1.2 String Theory
11
theories with a finite number of fields 5 —in order to keep the spin of a family in the
range where consistent actions exist:
• N max = 4 without gravity (−1 ≤ spin ≤ 1);
• N max = 8 with gravity (−2 ≤ spin ≤ 2).
This counting serves as a basis to determine the maximal number of supersymmetries in other dimensions (by relating them through dimensional reductions).
Let us turn our attention to the case of the two-dimensional worldsheet theory.
The number of supersymmetries of the closed left- and right-moving sectors can be
chosen independently, and the number of charges is written as (N L , N R ) (the index
is omitted when statements are made at the level of the CFT). The critical dimension
(absence of quantum anomaly for the Weyl invariance) depends on the number of
supersymmetry
D(N = 0) = 26,
D(N = 1) = 10.
(1.21)
Type II superstrings have (N L , N R ) = (1, 1) and come in two flavours called IIA
and IIB according to the chirality of the spacetime gravitini chiralities. A theory is
called heterotic if N L > N R ; we will mostly be interested in the case N L = 1 and
N R = 0. 6 In such theories, there cannot be open strings since both sectors must be
equal in the latter. Since the critical dimensions of the two sectors do not match, one
needs to get rid of the additional dimensions of the right-moving sector; this leads
to the next topic—gauge groups.
Gauge groups associated with spacetime gauge bosons appear in two different
places. In heterotic models, the compactification of the unbalanced dimensions
of the left sector leads to the appearance of a gauge symmetry. The possibilities
are scarce due to consistency conditions which ensure a correct gluing with the
right-sector. Another possibility is to add degrees of freedom—known as Chan–
Paton indices—at the ends of open strings: one end transforms in the fundamental
representation of a group G, while the other end transforms in the anti-fundamental.
The modes of the open string then reside in the adjoint representation, and the
massless spin-1 particles become the gauge bosons of the non-Abelian gauge
symmetry.
Finally, one can consider oriented or unoriented strings. An oriented string
possesses an internal direction, i.e. there is a distinction between going from the
left to the right (for an open string) or circling in clockwise or anti-clockwise
direction (for a closed string). Such an orientation can be attributed globally to the
spacetime history of all strings (interacting or not). The unoriented string is obtained
by quotienting the theory by the Z 2 worldsheet parity symmetry which exchanges
the left- and right-moving sectors. Applying this to the type IIB gives the type I
theory.
5 These conditions exclude the cases of free theories and higher-spin theories.
6 The case N L < N R is identical up to exchange of the left- and right-moving sectors.
11
theories with a finite number of fields 5 —in order to keep the spin of a family in the
range where consistent actions exist:
• N max = 4 without gravity (−1 ≤ spin ≤ 1);
• N max = 8 with gravity (−2 ≤ spin ≤ 2).
This counting serves as a basis to determine the maximal number of supersymmetries in other dimensions (by relating them through dimensional reductions).
Let us turn our attention to the case of the two-dimensional worldsheet theory.
The number of supersymmetries of the closed left- and right-moving sectors can be
chosen independently, and the number of charges is written as (N L , N R ) (the index
is omitted when statements are made at the level of the CFT). The critical dimension
(absence of quantum anomaly for the Weyl invariance) depends on the number of
supersymmetry
D(N = 0) = 26,
D(N = 1) = 10.
(1.21)
Type II superstrings have (N L , N R ) = (1, 1) and come in two flavours called IIA
and IIB according to the chirality of the spacetime gravitini chiralities. A theory is
called heterotic if N L > N R ; we will mostly be interested in the case N L = 1 and
N R = 0. 6 In such theories, there cannot be open strings since both sectors must be
equal in the latter. Since the critical dimensions of the two sectors do not match, one
needs to get rid of the additional dimensions of the right-moving sector; this leads
to the next topic—gauge groups.
Gauge groups associated with spacetime gauge bosons appear in two different
places. In heterotic models, the compactification of the unbalanced dimensions
of the left sector leads to the appearance of a gauge symmetry. The possibilities
are scarce due to consistency conditions which ensure a correct gluing with the
right-sector. Another possibility is to add degrees of freedom—known as Chan–
Paton indices—at the ends of open strings: one end transforms in the fundamental
representation of a group G, while the other end transforms in the anti-fundamental.
The modes of the open string then reside in the adjoint representation, and the
massless spin-1 particles become the gauge bosons of the non-Abelian gauge
symmetry.
Finally, one can consider oriented or unoriented strings. An oriented string
possesses an internal direction, i.e. there is a distinction between going from the
left to the right (for an open string) or circling in clockwise or anti-clockwise
direction (for a closed string). Such an orientation can be attributed globally to the
spacetime history of all strings (interacting or not). The unoriented string is obtained
by quotienting the theory by the Z 2 worldsheet parity symmetry which exchanges
the left- and right-moving sectors. Applying this to the type IIB gives the type I
theory.
5 These conditions exclude the cases of free theories and higher-spin theories.
6 The case N L < N R is identical up to exchange of the left- and right-moving sectors.