78
Topological defects
and hap(r, a) is a world-sheet metric of signature (+. -) where a = 0 and
1 refer to r and a respectively. This action displays two-dimensional worldsheet reparametrization invariance and also possesses conformal invariance under
a local rescaling of the world-sheet metric
~hafJ = A(r, a)hafJ
~xp. = O.
(3.57)
With the aid of world-sheet reparametrization invariance, the world-sheet metric
may be reduced to the form
hafJ(r, a) = e"('f,C1)TJafJ
(3.58)
where
TJafJ = diag(l. -1).
(3.59)
With the aid of conformal invariance. it may be further reduced to
hafJ = TJafJ·
(3.60)
The gauges (3.60) are referred to as 'covariant' gauges. There is still further gauge
freedom which we shall exploit shortly.
In a covariant gauge. the equations of motion of the string. obtained by
varying with respect to Xp. and hafJ (exercise 6). take the simple form
aaaaxp. = ( - a2 - - a 2 ) Xp. =
at"2
0
(3.61)
aa 2
with the constraints
axp. axp' + axp. axp' = 0
ar at"
(3.62)
80' aa
axp. axp' = o.
ar
(3.63)
aa
For a closed string loop. there is the boundary condition
XP.(r, 0'+ 1l') = Xp.(r, a).
(3.64)
The remaining gauge degrees of freedom may be used to choose the 'temporal'
gauge in which r is identified with XO == t (Minkowski time.) Then, the equations
of motion and constraints become
( ~-~)x=o
(3.65)
at 2
aa 2
ax . ax =0
(3.66)
at aa
(aa~Y + (~:Y =
(3.67)
1.