The general solution of (9.150) is given by:
e(z, 2) = I
d^vv -h f(z)
184
GAUGE FIELDS AND STRINGS
(9.151)
where /(z) must be adjusted so as to satisfy the boundary conditions
and G(z, w) is a Green function for the Laplacian.
The solution of this problem always exists, but is it unique? To
answer this question we have to look at the homogeneous equation:
d^e = 0
= Re(e~‘*£(e‘®)) = 0
(9.152)
It is easy to check that this has three linearly independent solutions:
= iz,
= 1 — z^,
= i(l -h z^)
(9.153)
which can be added to (9.151). These solutions are infinitesimal
conformal transformations of the SL(2, R) group which map the unit
disc onto itself. The finite version of these maps is given by:
► z = e*“
z — a
\ — az
(9.154)
(where a is a real phase and a a complex number).
Notice, that after the solution of (9.150) is found, then cU||(^o(5)) is
uniquely defined (modulo SL(2, R) transformations).
Our conclusion for the case of the unit disc is thus the following. The
conformal gauge is accessible provided we include diffeomorphisms
which reparametrize the boundary. This reparametrization, defined
modulo 5L(2, R) transformations, is determined by the original metric
habiO- Since in our original formulation of the functional integral (9.76)
we factored out the diffeomorphisms which become identical at the
boundary, we have to expect that the integration will be reduced to the
form:
^ /( i)
= ^
We shall compute the jacobian later, by now it is important to realize
that the integration over all metrics must include not only the (pintegration, but also integration over all possible reparametrizations
a(s). In a certain sense (a(s)} replaces the discrete set of parameters
which we had for closed surfaces with complicated topology.
e(z, 2) = I
d^vv -h f(z)
184
GAUGE FIELDS AND STRINGS
(9.151)
where /(z) must be adjusted so as to satisfy the boundary conditions
and G(z, w) is a Green function for the Laplacian.
The solution of this problem always exists, but is it unique? To
answer this question we have to look at the homogeneous equation:
d^e = 0
= Re(e~‘*£(e‘®)) = 0
(9.152)
It is easy to check that this has three linearly independent solutions:
= iz,
= 1 — z^,
= i(l -h z^)
(9.153)
which can be added to (9.151). These solutions are infinitesimal
conformal transformations of the SL(2, R) group which map the unit
disc onto itself. The finite version of these maps is given by:
► z = e*“
z — a
\ — az
(9.154)
(where a is a real phase and a a complex number).
Notice, that after the solution of (9.150) is found, then cU||(^o(5)) is
uniquely defined (modulo SL(2, R) transformations).
Our conclusion for the case of the unit disc is thus the following. The
conformal gauge is accessible provided we include diffeomorphisms
which reparametrize the boundary. This reparametrization, defined
modulo 5L(2, R) transformations, is determined by the original metric
habiO- Since in our original formulation of the functional integral (9.76)
we factored out the diffeomorphisms which become identical at the
boundary, we have to expect that the integration will be reduced to the
form:
^ /( i)
= ^
We shall compute the jacobian later, by now it is important to realize
that the integration over all metrics must include not only the (pintegration, but also integration over all possible reparametrizations
a(s). In a certain sense (a(s)} replaces the discrete set of parameters
which we had for closed surfaces with complicated topology.