Feynman Path Integral Formulation

50 1 Continuum Formulation
The superconformal gauge fixing implies a Faddeev-Popov determinant contribution
to the functional measure; after this contribution is properly taken into account
one finds an effective action given in terms of the direct supersymmetric extension
of the Liouville action,
I [φ, χ] = 10 − d ∫
d 2 σ [ 1
8π
2
(∂ a φ) 2 + 1 2 μ2 e 2φ + 1 2 i ¯χ (γ · ∂)χ + 1 2 μ ( ¯χγ 5χ)e φ ] ,
(1.235)
where d again is the number of components of the original X field, or, equivalently,
the embedding dimension of the supersymmetric string. The theory then describes a
two-dimensional renormalizable field theory, which is intended to reproduce a sum
over random surfaces with fermionic structure. The result of Eq. (1.235) implies
that the supersymmetric string only makes sense in ten dimensions, just like the
bosonic string was only consistent in ten dimensions. There is one important difference
though: the ground state of the supersymmetric string can be chosen so as
to avoid the tachyon (the requirement to achieve this are known as the GSO conditions).
Also, the previous discussion dealt with an extension of the original bosonic
string which included N = 1 world-sheet supersymmetry. It is possible though to
consider a wider range of string theories which have N = 2 (with a local SO(2) invariance)
and N = 4 world-sheet supersymmetry (with a local SU(2) invariance).
But there is one important physical aspect that is still missing in the tendimensional
supersymmetric theory, and that is the presence of non-abelian gauge
bosons, which are necessary, at least for a grand unified theory, in order to eventually
make contact with the real world. There are two approaches one can follow in
introducing gauge interactions in d = 10, which will be outlined here.
In the first approach the new internal symmetry charges are placed at the ends
of the string. In type I superstring has one supersymmetry (N = 1) in the tendimensional
sense, and therefore 16 supercharges. The unique feature of this theory
is that it is based on unoriented open and closed strings; since only type I superstring
theories contain open strings, only there this approach is possible. In type I strings
the symmetry group is SO(32).
The remaining string theories are based on oriented closed strings. The type
II string has N = 2 supersymmetry and cannot be consistently coupled to open
strings, which only allow at most N = 1 supersymmetry. Here one has two supersymmetries
in the ten-dimensional sense, giving 32 supercharges. There are in
fact two kinds of type II strings, the IIA and the IIB type. The main difference is
in the fact that the IIA theory massless fermion modes are such that the theory is
non-chiral and thus parity conserving, while the IIB theory is chiral and therefore
parity violating.
In the so-called heterotic string one again has supersymmetry and closed strings
only, but with separate right- and left-moving string modes. These models are based
therefore on a peculiar hybrid of the type I superstring and a bosonic string. There
are two kinds of heterotic strings which differ in their ten-dimensional gauge groups,
which can be either SO(32) or E 8 × E 8 (Gross, Harvey, Martinec and Rohm, 1985).
In these theories the charges are distributed on closed strings (as stated before, open
strings are not possible in the heterotic string scenario). Since it is a general feature