2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicswhere gauge groups come from

Where gauge fields come from — the group is read off from the number of branes, a lattice, the shape of a singularity

In four-dimensional Yang–Mills theory the gauge group $G$ is an input chosen by hand. The text of the prize problem (Jaffe–Witten) too says 'for any compact simple gauge group G', allowing any of the list in 1-04 (0-03). In theories with extra dimensions the group becomes an output, read off from something else Physics.

What any of these roads fixes is the set of fields and the group seen at low energy, not the existence of a four-dimensional quantum theory. Of the four, three (branes, lattice, singularity) are stories within supersymmetric settings (theories with a symmetry exchanging bosons and fermions), and what that means for pure Yang–Mills we look back on in §10.

In 7-06 §04 the rotation of the circle became U(1). If the group $\mathrm{Isom}(K)$ of transformations of the internal space $K$ that do not change distances (isometries) is large, each of its infinitesimal generators (Killing vector fields) gives one gauge field, and the commutators of the generators become the structure constants (1-03) Physics. For the generators of rotations of a sphere, $L_a=\varepsilon_{abc}x_b\partial_c$,

(the left formula was checked numerically Computed); $S^n$ gives $n(n+1)/2$ gauge fields, and $\mathbb{CP}^2$ (the set of all complex lines through the origin of $\mathbb C^3$; real dimension 4) gives the 8 gauge fields of SU(3) Known. An example of a seven-dimensional space whose isometry group is locally (as a Lie algebra) SU(3)×SU(2)×U(1) (3-07) is $\mathbb{CP}^2\times S^2\times S^1$ ($4+2+1=7$ dimensions), and together with four dimensions the whole is $4+7=11$-dimensional.

Witten (1981) studied the road of compactifying eleven-dimensional supergravity on seven-dimensional spaces, and wrote in the abstract 'It is possible to obtain an SU(3) × SU(2) × U(1) gauge group, but the proper fermion quantum numbers are difficult to achieve'. The seven-dimensional spaces Witten studied are not necessarily the same as the example above.

Impose on open strings a boundary condition attaching their ends to a subspace, and that subspace is a D-brane (Polchinski 1995: 'extended objects defined by mixed Dirichlet-Neumann boundary conditions in string theory') Physics. One with $p$ spatial dimensions is called a D$p$-brane. Labelling the two ends of a string with brane numbers $1,\dots,N$, the states have $N\times N$ matrices as coefficients, and the massless vectors form the adjoint representation of U(N) (1-04) (Polchinski's lectures of 1996: 'this global symmetry of the world-sheet theory is elevated to a gauge symmetry in spacetime'). There are $N^2$ kinds of strings, and $\dim U(N)=N^2$.

Separating the branes makes the strings between them heavy (Witten 1995b §3; $T$ is the string tension, $a_i$ the positions of the branes).

If they are stacked $n_1,n_2,\dots$ at a time, the remaining group is $U(n_1)\times U(n_2)\times\cdots$, and

(the same count appears in Polchinski's lectures). Separating branes is the Higgs mechanism seen from the side of geometry. More precisely, when the adjoint scalars $X^i$ of §04 take diagonal values $\mathrm{diag}(a_1,\dots,a_N)$, the off-diagonal gauge fields acquire the masses (2). This mechanism, in which the values of scalars make gauge fields heavy, is called the Higgs mechanism (3-07) Physics.

ComputedTable 1 was counted on this machine, and Figure 1 is drawn on the spot from (2)(3). If JavaScript is disabled, see Table 1.

The W masses of four separated branes are $d_1,d_2,d_3$ and their consecutive sums, 6 in all: the relation between the 3 simple roots and the 6 positive roots of SU(4) ($A_3$). In §07 the same structure appears as the areas of spheres.

The theory on stacked branes is 'the dimensional reduction of ten-dimensional supersymmetric Yang-Mills theory with gauge group U(n) from ten to p+1 dimensions' (Witten 1995b §3) Physics. Splitting the components of the ten-dimensional gauge field into directions along the branes and perpendicular to them gives

where $X^i$ are the $9-p$ scalars in the adjoint representation ($N\times N$ Hermitian matrices), and $V$ is their potential. Since $X$ is Hermitian, $[X^i,X^j]$ is anti-Hermitian and $\operatorname{tr}[X^i,X^j]^2\le0$. Therefore $V\ge0$. If $V=0$, the matrices can be diagonalised simultaneously, and Witten reads the diagonal entries as the positions of the branes. The off-diagonal entries are the strings between branes, and the eigenvalues of the map $\delta\mapsto\sum_k[X^k,[X^k,\delta]]$ are $|a_\lambda-a_\mu|^2$ (times $T^2$, the square of (2)). We checked agreement within $4\times10^{-15}$ on an example of two directions conjugated by a unitary matrix Computed.

On D3-branes the theory is four-dimensional $\mathcal N=4$ U(N) supersymmetric Yang–Mills with 6 scalars and fermions (the starting point of 7-08), not pure Yang–Mills. The matrix indices have the same structure as the double lines of 7-04.

The heterotic string is a closed string whose right-movers follow the ledger of the superstring (central charge 15) and whose left-movers follow that of the bosonic string (26) (7-05 §06, Gross–Harvey–Martinec–Rohm 1985). The left-movers are 16 short of the 10 of flat ten dimensions; Polchinski's lectures (1994), on the supersymmetric heterotic string, write 'an additional (0, 16) is needed from another unitary CFT', and go on to say that the only solutions are $E_8\times E_8$ or SO(32) Physics.

A 16-dimensional lattice

In the lattice picture this 16 is seen as a torus $\mathbb R^{16}/\Gamma$ on which 16 left-moving bosons are wound Physics. The momentum $P$ lies on the lattice $\Gamma$, and the states with $P\cdot P=2$ become massless vectors. On the heterotic string wound on a torus, Witten (1995 §4.6) writes 'The gauge symmetry is extended precisely when $V^\perp$ contains some P of $d_P=1$'. Witten's $V^\perp$ is the space of left-moving charges. Witten takes the sign of the inner product opposite to this chapter and writes $d_P=-(P,P)/2$, so $d_P=1$ corresponds to $P\cdot P=2$ in this chapter's sign. $\Gamma$ must be even (every $P\cdot P$ even) and self-dual (the points having integer inner product with all points are exactly $\Gamma$). This condition is said to come from the requirement that the one-loop amplitude, whose world sheet is a torus, not depend on how the shape of the torus is chosen (modular invariance), and Polchinski's lectures also write 'The constraints of modular invariance are strong'.

There are only two even self-dual lattices in 16 dimensions Known. Both can be written as "$x\in\mathbb Z^n\cup(\mathbb Z+\tfrac12)^n$ with $\sum x_i$ even": two copies of $n=8$ side by side give $E_8\oplus E_8$, and $n=16$ gives $D_{16}^+$. Both have 480 roots, and (5) is 496, but joining roots with non-zero inner product, the former splits into two sets of 240 ($E_8\times E_8$) and the latter is connected ($\mathfrak{so}(32)$) Computed. The ten-dimensional group is written precisely as $\mathrm{Spin}(32)/\mathbb Z_2$. $\mathrm{Spin}(32)$ is the double cover of SO(32), the same relation as SU(2)→SO(3) in 1-02 (the $\mathbb Z_2$ divided out is a different element from the one that returns to SO(32)). The lattice spanned by the roots of $D_{16}^+$ is a sublattice of index 2 in $D_{16}^+$, and this $/\mathbb Z_2$ is said to correspond to that difference.

Seen from eleven dimensions, Hořava–Witten proposed that the two $E_8$ sit one on each of the walls at the two ends of the segment $S^1/\mathbb Z_2$ (the 1995 abstract says 'We propose'; the 1996 abstract writes that the group is fixed to 'a copy of E8 for each boundary component', explicitly presupposing 'the hypothesis that there is a consistent quantum M-theory') Physics.

Detour: the same tower, different groups

To bring $E_8\times E_8$ down to four dimensions, compactify on a six-dimensional Calabi–Yau manifold $K$ (7-06 §05). The holonomy of $K$ (1-07) lies in SU(3), and in the standard construction the holonomy of the gauge field is said to be aligned with an SU(3) inside $E_8$ too Physics. A gauge transformation visible in four dimensions is a $g$ constant over all of $K$ that is unchanged by parallel transport along every closed path, that is, it must commute with every element of the holonomy Known.

Of the 240 roots of $E_8$, 72 are orthogonal to the roots of SU(3), giving $E_6$ (dimension 78) Computed. The abstract of Candelas–Horowitz–Strominger–Witten (1985) writes 'a model that in four dimensions has an E6 gauge group with four standard generations of fermions'.

ComputedTable 3 was counted by building the 240 roots of $E_8$ in coordinates; the types were fixed by rank, number of roots and the determinant of the Cartan matrix.

The remaining 162 are $2\times27\times3$ (pairs of the 27-dimensional representation of $E_6$ and the 3-dimensional one of SU(3), and their conjugates), and the four-dimensional fermions are said to come from here.

A K3 surface is a Calabi–Yau manifold of complex dimension two (real dimension four), with holonomy SU(2) (the left end of Table 3) Known. Witten (1995 §4.6), on the correspondence between the heterotic string wound on a torus and the IIA string wound on a K3 surface, writes 'Whenever the heterotic string on a four-torus gets an enhanced gauge group G, the corresponding K3 gets an orbifold singularity of type G' Physics. Then 'a configuration of r two-spheres $S_i$ (with an intersection matrix given by the Dynkin diagram of G) collapses to a point' (Dynkin diagrams are defined later in this section), and each sphere gives one U(1). The W bosons are two-dimensional membranes wrapped on the collapsing spheres (as described in Bershadsky–Sadov–Vafa 1996 §1). Their mass is said to be the tension of the membrane times the area of the sphere, the same form as the $T|a|$ of §03.

Intersection numbers and negative definiteness

Why only ADE? Look at the intersection numbers $M_{ij}=S_i\cdot S_j$ of the spheres. For two spheres this is the sum over intersection points counted with orientation signs $\pm1$. The intersection number of a sphere with itself (the self-intersection number) counts in the same way the intersection points with a slightly shifted copy, and for spheres in K3 it is $-2$ Known. Defining the "inner product" of classes of spheres by this intersection number, on K3 the inner product is positive in three directions and negative in nineteen (signature $(3,19)$) Known. According to Witten, in the collapsing limit the classes of the spheres lie in the subspace $V^\perp$ on which this inner product is negative definite, so $M$ is negative definite ($x^{\mathsf T}Mx<0$ for every $x\ne0$). The collapsing spheres $S_i$ can be taken to be holomorphic curves (curves of complex dimension one, locally the zero sets of holomorphic functions of complex coordinates), and intersection points of distinct holomorphic curves all count as positive because of the complex structure, so $M_{ij}\ge0$ Known. With $x=e_i+e_j$, $-4+2M_{ij}<0$ gives $M_{ij}\le1$. Hence $M=-(2I-A)$, where $A$ is the adjacency matrix of the graph whose vertices are the spheres and whose edges are the intersections. This graph is called the Dynkin diagram (those that appear here are of "simply laced" type, all edges single). We need the following.

If the graph is not connected we consider each component (the group is a product over components). If a connected graph has a cycle, the vector equal to 1 on the cycle makes the left-hand side at most 0, so it must be a tree. Examining the 201 trees with at most 10 vertices, only types A (a straight line), D (forked at one end) and $E_6,E_7,E_8$ remained (Table 4) Computed. That these are the only ones for any number of vertices is known as part of the classification of simple Lie algebras (1-04) Known.

ComputedThe numbers of Table 4 and Figure 2 were computed on this machine. The smallest eigenvalue agrees to 6 digits with $2-2\cos(\pi/h)$ ($h$ is the number of roots ÷ the rank), and the figure is drawn with this formula. One beyond $E_8$ there is an integer vector making (7) 0 (centre 6, arms 3 / 4, 2 / 5, 4, 3, 2, 1). If JavaScript is disabled, see Tables 4 and 5.

ADE are only the types whose roots have equal lengths. B, C, F and G come out when the singularities are further bundled in another direction. Bershadsky et al. (1996 §3) write that going once around over another parameter can permute the collapsing spheres by an automorphism of the Dynkin diagram (a permutation of vertices preserving edges), and that then 'the actual gauge group will be smaller', listing the following (the last is the cyclic permutation of three legs) Physics.

With this every type from A to G comes out of geometry. The number of orbits into which the permutation bundles the positive roots agrees with the number of positive roots of the folded algebra (Table 5) Computed.

ComputedTable 5 was counted by generating roots from the Cartan matrices ($A_3\to C_2$, $A_7\to C_4$ and $D_6\to B_5$ also agree).

But the same paper notes that 'a given A-D-E singularity may not correspond to an enhanced gauge symmetry'. Computing field theories by this mechanism is geometric engineering; Katz–Klemm–Vafa (1997) reduced the exact results of four-dimensional $\mathcal N=2$ (7-11) to the T-duality of strings (strings wound on a circle of radius $R$ and on a circle of radius $\alpha'/R$ give the same theory; $\alpha'$ is from (1) of 7-05), and write 'This construction is not based on any duality conjecture' Physics. What it can handle is supersymmetric theories with at least 8 supercharges (7-05 §07) (corresponding to four-dimensional $\mathcal N=2$ or more); within the range we searched we found no example handling the mass gap of non-supersymmetric pure Yang–Mills by geometric engineering.

Witten (1998 §4) describes a road from eleven dimensions down to four-dimensional pure Yang–Mills Physics. The starting point is the six-dimensional (2,0) theory on M5-branes (membranes of M-theory extending in five spatial dimensions) ((2,0) is the name of a type of six-dimensional supersymmetry; Witten calls it 'exotic'), which 'can apparently be interpreted' as M-theory on $AdS_7\times S^4$ ($AdS_7$ is a seven-dimensional spacetime of constant negative curvature, treated in 7-08) (according to Maldacena's conjecture). No description by a Lagrangian is said to be known, and there is no mathematical definition either. This is wrapped on two circles.

Making the fermions antiperiodic (changing sign once around) on $C_2$, the fermions acquire masses of order $1/R_2$, the scalars too 'very plausibly' become heavy at one loop, and 'If this is so, the low energy theory will be the pure SU(N) theory without supersymmetry'. In terms of the IIA string this is $N$ D4-branes wrapped on $\mathbb R^4\times S^1$, and the theory on the branes of §03–04 comes down to four dimensions.

Witten writes that he would like to keep $\eta=g_4^2N$ small, so that the coupling is weak at the scale of the circles and strong at much longer distances. A one-loop rough guide: putting $b_0=11N/(48\pi^2)$ (formula (1) of 7-01), $\mu=1/R_2$ and $g^2(\mu)=g_4^2=\eta/N$ into (3) of 7-01 gives

which is $4\times10^{-10}$ for $\eta=1$ and 0.12 for $\eta=10$ Computed. The coefficient changes with the convention for the coupling and the choice of $\mu=1/R_2$, so this is a guide to the order of magnitude. But supergravity can compute only 'at least if η is large', and Witten writes of connecting to small $\eta$ 'This looks like a tall order'. The order of limits is in 7-10, and how the gap appears as geometry in 7-09.

On every road the group comes with extra fields or supersymmetry (the $9-p$ scalars on the branes, the fields of mass $1/R_2$ from the two circles, and so on) Physics. And for the prize problem, the group is not where the difficulty lies. The problem statement allows any compact simple group; what is asked is that its quantum Yang–Mills theory exists on $\mathbb R^4$ satisfying the axioms (4-10) and has a mass gap (4-11). Geometry can produce all the names from A to G, but within supersymmetric settings, and the road to pure Yang–Mills (§09) relies on expectations and on the order of limits.

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Next chapter to read: 7-08 Holography — the $\mathcal N=4$ theory on branes, and the correspondence in which the radial direction becomes scale.

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