Gravity and Strings

406 String theory
String
Theory
Solitonic
States
Worldvolume
Actions
Effective
Action
Solitonic
Solutions
Fig. 14.1. Most of the recent progress in string theory has been based on the relations
represented in this diagram between different aspects of the theory. The implications of
duality symmetries in each box must be related to similar implications in other boxes.
The box in the center represents the worldvolume theories of all the extended objects of
string theory, including fundamental strings. They give, to start with, the perturbative definition
of string theory (the worldsheet formulation), represented by the box in the upper-lefthand
corner. This worldsheet formulation allows us to calculate the perturbative spectrum
and also find some non-perturbative states (the D-branes), which are represented by the
lower-left-hand corner of the diagram (their worldvolume actions are also represented by
the box in the center). The low-energy effective-field theories that describe the dynamics
of the massless states of the perturbative spectrum are represented in the upper-right-hand
corner. These theories are usually supergravity theories that have certain (on-shell) global
symmetries that transform and mix the fields that represent the string massless modes. The
equations of motion of these effective-field theories can be derived (a most important point)
from consistency conditions (conformal invariance or κ-symmetry) of worldvolume theories
(the box in the center) in general backgrounds. The effective field theories also admit
solitonic solutions, which are represented in the lower-right-hand corner. The solitonic solutions
can be excited and deformed and their effective dynamics is, yet again, given by
worldvolume actions.
The meaning of the ↓, →, and ↓ arrows and the arrows that come from the box in the
center of this diagram is clear. The progress made in this field comes from the realization
of the existence and use of the remaining arrows of the figure.
The main idea is that, generically, the global symmetries of the effective-field theories
correspond to dualities of the string theories. 1
Some of these dualities are essentially perturbative (in the string coupling constant, that
we will define later) and can be found and studied using the worldsheet approach. They are
1 This is the point of view proposed in [583], but a more precise statement that includes more cases would
be that the relations between different effective field theories correspond to dualities of the corresponding
string theories. Some of the relations can be described as global symmetries of a single effective action, but
in other cases there are relations between very different effective actions.