Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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which are to be solved for k
1
and k
2
. Note that
e
i(k
1
þk
2
)L
= 1.
Case 4: n = 3
The full power of the Bethe ansatz method becomes
evident for three particles. Here
¼
X
x<y<z
aðx; y; zÞj ðx; y; zÞi ½16
There are several cases to consi der:
1. y > x þ 1 and z > y þ 1, where
Eaðx; y; zÞ¼ðL 6Þaðx; y; zÞþaðx 1; y; zÞ
þ aðx; y 1; zÞþaðx; y; z 1Þ½17
By a(x 1, y, z), we mean a(x þ 1, y,z
)þ
a(x 1, y, z), etc.
2. y = x þ 1 and z > y þ 1, with
Eaðx; x þ 1; zÞ
¼ðL 4Þaðx; x þ 1; zÞþa ðx 1; x þ 1; zÞ
þ aðx; x þ 2; zÞþaðx; x þ 1; z 1Þ½18
3. y > x þ 1 and z = y þ 1, where
Eaðx;y;
y þ1Þ
¼ðL 4Þaðx;y; y þ1Þþaðx 1;y; y þ1Þ
þaðx; y 1; y þ1Þþaðx; y; y þ2Þ½19
4. y = x þ 1 and z = y þ 1, for which
Eaðx; x þ 1; x þ 2Þ¼ðL 2Þaðx 1; x þ 1; x þ 2Þ
þ aðx
; x þ 1; x þ 3Þ½20
Again, we must ensure that these equations are
compatible. This involves comparison of the last
three equations with [17]. The three equations to be
satisfied are
2aðx; x þ 1; zÞ¼aðx; x; zÞþaðx þ 1; x þ 1; zÞ½21
2aðx; y; y þ 1Þ¼aðx; y; yÞþaðx; y þ 1; y þ 1Þ½22
4aðx; x þ1; x þ 2Þ¼aðx; x;
x þ2Þþa ðx; x þ 1; x þ1Þ
þaðx; x þ 2; x þ2Þ
þaðx þ 1; x þ1; x þ2Þ½23
But note that setting z = x þ2in[21] and y = x þ1
in [22] leads to [23] being autom atically satisfied.
We are thus left with only two equations [21] and
[22]. Note the similarity between these two equa-
tions and the meeti ng condition [10] for the n = 2
case.
In this case the Bethe ansatz is
aðx; y; zÞ¼A
123
z
x
1
z
y
2
z
z
3
þ A
132
z
x
1
z
y
3
z
z
2
þ A
213
z
x
2
z
y
1
z
z
3
þ A
231
z
x
2
z
y
3
z
z
1
þ A
321
z
x
3
z
y
2
z
z
1
þ A
312
z
x
3
z
y
1
z
z
2
½24
in which z
j
= e
ik
j
. This is a sum over the 3!
permutations of the integers 1, 2, 3. Inserting this
ansatz into [17] gives
E ¼ L 6 þ 2ðcos k
1
þ cos k
2
þ cos k
3
Þ½25
To determine the k
j
, it is convenient to define
s
ij
¼ 1 2z
j
þ z
i
z
j
½26
Substitution of [24] into the meeting conditions [21]
and [22] then gives
s
12
A
123
þ s
21
A
213
þ s
13
A
132
þ s
31
A
312
þ s
23
A
231
þ s
32
A
321
¼ 0 ½27
s
23
A
123
þ s
32
A
132
þ s
13
A
213
þ s
31
A
231
þ s
21
A
321
þ s
12
A
312
¼ 0 ½28
These equations are assumed to be satisf ied in
permutation pairs, that is,
s
12
A
123
þ s
21
A
213
¼ 0
s
23
A
123
þ s
32
A
132
¼ 0; etc:
½29
Up to an overall constant, the relations [27] and [28]
are satisfied by
A
123
¼s
21
s
31
s
32
; A
132
¼s
31
s
21
s
23
A
312
¼s
13
s
23
s
21
; A
321
¼s
23
s
13
s
12
A
231
¼s
32
s
12
s
13
; A
213
¼s
12
s
32
s
31
½30
The boundary condition, a(y, z, x þ L) = a(x, y, z),
gives
z
L
1
A
321
A
132
z
x
1
z
y
3
z
x
2
þ z
L
2
A
312
A
231
z
x
2
z
y
3
z
x
1
þ z
L
1
A
231
A
123
z
x
1
z
y
2
z
x
3
þ z
L
3
A
213
A
321
z
x
3
z
y
2
z
x
1
þ z
L
2
A
132
A
213
z
x
2
z
y
1
z
x
3
þ z
L
3
A
123
A
312
z
x
3
z
y
1
z
x
2
¼ 0 ½31
This leads to the equations
z
L
1
¼
A
123
A
231
¼
A
132
A
321
¼
s
21
s
31
s
12
s
13
z
L
2
¼
A
213
A
132
¼
A
231
A
312
¼
s
12
s
32
s
21
s
23
z
L
3
¼
A
321
A
213
¼
A
312
A
123
¼
s
13
s
23
s
31
s
32
½32
which can be solved for the Bethe roots k
j
.
Bethe Ansatz 255
General n
The general Bethe ansatz is
aðx
1
; ...; x
n
Þ¼
X
P
A
p
1
;...;p
n
z
x
1
p
1
...z
x
n
p
n
½33
where the sum is over all n! permutations
P = {p
1
, ..., p
n
} of the integers 1, ..., n. The boundary
condition is
aðx
2
; x
3
; ...; x
n
; x
1
þ LÞ¼aðx
1
; x
2
; ...; x
n
Þ½34
leading to the Bethe equations
z
L
p
1
¼
A
p
1
;...;p
n
A
p
2
;...;p
n
;p
1
½35
for all permutations, with
A
p
1
;...;p
n
¼
P
Y
1i<jn
s
p
j
;p
i
½36
where
P
is the signature of the permutation. Finally,
z
L
p
1
¼ ðÞ
n1
Y
n
‘¼2
s
p
‘
;p
1
s
p
1
;p
‘
or z
L
j
¼ ðÞ
n1
Y
n
‘¼1
6¼j
s
‘;j
s
j;‘
½37
for j = 1, ..., n. The eigenvalues are given by
E ¼ L þ
X
n
j¼1
2 cos k
j
2
½38
Another form of the Bethe equations is obtained
by defining
e
ik
j
¼
u
j
ð1/2Þi
u
j
þð1/2Þi
½39
which gives
E ¼ L
X
n
j¼1
1
u
2
j
þ 1/4
½40
with u
j
satisfying
u
j
ð1/2Þi
u
j
þð1/2Þi
L
¼
Y
n
‘¼1
u
j
u
‘
i
u
j
u
‘
þ i
½41
for j = 1, ..., n.
All eigenvalues of the Heisenberg spin chain may
be obtained in terms of the Bethe ansatz solution.
For example, the distribution of roots u
j
for the
ground state are real and symmetric about the
origin. Excitations may involve complex roots.
Although obtained exactly in terms of the Bethe
roots, the Bethe ansatz wave function is
cumbersome.
We have thus seen how the Bethe ansatz works
for the Heisenberg spin chain. The underlying
mechanism is the way in which the collision or
meeting conditions can be handled in terms of two-
body interactions. To see this more clearly, the six
permutation pair equations [29] can be written in
the genera l form A
abc
= Y
ab
A
bac
and A
abc
= Y
bc
A
acb
,
where Y
ab
= s
ba
=s
ab
. Now there are two possible
paths to get from A
abc
to A
cba
, namely
A
cba
¼ Y
ab
Y
ac
Y
bc
A
abc
A
cba
¼ Y
bc
Y
ac
Y
ab
A
abc
½42
Both paths must be equivalent, with
Y
ab
Y
ba
¼ 1 and Y
ab
Y
ac
Y
bc
¼ Y
bc
Y
ac
Y
ab
½43
The latter is a condition of nondiffraction or
equivalently a manifestation of the Yang–Baxter
equation.
Historically, the next model to be exactly solved in
terms of the Bethe ansatz was the one-dimensional
model of N interacting bosons on a line of length L
defined by the Hamiltonian
H ¼
X
N
i¼1
@
2
@x
2
i
þ 2c
X
1i<jN
ðx
i
x
j
Þ½44
where c is a measure of the interaction strength. For
this model the Bethe ansatz wave function is of the
same form as [33] with the two-body interaction
term given by
s
ab
¼ k
a
k
b
þ ic ½45
The Bethe equations are given by
expðik
j
LÞ¼
Y
N
‘¼1
k
j
k
‘
þ ic
k
j
k
‘
ic
for j ¼ 1; ...; N ½46
The energy eigenvalue is
E ¼
X
N
j¼1
k
2
j
½47
For repulsive (c > 0) interactions, one can prove that
all Bethe roots are real.
The Bethe ansatz has been applied to a number of
other and more general models, both for discrete
spins and in the continuum. These include the
anisotropic Heisenberg (XXZ) spin chain, for
which the above working readily generalizes to
trigonometric functions. The underlying ansatz [33]
remains the same . One key generalization is the
nested Bethe ansatz, which arises, for example, in
the solution of the general N-state permutator
model, the Hubbard model, and the Gaudi n–Yang
model of interacting fermions. For such models the
nested Bethe ansatz involves an additional level of
work to determine the amplitudes appearing in the
256 Bethe Ansatz
wave function [33] due to higher symmetries. This
results in Bethe equations involving different types
or colors of roots.
The exactly solved one-dimensional quantum spin
chains may also be obtained from their two-dimen-
sional classical counterparts – the vertex models. For
example, the six-vertex model shares the same Bethe
ansatz wave function and Bethe equations as the
XXZ spin chain. The more general permutator
Hamiltonians are related to multistate vertex models.
One may also consider other spin-S models.
Thediscussioninthisarticlehascenteredonwhatis
known as the coordinate Bethe ansatz. Another
formulation is the algebraic Bethe ansatz, which was
developed for the systematic treatment of the higher-
spin models. In this formulation, operators create the
Bethe states by acting on a vacuum. The algebraic
Bethe ansatz goes hand-in-hand with the quantum
inverse-scattering method. In all of the exactly solved
Bethe ansatz models, it is possible to derive quantities
like the ground-state energy per site via the root density
method, which assumes that the Bethe roots form a
uniform distribution in the infinite-size limit. The
thermodynamics of the Bethe ansatz solvable models
may also be calculated in a systematic fashion.
Despite Bethe’s early optimism, the Bethe ansatz
has not been extended to higher-di mensional
systems.
See also: Affine Quantum Groups; Eight Vertex and Hard
Hexagon Models; Integrability and Quantum Field
Theory; Integrable Systems: Overview; Quantum Spin
Systems; Yang–Baxter Equations.
Further Reading
Baxter RJ (1983) Exactly Solved Models in Statistical Mechanics.
London: Academic Press.
Baxter RJ (2003) Completeness of the Bethe ansatz for the six-
and eight-vertex models. Journal of Statistical Physics
108: 1–48.
Bethe HA (1931) Zur Theorie der Metalle I. Eigenwerte und
Eigenfunktionen der linearen Atomkette. Zeitschrift fu¨ r Physik
71: 205–226.
Gaudin M (1967) Un Syste`me a` Une Dimension de Fermions en
Interaction. Physics Letters A 24: 55–56.
Gaudin M (1983) la Fonction d’onde de Bethe. Paris: Masson.
Korepin VE, Izergin AG, and Bogoliubov NM (1993) Quantum
Inverse Scattering Method and Correlation Functions.
Cambridge: Cambridge University Press.
Lieb EH and Liniger W (1963) Exact analysis of an interacting
Bose gas I. The general solution and the ground state. Physical
Review 130: 1605–1616.
Mattis DC (1993) The Many-Body Problem: An Encyclopaedia of
Exactly Solved Models in One-Dimension. Singapore: World
Scientific.
McGuire JB (1964) Study of exactly soluble one-dimensional
N-body problems. Journal of Mathematical Physics
5: 622–636.
Sutherland B (2004) Beautiful Models: 70 Years of Exactly Solved
Quantum Many–Body Problems. Singapore: World Scientific.
Takahashi M (1999) Thermodynamics of One-Dimensional
Solvable Models. Cambridge: Cambridge University Press.
Yang CN (1967) Some exact results for the many-body problem
in one-dimension with repulsive Delta-function interaction.
Physical Review Letters 19: 1312–1315.
BF Theories
M Blau, Universite
´
de Neucha
ˆ
tel, Neucha
ˆ
tel,
Switzerland
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
BF theories are a class of gauge theories with a
nontrivial metric-independent classical action. As
such these theories are candidate topological field
theories akin to the Chern–Simons theory in three
dimensions, but in contrast to the Chern–Simons
theory these exist and are well defined in arbitrary
dimensions.
The name ‘‘BF theories’’ derives from the fact
that, roughly (see [1] below and the subsequent
discussion for a more precise description), the action
of the BF theory takes the form
R
B ^ F
A
with F
A
the
curvature of a connection A and B a Lagrange
multiplier. The classical equations of motion imply
that A is flat, F
A
= 0, and thus BF theories are
topological gauge theories of flat connections.
Abelian BF theories and their relation to topolo-
gical invariants (the Ray–Singer torsion) were
originally discussed by Schwarz (1978, 1979).In
the context of the topological field t heory, non-
abelian BF theories w ere introduced in Horowitz
(1989)andBlau and Thompson (1989, 1991).
Since then, BF theories have attr acted a lot of
attention as simple toy-models of (topological)
gauge theories, and also because of their relation-
ships with the Chern–Simons theory, the Yang–Mills
theory, and gauge-theory formulations of gravity, as
well as because of the rather rich and intricate
structure of their quantum theori es.
The purpose of this article is to provide an
overview of these various features of BF theories.
The standard reference for the basic classical and
quantum properties of BF theories is Birmingham
et al. (1991).
BF Theories 257
Basic Classical Properties of BF Theories
Nonabelian BF Theories
The classical action and equations of motion Typi-
Typically, the classical action of the BF theory takes
the form
S
BF
ðA; BÞ¼
Z
M
tr
G
B ^ F
A
½1
where F
A
is the curvature of a connection A on a
principal G-bundle P ! M over an n-dimensional
manifold M, B is an ad-equivariant horizontal
(n 2)-form on P,andtr
G
(a trace) denotes an
ad-invariant nondegenerate scalar product on the
Lie algebra g of the Lie group G. Generalizations of
this are possible, in particular, for G abelian or for
n = 3 and are mentioned below.
We consider F
A
and B as forms on M taking
values in the bundle of Lie algebras ad P = P
ad
g
and refer to such objects as elements of
(M, g).
Then tr B ^ F
A
2
n
(M, R) is a volume form on M.
In order to simplify the exposition, in the following
we will mostly assume that G is compact semisimple
and that M is compact without a boundary (even
though relaxing any one of these conditions is
possible and also of interest in its own right).
Varying the action [1] with respect to A and B,
one obtains the classical equatio ns of motion
F
A
¼ 0; d
A
B ¼ 0 ½2
where
d
A
B ¼ dB þ½A; B½3
is the covariant exterior derivative. In particular,
therefore, the equations of motion imply that the
connection A is flat.
Gauge invariance For any n, the actio n [1] is
invariant under G gauge transformations (vertical
automorphisms of P) acting on A and B as
A ! g
1
Ag þ g
1
dg; B ! g
1
Bg ½4
(the latter is what is meant by the fact that B takes
values in ad P), because F
A
is also ad-equivariant,
F
A
! g
1
F
Ag
, and tr
G
is ad-invariant. The infinitesi-
mal version of this statement is that the action is
invariant under the variations
A ¼ d
A
; B ¼½B;½5
where 2
0
(M, g) can (formally) be thought of as
an element of the Lie algebra of the group of gauge
transformations.
Gauge-fixing this symmetry can proceed in the
usual way (via the Faddeev–Popov or Becchi–Rouet–
Stora–Tyupkin procedure), a typical gauge choice
being d
A
0
? (A A
0
) = 0whereA
0
is a reference
connection, and ? is the Hodge duality operator
corresponding to a choice of metric on M.
Local p-form symmetries For n = 2, the only local
symmetries of the BF action are the above G gauge
transformations. For n > 2, however, there are other
local symmetries associated with shifts of B
p
2
p
(M, g)withp = n 2 > 0. Indeed, integration by
parts using Stokes’ theorem and @M = 0showsthat[1 ]
is invariant under
A !A; B
p
!B
p
þd
A
p1
;
p1
2
p1
ðM; gÞ½6
For p = 1, is a 0-form and the invariance follows.
For p > 1, however, the gauge parameter has, in
some sense, its own gauge invariance. Namely,
under the shift
p1
!
p1
þ d
A
p2
½7
one has
d
A
p1
! d
A
p1
þ½F
A
;
p2
½8
Thus for F
A
= 0, the shift [7] has no effect on the
local symmetry [6]. Likewise, for p > 2 the parameter
p2
itself has a similar invariance, etc. Since F
A
= 0
is one of the classical equations of motion, the shift
symmetry [6] is what is called an ‘‘on-shell reducible
symmetry.’’ Gauge-fixing such symmetries is not
straightforward, and one generally appeals to the
Batalin–Vilkovisky formalism to accomplish this.
Diffeomorphisms and local symmetries One mani-
festation of the general covariance of the BF action
[1] is the on-shell equivalence of (infinitesimal)
diffeomorphisms and (infinitesimal) local symme-
tries. Diffeomorphisms are generated by the Lie
derivative L
X
along a vector field X. The action of
L
X
on differential forms is given by the Cartan
formula L
X
= di
X
þ i
X
d, where i
(.)
is the operation
of contraction. The action of the Lie derivative on
A and B can be written in gauge covariant form as
L
X
A ¼ i
X
F
A
þ d
A
ðXÞ;
L
X
B ¼ i
X
d
A
B þ½B;ðXÞ þ d
A
0
ðXÞ
½9
where (X) = i
X
A and
0
(X) = i
X
B. This s hows that
on-shell diffeomorphisms are equivalent to field-
dependent gauge and p-form symmetries of the
BF action.
The classical moduli space The classical moduli
space C= C(P, M, G) is the space of solutions to the
classical equations of motion modulo the local
symmetries of the action. Since the field content
258 BF Theories
and the nature of the local symmetries of the BF
theory depend strongly on the dimension n of M, the
structure and interpretation of the classical moduli
space also depend on n.
For n = 2, by [5] the equation of motion [2] for
B 2
0
(M, g) says that A is invariant under the
infinitesimal gauge transformation generated by B.
Thus if A is ‘‘irreducible,’’ there are no nontrivial
solutions for B and, away from reducible flat
connections, the classical moduli space is just the
moduli space of flat connections on P ! M over the
surface M:
C
n¼2
¼M
flat
ðP; GÞ½10
This space may or may not be empty, depending on
whether P admits flat connections or not.
For n = 3, the equation of motion [2] for
B 2
1
(M, g) says that B is a tangent vector to the
space of flat connections at the flat connection A,in
the sense that under the variation A = B, one has
F
A
¼ d
A
B ¼ 0 ½11
The local G gauge symmetry and the 1-form symmetry
[6] now imply that the moduli space of classical
solutions can be identified with the (co-)tangent bundle
of the moduli space of flat connections on P ! M
over the 3-manifold M:
C
n¼3
¼ TM
flat
ðP; GÞ½12
In higher dimensions there appears to be less
geometrical structure associated with BF theories,
and all that can be said in general is that the tangent
space to C
n
at a solution (A, B) of the equations of
motion [2] is the vector space:
T
ðA;BÞ
C
n
¼ H
1
A
ðM; gÞH
n2
A
ðM; gÞ½13
where H
k
A
(M, g) are the cohomology groups of the
deformation complex
d
A
:
ðM; gÞ!
þ1
ðM; gÞ½14
associated with the flat connection A, F
A
= (d
A
)
2
= 0.
When M is topologically of the form M = R
(where one can think of R as time), one has
T
ðA;BÞ
C
n
¼ H
1
A
ð; gÞH
n2
A
ð; gÞ½15
This is naturally a symplectic vector space (necessary
for a phase space), the nondegenerate antisymmetric
pairing being given by Poincare´ duality:
!ð½a
1
;½b
1
;½a
2
;½b
2
Þ¼
Z
tr
G
ða
1
^b
2
a
2
^b
1
Þ½16
Metric independence Perhaps the most important
property of the action [1] is that, in contrast to,
for example, the usual Yang–Mills action for
nonabelian gauge fields
S
YM
¼
1
4g
2
Z
M
tr
G
F
A
^ ?F
A
½17
it does not require a metric (or the corresponding
Hodge duality operator ?) for its formulation. This
makes it a candidate action for a ‘‘topological field
theory,’’ this term loosely referring to field theories
which, in a suitable sense, do not depend on
additional structures imposed on the underlying
space(-time) manifold M, in this case a Riemannian
structure.
To establish that BF theories are ‘‘topological
quantum field theories,’’ one needs to show that
the partition function (and correlation functions)
of the quantized BF theory are also metric
independent. This is not completely automatic as
typically the metric enters in the gauge fixing of
the local symmetries of the action which is
required to make the quantum theory well defined.
The usual lore is that since the metric only enters
through the gauge fixing and since the quantum
theory should be independent of the choice of
gauge, it should also be metric independent. In the
case of nonabelian BF theories, the complexity of
their local symmetries complicates the analysis
somewhat, but it can nevertheless be shown that
BF theories indeed define topological field theories
also at the quantum level.
Special Features of Abelian BF Theories
All the features of nonabelian BF theories discussed
above are, of course, also valid when G is abelian
(with some obvious modifications and simplifica-
tions). However, when G is abelian, a more general
action than [1] is possible. Indeed, although there is
no obvious higher p-form analog of nonabelian
gauge fields, in the abelian case G = U(1) or G = R,
and the condition F
A
2
2
(M, R) can be relaxed. In
particular, one can consider the actions
Sðn; pÞSðB
p
; C
np1
Þ¼
Z
M
B
p
^ dC
np1
½18
with B
p
2
p
(M, R), C
np1
2
np1
(M, R), and
F
C
= dC; its (n p)-form field strength. More gen-
erally, one can also consider the hybrid action
S
A
ðn; pÞ¼
Z
M
B
p
^ d
A
C
np1
½19
where A is a fixed (nondynamical) flat G-connection,
d
2
A
= 0, and B and C take values in the corresponding
adjoint bundle. This action can be considered as the
linearization of the nonabelian BF action [1] around
BF Theories 259
the flat connection A, and it reduces to the abelian BF
action [18] for g = R.
The action is invariant under the (reducible) local
symmetries
B
p
! B
p
þ d
A
p1
C
np1
! C
np1
þ d
A
0
np2
½20
The space of solutions to the equations of motion
d
A
C = d
A
B = 0 modulo gauge symmetries is (cf. [13])
the finite-dimensional vector space
C
n; p
¼ H
p
A
ðM; gÞH
np1
A
ðM; gÞ½21
which is naturally symplectic for M = R.
Uses and Applications of Quantum
Abelian BF Theories
Quantization of Abelian BF Theories and the
Ray–Singer Torsion
We will now show that the partition function of
the abelian BF theory (actually more generally that
of the linearized nonabelian BF action [19])is
related to the Ray–Singer torsion of M.This
requires some preparatory material on Gaussian
path integrals, determinants, and gauge fixing that
we present first.
In order to simplify the exposition, we assume
that there are no harmonic modes, either because
they have been gauged away or because the
cohomology groups of d
A
are trivial, H
k
A
(M, g) = 0,
that is, the deformation complex [14] is ‘‘acyclic.’’
Laplacians, determinants, and the Ray–Singer
torsion Choosing a Riemannian metric g (and
Hodge duality operator ?)onM, the twisted
Laplacian on p-forms is
ðpÞ
A
¼ðd
A
þ d
?
A
Þ
2
¼ d
A
d
?
A
þ d
?
A
d
A
½22
where d
?
A
= ? d
A
? is the adjoint of d with respect to
the scalar product on p-forms defined by ?. This is an
elliptic operator whose determinant can be defined, for
example, by a -function regularization. Denoting the
(nonzero) eigenvalues of
(p)
A
by
(p)
k
,its-function is
ðpÞ
ðsÞ¼
X
k
ðpÞ
k
s
½23
This converges for Re(s) sufficiently large and can be
analytically continued to a meromorphic function of
s analytic at s = 0, so that
det
ðpÞ
A
:¼ e
ðpÞ0
ð0Þ
½24
is well defined. The Ray–Singer torsion of (M, g)
(with respect to the flat connection A) is then
defined by
T
A
ðMÞ¼
Y
n
p¼0
det
ðpÞ
A
ð1Þ
p
p=2
½25
Even though this definition depends strongly on the
metric g on M, the Ray–Singer torsion has the
remarkable property of being independent of g. The
Ray–Singer torsion can be shown to be trivial
(essentially =1 modulo zero-mode contributions)
in even dimensions, but is a nontrivial topological
invariant in odd dimensions. Henceforth, we will
suppress the dependence on M and denote the
n-dimensional Ray–Singe r torsion by T
A
(n).
Gaussian path integrals and determinants The path
integral for abelian BF theories is modeled on the
usual formula for a -function
n
ðxÞ¼
1
ð
ffiffiffiffiffiffi
2
p
Þ
n
Z
R
n
d
n
e
ix
½26
from which one deduces the Gaussian integral
formula
1
ð
ffiffiffiffiffiffi
2
p
Þ
n
Z
R
n
R
n
d
n
d
n
x e
iDxþiKxþiJ
¼
Z
R
n
d
n
x
n
ðDx þ JÞe
iKx
¼
1
det D
e
iK:D
1
J
½27
Here, we have assumed that the operator (matrix ) D
is invertible. The model that one uses in the path
integral is that
Z
d½d½e
i
R
M
?D
¼ðdet DÞ
1
½28
where is a set of fields and the are a set of dual
fields with D again a nondegenerate operator. The
inverse determinant arises for Grassmann even fields
(as in [27]), while it is the determinant that appears
for Grassmann odd fields.
Gauge fixing – the Faddeev–Popov trick If the
action [19], S
A
(n, p) =
R
B
p
d
A
C
np1
, were non-
degenerate, its partition function could be defined
directly by [28]. However, because of gauge invariance
of the action, the kinetic term is degenerate and one
needs to eliminate the gauge freedom to obtain an (at
least formally) well-defined expression for the partition
function. Concretely, this degeneracy can be seen by
260 BF Theories
recalling that, when there are no harmonic forms (as we
have assumed), there is a unique orthogonal Hodge
decomposition of a p-form B
p
2
p
(M, g)intoasumof
a d
A
-exact and a d
A
-coexact form:
B
p
¼ d
A
p1
þ d
?
A
pþ1
½29
(and likewise for C). Evidently, the exact (longitudinal)
parts d
A
of B and C do not appear in the action, and
these are precisely the gauge-dependent parts of B and
C under the gauge transformation [20]. Gauge fixing
amounts to imposing a condition F(B
p
) = 0onB
p
that
determines the longitudinal part uniquely in terms of
the transversal part d
?
A
. A natural condition is
d
A
p1
¼ 0 ,FðB
p
Þ¼d
?
A
B
p
¼ 0 ½ 30
A gauge-fixing condition independent of the partition
function results from inserting ‘‘1’’ in the form of
1 ¼
Z
G
d½gðFðB
g
ÞÞ
F
ðBÞ½31
into the functional integral (the Faddeev–Popov
trick), where G is the gauge group. This defines the
Faddeev–Popov determinant
F
, and the functional
properties of the delta functional imply that
F
is
the determinant of the operator that one obtains
upon gauge variation of F(B).
In the general case of reduci ble gauge symmetries,
the nature of the gauge group is complicated and
requires some more thought. In the irreducible case,
however, that is, for p = 1, the Lie algebra of the
gauge group can be identified with
0
(M, g), and
F
is the determinant of the operator:
F
B
d
A
:
0
ðM; gÞ!
0
ðM; gÞ½32
For [30], this is simply the Laplacian on 0-forms,
and thus
F
¼ det
ð0Þ
A
½33
The partition function Following the finite-dimen-
sional model, both the -function implementing the
gauge-fixing condition and the Faddeev–Popov
determinant can be lifted into the exponential, the
former by a Lagrange multiplier [26], a Grassmann
even 0-form, and the latter by a pair of Grassmann
odd 0-forms c and c
¯
[28], the ghost and antighost
fields, respectively. The sum of the classical action
and these gauge-fixing and ghost terms defines the
(BRST-invariant) ‘‘quantum action’’ S
q
A
(n, p), and the
partition function is
Z
A
ðn; pÞ¼
Z
d½e
iS
q
A
ðn;pÞðÞ
½34
where denotes collectively all the fields. Concre-
tely, when n = 2andp = 0 (or, equivalently, p = 1),
the quantum action is
S
q
A
ð2; 0Þ¼
Z
B
0
d
A
C
1
þ d
A
? C
1
þ
c ?
ð0Þ
A
c ½ 35
Likewise, for n = 3 and p = 1 (the only other case
when the gauge symmetry is indeed irr educible),
both B
1
and C
1
require separate gauge fixing, and
the quantum action is
S
q
A
ð3; 1 Þ¼
Z
B
1
d
A
C
1
þ d
A
? C
1
þ
c ?
ð0Þ
A
c
þ
0
d
A
? B
1
þ
c
0
?
ð0Þ
A
c
0
½36
Formally, therefore, the two-dimensional partition
function is
Z
A
ð2; 0Þ¼
det
ð0Þ
det D
A
½37
where D
A
is the operator:
D
A
¼
?d
A
?d
A
?
:
1
ðM; gÞ
!
0
ðM; gÞ
0
ðM; gÞ½38
One can define the determinant of this operator as
the square root of the determinant of the operator
D
?
A
D
A
=
(1)
A
, and therefore the partition function
Z
A
ð2; 0Þ¼det
ð0Þ
ðdet
ð1Þ
Þ
1=2
¼ T
A
ð2Þ½39
is equ al to the two-dimensional Ray–Singer torsion
[25]. In this case, it is easy to see directly that the
even-dimensional Ray–Singer torsion is trivial, as
one could have equally well defined the determinant
of D
A
as the square root of the operator
D
A
D
?
A
=
(0)
A
(0)
A
, which implies Z
A
(2, 0) = 1.
In three dimensions, the two pairs of ghosts each
contribute a det
(0)
A
, and thus
Z
A
ð3; 1Þ¼
ðdet
ð0Þ
Þ
2
det D
A
½40
where
D
A
¼
?d
A
d
A
d
A
? 0
!
:
0
ðM; gÞ
1
ðM; gÞ
!
0
ðM; gÞ
1
ðM; gÞ½41
is the operator acting on the fields (B
1
, C
1
, ,
0
). As
before, this operator can be diagonal ized by squar-
ing it, D
A
D
A
=
(0)
(1)
, and thus
Z
A
ð3; 1Þ¼ðdet
ð0Þ
A
Þ
3=2
ðdet
ð1Þ
A
Þ
1=2
¼ T
A
ð3Þ
1
½42
BF Theories 261
is again related to the (this time genuinely nontrivial)
Ray–Singer torsion.
In spite of the complications caused by reducible
gauge symmetries, it can be shown that all of the
above generalizes to arbitrary n and p, with the
result that (for n odd)
Z
A
ðn; p Þ¼T
A
ðnÞ
ð1Þ
p
½43
confirming the topological nature of BF theories.
In the nonabelian case, the situation is significantly
more complica ted because of the complexity of the
classical moduli space, the (higher cohomology) zero
modes, and the on-shell reducibility of the gauge
symmetries. Nevertheless, ignoring all the zero modes
except those of A, that is, except the moduli m of flat
connections A(m), the result is similar to that in the
abelian case, in that the partition function reduces to an
integral over the moduli space of flat connections, with
measure determined by the Ray–Singer torsion T
A(m)
.
Linking Numbers as Observables of Abelian
BF Theories
With the exception of p = 0, there are no interesting
‘ ‘local’’ observables (gauge-invariant functionals of the
fields C and B)intheabelianBFtheory,sincethegauge-
invariant field strengths dC and dB vanish by the
equations of motion. (For p = 0, B is a gauge-invariant
0-form and hence B(x) is a good local observable.)
However, as in the Chern–Simons and Yang–Mills
theories, certain (weakly) nonlocal observables such as
Wilson loops are also of interest. In the case at hand (eqn
[18]), we have abelian Wilson surface operators
W
S
½B¼
Z
S
B; W
S
0
½C¼
Z
S
0
C ½44
associated with p- and (n p 1)-dimensional sub-
manifolds S and S
0
of M, respectively. These operators
are gauge invariant , that is, invariant under the local
symmetries [20] provided that @S = @S
0
= 0, so that S
and S
0
represent homology cycles of M.
For M = R
n
, correlation functions of these opera-
tors are related to the topological linking number of
S and S
0
. We choose S = @ and S
0
= @
0
to be
disjoint compact-oriented boundaries of oriented
submanifolds and
0
of R
n
. We also introduce
de Rham currents
and
S
(essentially distribu-
tional differential forms with -function support on
or S, respectively), characterized by the properties
Z
S
!
p
¼
Z
M
S
^ !
p
Z
!
pþ1
¼
Z
M
^ !
pþ1
½45
for all !
k
2
k
(M, R) (and likewise for S
0
and
0
).
Since the dimension of is equal to the codimen-
sion of S
0
= @
0
, and S
0
will generically intersect
transversally at isolated points, and we define the
‘‘linking number’’ of S and S
0
to be the intersection
number of and S
0
, expressed in terms of de Rham
currents as
LðS; S
0
Þ¼
Z
S
0
¼
Z
M
S
0
½46
In terms of de Rham currents, the Wilson surface
operators can be written as W
S
[B] =
R
M
S
^ B, etc.
Thus, the generating functional for correlation
functions of Wilson surface operat ors
he
iW
S
½B
e
iW
S
0
½C
i
¼
Z
D½CD½Be
i
R
M
ðB d Cþ
S
0
Cþ
S
BÞ
½47
is simply a Gaussian path integral. Using the
defining properties of de Rham currents, this can
be formally evaluated (using [27]) to give
he
iW
S
½B
e
iW
S
0
½C
i¼e
iLðS;S
0
Þ
½48
As expected, correlation functions of these topolog-
ical field theories encode topolog ical information.
Uses and Applications of Classical
Nonabelian BF Theories
Low-dimensional BF theories are closely related to
other theories of interest, for example, the Yang–
Mills theory, the Chern–Simons theory, and gravity.
Here, we briefly review some of these relationships.
In order to avoid the complexities of quantum
nonabelian BF theories, we focus on their classical
features . Brie f sugges tions for further reading are
provided at the end of each subsection.
Relation with Yang–Mills Theory
In any dimension, the nonabelian BF action can be
regarded as the zero-coupling limit g
2
! 0 of the
Yang–Mills theory since the Yang–Mills action [17]
can be written in first-order form as
1
4g
2
Z
M
tr
G
F
A
^ ?F
A
Z
M
tr
G
½iB
n2
^ F
A
þ g
2
B
n2
^ ?B
n2
½49
However, whereas for n 3 the B
2
-term breaks the
p-form gauge invariance of the BF action (and thus
liberates the physical Yang–Mills degrees of free-
dom), this limit is nonsingul ar in two dimensions
where this p-form symmetry is absent and, indeed,
both theories have zero physical degrees of freedom.
262 BF Theories
A nonsingular BF-like zero coupling limit of
the Yang–Mills theory for n 3 can be obtai ned
by introducing an auxiliary (Stu¨ ckelberg) field
2
n3
(M, g) which restores the p-form gauge
invariance. The resulting BF Yang– Mills action is
S
BFYM
¼
Z
M
tr
G
iB
n2
^ F
A
þ g
2
B
n2
1
ffiffiffi
2
p
g
d
A
^ B
n2
1
ffiffiffi
2
p
g
d
A
½50
This action is not only invariant under ordinary G
gauge transformations, but also under the p-form
gauge symmetry B ! B þ d
A
[6] provided that
transforms as ! þ
ffiffiffi
2
p
g. Thus, this shift can be
used to set to zero, upon which one recovers the
first-order form of the Yang–Mills action. More-
over, in the zero-coupling limit all that survives is a
standard (and nontopological) minimal coupling of
to the BF action:
lim
g
2
!0
S
BFYM
¼
Z
M
tr
G
iB
n2
^ F
A
þ
1
2
d
A
^d
A
½51
accounting for the correct number of degrees of
freedom of the Yang–Mills theory (the (n 3)-form
being absent for n = 2).
Two-dimensional quantum BF and Yang–Mills
theories have a variety of interesting topological
properties. An account of some of them can be found
in Blau and Thompson (1994) and Witten (1991).For
a detailed discussion of the gauge symmetries and gauge
fixing of the BFYM action, see Cattaneo et al. (1998).
Chern–Simons Theory, Gravity, and (Deformed)
BF Theory
The Chern–Simons theory is a three-dimensional
gauge theory. The Chern–Simons action for an
H-connection C, H the gauge group, is
S
CS
ðCÞ¼
Z
M
tr
H
C ^ dC þ
2
3
C ^ C ^ C
½52
It is invariant under the infinitesimal gauge transforma-
tions C = d
C
, 2
0
(M, h), and the gauge-invariant
equation of motion is the flatness condition F
C
= 0.
Now let H = TG be the tangent bundle group
TG G
s
g. This is a semidirect product group
with G acting on g via the adjoint and g regarded
as an abelian Lie algebra of translations. Thus, in
terms of generators (J
a
, P
a
), where the J
a
are
generators of G, the commutation relations are
[J
a
, J
b
] = f
c
ab
J
c
,[J
a
, P
b
] = f
c
ab
P
c
and [P
a
, P
b
] = 0, and
the curvature of the TG-connection C = J
a
A
a
þ P
a
B
a
is
F
C
¼ J
a
F
a
A
þ P
a
d
A
B
a
½53
Thus, the equations of motion of the TG Chern–
Simons theory are equivalent to the equations of
motion [2] of the BF theory with gauge group G.
This equivalence also holds at the level of the action:
1
2
S
CS
ðCÞ¼S
BF
ðA; BÞ½54
provided that one chooses the nondegenerate invar-
iant scalar product to be
tr
TG
ðJ
a
P
b
Þ¼tr
G
ðJ
a
J
b
Þ
tr
TG
ðJ
a
J
b
Þ¼tr
TG
ðP
a
P
b
Þ¼0
½55
For G = SO(3), TG is the Euclidean group of
isometries of R
3
and for G = SO(2, 1), TG is the
Poincare´ group of isometries of the three-dimensional
Minkowski space R
2, 1
. For these gauge groups, the BF
action takes the form of the three-dimensional
(Euclidean or Lorentzian) Einstein–Hilbert action,
with the interpretation of B = e as the dreibein and
A = ! as the spin connection. The equations of motion
for e and ! express the vanishing of the torsion
and the Riemann tensor (equivalent to the vanishing
of the Ricci tensor for n = 3), respectively. This
Chern–Simons interpretation of three-dimensional
gravity extends to gravity with a cosmological
constant, with H the appropriate de Sitter or anti-de
Sitter isometry group (SO(4), SO(3, 1), or SO(2, 2),
depending on the signature and the sign of the
cosmological constant). In terms of the BF interpreta-
tion, this corresponds to the simple topological
deformation
S
BF
ðA; BÞ¼
Z
M
tr
G
B ^ F
A
þ
1
3
B ^ B ^ B
½56
of the BF action, which ha s the deformed local
symmetries (cf. [5] and [6])
A ¼ d
A
þ ½B;
0
;B ¼½B;þd
A
0
½57
A simple way to understand these symmetries is to
note that the action can be written as the difference
of two Chern–Simons actions:
S
CS
ðA þ
ffiffiffi
p
BÞS
CS
ðA
ffiffiffi
p
BÞ
¼ 4
ffiffiffi
p
S
BF
ðA; BÞ½58
whose evident standard local gauge symmetries
(A
ffiffiffi
p
B) = d
A
ffiffi
p
B
are equivalent to [57] for
=
ffiffiffi
p
0
.
A detailed account of three-dimensional classical
and quantum gravity can be found in Carlip
(1998).
BF Theories 263
Relation with Gravity
Theories of two-dimensional gravity and topological
gravity also have a BF formulation (Blau and
Thompson 1991, Birmingham et al. 1991) which
resembles the Chern–Simons BF formulation of
three-dimensional gravity described above, the nat-
ural gauge group now being SO(2, 1) or SO(3) or
one of its contractions.
In the first-order (Palatini) formulation, the
Einstein–Hilbert action for four-dimensional gravity
can be written as
S
EH
¼
Z
trðe ^ e ^ F
!
Þ½59
where e is the vierbein and ! is the spin
connection. This action has the general form of a
BF action with a constraint that B = e ^ e be a
simple bi(co-)vector. Thus, four-dimensional
general relativity can be regarded as a constrained
BF theory. A lthough this constraint drastically
changes the number of physical degrees of freedom
(BF theory has zero degrees o f freedom, while
four-dimensional gravity has two), this is never-
theless a fruitful a nalogy which also lies at the
heart of the spin-foam quantization approach to
quantum gravity. This constrained BF description
of gravity is also available for higher-dimensional
gravity theories.
For further details, and references, see Freidel et al.
(1999) and the review article (Baez 2000).
Knot and Generalized Knot Invariants
The known relati onship between Wilson loop
observables of the Chern–Simons theory with
a compact gauge group and knot invariants
(Witten 1989), and the interpretation of the three-
dimensional BF theory as a Chern–Simons theory
with a noncompact gauge group raise the question of
the relation of observables of an n = 3 BF theory to
knot invariants, and suggest the possibility of using
an n 4 BF theory to define higher-dimensional
analogs of knot invariants. It turns out that an
appropriate observable of n = 3 BF theory for
G = SU(2) is related to the Alexander–Conway
polynomial. The analysis of higher-dimensional BF
theories requires the full power of the Batalin–
Vilkovisky (BV) formalism. BV observables general-
izing Wilson loops have been shown to give rise to
cohomology classes on the space of imbedded curves.
For a detailed discussio n of these issues, see
Cattaneo and Rossi (2001) and references therein.
A relation between the algebra of generalized
Wilson loops and string topology has been investi-
gated in Cattaneo et al. (2003).
See also: Batalin–Vilkovisky Quantization; BRST
Quantization; Chern–Simons Models: Rigorous Results;
Gauge Theories From Strings; Knot Invariants and
Quantum Gravity; Loop Quantum Gravity; Moduli
Spaces: An Introduction; Nonperturbative and
Topological Aspects of Gauge Theory; Schwarz-Type
Topological Quantum Field Theory; Spin Foams;
Topological Quantum Field Theory: Overview.
Further Reading
Baez J (2000) An introduc tion to spin foam models of
quantum gravity and BF theory. Lecture Notes in Physics
543: 25–94.
Birmingham D, Blau M, Rakowski M, and Thompson G (1991)
Topological field theory. Physics Reports 209: 129–340.
Blau M and Thompson G (1989) A New Class of Topological
Field Theories and the Ray–Singer Torsion. Physics Letters B
228: 64–68.
Blau M and Thompson G (199 1) Topological gauge theories
of antisymmetric tensor fields. Annals of Physics
205: 130–172.
Blau M and Thompson G (1994) Lectures on 2d gauge theories:
topological aspects and path integral techniques. In: Gava E,
Masiero A, Narain KS, Randjbar–Daemi S, and Shafi Q (eds.)
Proceedings of the 1993 Trieste Summer School on High
Energy Physics and Cosmology, pp. 175–244. Singapore:
World Scientific.
Carlip S (1998) Quantum Gravity in 2 þ 1 Diemensions. Cambridge:
Cambridge University Press.
Cattaneo A and Rossi C (2001) Higher-dimensional BF theories in
the Batalin–Vilkovisky formalism: the BV action and general-
ized Wilson loops. Communications in Mathematical Physics
221: 591–657.
Cattaneo A, Cotta-Ramusino P, Fucito F, Martellini M, and
Rinaldi M, et al. (1998) Four-dimensional Yang–Mills theory
as a deformation of topological BF theory. Communications in
Mathematical Physics 197: 571–621.
Cattaneo A, Pedrini P, and Fro¨ hlich J (2003) Topological field
theory interpretation of string topology. Communications in
Mathematical Physics 240: 397–421.
Freidel L, Krasnov K, and Puzio R (1999) BF description of
higher-dimensional gravity theories. Advances in Theoretical
and Mathematical Physics 3: 1289–1324.
Horowitz GT (1989) Exactly soluble diffeomorphism invariant
theories. Communications in Mathematical Physics
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