182
GAUGE FIELDS AND STRINGS
Recalling at last that a sphere has Euler character ^ = 2 we obtain:
N o(L )-N o(L ^ ) = 3x
(9.141)
Actually, this result can be strengthened. Let us show, that for < 0 (a
sphere with more than one handle) Nq(L) = 0. For this we compute:
(L^Lco), = -VW.co, +
= - V X -[V ^ V J o ;,
= -(V^ + ^R)co,
(9.142)
For the manifolds with ;c < 0 one can consider a metric with constant
R < 0 everywhere. For such a metric we have:
{Leo, Leo) =
- \R{(o^, coj > 0
(9.143)
(where scalar products are understood in the same way as above). From
this inequality we conclude that Nq{L) = 0. This means that:
No(L^)= - 3 x = 6 g -6 (iorg>2)
(9.144)
where g is the number of handles. It is easy to check directly, that on a
torus (^ = 1, = 0) No(L) = N^iL^) = 2.
So, we have found that on a sphere we can always introduce a
conformal gauge, which is defined modulo SL(2, C) transformations
(9.139), which require extra gauge fixing. In the case of manifolds with
higher topologies we have topological obstructions for the conformal
gauge. The best thing which can be done is the following choice of
gauge:
(9.145)
where
is some metric, which can be chosen to have constant
negative curvature and which depends on 6g — 6 extra parameters.
Integration over all metrics must include not only functional integration over (p{^) but also the 6g — 6 dimensional integral over {tJ. The
theory of such integrations is not well developed, but we shall explain
here a qualitative meaning of these extra parameters.
Let us first consider a torus. It can be represented as a parallelogram
in the (^-plane for which opposite sides are identified. In general, this
figure can be mapped by a conformal transformation onto any other,
say onto a square. However, the identification of the opposite sides will
be lost. If we insist on preserving the identifications, the only options for
a conformal map onto another parallelogram will be rigid rotations
and scale transformations.Hence each parallelogram representing the
torus can be characterized by the two conformal invariants—the ratio
GAUGE FIELDS AND STRINGS
Recalling at last that a sphere has Euler character ^ = 2 we obtain:
N o(L )-N o(L ^ ) = 3x
(9.141)
Actually, this result can be strengthened. Let us show, that for < 0 (a
sphere with more than one handle) Nq(L) = 0. For this we compute:
(L^Lco), = -VW.co, +
= - V X -[V ^ V J o ;,
= -(V^ + ^R)co,
(9.142)
For the manifolds with ;c < 0 one can consider a metric with constant
R < 0 everywhere. For such a metric we have:
{Leo, Leo) =
- \R{(o^, coj > 0
(9.143)
(where scalar products are understood in the same way as above). From
this inequality we conclude that Nq{L) = 0. This means that:
No(L^)= - 3 x = 6 g -6 (iorg>2)
(9.144)
where g is the number of handles. It is easy to check directly, that on a
torus (^ = 1, = 0) No(L) = N^iL^) = 2.
So, we have found that on a sphere we can always introduce a
conformal gauge, which is defined modulo SL(2, C) transformations
(9.139), which require extra gauge fixing. In the case of manifolds with
higher topologies we have topological obstructions for the conformal
gauge. The best thing which can be done is the following choice of
gauge:
(9.145)
where
is some metric, which can be chosen to have constant
negative curvature and which depends on 6g — 6 extra parameters.
Integration over all metrics must include not only functional integration over (p{^) but also the 6g — 6 dimensional integral over {tJ. The
theory of such integrations is not well developed, but we shall explain
here a qualitative meaning of these extra parameters.
Let us first consider a torus. It can be represented as a parallelogram
in the (^-plane for which opposite sides are identified. In general, this
figure can be mapped by a conformal transformation onto any other,
say onto a square. However, the identification of the opposite sides will
be lost. If we insist on preserving the identifications, the only options for
a conformal map onto another parallelogram will be rigid rotations
and scale transformations.Hence each parallelogram representing the
torus can be characterized by the two conformal invariants—the ratio