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GAUGE FIELDS AND STRINGS
critical properties of the Ising model (such as the singularity in the
specific heat) are shared by many statistical systems in Nature. How>
ever if we are far from the phase transition point no lessons for other
systems can be extracted.
In the present section we shall try to argue that in the continuum
limit (i.e. in the vicinity of the phase transition point) the threedimensional Ising model can be reduced to an exactly solvable system—the supersymmetric string. This result should allow in principle
the determination of the critical behaviour, but it is still an unsolved
problem.
In order to explain our ideas we have to recall the situation in the 2d
Ising model, solved by Onsager. There are two kinds of basic variables
in this model. First are spin variables cr^ (or order parameters) taking
the values ± 1. Second, there exist disorder variables, first introduced
by Kadanoff and Ceva which are defined as the endpoints of dislocation
lines. These two sets of variables are dual to each other in the same way
as electric and magnetic charges. It is remarkable that although
equations of motion for
and
separately are complicated, the
product variable ij/ = afi satisfies a linear equation. Moreover, in the
continuum limit (near the phase transition point) this linear equation is
reduced to the two-dimensioñal Euclidean Dirac equation, implying
that the i/^-variable describes a fermionic excitation. So, we can say that
the two-dimensional Ising model is equivalent to a system of relativistic,
noninteracting Fermi particles.
Let us turn now to the three-dimensional case. The basic difference
here is connected with the nature of the disorder variables. The
dislocation lines (surrounding the regions of the reversed spins) of the
two-dimensional model are now replaced by dislocation surfaces. The
boundary of these surfaces is now the argument on which the fi variable
depends. So, instead of variables /i(x) of the 2d model we have now the
contour variable fi(C) (here C is a closed loop). Again, in order to
obtain simple equations, we have to form the product of ju(C) and
n , a ( x ¡ ) where are the points adjacent to the loop C. If we draw small
normals to the loop C, connecting it with the points (x j, we obtain a
string with pseudospins living on it, an object which resembles barbed
wire. These “barbed wire” variables satisfy a linear equation in the loop
space. The meaning of this equation is that each small piece of the
barbed wire propagates exactly as an Ising fermion of the two dimensional model, if we consider the plane, orthogonal to the piece under
consideration. Just as a Fermi particle propagator can be represented
by a sum over all possible paths with a certain fermionic structure on
GAUGE FIELDS AND STRINGS
critical properties of the Ising model (such as the singularity in the
specific heat) are shared by many statistical systems in Nature. How>
ever if we are far from the phase transition point no lessons for other
systems can be extracted.
In the present section we shall try to argue that in the continuum
limit (i.e. in the vicinity of the phase transition point) the threedimensional Ising model can be reduced to an exactly solvable system—the supersymmetric string. This result should allow in principle
the determination of the critical behaviour, but it is still an unsolved
problem.
In order to explain our ideas we have to recall the situation in the 2d
Ising model, solved by Onsager. There are two kinds of basic variables
in this model. First are spin variables cr^ (or order parameters) taking
the values ± 1. Second, there exist disorder variables, first introduced
by Kadanoff and Ceva which are defined as the endpoints of dislocation
lines. These two sets of variables are dual to each other in the same way
as electric and magnetic charges. It is remarkable that although
equations of motion for
and
separately are complicated, the
product variable ij/ = afi satisfies a linear equation. Moreover, in the
continuum limit (near the phase transition point) this linear equation is
reduced to the two-dimensioñal Euclidean Dirac equation, implying
that the i/^-variable describes a fermionic excitation. So, we can say that
the two-dimensional Ising model is equivalent to a system of relativistic,
noninteracting Fermi particles.
Let us turn now to the three-dimensional case. The basic difference
here is connected with the nature of the disorder variables. The
dislocation lines (surrounding the regions of the reversed spins) of the
two-dimensional model are now replaced by dislocation surfaces. The
boundary of these surfaces is now the argument on which the fi variable
depends. So, instead of variables /i(x) of the 2d model we have now the
contour variable fi(C) (here C is a closed loop). Again, in order to
obtain simple equations, we have to form the product of ju(C) and
n , a ( x ¡ ) where are the points adjacent to the loop C. If we draw small
normals to the loop C, connecting it with the points (x j, we obtain a
string with pseudospins living on it, an object which resembles barbed
wire. These “barbed wire” variables satisfy a linear equation in the loop
space. The meaning of this equation is that each small piece of the
barbed wire propagates exactly as an Ising fermion of the two dimensional model, if we consider the plane, orthogonal to the piece under
consideration. Just as a Fermi particle propagator can be represented
by a sum over all possible paths with a certain fermionic structure on