Table Of ContentIFUP-TH/2013-03
LAPTH-003/13
Three-manifold invariant
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from functional integration
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J
7
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] Enore Guadagninia and Frank Thuillierb
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a a Dipartimento di Fisica “E. Fermi” dell’Universit`a di Pisa and INFN, Sezione di Pisa, Italy.
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b LAPTH, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux cedex, France.
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v Abstract
i
X
r We give a precise definition and produce a path-integral computation of the
a
normalizedpartitionfunctionoftheabelianU(1)Chern-Simonsfieldtheorydefined
in a general closed oriented 3-manifold. We use the Deligne-Beilinson formalism,
we sum over the inequivalent U(1) principal bundles over the manifold and, for
each bundle, we integrate over the gauge orbits of the associated connection 1-
forms. The result of the functional integration is compared with the abelian U(1)
Reshetikhin-Turaev surgery invariant.
1
1. Introduction
Gauge quantum field theories play a fundamental role in the description of physical
phenomena. Most of the models that have been considered so far are defined in Minkowski
space. But one can imagine that, in certain conditions, it will become important to study
a quantum gauge theory defined in a topological nontrivial manifold. In this paper we will
consider a quantum field theory with a U(1) local gauge symmetry which is defined in a
general connected closed oriented 3-manifold M; the action is given by the Chern-Simons
functional and the observables of this model represent topological invariants [1,2,3,4,5,6].
In the present article we shall concentrate on the quantum field theory aspects which
are related with the path-integral definition and with the computation of the normalized
partition function of the theory, which represents a topological invariant of the 3-manifold
M. By using the Deligne-Beilinson formalism, it turns out that the result of the functional
integration for the normalized partition function of the U(1) Chern-Simons theory is strictly
related with the Reshetikhin-Turaev U(1) surgery invariants of 3-manifolds [7,8,9].
1.1 Summary and results
Let us give a short description of the content of our paper and a presentation of the
main results. In the Deligne-Beilinson (DB) formalism [5,10,11,12,13], each gauge orbit A
of a U(1) connection on the 3-manifold M is a class belonging to the DB cohomology space
H1(M). The Chern-Simons action S is given [5] by
D
S[A] = 2πk A A , (1.1)
∗
ZM
where the -product denotes the pairing H1(M) H1(M) H3(M) R/Z which is
∗ D × D → D ∼
associated with the canonical DB product [13]. A modification of the orientation of the
manifold M is equivalent to a change in the sign of the integer coupling constant k, so one
can choose k > 0. Let Ω1(M) be the space of the 1-forms on M and Ω1(M) the subspace
Z
of closed forms with integral periods. The space Ω1(M) corresponds to the set of gauge
Z
transformations. A presentation of H1(M) is given [10,11,12,13,14] by the following exact
D
sequence
0 Ω1(M)/Ω1(M) H1(M) H2(M) 0 , (1.2)
→ Z → D → →
inwhichH2(M)denotesthesecondintegralcohomologygroupofM and, becauseofPoincar´e
duality, H2(M) H (M) where H (M) stands for the first homology group of M. Thus
1 1
≃
H1(M) can be understood as an affine bundle over H (M) for which Ω1(M)/Ω1(M) acts as
D 1 Z
a translation group on the fibres. More precisely, each fibre is characterized by an element
2
of H (M); a generic DB class A that belongs to the fibre over γ H (M) can be written as
1 1
∈
A = A +ω , (1.3)
γ
where ω Ω1(M)/Ω1Z(M). The element Aγ bjust fixes an origin on the fibre over γ and any
∈
other element of this fibre can be obtained from A by means of a translationwith the 1-form
γ
b
ω modulo closed forms of integral periods. For each fibre, the choice of the corresponding
origin class A is not unique. On the fibre overbthe trivial element of H (M) one can take
γ 1
as canonical origin the zero class, A = 0, which is precisely the gauge orbit of the vanishing
0
connection. b
Each DB class A H1(M)bdescribes a U(1) principal bundle with connection (up
∈ D
to gauge transformations), and equation (1.2) shows that the inequivalent principal U(1)
bundles can be labelled by H (M). Let us assume [5] that the functional integration consists
1
ofa sum over theinequivalent principal bundles and, for each bundle, ofa sum over the gauge
orbits of the corresponding connection 1-forms. According to equation (1.3), this means that
the path-integral is given by DAeiS[A] = DωeiS[Aγ+ω] where one has a sum
γ∈H1(M)
over all the elements of the homology group of the manifold.
R P R
b
Let us define the normalized partition function Z (M) as
k
Dω eiS[Aγ+ω]
Z (M) = γ∈H1(M) . (1.4)
k
Dω eiS[ω]
P R
b
R
The normalization factor DωeiS[ω] = DωeiS[A0+ω] just corresponds to the functional in-
tegralassociatedwiththefibreofH1(M)overthetrivialelement0 H (M); i.e. DωeiS[ω]
R D R ∈ 1
b
represents the integral over the gauge orbits of the connection 1-forms of the trivial principal
R
bundle over M.
Remark 1.1. In quantum field theories one is really concerned with distributional fields, so
one may be interested in the possible modifications of sequence (1.2) under rough extensions
of the fields space. Quite remarkably, the basic structure of the configuration space —as
described by sequence (1.2)— is stable under the inclusion of distributional configurations.
Indeed there is a natural inclusion [5,6] of H1 (M) and of the space Z (M) of 1-cycles in
D 1
M into the Pontrjagyn dual Hom(H1(M),S1) of H1(M). These inclusions are ensured by
D D
the canonical DB product and the R/Z-valued integration over 1-cycles of M. This dual
space contains generalized (i.e. distributional) connections and it is embedded into the exact
sequence
0 → Hom(Ω2Z(M),S1) → Hom(HD1 (M),S1) → H2(M) → 0 . (1.5)
Let us introduce the simplified notation
H1(M)∗ Hom(H1 (M),S1) , Ω2(M)∗ Hom(Ω2(M),S1) ; (1.6)
D ≡ D Z ≡ Z
3
note that there also is a natural inclusion
Ω1(M)
֒ Ω2(M)∗ . (1.7)
Z
Ω1(M) →
Z
Equation (1.3) admits a distributional extension in which A H1(M)∗ and ω Ω2(M)∗.
γ ∈ D ∈ Z
Ingeneral, theabelianhomologygroupH (M)canbedecomposedas H (M) = F(M)
1 1
⊕
b
T(M), where F(M) is freely generated and the torsion component T(M) can be written as
a direct sum of Z Z/pZ factors. For torsion-free manifolds, when the torsion component
p
≡
T(M) is trivial, the main properties of the path-integral have been studied in Ref.[5]. In the
present article we shall concentrate on the pure torsion case, in which the freely generated
component F(M) is trivial and then H (M) is a finite group
1
H (M) = T(M) = Z Z Z , (1.8)
1 p1 ⊕ p2 ⊕···⊕ pw
in which the torsion numbers p ,p ,...,p are fixed by the convention that p divides p .
1 2 w i i+1
{ }
Some preliminary results on the pure torsion case have been discussed in Ref.[6]. The action
(1.1) is a quadratic function of the fields and then the result of the functional integration
(1.4) does not depend on the particular choice of the origin class A for each γ H (M).
γ 1
∈
Proposition 1. For each torsion element γ T(M), one can sbelect the origin class Aγ
∈
to correspond to a stationary point of the action, i.e. A can be chosen to be equal to the
γ
gauge orbit A0 of a flat connection. Therefore the normalized partition function (1.4) canbbe
γ
written as a sum over the gauge orbits of flat connectionbs
0
Dω eiS[Aγ+ω]
Zk(M) = γ∈H1(M) = eiS[A0γ] . (1.9)
Dω eiS[ω]
P R
γ∈HX1(M)
R
Indeed, S[A0 + ω] = S[A0] + S[ω] + 2πk A0 ω but since A0 is the class of a flat
γ γ γ ∗ γ
connection and ω is globally well defined in M, the last term is vanishing and therefore
R
S[A0 +ω] = S[A0]+S[ω]. Consequently
γ γ
0
Dω eiS[Aγ]eiS[ω]
Z (M) = γ∈H1(M) =
k
Dω eiS[ω]
P R
(1.10)
= eRiS[A0γ] Dω eiS[ω] = eiS[A0γ] .
Dω eiS[ω]
R
γ∈HX1(M) γ∈HX1(M)
R
On the other hand, since the value of the path-integral does not depend on the choice of the
origins in H1(M)∗, for each γ one finds
D
0 Dω eiS[Aγ+ω]
eiS[Aγ] = . (1.11)
Dω eiS[ω]
R
b
R
4
If A satisfies S[A ] = 0 mod Z, then
γ γ
b b 0 Dω eiS[ω]e4πik ω∗Aγ
eiS[Aγ] = , (1.12)
Dω eiS[ωR]
R
b
R
0
and by means of the path-integral (1.12) one can compute the amplitude eiS[Aγ].
Let us introduce a set of generators h ,h ,...,h for H (M); the element h is a
1 2 w 1 i
{ }
generator for Z , with p h = 0. A generic element γ H (M) can be described by means
pi i i ∈ 1
of the sum γ = wi=1nihi with integers {ni}. Each term eiS[A0γ] can now be written as
P
eiS[A0γ] = e2πik ijninjQij , (1.13)
P
where the matrix Q determines a Q/Z-valued quadratic form Q on the torsion group T(M).
ij
Although Q only depends [16,17,18] on the manifold M, in order to describe the result of
the functional integration (1.12), it is useful to consider a surgery presentation [19] of M in
S3.
Let = S3 beaframedsurgerylink—associatedwithaDehnsurgery
1 2 m
L L ∪L ···∪L ⊂
presentationofM inS3—withintegersurgerycoefficientsandletLdenotethecorresponding
linking matrix. When the homology group is given by equation (1.8), one can always find
a surgery presentation in which the linking matrix L is non-degenerate (invertible), so we
assume that this is indeed the case. For each link component (with t = 1,2,...,m), let G
t t
L
be a simple small circle linked with which can be taken as a generator of the homology
t
L
of the complement of in S3; then G ,...,G is a set of generators for the homology of
t 1 m
L { }
S3 . The homology group H (M) admits the presentation
1
−L
H (M) = G ,...,G [ ] = 0,[ ] = 0,...,[ ] = 0 , (1.14)
1 1 m 1f 2f mf
h | L L L i
where [ ] is the homology class (in S3 ) of the framing of the component
tf tf t
L −L L L
m
[ ] = L G . (1.15)
tf ts s
L
s=1
X
Each generator h of H (M) can be written as a linear combination of the G generators
i 1 t
{ }
m
h = B G , (i = 1,2...,w), (1.16)
i it t
t=1
X
with integer coefficients B .
it
5
Corollary 1. In the basis defined by the generators h , the matrix elements Q of the
i ij
{ }
quadratic form on the torsion group T(M) are given by
m
Q = B B L−1 , (1.16)
ij it js ts
t,s=1
X
where L−1 represents the inverse in Rmof the linking matrix; the normalized partition func-
tion Z (M) takes the form
k
p1−1p2−1 pw−1
Zk(M) = e2πik ijninjQij . (1.17)
···
nX1=0nX2=0 nXw=0 P
Moreover,
1/2
Z (M) = (p p p ) I (M) , (1.18)
k 1 2 w k
···
where I (M) denotes the value of the Reshetikhin-Turaev U(1) surgery invariant of the 3-
k
manifold M.
Asa matteroffacts, inthedefinitionand inthe computationofthe normalizedpartition
function Z (M) of the U(1) Chern-Simons theory there is no need of introducing a metric
k
in the 3-manifold M; moreover, Z (M) has nothing to do with the perturbative gauge-fixing
k
procedure.
The paper is organized as follows. The basic rules which are used in the computation
of the field theory path-integrals are listed in Section 2. Section 3 contains a proof of
Proposition 1 and Corollary 1 together with a path-integral derivation of expression (1.17)
of the normalized partition function. One illustrative example is presented in Section 4.
2. Computation rules
In certain expressions of the previous section, ratios of functional integrations —as
indicated for instance in equations (1.4) and (1.12)— appear. This notation belongs to
the set of standard conventions which are employed in physics, in which any meaningful
quantity takes the form of a ratio of regularized functional integrations in the limit in which
the regularization is removed.
Remark 2.1. Each functional integration, which formally involves an infinite number of
integration variables, can be approximated or regularized by restricting the integral to a
finite number N of variables; this regularization is removed in the N limit. The ratio
→ ∞
of two path-integrals means: (1) introduce a regularization in the numerator and in the
6
denominator simultaneously (with the same finite N), (2) for each fixed N, the regularized
ratio —that is the ratio of the two regularized integrals— is well defined and depends on N,
(3) finally consider the N limit of the regularized ratio. For the ratios of functional
→ ∞
integrations considered in quantum field theory, this limit normally exists. For example, all
the perturbative computations in quantum electrodynamics or in the SU(3) SU(2)
c L
× ×
U(1) StandardModeloftheparticlesinteractionsarebasedpreciselyontheexistenceofthis
Y
limitfor theappropriate ratiosoffunctional integrations. In any path-integral expression one
must specify the so-called overall normalization, i.e. the choice of the functional integration
which appears in the denominator, because different normalizations generally give rise to
different results. The path-integrals in which the normalization is not specified are not well
defined.
The limit procedure which has been mentioned in Remark 2.1 ensures the validity of
the following two properties.
(P1) Linearity. If, in a given quantum field theory, the functional integration region R is the
union of two disjoint parts, R = R R , then the path-integral over R is the sum of
1 2
∪
the path-integrals over R and over R .
1 2
(P2) Translation invariance. Suppose that, in a given quantum field theory, any field config-
uration φ(x) can be written as
φ(x) = φ (x)+ψ(x) ,
0
where φ (x) is fixed and the variable ψ(x) can fluctuate. When the action S[φ] is a
0
quadraticfunctionofthefield variables, thefunctional integrationisinvariant [20]under
translation
Dφ eiS[φ]X(φ) Dψ eiS[φ0+ψ]X(φ +ψ)
0
X(φ) = .
h i ≡ Dφ eiS[φ] Dφ eiS[φ]
R R
R R
The basic properties (P1) and (P2) can also be understood as defining relations because all
our functional integral computations are based precisely on these two properties exclusively.
For instance, properties (P1) and (P2) have been used to write equation (1.4).
Each gauge orbit A can be represented by a field configuration which admits a Cˇech-de
Rham representation, i.e. a representative of the class A can be described by a collection of
local variables which, in a good covering of M, are given by
a
{U }
A va,λab,nabc . (2.1)
↔
(cid:0) (cid:1)
va denotes a 1-form locally defined in the open set ; λab represents a 0-form in the inter-
a
U
section such that va vb = dλab, and the integer nabc is defined in with
a b a b c
U ∩U − U ∩U ∩U
7
the property λab + λbc + λca = nabc. In our notations, a particular representative element
of the DB class, which appears on the left-hand-side of the arrow , is described by the
↔
collection of Cˇech-de Rham field components that are shown on the right-hand-side of .
↔
In particular a representative of a class ω Ω1(M)/Ω1(M) fulfills
Z
∈
ω (ωa,0,0) , (2.2)
↔
where ωa are the restrictions in the open sets of a 1-form on M also denoted by ω.
a
{ } {U }
Vice versa, given a 1-form α Ω1(M), the DB class associated with α has Cˇech-de Rham
∈
representation
α (αa,0,0) , (2.3)
↔
where αa are the restrictions of α Ω1(M) in the open sets . Note that if ω
a
{ } ∈ {U } ∈
Ω1(M)/Ω1(M) then ω ω = ω dω mod Z.
Z
∗ ∧
In the U(1) Chern-Simons field theory, a typical path-integral computation —that will
R R
appear below— takes the form
2πik ω∗ω 2πi ω∗α
Dω e e
2πi ω∗α
e , (2.4)
≡ R2πik ω∗ωR
R Dω e
DD R EE
R
where ω Ω2(M)∗ represents the integratioRn variable, whereas α can be interpreted as a
Z
∈
given classical external source. Let us first consider the case in which α is the DB class
associated with a 1-form α Ω1(M). Let α′ be the DB class such that
∈
1
α′ αa,0,0 . (2.5)
↔ 2k
(cid:18) (cid:19)
One can put
ω = α′ +ω , (2.6)
−
where α′ is fixed and the variable ω Ω2(M)∗ can fluctuate. Since Dω = Dω (property
Z
∈ e
(P2)) and [5]
k ω ω + eω α = k ω ω k α′ α′ , e (2.7)
∗ ∗ ∗ − ∗
Z Z Z Z
one finds
e e
2πik ω∗ω
′ ′ Dω e ′ ′
2πi ω∗α −2πik α ∗α −2πik α ∗α
e = e = e . (2.8)
2πikR ω∗ω
R Dω e
DD R EE R e e R
e
R
Because α is globally defined in M, one finalRly obtains [5]
′ ′
−2πik α ∗α −(2πi/4k) α∧dα
e = e . (2.9)
R R
8
This procedure can also be applied when α is a 1-current. In particular, for each oriented
knot C which belongs to a 3-ball inside M, one can find a Seifert surface Σ M such
B ⊂
that C = ∂Σ. This equation can be written in terms of currents: j = dα , where j is the
C Σ C
2-current of the knot C and α is the 1-current of Σ. The 1-current α can be understood as
Σ Σ
distributional limit of 1-forms in M. By construction one has ω dα = ω j = ω.
∧ Σ ∧ C C
The DB class η H1(M)∗ defined by
C ∈ D R R H
η (αa,0,0) (2.10)
C ↔ Σ
only depends on the knot C. Then for any ω Ω2(M)∗ one has
Z
∈
ω η = ω dα = ω mod Z . (2.11)
C Σ
∗ ∧
Z Z IC
For a two components oriented link C C M, the value of the linking number of
1 2
∪ ⊂ B ⊂
C and C is given by
1 2
ℓk(C ,C ) = α dα ,
1 2 Σ1 ∧ Σ2
Z
with C = ∂Σ and C = ∂Σ . For a single oriented framed knot C M, the integral
1 1 2 2
⊂ B ⊂
α dα represents the self-linking number of C = ∂Σ which is defined to be the linking
Σ Σ
∧
number of C and its framing C , α dα ℓk(C,C ) = α dα .
R f Σ ∧ Σ ≡ f Σ ∧ Σf
Consider now equation (2.8) in the case α = η = q η where C are the com-
R L jRj Cj { j}
ponents of a framed oriented colored link L = C C M and q denotes the
1 n j
∪···∪P ⊂ B ⊂
color of C ; in the DB formalism each color (or charge) q must assume [5] integer values.
j j
One obtains
e2πi nj=1qj Cj ω = e−(2πi/4k) nij=1qiqjLij , (2.12)
(cid:28)(cid:28) P H (cid:29)(cid:29) P
where the integers L are the matrix elements of the linking matrix associated with L. The
ij
result (2.12) can also be obtained by taking α Ω1(M) and considering the α η limit in
L
∈ →
equation (2.8).
Equation (2.12) also gives the complete solution [5] of the U(1) Chern-Simons quantum
field theory defined in M = S3 because, in this case, any link belongs to a 3-ball.
The concluding remarks of this section concern some general properties [5] of the ex-
pectation values in the abelian Chern-Simons theory.
Remark 2.2. Let the colored oriented and framed link L′ = L U M be the union of
∪ ⊂
the link L with the unknot U. If U belongs to a 3-ball which is disjoint from L, and U has
trivial framing —i.e. its framing U satisfies ℓk(U,U ) = 0— then the expectation value of
f f
the holonomy associated with L′ is equal to the expectation value of the holonomy associated
with L. Indeed the expectation value of the holonomy associated with L′ is the product [5]
9
of the expectation values associated with L and with U (this feature can be understood as
the topological version of the cluster property of ordinary quantum field theories), and the
expectation value of the holonomy associated with the unknot U is equal to the expectation
value of the identity.
C
1
C
2
C #C
1 2
Figure 2.1. Sum of knots.
Definition 2.1. Let C and C be two oriented (and possibly framed) knots in M. By
1 2
joining C and C in the way shown in Figure 2.1, one obtains the knot C #C , that is
1 2 1 2
called the (band connected) sum of C and C . The dashed lines in Figure 2.1 refer to the
1 2
framings; by construction, the framing (C #C ) of C #C is just the sum of the framings
1 2 f 1 2
C #C . When M = S3 —which is of particular interest in the Dehn surgery presentation
1f 2f
in S3 of generic 3-manifold— one has
ℓk((C #C ) ,C #C ) = ℓk(C ,C )+ℓk(C ,C )+2ℓk(C ,C ) . (2.13)
1 2 f 1 2 1f 1 2f 2 1 2
The knot C with modified orientation is indicated by C ; one finds
2 2
−
ℓk((C #( C )) ,C #( C )) = ℓk(C ,C )+ℓk(C ,C ) 2ℓk(C ,C ) . (2.14)
1 2 f 1 2 1f 1 2f 2 1 2
− − −
For colored knots C and C , the sum C #C is well defined when C and C have the
1 2 1 2 1 2
same color. Since the linking number can be interpreted as an intersection product for
which # plays the role of the standard sum, equations (2.13) and (2.14) also have a natural
homological interpretation. The sum of knots enters the Kirby calculus [21].
Remark 2.3. If the knots C and C have the same color, the expectation value of the
1 2
holonomy associated with the link L = C C C C is equal to the expectation
1 2 3 n
∪ ∪ ∪···∪
value of the holonomy associated with L′ = C #C C C . Indeed the expectation
1 2 3 n
∪ ∪···∪
values of the link holonomies are invariant under the addition in the link of a (trivially-
framed) unknot which belongs to a 3-ball M (Remark 2.2); on the other hand, the band
B ⊂
