The non-critical string theory describes the relativistic string without enforcing the critical dimension. Although this allows the construction of a string theory in 4 spacetime dimensions, such a theory usually does not describe a Lorentz invariant background. However, there are recent developments which make possible
Lorentz invariant quantization
of string theory in 4-dimensional Minkowski space-time.
There are several applications of the non-critical string. Through the
AdS/CFT correspondence
In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter ...
it provides a holographic description of gauge theories which are asymptotically free.
It may then have applications to the study of the
QCD
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type o ...
, the theory of strong interactions between
quarks
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All common ...
.
Another area of much research is two-dimensional string theory which provides simple
toy models of
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
. There also exists a
duality to the 3-dimensional
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
.
The critical dimension and central charge
In order for a
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
to be consistent, the
worldsheet
In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind as a direct generalization of the world line concept for a point particle in special and ...
theory must be conformally invariant. The obstruction to
conformal symmetry
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetr ...
is known as the
Weyl anomaly and is proportional to the
central charge of the worldsheet theory. In order to preserve conformal symmetry the Weyl anomaly, and thus the central charge, must vanish. For the
bosonic string this can be accomplished by a worldsheet theory consisting of 26
free bosons. Since each boson is interpreted as a flat spacetime dimension, the critical dimension of the bosonic string is 26. A similar logic for the
superstring results in 10 free bosons (and 10 free
fermions as required by worldsheet
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
). The bosons are again interpreted as spacetime dimensions and so the critical dimension for the superstring is 10. A string theory which is formulated in the critical dimension is called a critical string.
The non-critical string is not formulated with the critical dimension, but nonetheless has vanishing Weyl anomaly. A worldsheet theory with the correct central charge can be constructed by introducing a non-trivial target space, commonly by giving an
expectation value to the
dilaton which varies linearly along some spacetime direction. (From the point of view of the worldsheet CFT, this corresponds to having a
background charge.)
For this reason non-critical string theory is sometimes called the linear dilaton theory. Since the dilaton is related to the string
coupling constant
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
, this theory contains a region where the coupling is weak (and so perturbation theory is valid) and another region where the theory is strongly coupled. For dilaton varying along a
spacelike direction, the dimension of the theory is less than the critical dimension and so the theory is termed subcritical. For dilaton varying along a
timelike direction, the dimension is greater than the critical dimension and the theory is termed supercritical. The dilaton can also vary along a
lightlike direction, in which case the dimension is equal to the critical dimension and the theory is a critical string theory.
Two-dimensional string theory
Perhaps the most studied example of non-critical string theory is that with two-dimensional target space. While clearly not of phenomenological interest, string theories in two dimensions serve as important toy models. They allow one to probe interesting concepts which would be computationally intractable in a more realistic scenario.
These models often have fully non-perturbative descriptions in the form of the quantum mechanics of large matrices. Such a description known as the c=1 matrix model captures the dynamics of
bosonic string theory in two dimensions. Of much recent interest are matrix models of the two-dimensional
Type 0 string theories. These "matrix models" are understood as describing the dynamics of
open strings lying on
D-branes
In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polc ...
in these theories. Degrees of freedom associated with
closed string
In physics, a string is a physical entity postulated in string theory and related subjects. Unlike elementary particles, which are zero-dimensional or point-like by definition, strings are one-dimensional extended entities. Researchers often h ...
s, and
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
itself, appear as emergent phenomena, providing an important example of open string
tachyon condensation in string theory.
See also
*
String theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
, for general information about critical superstrings
*
Weyl anomaly
*
Central charge
*
Liouville gravity
References
*
Polchinski, Joseph (1998). ''String Theory'', Cambridge University Press. A modern textbook.
** Vol. 1: An introduction to the bosonic string. .
** Vol. 2: Superstring theory and beyond. .
*
*
*
rratum-ibid. 48 (1982) 1768
*
{{String theory topics , state=collapsed
String theory