String Theory and M-Theory

7.3 Toroidal compactification 285
Fig. 7.3. Fundamental domain of the torus displaying the discrete identifications.
Points in the τ plane where enhanced symmetries appear for ρ = i are displayed.
Suppose that ρ1 = τ1 = 0, so that B = 0 and
ρ2τ2
G =
0
0
ρ2/τ2
.
If ρ2 = τ2 or ρ2 = 1/τ2 then one of the two entries is one, which means that
one of the two circles is at the self-dual radius, and there is an enhanced
SU(2) gauge symmetry for both left-movers and right-movers. These two
relations are satisfied simultaneously if
(τ, ρ) = (i, i).
In this case both circles are at the self-dual radius and the enhanced symmetry
is SU(2) × SU(2) for both left-movers and right-movers giving SU(2) 4
altogether.
Another point of enhanced symmetry appears when
In this case B = 0 and
(τ, ρ) = (− 1
+ i
2
G = 1 √ 3
2 −1
−1 2
√
3
, i).
2
Here G is proportional to the Cartan matrix of SU(3). (For an introduction
to the theory of roots and weights of Lie algebras see the review article by
Goddard and Olive.) As a consequence both the left-movers and the rightmovers
contain the massless vectors required for SU(3) enhanced gauge
.