12
1 Introduction
Table 1.1 List of the consistent tachyon-free (super)string theories. The bosonic theory is added
for comparison. There are additional heterotic theories without spacetime supersymmetry, but they
contain a tachyon and are thus omitted
Worldsheet
Spacetime Gauge
Open
susy
D susy
group
string Oriented Tachyon
Bosonic
(0, 0)
26 0
Any a
Yes
Yes/no
Yes
Type I
(1, 1)
10 (1, 0)
SO(32)
Yes
No
No
Type IIA
(1, 1)
10 (1, 1)
U(1)
(Yes) b Yes
No
Type IIB
(1, 1)
10 (2, 0)
None
(Yes) b Yes
No
Heterotic SO(32) (1, 0)
10 (1, 0)
SO(32)
No
Yes
No
Heterotic E 8
(1, 0)
10 (1, 0)
E 8 × E 8
No
Yes
No
Heterotic SO(16) (1, 0)
10 (0, 0)
SO(16) × SO(16) No
Yes
No
a UV divergences beyond the tachyon (interpreted as closed string dilaton tadpoles) cancel only
for the unoriented open plus closed strings with gauge group SO(2 13 ) = SO(8192)
b The parenthesis indicates that type II theories do not have open strings in the vacuum: they
require a D-brane background. This is expected since there is no gauge multiplet in d = 10
(1, 1) or (2, 0) supergravities (the D-brane breaks half of the supersymmetry)
The tachyon-free superstring theories together with the bosonic string are
summarized in Table 1.1.
1.2.3 Interactions
Worldsheet and Riemann Surfaces
After having described the spectrum and the general characteristics of string theory
comes the question of interactions. The worldsheets obtained in this way are
Riemann surfaces, i.e. one-dimensional complex manifolds. They are classified by
the numbers of handles (or holes) g (called the genus) and external tubes n. In the
presence of open strings, surfaces have boundaries: in addition to the handles and
tubes, they are classified by the numbers of disks b and of strips m. 7 A particularly
important number associated to each surface is the Euler characteristics
χ = 2 − 2g − b ,
(1.22)
which is a topological invariant. It is remarkable that there is a single topology at
every loop order when one considers only closed strings, and just a few more in the
presence of open strings. The analysis is greatly simplified in contrast to QFT, for
which the number of Feynman graphs increases very rapidly with the number of
loops and external particles.
7 We ignore unoriented strings in this discussion. The associated worldsheets can have cross-caps
which make the surfaces non-orientables.
1 Introduction
Table 1.1 List of the consistent tachyon-free (super)string theories. The bosonic theory is added
for comparison. There are additional heterotic theories without spacetime supersymmetry, but they
contain a tachyon and are thus omitted
Worldsheet
Spacetime Gauge
Open
susy
D susy
group
string Oriented Tachyon
Bosonic
(0, 0)
26 0
Any a
Yes
Yes/no
Yes
Type I
(1, 1)
10 (1, 0)
SO(32)
Yes
No
No
Type IIA
(1, 1)
10 (1, 1)
U(1)
(Yes) b Yes
No
Type IIB
(1, 1)
10 (2, 0)
None
(Yes) b Yes
No
Heterotic SO(32) (1, 0)
10 (1, 0)
SO(32)
No
Yes
No
Heterotic E 8
(1, 0)
10 (1, 0)
E 8 × E 8
No
Yes
No
Heterotic SO(16) (1, 0)
10 (0, 0)
SO(16) × SO(16) No
Yes
No
a UV divergences beyond the tachyon (interpreted as closed string dilaton tadpoles) cancel only
for the unoriented open plus closed strings with gauge group SO(2 13 ) = SO(8192)
b The parenthesis indicates that type II theories do not have open strings in the vacuum: they
require a D-brane background. This is expected since there is no gauge multiplet in d = 10
(1, 1) or (2, 0) supergravities (the D-brane breaks half of the supersymmetry)
The tachyon-free superstring theories together with the bosonic string are
summarized in Table 1.1.
1.2.3 Interactions
Worldsheet and Riemann Surfaces
After having described the spectrum and the general characteristics of string theory
comes the question of interactions. The worldsheets obtained in this way are
Riemann surfaces, i.e. one-dimensional complex manifolds. They are classified by
the numbers of handles (or holes) g (called the genus) and external tubes n. In the
presence of open strings, surfaces have boundaries: in addition to the handles and
tubes, they are classified by the numbers of disks b and of strips m. 7 A particularly
important number associated to each surface is the Euler characteristics
χ = 2 − 2g − b ,
(1.22)
which is a topological invariant. It is remarkable that there is a single topology at
every loop order when one considers only closed strings, and just a few more in the
presence of open strings. The analysis is greatly simplified in contrast to QFT, for
which the number of Feynman graphs increases very rapidly with the number of
loops and external particles.
7 We ignore unoriented strings in this discussion. The associated worldsheets can have cross-caps
which make the surfaces non-orientables.