a four-dimensional theory in this sense either.

What this chapter can say

What is said	What is not said
KnownThe counting of (3)(5) (as a two-dimensional world-sheet computation in light cone).	A non-perturbative definition of string theory. The existence of M-theory.
PhysicsThe reading of (4) (reading the first excited state as massless) and the balance of (8)(9) (the central charge of the ghosts and the anomaly argument).	The treatment of the non-critical string (the Liouville field) for $D\ne26$ .
ComputedTables 1 and 2, Figures 1 and 2, 576, 496, 128, the Hagedorn temperature (the $1/L_H$ of 7-03), $\alpha'$ .	Why the Bessel form of Table 2 fits so well.
PhysicsThe readings of the graviton and the tachyon, the relation between 11 dimensions and IIA, Hořava–Witten (a proposal).	Checking "11 is the upper limit" in the original source.
PhysicsThat the effective theory of the four-dimensional tube is consistent with $c=26$ (to second order in perturbation). The breakdown of the dimensions of the models.	Consistency of the effective string at all orders. That a tube exists in four-dimensional Yang–Mills. Constructing a four-dimensional theory from extra dimensions. This site shows neither the mass gap nor the construction of the continuum limit.

Sources and reproduction

Item	Kind	Source / tools
$J_{\max}$ , (2)–(9), each quotation	Known, Physics	D. Tong, "String Theory", arXiv:0908.0333 §2.2–2.4, §3.1, §4.4.1, §5.3 (including the non-critical string of §5.3.2). Text checked.
Light-cone quantisation and $D=26$ (the original source of (4)(5)) / 'At D = 26 ...'	Known, Physics	P. Goddard, J. Goldstone, C. Rebbi, C. B. Thorn, Nucl. Phys. B56 (1973) 109 / A. M. Polyakov, Phys. Lett. B103 (1981) 207. Abstracts checked.
Meson models, NS and R / the rule removing the tachyon	Known, Physics	G. Veneziano, Nuovo Cim. A57 (1968) 190 (bibliographic data only; its standing as the starting point of dual resonance models not checked) / A. Neveu, J. H. Schwarz, Nucl. Phys. B31 (1971) 86 / P. Ramond, Phys. Rev. D3 (1971) 2415 (abstracts checked) / F. Gliozzi, J. Scherk, D. Olive, Nucl. Phys. B122 (1977) 253 (abstract only; the attribution of the projection not checked).
The little group / formula (7)	Known	E. P. Wigner, Ann. of Math. 40 (1939) 149 / H. W. J. Blöte, J. L. Cardy, M. P. Nightingale, PRL 56 (1986) 742 / I. Affleck, PRL 56 (1986) 746. Bibliographic data only (content checked in Tong).
Anomaly cancellation / the heterotic string	Known	M. B. Green, J. H. Schwarz, Phys. Lett. B149 (1984) 117 / D. J. Gross, J. A. Harvey, E. J. Martinec, R. Rohm, PRL 54 (1985) 502. Abstracts checked.
11-dimensional supergravity / the upper limit	Known	E. Cremmer, B. Julia, J. Scherk, Phys. Lett. B76 (1978) 409 (abstract checked) / W. Nahm, Nucl. Phys. B135 (1978) 149 (abstract only; "11 is the upper limit" not checked).
(10), KK masses, Hořava–Witten	Physics	E. Witten, Nucl. Phys. B443 (1995) 85, hep-th/9503124 (text checked; formulas (2.13)(2.14)) / P. Hořava, E. Witten, Nucl. Phys. B460 (1996) 506, hep-th/9510209 (manuscript 1995-10) and Nucl. Phys. B475 (1996) 94, hep-th/9603142 (abstracts checked).
The four-dimensional tube and $c=26$	Physics	J. Polchinski, A. Strominger, PRL 67 (1991) 1681. Abstract checked.
Asymptotic form of the partition numbers	Known	G. H. Hardy, S. Ramanujan, Proc. London Math. Soc. (2) 17 (1918) 75. Bibliographic data only.
Breakdown of the models / the problem statement	Physics, Known	J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/9711200 / E. Witten, Adv. Theor. Math. Phys. 2 (1998) 505, hep-th/9803131 §4 / U. Gürsoy, E. Kiritsis, JHEP 02 (2008) 032, arXiv:0707.1324 (abstract) / A. Jaffe, E. Witten, "Quantum Yang–Mills Theory" (the Clay problem statement, 2000). Witten 1998 §4 and the problem statement: text checked.
Tables, figures, arithmetic	Computed	Python 3 on this machine. zero_point.py (Table 1), levels.py (Table 2, $SO(D-1)$ , Hagedorn, $\alpha'$ ), counts.py (576, 128, 496, the balance).

Next chapter to read: 7-06 How extra dimensions become invisible — why extra dimensions that can be read as geometry do not show up in four-dimensional experiments.

Revised 2026-10-01: first version.