ap-3-10.dvi
Acta Polytechnica Vol. 50 No. 3/2010
From Gauge Anomalies to Gerbes and Gerbal Representations:
Group Cocycles in Quantum Theory
J. Mickelsson
Abstract
In this paper I shall discuss the role of group cohomology in quantummechanics and quantumfield theory. First, I recall how
cocycles of degree 1 and 2 appear naturally in the context of gauge anomalies. Then we investigate how group cohomology
of degree 3 comes from a prolongation problem for group extensions and we discuss its role in quantumfield theory. Finally,
we discuss a generalization to representation theory where a representation is replaced by a 1-cocycle or its prolongation by
a circle, and point out how this type of situations come up in the quantization of Yang-Mills theory.
1 Introduction
A projective bundle over a base M is completely de-
termined, up to equivalence, by the Dixmier-Douady
class, which is an element of H3(M,Z). This is the
origin of gerbes in quantum field theory: A standard
example of this type of situation is the case when M
is the moduli space of gauge connections in a vector
bundle over a compact spin manifold, [5]. Topologi-
cally a gerbe on a space M is just an equivalence class
of P U(H)= U(H)/S1 bundles over M. Here U(H) is
the (contractible) unitary group in a complex Hilbert
space H. In terms of Čech cohomology subordinate
to a good cover {Uα} of X, the gerbe is given as a
C×-valued cocycle {fαβγ},
fαβγ f
−1
αβδfαγδf
−1
βγδ =1
on intersections Uα ∩Uβ ∩Uγ ∩Uδ. This cocycle arises
from the lifting problem: A P U(H) bundle is given
in terms of transition functions gαβ with values in
P U(H). After lifting these to U(H) one gets a family
of functions ĝαβ which satisfy the 1-cocycle condition
up to a phase,
ĝαβĝβγ ĝγα = fαβγ1.
Thenotion of gerbal representationwas introduced
in recent paper [7]. This is to be viewed as the next
level after projective actions related to central exten-
sions of groups and is given in terms of third group
cohomology. One can view this setting as a categori-
fication of the representation theory of central exten-
sions of groups. We shall not discuss the problem in
this generality since our categories are of special kind:
A category of groups for us is just a principal bundle
over a base M. Each fiber can be identified as a group
G, but only after fixing a point in the fiber and call-
ing the chosen point the unit element in G. Fixing a
representation of G defines a standard way a vector
bundle over M. However, if the representation is only
a projective representation we obtain in general only
a projective vector bundle over M. We are now in the
setting for gerbes and we have a characteristic class
in H3(M,Z). But there is a role also for third group
cohomology.
In fact, the appearance of third group cohomology
in this context is not new and is related to group ex-
tensions as explained in [9]. In the simplest form, the
problem is the following. Let F be an extension of G
by the group N,
1 → N → F → G → 1
an exact sequence of groups. Suppose that 1 → a →
N̂ → N → 1 is a central extension by the abelian
group a. Then one can ask whether the extension F
of G by N can be prolonged to an extension of G by
the group N̂. The obstruction to this is an element in
the group cohomology H3(G, a) with coefficients in a.
In the case of Lie groups, there is a corresponding Lie
algebra cocycle representing a class in H3(g, a), where
a is the Lie algebra of a. We shall demonstrate this in
detail for an example arising from the quantization of
gauge theory. It is closely related to the idea in [2], fur-
ther elaborated in [3], which in turnwas a response to
a discussion in 1985 on breaking of the Jacobi identity
for the field algebra in Yang-Mills theory [6].
The paper is organized as follows. In Section 2 we
recall the basics about the role of group cohomology
of degree 1 and 2 in quantum theory. In Section 3 we
then explain how3-cocycles come from a prolongation
problem for group extensions. In Section 4 we take as
an example the gauge group extensions arising from
the gauge action on bundles of fermionic Fock spaces
over background gauge fields and the corresponding
Lie algebra cocycles. In Sections 5 and 6 we explain
the use of 1-cocycles as generalized representations,
with an example from quantum field theory.
2 Cocycles of degree 1 and 2
in quantum theory
In quantum mechanics a symmetry group G (e.g., the
group of Galilean symmetries) acts on Schrödinger
wave functions as
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Acta Polytechnica Vol. 50 No. 3/2010
(T(g)ψ)(x)= ω(g;x)ψ(g−1x)
where ω(g;x), for g ∈ G, is a (matrix valued) phase
factor. In order that the group multiplication rule is
preserved ithas to satisfyas consistencythe1-cocycle
condition
ω(g1;x)ω(g2;g
−1
1 x)ω(g1g2;x)
−1 =1.
It may happen that the cocycle condition does not
hold, for example in the case of the Galilean transfor-
mations onwave functions ofmassive particles; in this
case the left-hand-side defines a S1 valued (it does not
depend on the coordinate x) 2-cocycle. The represen-
tation of theGalilei group is nowprojective but it can
still be viewed as a true representation for a central
extension Ĝ of G, [1].
1-cocycles appear also in the context of symmetry
breaking in QFT. Classically, one might expect that
the (exponentiated) quantum action is invariant un-
der a group G (group of gauge symmetries or group of
diffeomorphisms of space-time),
Z(A)= Z(Ag)
where A denotes a set of fields; in the case of a gauge
action the right action is Ag = g−1Ag + g−1dg. But
in case of chiral anomaly, for example,
Z(Ag)= ω(g;A)Z(A)
where ω is a phase factor. Consistency requires again
that ω is a 1-cocycle. However, unlike in the case of
the Galilei group, the nontrivial 1-cocycle has serious
physical consequences: It signals the breakingof gauge
symmetry. Nontrivialitymeans that there is noway to
modify the quantum effective action by a multiplica-
tive phase, Z(A) �→ Z′(A)= η(A)Z(A), such that the
modified action Z′ would be gauge invariant. This
means that
ω(g;A) �= η(Ag)η(A)−1
for anyphase function η. In case of an equality, we say
that the 1-cocycle ω is a coboundary of the 0-cochain
η.
Denote by A the space of all (smooth) fields A. If
G acts smoothly and freely on A then X = A/G is
a manifold and the cocycle ω defines a complex line
bundle L. Sections of L are complex functions on A
satisfying
ψ(Ag)= ω(g;A)ψ(A).
The complex line bundle hasChern class c ∈ H2(X,Z)
which is obtained by transgression from ω. In the case
of the chiral anomaly, the action Z is thus a section of
the complex line bundle L, which is called the Dirac
determinant bundle. Indeed, the function Z(A) can
be viewed as a regularizeddeterminant of themassless
Weyl-Dirac operator D+A which is a linear map from
the left-handed spinor fields to the right-handed sec-
tor. To define the determinant, one fixes a map D−A0
from right-handed sector to the left-handed sector, by
fixing abackgroundpotential A0, and then one applies
the zeta function regularization to the determinant of
the operator D−A0D
+
A.
In Hamiltonian quantization the breaking of
(gauge, diffeomorphism) symmetry is best seen in the
modified commutation relations in the Lie algebra g
of G,
[X, Y ] �→ [X, Y ]+ c(X, Y )
where c takes values in an abelian ideal a; in the
simplest case a = C and c satisfies the Lie algebra
2-cocycle condition
c(X, [Y, Z])+ c(Y, [Z, X])+ c(Z, [X, Y ]) = 0
A famous example is given by the central extension of
the loop algebra Lg of smooth functions on the unit
circle with values in a semisimple Lie algebra g,
c(X, Y )=
k
2πi
∫
S1
〈X(φ), Y ′(φ)〉dφ
where k is a constant (“level” of a representation),
which is equal to a nonnegative integer in a positive
energy representationwhen the invariantbilinear form
〈·, ·〉 on g is properly normalized.
Given a (central) extension of a Lie algebra one
expects that there is a central extension of the corre-
sponding group. In case of Lg the group is the loop
group LG of maps from S1 to a (compact) group G.
A central extension of LG would then be given by a
circle valued function Ω:LG × LG → S1 with group
2-cocycle property
Ω(g1, g2)Ω(g1g2, g3)=Ω(g1, g2g3)Ω(g2, g3).
However, in case of LG there is a topological ob-
struction, Ω is defined only in an open neighborhood
of theunit element. Theobstruction is givenbyanele-
ment in H2(LG,Z)whose deRham representative is a
left invariant 2-formfixed by the Lie algebra 2-cocycle
c. In case of a compact simple simply connected Lie
group the cohomology H2(LG,Z) is equal to H3(G,Z)
is equal to Z. The Lie algebra cocycle for Lg can be
viewed as a left invariant 2-formon the group LG. For
a correct choice of normalization of the bilinear form
〈, ·, ·〉 on g the generator in H2(LG,Z) corresponds to
the basic extension with level k =1.
3 3-cocycles
Group andLie algebra cohomologywith coefficients in
anabeliangroup (Lie algebra) is defined inanydegree.
Sowhat about degree 3? And the relation to deRham
cohomology in dimension 3?
First, let us recall the basic definitions. Assume
that a group G acts as automorphisms of an abelian
group A. A map f:G× G× . . . G:→ A (n arguments)
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Acta Polytechnica Vol. 50 No. 3/2010
is a n-cocycle if δf =0where the coboundary operator
δ is defined by
(δf)(g1, g2, . . . , gn+1)=
i=n∑
i=1
(−1)if(g1, . . . gigi+1, . . . , gn+1)+
(−1)n+1f(g1, . . . , gn)+ g1 · f(g2, . . . , gn+1).
Cocycles of type δf are exact cocycles. The group co-
homology in degree n is then defined as the abelian
group Hn(G;A) of n-cocycles modulo exact cocycles.
In case of a Lie algebra g the cochains are alternat-
ing multilinear maps f:g × . . . × g → a. The cocycles
are elements in the kernel of the Lie algebra cobound-
ary operator, which is now defined as
(δf)(x1, . . . , xn+1)=∑
i<j
(−1)i+j+1f([xi, xj], x1, . . . x̂i . . . x̂j . . . xn+1)+
i=n+1∑
i=1
(−1)ixi · f(x1, . . . x̂i . . . xn+1),
where the argument under the hat is deleted. Here
theLie algebra g acts as endomorphisms of the abelian
Lie algebra a. The Lie algebra cohomology Hn(g;a)
is now the abelian group of n-cocycles modulo exact
n-cocycles.
Let B be an associative algebra and G a group.
Assume that we have a group homomorphism s:G →
Out(B) where Out(B) is the group of outer automor-
phisms of B, that is, Out(B) = Aut(B)/In(B), all au-
tomorphismsmodulo thenormal subgroupof inner au-
tomorphisms.
If one chooses any lift s̃:G → Aut(B) then we can
write
s̃(g)s̃(g′)= σ(g, g′) · s̃(gg′)
for some σ(g, g′) ∈ In(B). From the definition follows
immediately the cocycle property
σ(g, g′)σ(gg′, g′′)= [s̃(g)σ(g′, g′′)s̃(g)−1]σ(g, g′g′′)
Let next H be any central extension of In(B) by an
abelian group a. That is, we have an exact sequence
of groups,
1 → a → H → In(B) → 1.
Let σ̂ be a lift of the map σ:G × G → In(B) to a map
σ̂:G × G → H (by a choice of a section In(B) → H).
We have then
σ̂(g, g′)σ̂(gg′, g′′)= [s̃(g)σ̂(g′, g′′)s̃(g)−1]
×σ̂(g, g′g′′) · α(g, g′, g′′) for all g, g′, g′′ ∈ G
where α:G × G × G → a.
Here the action of the outer automorphism s(g)
on σ̂(∗) is defined by s̃(g)σ̂(∗)s̃(g)−1 = the lift of
s̃(g)σ(∗)s̃(g)−1 ∈ In(B) to an element in H. One can
show that α is a 3-cocycle, [9],
α(g2, g3, g4)α(g1g2, g3, g4)
−1α(g1, g2g3, g4)
×α(g1, g2, g3g4)−1α(g1, g2, g3)= 1,
wherewe have used themultiplicative notation for the
groupproduct in a (instead of the additive notation in
the definition of the coboundary operator). The group
G acts trivially on a.
Remark If we work in the category of topological
groups (or Lie groups) the lifts above are in general
discontinuous; normally, we can require continuity (or
smoothness) only in an open neighborhood of the unit
element.
4 A QFT example
Nextwe construct an example fromquantumfield the-
ory. Let G be a compact simply connected Lie group
and P the spaceof smoothpaths f: [0,1]→ G with ini-
tial point f(0)= e, the neutral element, andquasiperi-
odicity condition f −1df a smooth function.
P is a group under point-wisemultiplication but it
is also a principal ΩG bundle over G. Here ΩG ⊂ P
is the based loop group with f(0) = f(1) = e and
π:P → G is the projection to the end point f(1). Fix
an unitary representation ρ of G in CN and denote
H = L2(S1,CN).
For each polarization H = H− ⊕ H+ we have a
vacuum representation of the canonical anticommuta-
tion relations algebra (CAR) B(H) in a Hilbert space
F(H+). Denote by C the category of these represen-
tations. Denote by a(v), a∗(v) the generators of B(H)
corresponding to a vector v ∈ H,
a∗(u)a(v)+ a(v)a∗(u)= 2〈v, u〉 ·1
and all the other anticommutators equal to zero.
Any element f ∈ P defines a unique automorphism
of B(H) with φf(a∗(v)) = a∗(f · v), where f · v is the
function on the circle defined by ρ(f(x))v(x). These
automorphisms are in general not inner except when
f is periodic.
Wehavenowamap s:G → Aut(B)/In(B) givenby
g �→ F(g)where F(g) is anarbitrarysmoothquasiperi-
odic function on [0,1] such that F(g)(1)= g.
Any two such functions F(g), F ′(g) differ by an el-
ement σ of ΩG, F(g)(x)= F ′(g)(x)σ(x). Now σ is an
inner automorphism through a projective representa-
tion of the loop group ΩG in F(H+).
In an open neighborhood U of the neutral element
e in G we can fix in a smooth way for any g ∈ U a
path F(g) with F(g)(0)= e and F(g)(1)= g.
Of course, for a connected group G we can make this
choice globally on G but then the dependence of the
path F(g) would not be a continuous function of the
end point. For a pair g1, g2 ∈ G we have
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Acta Polytechnica Vol. 50 No. 3/2010
σ(g1, g2)F(g1g2)= F(g1)F(g2)
with σ(g1, g1) ∈ ΩG.
For a triple of elements g1, g2, g3 we have now
F(g1)F(g2)F(g3) = σ(g1, g2)F(g1g2)F(g3)
= σ(g1, g2)σ(g1g2, g3)F(g1g2g3).
In the same way,
F(g1)F(g2)F(g3)= F(g1)σ(g2, g3)F(g2g3)=
[g1σ(g2, g3)g
−1
1 ]F(g1)F(g2g3)=
[g1σ(g2, g3)g
−1
1 ]σ(g1, g2g3)F(g1g2g3)
which proves the 2-cocycle relation for σ.
Lifting the loop group elements σ to inner auto-
morphisms σ̂ through a projective representation of
ΩG we can write
σ̂(g1, g2)σ̂(g1g2, g3)=
Aut(g1)[σ̂(g2, g3)]σ̂(g1, g2g3)α(g1, g2, g3),
where α:G×G×G → S1 is somephase functionarising
from the fact that the projective lift is not necessarily
a group homomorphism.
Since (in the case of a Lie group) the function F(·)
is smooth only in a neighborhood of the neutral ele-
ment, the same is true also for σ and finally for the
3-cocycle α.
An equivalent point of view to the construction of
the 3-cocycle α is this: We are trying to construct
a central extension P̂ of the group P of paths in G
(with initial point e ∈ G) as an extension of the cen-
tral extension over the subgroup ΩG. The failure of
this central extension ismeasured by the cocycle α, as
an obstruction to associativity of P̂.
On the Lie algebra level, we have a corresponding
cocycle c3 = dα which is easily computed. The cocycle
c of Ωg extends to the path Lie algebra P g as
c(X, Y )=
k
4πi
∫
[0,2π]
(〈X,dY 〉−〈Y,dX〉).
This is an antisymmetric bilinear form on P g but it
fails to be a Lie algebra 2-cocycle. The coboundary is
given by
(δc)(X, Y, Z)=
c(X, [Y, Z])+ c(Y, [Z, X])+ c(Z, [X, Y ])=
−
k
4πi
〈X, [Y, Z]〉|2π = dα(X, Y, Z).
Thus δc reduces to a 3-cocycle of the Lie algebra g
of G ontheboundary x =2π. Assumingthat thebilin-
ear form is normalized as 〈X, Y 〉 =trXY , trace in the
defining representation of G, then the above 3-cocycle
defines by left translations on G the left-invariant de
Rham form −
1
12πi
tr(g−1dg)3; this is normalized as
2πi times an integral 3-formon G and is the generator
of H3(G,Z) for G = SU(n).
5 Cocycles and associated
vector bundles
Letus recall the standardconstructionof avectorbun-
dle associated to a principal G bundle π:P → M over
a base M. Fix a representation ρ:G → Aut(V ), in
a vector space V . Define the total space of a vector
bundle as
E = P ×ρ V,
with the equivalence
(p, v) ≡ (pg, ρ(g)−1v).
The projection onto the base M is (p, v) �→ π(p). Con-
sider the following generalization of this construction:
Fix a 1-cocycle ω:P × G → Aut(V ) and set
E = P ×ω V, with (pg, ω(p;g)−1v) ≡ (p, v).
The transitivity of the relation is given by the cocycle
condition
ω(p;gg′)= ω(p;g)ω(pg;g′).
An example of this construction was already given in
the construction of the determinant bundle over the
gauge orbit space A/G.
Let G be a topological group and f:G → H a
homotopy to another topological group H. Morally,
representation theory of H should encode information
about representations of G. However, it can happen
that H has a good representation theory but G lacks
unitary faithful Hilbert space representations. But we
can define a cocycle
ω(b;g)= f(b)−1f(bg)
with values in H. Selecting a representation ρ of H in
V we obtain a 1-cocycle ρ(ω(b;g)) for the right action
of G on itself, with values in Aut(V ). We can view
this as a representation of G in a group of matrices
with entries in the algebra of complex functions on G
(but with an action of G on functions through right
translation).
Example 1 The loop group LG of smooth maps
f:S1 → G, G a compact Lie group, has a beauti-
ful theory of projectivehighestweight representations,
[11, 10]. These are representations of a central exten-
sion L̂G. On the Lie algebra level, the central exten-
sion is given as in Sect. 3.
One can show that LG is homotopy equivalent to
the Banach-Lie group LcG of continuous loops, [4].
However, no representations of LcG analogous to the
highest weight representations are known. Instead, we
can use the cocycle
ω:LcG × LcG → LG
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Acta Polytechnica Vol. 50 No. 3/2010
to tranfer representations of LG to Hilbert space op-
erator cocycles on LcG, [8].
Example 2 Let H = H− ⊕ H+ be a polarized
Hilbert space and Up the group of unitaries in H such
that the off-diagonal blocks with respect to the polar-
ization are in the Schatten ideal Lp of bounded oper-
ators A with tr|A|p < ∞. The case p =2 is important
since the highest weight representations of L̂G can be
constructed fromrepresentationsof a central extension
Û2, [11]. The Lie algebra central extension is defined
by the 2-cocycle
c(X, Y )=
1
2
trc X[�, Y ]
where � is the grading operator in H and the condi-
tion trace trc is defined as trc X =
1
2
trtr(X + �X�).
The groups Up are important because one has an em-
bedding M ap(M, G) ⊂ Up when M is a compact spin
manifold and G compact Lie group, for p > dimM,
[15]. According to Richard Palais, Up is homotopy
equivalent to U2 for all p ≥ 1, [16], so we can define
generalized representations of M ap(M, G) from this
equivalence and the embebding to Up.
6 Application to gauge theory
Let DA Dirachamiltonianonanodddimensional com-
pact spin manifold coupled to a gauge potential A.
The quantization D̂A of DA acts in a fermionic Fock
space. For different potentials the representations of
the fermion algebra are inequivalent [17]. In scatter-
ing problems one would like to realize the operators
D̂A in a single Fock space F, the Fock space of free
fermions, A =0. This can be achieved by choosing for
each A a unitary operator TA which reduces the off-
diagonal blocks of DA to Hilbert-Schmidt operators,
for the ‘free’ polarization � = D0/|D0|. Then each
D′A = T
−1
A DATA can be quantized in the free Fock
space, [12], [13].
This has a consequence for the implementation of
the gauge action A �→ Ag = g−1Ag + g−1dg in the
Fock space. In the 1-particle space the action of g is
replaced by ω(A;g)= T −1A gTAg with
ω(A;gg′)= ω(A;g)ω(Ag;g′).
Now the Shale-Stinespring condition [�, ω(A;g)] ∈
L2 is satisfied and we can quantize in F,
ω(A;g) �→ ω̂(A;g′).
For the Lie algebra of the gauge groupwe have the Lie
algebra cocycle
dω(A;X)= T −1A XTA + T
−1
A LX TA
with quantization d̂ω(A;X).
Let X be an element in the Lie algebra M ap(M, g)
of the gauge group G = M ap(M, G). Its quantization
is then
GX = LX + d̂ω(A;X)
where LX is the Lie derivative in the direction of
X, corresponding to the infinitesimal gauge action
A �→ [A, X] + dX. The commutation relations are
now modified by a cocycle c,
[GX , GY ] = G[X,Y ] + c(A;X, Y )
LX c(A; [Y, Z])+c(A; [X, [Y, Z]])+ cyclic combin. = 0.
The Lie derivative LX is needed since the Fock spaces
F depend on the external gauge field A: although as
Hilbert spaces FA are all the same, the gauge action
explicitly depends on A. The 2-cocycle property quar-
antees the Jacobi identity for the extension
Lie(Ĝ)= M ap(M, g)⊕ M ap(A,C).
In the casewhen M = S1 one can take TA =1and
we obtain the standard central extension of the loop
algebra M ap(S1, g). In the case when dimM =3 one
can show that the 2-cocycle c is equivalent to the local
form
c ≡ const.
∫
M
trA[dX,dY ]
where the trace under the integral sign is computed in
afinite-dimensional representationof g. This represen-
tation is the same defined by the G-action on fermions
in the 1-particle space. Actually, the coefficient in the
front of the integral is nonzero only for chiral fermions
(the Schwinger terms from left and right chiral sectors
cancel).
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Jouko Mickelsson
Department of Mathematics and Statistics
University of Helsinki
Department of Theoretical Physics
Royal Institute of Technology, Stockholm
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