Nonabelian gauge symmetry

It is interesting to check how nonabelian gauge symmetries that arise in the

heterotic string theory are understood from the M-theory point of view. We

learned in Chapter 7 that the generic U (1)22 abelian gauge symmetry of the

heterotic string compacti¬ed on T 3 is enhanced to nonabelian symmetry at

singularities of the Narain moduli space, which exist due to the modding out

by the discrete factor SO(19, 3; ). It was demonstrated in examples that

at such loci certain spin-one particles that are charged with respect to the

U (1) s and massive away from the singular loci become massless to complete

the nonabelian gauge multiplet. The nonabelian gauge groups that appear

in this way are always of the type An = SU (n + 1), Dn = SO(2n), E6 , E7 ,

E8 , or semisimple groups with these groups as factors. The ADE groups in

the Cartan classi¬cation are the simple Lie groups with the property that

all of their simple roots have the same length. Such Lie groups are called

simply-laced. Given the duality that we have found, these results should be

explainable in terms of M-theory on K3.

Generically, K3 compacti¬cation of M-theory gives 22 U (1) gauge ¬elds

9.11 K3 compacti¬cations and more string dualities 423

in seven dimensions. These one-forms arise as coe¬cients in an expansion

of the M-theory three-form A3 in terms of the 22 linearly independent har-

monic two-forms of K3. The three gauge ¬elds associated with the self-dual

two-forms correspond to those that arise from right-movers in the heterotic

description and belong to the supergravity multiplet. Similarly, the 19 gauge

¬elds associated with the anti-self-dual two-forms correspond to those that

arise from left-movers in the heterotic description and belong to the vector

supermultiplets.

The singularities of the Narain moduli space correspond to singularities

of the K3 moduli space. So we need to understand why there should be

nonabelian gauge symmetry at these loci. Each of these singular loci of

the K3 moduli space correspond to degenerations of a speci¬c set of two-

cycles of the K3 surface. When this happens, wrapped M2-branes on these

cycles give rise to new massless modes in seven dimensions. In particular,

these should provide the charged spin-one gauge ¬elds for the appropriate

nonabelian gauge group.

The way to tell what group appears is as follows. The set of two-cycles

that degenerate at a particular singular locus of the moduli space has a

matrix of intersection numbers, which can be represented diagrammatically

by associating a node with each degenerating cycle and by connecting the

nodes by a line for each intersection of the two cycles. Two distinct cycles

of K3 intersect either once or not at all, so the number of lines connecting

any two nodes is either one or zero.

The diagrams obtained in this way look exactly like Dynkin diagrams,

which are used to describe Lie groups. However, the meaning is entirely

di¬erent. The nodes of Dynkin diagrams denote positive simple roots, whose

number is equal to the rank of the Lie group, and the number of lines

connecting a pair of nodes represents the angle between the two roots. For

example, no lines represents π/4 and one line represents 2π/3. For simply-

laced Lie groups these are the only two cases that occur.

Mathematicians observed long ago that the intersection diagrams of de-

generating two-cycles of K3 have an ADE classi¬cation, but it was com-

pletely mysterious what, if anything, this has to do with Lie groups. M-

theory provides a beautiful answer. The diagram describing the degener-

ation of the K3 is identical to the Dynkin diagram that describes the re-

sulting nonabelian gauge symmetry in seven dimensions. The ADE Dynkin

diagrams are shown in Fig. 9.10. The simplest example is when a single

two-cycle degenerates. This is represented by a single node and no lines,

which is the Dynkin diagram for SU (2). This case was examined in detail

from the heterotic perspective in Chapter 7. A somewhat more complicated

424 String geometry

example is the degeneration corresponding to the T 4 / 2 orbifold discussed

in Section 9.3. In this case 16 nonintersecting two-cycles degenerate, which

gives [SU (2)]16 gauge symmetry (in addition to six U (1) factors). Similarly,

the 3 orbifold considered in Exercise 9.2 gives [SU (3)]9 gauge symmetry

(in addition to four U (1) factors). The number of U (1) factors is determined

by requiring that the total rank is 22.

An

Dn

E6

E7

E8

Fig. 9.10. The Dynkin diagrams of the simply-laced Lie algebras.

Type IIA superstring theory on K3

Compacti¬cation of the type IIA theory on K3 gives a nonchiral theory with

16 unbroken supersymmetries in six dimensions. This example is closely re-

lated to the preceding one, because type IIA superstring theory corresponds

to M-theory compacti¬ed on a circle. Compactifying the seven-dimensional

theory of the previous section on a circle, this suggests that the type IIA

theory on K3 should be dual to the heterotic theory on T 4 . A minimal spinor

in six dimensions has eight components, so this is an N = 2 theory from

the six-dimensional viewpoint. Left“right symmetry of the type IIA theory

implies that the two six-dimensional supercharges have opposite chirality,

which agrees with what one obtains in the heterotic description.

Let us examine the spectrum of massless scalars (moduli) in six dimensions

from the type IIA perspective. As in the M-theory case, the metric tensor

9.11 K3 compacti¬cations and more string dualities 425

gives 58 moduli. In addition to this, the dilaton gives one modulus and the

two-form B2 gives 22 moduli, since b2 (K3) = 22. The R“R ¬elds C1 and C3

do not provide any scalar zero modes, since b1 = b3 = 0. Thus, the total

number of moduli is 81. The heterotic string compacti¬ed on T 4 also has

an 81-dimensional moduli space, obtained in Chapter 7,

+

— M20,4 . (9.182)

¡

Thus, this should also be what one obtains from compactifying the type

IIA superstring theory on K3. The + factor corresponds to the heterotic

¡

dilaton or the type IIA dilaton, so these two ¬elds need to be related by the

duality.

We saw above that the 58 geometric moduli contain 38 complex-structure

moduli and 20 K¨hler-structure moduli. Of the 22 moduli coming from

a

B2 the 20 associated with (1, 1)-forms naturally combine with the 20 ge-

ometric K¨hler-structure moduli to give 20 complexi¬ed K¨hler-structure

a a

moduli, just as in the case of Calabi“Yau compacti¬cation described earlier.

Altogether the 80-dimensional space M20,4 is parametrized by 20 complex

K¨hler-structure moduli and 20 complex-structure moduli. There is a mir-

a

ror description of the type IIA theory compacti¬ed on K3, which is given by

type IIA theory compacti¬ed on a mirror K3 in which the K¨hler-structure