Carlip S. Quantum Gravity in 2+1 Dimensions
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246 Appendix C
If a manifold M is flat - that is, if the curvature tensor
R
P
VGT
vanishes
everywhere - and M is also topologically trivial, then M is necessarily a
region of Euclidean (or Minkowski) space. If n\{M)
^=
0, on the other
hand, the vanishing of the curvature does not uniquely determine the
geometry: a noncontractible curve y does not enclose a surface, and we
cannot use Stokes' theorem to evaluate its holonomy.
C.3
Frames
and
spin connections
It is often useful to introduce a local frame, that is, a set of orthonormal
vectors e^
a
that satisfy
g
flv
e,
a
e
v
b
= r,
a
\
e,
a
e
v
b
r,
ab
= g^
v
. (C.15)
In three dimensions, such a frame is often called a triad or dreibein. In
general, the e^ cannot have vanishing covariant derivatives, since this
would imply flatness. Rather, we can write
where co^b is the spin connection. If the connection F£
v
is torsion-free, it
is easy to see that
D
w
e
a
= de
a
+
co
a
b
A e
b
= 0, (C.17)
where e
a
= e^dx
11
is the frame one-form and D^, defined by (C.17), is the
gauge-covariant exterior derivative. If F£
v
is not torsion-free, it may be
checked that D^e
a
is proportional to the torsion. Equation (C.16) can be
solved for co^b in terms of the metric and the torsion; in 2+1 dimensions,
the solution for the torsion-free case is given by equation (2.63).
The curvature may be obtained in terms of the spin connection by
computing the commutator
[V
M
,V
v
]^
a
.
A simple calculation gives
R
a
b
=
e^e
v
bR^
P
adxP
A
dx
G
=
dco
a
h
+
co
a
c
A
a>
c
b
,
where
co
a
b
= co^iydx^ is the spin connection one-form. Equations (C.17)
and (C.18) are the Cartan structure equations.
C.4
The
tangent bundle
Many elements of differential geometry and gauge theory are best ex-
pressed in the language of fiber bundles. The archetypical fiber bundle
is the tangent bundle TM of an n-manifold M. A point in TM consists
of a pair
(p,v
p
),
where p is a point in M and v
p
is a tangent vector at p.
There is a natural projection
%
: TM
—>
M that takes (p,v
p
) to p. The
Cambridge Books Online © Cambridge University Press, 2009
Differential geometry and fiber bundles 247
inverse image n~
1
(p) is known as the fiber at p; for the tangent bundle, it
is just the tangent space T
P
M. Locally, TM looks like U x R", where U
is an open set in M. This product structure generally cannot be extended
globally, however; that is, TM as a whole is typically not a product
manifold M x R".
The cotangent bundle T*M has a similar definition; its points are pairs
(p,
(Op),
where co
p
is a covariant vector (or one-form) at p. Similarly, the
frame bundle consists of pairs
(p,
e
p
), where
e
a
v
is a frame at p.
Much of the differential geometry summarized in the preceding sections
has a natural description in terms of TM. In particular, parallel transport
can be understood as a procedure for lifting curves from M to TM.
Consider a curve y in M and a tangent vector v at y(0). By definition,
the pair (y(0),v) is a point in TM. If we now parallel transport v along
y,
we obtain a new vector U(s,0)v at each point 7(5), and thus a curve
(y(s),
U(s,0)v) in TM. In contrast to equation (C.I), this lifting' description
does not require an explicit choice of coordinates, and is thus useful for
analyzing global issues.
C.5 Fiber bundles and connections
A fiber bundle is a manifold E with a local product structure that gener-
alizes the properties of the tangent bundle. More precisely, a fiber bundle
with total space £, base space M, and fiber F consists of a space E and a
projection n : E
—•
M that is locally trivial - for each point x G M, there
is a neighborhood U a M and a homeomorphism O:[/xF^
TC"
1
^)
such that
7i o ®(x,y) = x, x G 17, y G F. (C.19)
In other words, the region n~
{
(U) looks like a product [/ x F, with each
fiber 7r
-1
(x) homeomorphic to F, but this product description need not
be globally valid.
Given two overlapping neighborhoods Ut and Uj in M, one can form
the transition function
<&..
= (Dr
1
o
<D
7
-
: ([/,- n [/,-) xF^ ([/,- n Uj) x F. (C.20)
For each fixed x G [/,• n (7,, Oy maps F to
itself.
We require that the
transition functions be elements of some group G of transformations of
the fiber F; G is called the structure group. If F is a vector space and G
is a group of linear maps, then E is called a vector bundle. If F is itself
a group G, and the transition functions $ G G act by left multiplication,
then £ is a principle bundle. The tangent bundle TM is a vector bundle;
the frame bundle can be viewed as a principle GL(n, R) bundle.
Cambridge Books Online © Cambridge University Press, 2009
248 Appendix
C
A section
(or
cross-section)
of
a
bundle
is
a
map
a
:
M
—•
E
such that
the image
of
each point
x e M
lies
in the
fiber n~
1
(x) over
x,
that
is,
n
o o{x)
=
x,
x e
M.
(C.21)
Vector fields
are
thus sections
of
the tangent bundle, while charged fields
in quantum electrodynamics
are
sections
of
(7(1) bundles.
There
are
a
number
of
equivalent definitions
of a
connection
on
a
fiber
bundle,
but for
the
purposes
of
this book
a
fairly simple approach
is
sufficient.
Let
E
be
a
vector bundle,
and let
U
be
a
neighborhood
in M
such that 7T~
1
(t7)
« U x
F. We can
choose
a
trivialization
O
and use
it
to identify n~
l
(U) with
U
x
F;
this
is the
analog
of
choosing
a
particular
coordinate system
in
elementary differential geometry.
The
fiber
F is a
vector space,
and
carries
a
representation
of
the
structure group
G. If
we choose
a
basis
e,-
of
F,
points
in
n~
l
(U)
can be
represented
as
pairs
(x^,f*),
and the
generators
of
G will
be
matrices (T
a
)
l
j.
To define parallel transport
in £, we
again
let x^ =
x^(s)
denote
a
curve
in M, and
choose
an
initial vector
v
l
at
x^(0). The parallel transport
equation analogous
to
(C.I)
is
then
where
A^ is
again called
a
connection. Once again, parallel transport
defines
a
lifting
of
the curve x^(s) from
M
to E. As in
section
1,
equation
(C.22)
can be
solved
in
terms
of
a
path-ordered exponent,
v(s) =U(s,0)v(0)
t7(si,s
2
)=Pexp|-jf AfTa—dsj,
(C.23)
where
I
have suppressed
the
fiber indices.
The
concepts
of
covariant
derivative, holonomy,
and
curvature
all
generalize
in a
straightforward
manner.
In
particular,
if
p
t
p
(C.24)
is
the
covariant derivative,
it
is
easy
to
check that
y*
- d
v
V
+ c^A^AA
T
a
v =
F^TaV,
(C.25)
where Cb
c
a
are the
structure constants
of
the
group
G.
Equation (C.25)
shows that
the
familiar gauge field strength tensor
can
be
interpreted
as
the
curvature
of a
bundle connection. Moreover,
if y :
[0,1]
—>
M
Cambridge Books Online © Cambridge University Press, 2009
Differential geometry
and fiber bundles 249
is a closed curve, the parallel transport matrix
1/(1,0)
again gives the
holonomy, and the exponential (C.23) may be recognized as the 'Wilson
loop'
(4.60). The idea that a holonomy may be nonzero even when F^
v
vanishes is familiar in gauge theories: the holonomy is then precisely the
Aharonov-Bohm phase shift [6].
Cambridge Books Online © Cambridge University Press, 2009
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