4.6 Fiber bundles and flat connections 79
which is essentially the same as the holonomy of the geometric structure
defined in section 2. (The expression (4.52) is actually the inverse of the
holonomy of the geometric structure found previously, but the difference
is merely one of convention; traversing y in the opposite direction would
invert the holonomy of the connection.)
We can now specialize to the case G = ISO (2,1). We saw above that
a solution of the Einstein equations in 2+1 dimensions is given by a
Lorentzian structure on a spacetime manifold M. It is now apparent
that we can equally well specify a flat 750(2,1) connection on M. For
the topologies we are considering, in which the holonomy determines the
geometric structure, the two approaches are completely equivalent, thus
confirming the equivalence of the metric and Chern-Simons formulations
of the field equations.
This equivalence can be made rather concrete: given a flat ISO(2,1)
connection (e,co), we can write down an explicit differential equation for
the developing map D of page 66 [3]. Let U be an open set in a spacetime
M, and consider a function q from U to Minkowski space R
2
'
1
that
satisfies
dq
a
+
e
abc
co
b
q
c
+ e
a
= 0. (4.53)
It is easy to check that the integrability conditions for this equation are
the first-order field equations of chapter 2,
T
a
= de
a
+
e
abc
co
b
e
c
= 0
R
a
=
dco
a
+
U
abc
co
b
co
c
= 0. (4.54)
If we choose a gauge such that co
a
= 0 in U - such a choice is always
possible for a flat connection - then (4.53) implies that
v
q
b
,
(4.55)
so the q
a
can be viewed as local coordinates in a patch of Minkowski
space. The conditions (4.54) guarantee only local integrability, but if we
lift e and
a>
from M to its universal covering space M, there are no
obstructions to globally integrating (4.53). We can thus treat q as a map
q : M -> R
2
'
1
.
To understand the global properties of this map, we must investigate
its behavior under gauge transformations. The infinitesimal ISO (2,1)
transformations of
(e,co)
were given in equations (2.66)-(2.67); for a finite
transformation (A,b) € ISO
(2,1),
the integrated version is
e
a
-+
A V
- db
a
+
(dA
a
c
)A
b
c
b
b
+
e
abc
A
cd
b
b
co
d
co
a
->
A
a
b
co
b
+
l
-e
abc
{dA
b
d
)A
cd
.
(4.56)
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