ATTEMPT AT A SYNTHESIS
275
them, described in Section 10.3.1, the solution of the loop equation is
found to be equal to a sum over surfaces with extra fermionic structure.
This is described in Section 10.3.2.
The strategy which follows will be to find a continuous string model,
such that the string moves piecewise as a free Dirac fermion. The
natural candidate is the Neveu-Schwarz-Ramond string.
10.3.1 The Dirac Equation in the Two-Dimensional Ising Model
The two-dimensional Ising model is described by the partition function:
Z =
{« T . }
= - Z
+ «
x,6
(10.54)
Here j? is the inverse temperature, the points {jc} belong to the two
dimensional rectangular lattice,
±1, and {8} are the two possible
unit lattice vectors. Important physical information is condensed in the
correlation functions
<<7(jTi) • • • (T(Xff)y
(10.55)
defined in the obvious way. It is possible to obtain a Schwinger-Dyson
chain of equations for G^(xi,..., jc^). However, this way of proceeding
would obscure the exact solvability of the model. The appropriate way
is to introduce the so-called disorder variables. They are defined as
follows. Consider a point of the dual lattice (formed by the centres of
the faces of the original lattice) and draw some path P on the dual
lattice, leading from
to infinity. Change the sign of P on all bonds
intersected by the path. Define the distorted partition function,
Z(jc^, P). Then, the disorder variable
(defined through its Green
functions) is given by:
P )
Analogously one defines
(10.56)
(10.57)
Now, a simple argument shows that (10.57) does not depend on the
paths leading from infinity to
but only on
themselves. To show
this, consider a closed path which surrounds some two-dimensional
region, (a “drop”) and change the sign of the coupling on all the bonds