Introduction to String Theory and D–Branes - School of Natural ...

In general, the complete theory is often thought of as a conformal field theory of central charge c coupled
to two dimensional gravity. The gravity here refers to the physics of the two dimensional metric. This nomenclature
is consistent with the fact that the Riemann tensor in d dimensions has d 2 (d 2 − 1)/12 independent
components, so that in d = 2 there is one degree of freedom, that represented by ϕ.
For example, with just one ordinary scalar X, (which has c = 1), we have two dimensional string theory.
The spacetime picture that is kept in mind for this model is that ϕ plays the role of a spatial coordinate,
and time is recovered by Wick rotating X. Notice that the spacetime picture meshes very nicely with
the σ–model approach we worked with in equation (93). Comparing our Liouville theory Lagrangian with
that of the sigma model we see that the dilaton is set by ϕ as Φ = Qϕ/( √ 2α ′ ). Choose the spacetime
metric as flat: Gµν = ηµν. This “linear dilaton vacuum” is a solution to the β–function equations (96) if
Q = (26 − D)/3 = (25 − c)/3, as we saw above (where we have D = 2 for the case in hand). Since the
string coupling is set by the dilaton via gs = e Φ , we see that our earlier statement that ϕ → −∞ is weak
coupling is indeed correct. Further to this, we see that the case of larger ϕ is at stronger coupling. Note
that µ sets a natural scale where the cosmological term is of order one and the theory is firmly in the strong
coupling regime, at ϕ ∼ log(1/µ). This is often referred to the “Liouville Wall”, the demarcation between
the strong and weak coupling regimes.
Turning to other cases, the most well–studied conformal field theories are the c < 1 minimal models, an
infinite family indexed by two integers (p, q) with
c = 1 −
6(p − q)2
pq
. (173)
These models are distinguished by (among other things) having a finite number of primary fields. They
are unitary when |p − q| = 1. The trivial model is the (3, 2), which has c = 0. Two other famous unitary
members of the series are (4, 3), which has c = 1/2 and (5, 4), which has c = 7/10 are the critical Ising model
and the tricritical Ising model.
For the (3, 2) coupled to Liouville, we see that the model is just one with no extra embedding for the
string at all, just the Liouville dimension. It is often referred to as “pure gravity”, since the (non–Liouville)
conformal field theory is trivial. In general, the study of these “non–critical string theories” is said to be the
study of strings in D ≤ 2.
Remarkably, the path integral for these models can be supplied with a definition using the techniques
called “Matrix Models”. Basically, the sum over world–sheet metrics is performed by studying the theory of
an N×N matrix valued field, where N is large. The matrix integral can be expanded in terms of Feynmann
diagrams, and for large N, the diagrams can be organized in terms of powers of 1/N. This expansion in
1/N is the same as the gs topological expansion of string perturbation theory, and the Feynmann diagrams
act as a sort of regularized representation of the string world–sheets. We don’t have time or space to study
these models here, but it is a remarkable subject. In fact, it is in these simple string models that a lot of
the important modern string ideas have their roots, such as non–perturbative string theory, the behaviour
of string theory at high orders in perturbation theory, and non–perturbative relations between open and
closed strings 7 . Recently, this area has been revisited, as it has become increasingly clear that there may be
more lessons to be learned about the above topics and more, such as holography, tachyon condensation, and
open–closed transitions 8 .
There has yet to be presented a satisfactory interpretation for the physics for a vast range of dimensions,
however. This is because it is only for the case 0 ≤ c ≤ 1 that Q and γ are both real. Outside this range,
the above formulae of non–critical string theory await a physical interpretation.
7 For a review of the subject, see for example refs.[10]
8 There is no complete review of this area currently available (although see the last of refs.[10]), so it is recommended that
refs.[11] be consulted, as their opening sections give a good guide to the literature, old and new.
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