Gauge Invariant Factorisation and Canonical Quantisation of
  Topologically Massive Gauge Theories in Any Dimension
CP3-06-05
ICMPA-MPA/2006/35
April 2007
Gauge Invariant Factorisation and Canonical Quantisation of
Topologically Massive Gauge Theories in Any Dimension
Bruno Bertrand
Center for Particle Physics and Phenomenology (CP3)
Institut de Physique Nucléaire
Département de Physique, Université catholique de Louvain (U.C.L.)
2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium
E-mail: Bruno.Bertrand@fynu.ucl.ac.be
Jan Govaerts1
Institute of Theoretical Physics
Department of Physics, University of Stellenbosch
Stellenbosch 7600, Republic of South Africa
UNESCO International Chair in Mathematical Physics and Applications (ICMPA)
University of Abomey-Calavi
072 B.P. 50, Cotonou, Republic of Benin
Abstract
Abelian topologically massive gauge theories (TMGT) provide a topological mechanism to gen-
erate mass for a bosonic p-tensor field in any spacetime dimension. These theories include the
2+1 dimensional Maxwell-Chern-Simons and 3+1 dimensional Cremmer-Scherk actions as par-
ticular cases. Within the Hamiltonian formulation, the embedded topological field theory (TFT)
sector related to the topological mass term is not manifest in the original phase space. However
through an appropriate canonical transformation, a gauge invariant factorisation of phase space
into two orthogonal sectors is feasible. The first of these sectors includes canonically conjugate
gauge invariant variables with free massive excitations. The second sector, which decouples
from the total Hamiltonian, is equivalent to the phase space description of the associated non
dynamical pure TFT. Within canonical quantisation, a likewise factorisation of quantum states
thus arises for the full spectrum of TMGT in any dimension. This new factorisation scheme also
enables a definition of the usual projection from TMGT onto topological quantum field theories
in a most natural and transparent way. None of these results rely on any gauge fixing procedure
whatsoever.
1On sabbatical leave from the Center for Particle Physics and Phenomenology (CP3), Institut de Physique
Nucléaire, Université catholique de Louvain (U.C.L.), 2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium,
E-mail: Jan.Govaerts@fynu.ucl.ac.be.
http://arxiv.org/abs/0704.1512v1
1 Introduction
Topological field theories (TFT, see [1] for a review) have played an important role in a wide range
of fields in mathematics and physics ever since they were first constructed by A. S. Schwarz [2] and
E. Witten [3]. These theories actually possess so large a gauge freedom that their physical, namely
their gauge invariant observables solely depend on the topology (more precisely, the diffeomorphism
equivalence class) of the underlying manifold. Another related feature of TFT is the absence of
propagating physical degrees of freedom. Upon quantisation, these specific properties survive,
possibly modulo some global aspects related to quantum anomalies. As a consequence, topological
quantum field theories (TQFT) often have a finite dimensional Hilbert space and are quite generally
solvable, even though their formulation requires an infinite number of degrees of freedom. There
exists a famous classification scheme for TQFT, according to whether they are of the Schwarz or
of the Witten type [1].
As a class of great interest, TFT of the Schwarz type have a classical action which is explicitly
independent of any metric structure on the underlying manifold and does not reduce to a total
divergence or surface term. The present work focuses on all such theories defined by a sequence
of abelian B ∧ F theories for manifolds M of any dimension (d+ 1) [2, 4, 5]. Given a real valued
p-form field A in Ωp(M) and a real valued (d−p)-form field B in Ωd−p(M), the general TFT action
of interest is of the form
SB∧F [A,B] = κ
(1− ξ)F ∧B − (−1)p ξ A ∧H, (1)
κ being some real normalisation parameter of which the properties are specified throughout the
discussion hereafter. This action is invariant under two independent classes of finite abelian gauge
transformations acting separately in either the A- or B-sector,
A′ = A+ α, B′ = B + β, (2)
where α and β are closed p- and (d− p)-forms on M, respectively. The derived quantities F = dA
and H = dB are the gauge invariant field strengths associated to A and B. The arbitrary real
variable ξ introduced in order to parametrise any possible surface term is physically irrelevant for
an appropriate choice of boundary conditions on M. Given the definition of the wedge product, ∧,
the integrand in (1) is a (d+ 1)-form, the integration of which over M does not require a metric.
In the particular situation when the number of spatial dimensions d is even and such that d = 2p
with p itself being odd, in addition to the B ∧ F theories defined by (1) there exist TFT of the
Schwarz type involving only the single p-form field A with the following action1,
SA∧F [A] = κ
A ∧ F. (3)
These theories are said to be of the A ∧ F type. They include the abelian Chern-Simons theory in
2+1 dimensions [2, 6].
This sequence of TFT of the Schwarz type formulated in any dimension, and related to one
another through dimensional reduction [7], possesses some fascinating properties. First, the space
of gauge inequivalent classical solutions is isomorphic to Hp(M)×Hd−p(M), Hp(M) being the pth
cohomology group of the manifold M. Second, the types of topological terms contributing to these
actions define generalisations to arbitrary dimensions of ordinary two-dimensional anyons. Namely,
non local holonomy effects give rise to exotic statistics for the extended objects which may be
coupled to the higher order tensor fields [8, 9]. Third, these types of quantum field theories display
1If p is even with d = 2p, this action reduces to a surface term.
profound connections between mathematics and physics for what concerns topological properties
related, say, to the motion group, the Ray-Singer torsion and link theory. These connections appear
within the canonical quantisation2 of these systems [10, 11].
Furthermore, within the context of dynamical relativistic (quantum) field theories in any
spacetime dimension, which is a general framework of potential relevance to fundamental particle
physics as well as mathematical investigations for their own sake, such topological B ∧ F terms
may be considered to define couplings between two independent tensor fields whose dynamics is
characterised by the following action, provided the spacetime manifold M is endowed now with
a Lorentzian metric structure (of mostly negative signature) allowing for the introduction of the
associated Hodge ∗ operator,
STMGT[A,B] =
(−1)p F ∧ ∗F + 1
(−1)d−p H ∧ ∗H
(1− ξ)F ∧B − (−1)p ξ A ∧H. (4)
The notations are those introduced previously. Given a choice of units such that c = 1, the phys-
ical dimensions of A and B are L−p and L−d+p, respectively, whereas the multiplicative constant
κ possesses the same physical dimension as the action. The parameters e and g are arbitrary real
constants corresponding to coupling constants when matter fields coupled to A and B are intro-
duced. Without loss of generality for the present analysis, the parameters e and g are assumed to
be strictly positive. In 3+1 dimensions, one recovers the famous Cremmer-Scherk action [12, 13]
and in 2+1 dimensions, the doubled Chern-Simons theory [14]. It is well known that the topo-
logical terms generate a mass for the dynamical tensor fields without breaking gauge invariance.
Introducing an appropriate choice of gauge fixing, it is possible to render one of the tensor fields
massive through a combination with the other tensor field [13]. In the particular circumstance that
d = 2p with p odd, a topological term of the A∧F type (3) generates also a mass even though the
action involves a single p-form field A,
STMGT[A] =
F ∧ ∗F +
A ∧ F. (5)
In 2+1 dimensions, this action defines the well-known Maxwell-Chern-Simons theory [15].
The outline of the paper is as follows. Section 2 discusses a new property of the abelian TMGT
valid whatever the number of space dimensions and the value of 0 ≤ p ≤ d for the p- and (d − p)-
form fields: the “Physical-Topological” factorisation of their degrees of freedom. This result is
achieved within the Hamiltonian formulation through a canonical transformation of classical phase
space leading to two independent and decoupled sectors3. The first of these sectors, namely the
“physical” one, consists of gauge invariant variables which are canonically conjugate and describe
free massive propagating physical degrees of freedom. The second sector, namely the “topological”
one, consists of canonically conjugate gauge variant variables which are decoupled from the total
Hamiltonian and, hence, are non dynamical. This sector is equivalent to a pure TFT of the A∧ F
or B ∧ F type. This factorisation enables the identification of a mass generating mechanism for
any p-form (or, by dualisation, any (d− p)-form) without introducing any gauge fixing conditions
or second-class constraints whatsoever as has heretofore always been the case in the literature.
Section 3 addresses the Dirac quantisation of these systems, with the identification of the spectrum
of physical states through a likewise factorisation extended to the space of quantum states. Finally,
Sect. 4 discusses how the factorisation leads to a most transparent understanding of the projection
2When M = R× Σ, the physical Hilbert space is the set of square integrable functions on Hp(Σ) [4].
3In other words, the Poisson brackets of variables belonging to the two distinct sectors vanish identically.
from TMGT onto TQFT in whatever spacetime dimension in the limit of an infinite topological
mass.
2 Gauge Invariant Factorisation of the Classical Theory
2.1 Hamiltonian formulation
Because of the built-in gauge invariances of these systems, the analysis of the constraints [16, 17] of
topologically massive gauge theories is required in order to identify their Hamiltonian formulation.
Given the total action (4) written out in component form the associated Lagrangian density reads,
LTMGT =
(−1)p
(p + 1)!
Fµ1···µp+1 F
µ1···µp+1 +
(−1)d−p
(d− p+ 1)!
Hν1···νd−p+1 H
ν1···νd−p+1
(1− ξ)
(1 + p)! (d − p)!
ǫµ1···µp+1ν1···νd−p Fµ1···µp+1 Bν1···νd−p
− κ ξ (−1)
p! (d− p+ 1)!
ǫµ1···µpν1···νd−p+1 Aµ1···µp Hν1···νd−p+1 , (6)
where Greek indices, µ, ν = 0, 1, . . . , d, denote the coordinate indices of the spacetime manifold
M while h is the absolute value of determinant of the metric. According to our conventions, the
components of the field strength tensors are given as
Fµ1···µp+1 =
∂[µ1 Aµ2···µp+1], Hν1···νd−p+1 =
(d− p)!
∂[ν1 Bν2···νd−p+1]
, (7)
where square brackets on indices denote total antisymmetrisation. The above expression with the
single parameter κ multiplying each of the topological B∧F and A∧H terms while ξ parametrises
a possible surface term does not entail any loss of generality. Had two independent parameters κ
and λ multiplying each of the topological terms been introduced, only their sum, (κ + λ), would
have been physically relevant, the other combination corresponding in fact to a pure surface term.
In order to proceed with the Hamiltonian analysis, the spacetime manifold M is now taken
to have the topology of M = R × Σ where Σ is a compact orientable d-dimensional Riemannian
space manifold without boundary. Adopting then synchronous coordinates on M, the spacetime
metric takes the form ds2 = dt2 − h̃ij dxi dxj, h̃ij(~x ) being the Riemannian metric on Σ. Here
Latin indices, i = 1, . . . , d, label the space directions in Σ. The configuration space variable A(t, ~x )
may then be separated into its temporal component dt∧A0(t, ~x ) with A0(t, ~x ) being a (p−1)-form
on Σ, and its remaining components Ã(t, ~x ) restricted to Ωp(Σ),
A0(t, ~x ) =
(p − 1)!
A0i1···ip−1(t, ~x ) dx
i1 ∧ . . . ∧ dxip−1 ,
Ã(t, ~x ) =
Ai1···ip(t, ~x ) dx
i1 ∧ . . . ∧ dxip . (8)
A similar decomposition applies to the (d− p)-form B(t, ~x ).
The actual phase space variables are then the spatial components Ã and B̃ along with their
conjugate momenta P̃ and Q̃ defined to be the following differential forms on Σ,
h̃i1j1 . . . h̃ipjp P
i1···ip dxj1 ∧ . . . ∧ dxjp ,
(d− p)!
h̃i1j1 . . . h̃id−pjd−p Q
i1···id−p dxj1 ∧ . . . ∧ dxjd−p , (9)
of which the pseudo-tensorial space components are P i1···ip and Qi1···id−p . Expressed in terms of
the configuration space variables, these latter quantities are given as
P i1···ip =
F0j1···jp h̃
i1j1 . . . h̃ipjp + κ
(1− ξ)
(d− p)!
ǫi1···ipj1···jd−p Bj1···jd−p ,
Qi1···id−p =
H0j1···jd−p h̃
i1j1 . . . h̃id−pjd−p − κ
(−1)p(d−p) ǫi1···id−pj1···jpAj1···jp, (10)
while the symplectic structure of Poisson brackets is characterised by the canonical brackets
Ai1···ip(t, ~x ), P
j1···jp(t, ~y )
. . . δ
δ(d)(~x− ~y ),
Bi1···id−p(t, ~x ), Q
j1···jd−p(t, ~y )
. . . δ
id−p]
δ(d)(~x− ~y ). (11)
A priori , phase space also includes the canonically conjugate variables A0 and P
0, and B0 and Q
The Legendre transform of the Lagrangian (6) leads to the total gauge invariant Hamiltonian,
∗P̃ − κ (1 − ξ) B̃
u, P 0
∗Q̃+ κ ξ (−1)p(d−p)Ã
+ (surface term)
(−1)p (u′ +A0) ∧ d
∗P̃ + κ ξ B̃
(−1)d−p(v′ +B0) ∧ d
∗Q̃− κ (1 − ξ) (−1)p(d−p) Ã
. (12)
In this expression as well as throughout hereafter, the Hodge ∗ operation is now considered only
on the space manifold Σ endowed with the Riemannian metric h̃ij . In (12) the inner product on
Ωk(Σ)× Ωk(Σ) is constructed as
2 = (ωk, ωk) with (ωk, ηk) =
ωk ∧ ∗ηk. (13)
The quantities u′ and v′ are Lagrange multipliers for the two first-class constraints associated to
abelian gauge symmetries while u and v are those for the first-class constraints P 0 = 0 and Q0 = 0
arising because the fields A0 and B0 are auxiliary degrees of freedom of which the time derivatives
do not contribute to the action. Upon reduction to the basic layer of the Hamiltonian nested
structure [16], P 0 and Q0 decouple from the system whereas A0 and B0 play the role of Lagrange
multipliers enforcing the two Gauss laws. These constraints generate those gauge transformations
in (2) which are continuously connected to the identity transformation, namely the so-called small
gauge symmetries, one generated by the fields P̃ and B̃ and the other by Ã and Q̃, respectively.
Note that given Hodge duality, the phase space variables are associated to isomorphic spaces,
Ωp(Σ) ≡ Ωd−p(Σ). Hence at any given spacetime point, phase space has dimension 4Cpd .
2.2 The Physical-Topological (PT) factorisation
The above results are well-known. However the fields used to construct the theory do not necessarily
create physical states since these are not gauge invariant variables. Therefore, let us now introduce
the new Physical-Topological factorisation of the classical theory, by also requiring that these
field redefinitions are canonical and preserve canonical commutation relations. First consider the
quantities
A = −
(−1)p(d−p) ∗ Q̃+ (1− ξ) Ã, B =
∗ P̃ + ξ B̃, (14)
defined on the dual sets Ωp(Σ) and Ωd−p(Σ). This choice is made in such a way that the two Gauss
laws are expressed in term of these variables only, as is the case for a topological B ∧ F theory,
κ (−1)p(d−p) dA = 0, (−1)p κdB = 0. (15)
As a matter of fact, these variables are canonically conjugate,
Ai1···ip(t, ~x ),Bj1···jd−p(t, ~y )
ǫi1···ipj1···jd−p δ
(d)(~x− ~y ). (16)
The two finite gauge transformations in (2) act on these new variables according to the relations,
A′ = A+ α, B′ = B + β. (17)
At a given spacetime point, these canonically conjugate variables carry 2C
degrees of freedom. The
remaining 2C
degrees of freedom are associated to the following pair of gauge invariant variables,
G = Q̃+ κ ξ ∗ Ã, E = P̃ − κ (1 − ξ) (−1)p(d−p) ∗ B̃. (18)
Their pseudo-tensor Lorentz components are defined as in (9) while they possess the following non
vanishing canonical Poisson brackets,
Ei1···ip(t, ~x ), Gj1···jd−p(t, ~y )
= −κ ǫi1···ipj1···jd−p δ(d)(~x− ~y ). (19)
When considered in combination with the equations of motion, these variables correspond to the
non commutative electric fields associated, respectively, to the field strength tensors of A and B,
see (7). Consequently, we have achieved a coherent reparametrisation of phase space which, in fact,
factorises the system into two orthogonal sectors, namely sectors of which mutual Poisson brackets
vanish identically,
Ai1···ip(t, ~x ), Ej1···jp(t, ~y )
Ai1···ip(t, ~x ), Gj1···jd−p(t, ~y )
Bi1···id−p(t, ~x ), E
j1···jp(t, ~y )
Bi1···id−p(t, ~x ), G
j1···jd−p(t, ~y )
Finally in order to obtain the basic nested Hamiltonian formulation [16] within the factorised
parametrisation, the Lagrange multipliers in (12) may be redefined in a convenient way as
u = Ȧ0, A0 = A0 + u′ +
(−1)(p−1)(d−p)
2 g2 κ2
∗ d (κB − 2 ∗ E) ,
v = Ḃ0, B0 = B0 + v′ +
(−1)p
2 e2 κ2
κA + (1)p(d−p) 2 ∗G
where a dot stands for differentiation with respect to the time coordinate, t ∈ R. Consequently the
basic total first-class Hamiltonian of the system reads,
H[E,G,A,B] = e
2κ2 g2
2 e2 κ2
(−1)p A0 ∧ dB − (−1)(p+1)(d−p) B0 ∧ dA, (20)
where d† = ∗d∗ is the coderivative operator. Obviously, A0 and B0 are Lagrange multipliers
enforcing the first-class constraints which generate the small gauge transformations in (17),
G(1) = dA G(2) = dB. (21)
When restricted to the physical subspace for which these constraints are satisfied, the above gauge
invariant Hamiltonian reduces to a functional depending only on the dynamical physical sector,
given by the expression in the first line of (20).
These redefinitions of the phase space variables have indeed achieved the announced fac-
torisation. A first sector is comprised of the variables constructed in (14), which decouple from
the physical Hamiltonian and are therefore non propagating degrees of freedom. Furthermore, the
canonically conjugate variables A and B actually share the same Poisson brackets, Gauss law con-
straints and gauge transformations as the phase space description of a pure B ∧F topological field
theory constructed only from the topological terms in the action (6). Hence this “topological field
theory (TFT) sector” accounts for the B∧F theory embedded into the topologically massive gauge
theory.
Physical and non physical degrees of freedom are mixed in the original phase space. Our
redefinition of fields deals with the original degrees of freedom in such a way that within the
Hamiltonian formalism, non propagating (and gauge variant) degrees of freedom are decoupled
from the dynamical sector. This latter sector describes only physical degrees of freedom, namely
the gauge invariant canonically conjugate electric fields, which diagonalise the physical Hamiltonian
(20) in such a way that they acquire a mass through a mixing of the original variables (18). However
the Poisson bracket structure remains unaffected since these field redefinitions define merely a
canonical transformation. On account of Hodge duality between Ωp(Σ) and Ωd−p(Σ), one readily
identifies in the dynamical sector the Hamiltonian of a massive p-form field of mass m = ~µ,
H[C,E,A,B] =
+HTFT[A,B].
In comparison with (20) the following identifications have been applied,
µ = |κ e g| E →
∗G = e κ (−1)p(d−p) C,
where C is a p-form field of which the Lorentz components are covariant in the manner of (8).
Physical phase space is then endowed with the elementary Poisson brackets
Ci1···ip(t, ~x ), E
j1···jp(t, ~y )
. . . δ
δ(d)(~x− ~y ) .
Alternatively one may also obtain the Hamiltonian of a massive (d− p)-form field of mass m = ~µ,
H[C,G,A,B] = µ
+HTFT[A,B],
in which, in comparison with (20), the following identifications have been applied,
∗E = −g κC.
In this case, C is a (d− p)-form field with covariant Lorentz components as in (8). The elementary
Poisson brackets for these physical phase space variables are
Ci1···id−p(t, ~x ), G
j1···jd−p(t, ~y )
. . . δ
id−p]
δ(d)(~x− ~y ).
To conclude this discussion of the factorised Hamiltonian formulation of these TMGT, let us em-
phasize once more that no gauge fixing procedure whatsoever was applied, in contradistinction to
all discussions available until now in the literature leading to an identification of the physical con-
tent of these theories. Through the present approach, the TFT content of TMGT is made manifest
in a most transparent and simple manner, with in addition a decoupling of the actual physical and
dynamical sector of the system from its purely topological one, the latter carrying only topological
information characteristic of the underlying spacetime manifold.
2.3 Hodge decomposition
The space manifold Σ having been assumed to be orientable and compact, let us now consider the
consequences of its cohomology group structure, especially in the case when the latter could be
non trivial. Throughout the discussion it is implicitly assumed that the p- and (d − p)-form fields
A and B are globally defined differentiable forms in Ωp(M) and Ωd−p(M). When parametrising
the theory in terms of the PT factorised variables, the latter assumption of a topological character
concerns only the TFT sector. The variables of the dynamical sector are already globally defined
whatever the topological properties of the original variables. By virtue of the Hodge theorem [18],
the phase space variables of the TFT sector, thus globally defined on Σ itself endowed with the
Riemannian metric h̃ij , may uniquely be decomposed for each time slice into the sum of an exact,
a co-exact and a harmonic form with respect to the inner product specified in (13),
A = Ae +Ac +Ah, B = Be +Bc +Bh. (22)
A likewise decomposition applies to the dynamical sector.
Such a decomposition amounts to a split of the fields into a longitudinal part (subscript L),
a transverse part (subscript T ) and a “global” part. The transverse and longitudinal parts are
associated to idempotent orthogonal projection operators,
ΠT(p) =
(p+1)
d(p), Π
(p) =
d(p−1)d
ΠT(p) : Ω
p(Σ) → (Z†
)p(Σ), ΠL(p) : Ω
p(Σ) → Zp
(Σ), (23)
where △⊥
is the Laplacian operator acting on the space Ω
(Σ) of p-forms from which the kernel
ker△(p) of the Laplacian △(p) has been subtracted, while (Z
)p (resp. Z
) is the space of co-closed
(resp. closed) p-forms non cohomologous to zero. One therefore has the following properties,
(−1)p(d−p)ΠT(p) = ∗Π
(d−p)∗, Π
(p) +Π
(p) = Id
where ∗ is the Hodge star operator on Σ and Id⊥
the identity operator on the subspace Ω
In order that the longitudinal and transverse components possess the same physical dimen-
sions as the original fields, the Hodge decomposition of fields may be expressed in terms of a
convenient normalisation,
△⊥A = dAL + d†AT ,
△⊥B = dBL + d†BT . (24)
Let us then define a new set of variables in the TFT sector, using the projection operators (23),
ϕ = ΠT(p−1)AL, ∗Qϑ = Π
(p+1)AT ,
ϑ = ΠT(d−p−1)BL, ∗Pϕ = Π
(d−p+1)BT , (25)
where the components of ∗Pϕ and ∗Qϑ are pseudo-tensors defined in a manner analogous to the
conjugate momenta in (9). In terms of these new variables the non vanishing Poisson brackets are
ϕi1···ip−1(t, ~x ), P
j1···jp−1
ϕ (t, ~y )
)j1···jp−1
i1···ip−1
δ(d)(~x− ~y ),
ϑi1···id−p−1(t, ~x ), Q
j1···jd−p−1
(t, ~y )
)j1···jd−p−1
i1···id−p−1
δ(d)(~x− ~y ).
In conclusion, in the TFT sector, rather than working in terms of the phase space variables A and
B one may parametrise these degrees of freedom in terms of the “longitudinal” fields ϕ and ϑ as
well as their conjugate momenta, namely the “transverse” fields Pϕ and Qϑ, to which the harmonic
components Ah and Bh of A and B must still be adjoined. The same procedure may be applied to
the variables of the dynamical sector. The Hamiltonian (20) then decomposes into a transverse, a
longitudinal and a harmonic contribution from these latter variables only.
A natural consequence of the Hodge decomposition is the isomorphism between the pth de
Rham cohomology group, Hp(Σ,R), and the space of harmonic p-forms, ker△(p). This means that
each equivalence class of Hp(Σ,R) has an unique harmonic p-form representative identified through
the inner product (13). It is possible to choose a basis for ker△(p) in such a way that the harmonic
component of any p-form is expressed in a topological invariant way. This may be achieved by
defining a topological invariant isomorphism between the components of an equivalence class of the
pth (singular) homology group Hp(Σ,R) and the components of a form in ker△(p) (the p-homology
group is the set of equivalence classes of p-cycles differing by a p-boundary). Thus, instead of
constructing the basis from the Hodge decomposition inner product (13), one uses the bilinear, non
degenerate and topological invariant inner product Λ defined by
Λ : Hp(Σ)×Hp(Σ) → R : Λ ([Γ] , [ω]) =
ω, (26)
making explicit the Poincaré duality between homology and cohomology groups [18]. Given the
Hodge theorem, this inner product naturally induces a topological invariant inner product between
the equivalent classes of Hp(Σ) and the elements of ker△(p). Therefore, if one introduces generators
of the free abelian part of the pth singular homology group of rank Np,
, a convenient
dual basis {Xγ} of ker△(p) may be chosen such that
Σα(p)
= δαβ .
Using the duality (26), the harmonic component Ah of the p-form variable A is thus decomposed
according to
These components of Ah in the basis {Xγ} are topological invariants because they express the
periods of A over the cycle generators of Hp(Σ). This is thus nothing other than the classical
Wilson loop argument over these generators,
Ah . (27)
In other words, the variables aγ(t) specify the complete set of remaining “global” degrees of freedom
in the TFT sector for the field A,
Ah(t, ~x ) =
aγ(t)Xγ(~x ).
In a likewise manner, the harmonic component of the (d− p)-form variable B may be decomposed
according to
(d−p)
Y γ ,
where {Y γ} is the dual basis of the cycle generators in Hd−p(Σ),
(d−p)
. Hence, the compo-
nents of harmonic (d− p)-forms are expressed as
(d−p)
Bh, (28)
leading to a similar decomposition of the “global” degrees of freedom for the dual field B,
Bh(t, ~x ) =
bγ(t)Y γ(~x ).
The Poisson brackets between the above global variables are topological invariants,
aγ , bγ
, (29)
namely the signed intersection matrix of which each entry is the sum of the signed intersections of
the generators of Hp(Σ) and Hd−p(Σ),
(d−p)
. (30)
Within our approach, we recover the results of [19, 11, 10] in 2+1 (on the torus), 3+1 and d+1
dimensions, respectively.
2.4 Large and small gauge transformations
Only the TFT sector is not gauge invariant. Its phase space variables transform exactly like in a
pure B ∧ F theory, see (17). Let us recall that in (17), α and β are, respectively, closed p- and
(d− p)-forms on Σ. In the case of a homologically trivial space Σ any closed form is also exact. In
the case of a homologically non trivial space Σ, according to the Hodge theorem any closed form
α or β may uniquely be decomposed (for a given metric structure) into the sum of an exact and
a harmonic form. The exact parts of α and β define small gauge transformations, generated by
the two Gauss law first-class constraints (15). Given the Hodge decompositions in the TFT sector
(24) and (25), these constraints, which require that the phase space variables A and B of the TFT
sector be closed forms, reduce to
G(1) =
△⊥ ∗Qϑ, G(2) =
△⊥ ∗ Pϕ. (31)
Small gauge transformations act only on the exact part of the TFT sector fields by translating
them, namely in terms of the longitudinal (p− 1)- and (d− p− 1)-form fields defined in (25),
ϕ′ = ϕ+ αL, ϑ
′ = ϑ+ βL,
where αL and βL are, respectively, the longitudinal (p−1)- and (d−p−1)-forms defining the exact
components of the gauge transformation forms α and β through a construction similar to that in
(25). The harmonic components of α and β define the associated large gauge transformations.
The physical classical phase space in the TFT sector is the set of all field configurations A
and B obeying the first-class constraints setting to zero their transverse degrees of freedom, see
(31), and identified modulo the action of all gauge transformations, whether small or large. Since
under small transformations the longitudinal modes ϕ and ϑ are gauge equivalent to the trivial
configuration of vanishing longitudinal fields, like in any pure B ∧F TFT the physical phase space
of the TFT sector, so far for what concerns small gauge symmetries, is thus finite dimensional and
isomorphic to the ensemble of harmonic forms defined modulo exact forms,
P = Hp(Σ,R)⊕Hd−p(Σ,R), (32)
where Hp(Σ,R) is the pth de Rham cohomology group. Let us recall that according to Poincaré
duality, Hp(Σ) is isomorphic to Hd−p(Σ). Hence, whether one considers functionals of harmonic
p-forms or (d − p)-forms is of no consequence. The finite dimension of this group is given by the
corresponding Betti number Np (for example in the case of the torus, Σ = Td, Np = C
d ). The
physical phase space of the TFT sector is thus spanned by the global degrees of freedom aγ(t) and
bγ(t), which are indeed obviously invariant under all small gauge transformations. However, this
phase space is subjected to further restrictions still, stemming from large gauge transformations.
In a manner similar to the above characterisation of the physical phase space in the TFT
sector, the modular group is the quotient of the full gauge group by the subgroup of small gauge
transformations generated by the first-class constraints, namely essentially the set of large gauge
transformations (LGT) defined modulo small gauge transformations. Large gauge transformations
cannot be built from a succession of infinitesimal transformations. They correspond to the co-
homologically non trivial, namely the harmonic components of α and β. Rather than requiring
strict invariance of the global phase space variables aγ and bγ under large gauge transformations,
having in mind compact U(1) abelian gauge symmetries defined in terms of univalued pure imag-
inary exponential phase factors, the global physical observables to be considered and thus to be
required to remain invariant under large gauge transformations are the holonomy or Wilson loop
operators of the TFT sector around compact orientable submanifolds Σp and Σd−p in Σ. The only
non trivial Wilson loops are those around homotopically non trivial cycles, namely elements [Γp]
of Hp(Σ,Z) which may be decomposed in the basis
. Consequently, given the basis of
ker△(p) constructed from (26) one has the following set of global Wilson loop observables
W [Γ(p)] = exp
= exp
σγ aγ
W [Γ(d−p)] = exp
(d−p)
= exp
σ̃γ bγ
where σγ , σ̃γ are arbitrary integers. Large gauge transformations associated to closed forms α and
β act on the global variables aγ and bγ according to
a′γ = aγ + αγ , b′γ = bγ + βγ , (33)
where αγ and βγ are given by
α, βγ =
(d−p)
Although the Wilson loops are constructed on the free abelian homology group Hp(Σ,Z),
the cohomology group including the large gauge transformation parameters is dual to the singular
homology group Hp(Σ,R). Hence, the only allowed large gauge transformations correspond to
components of the harmonic content of the forms α and β which are discrete and quantised,
α = 2π ℓ
, βγ =
(d−p)
β = 2π ℓ
(d−p)
. (34)
Here ℓ
and ℓ
(d−p)
are integers which characterise the winding numbers of the large gauge transfor-
mations, namely the periods of these transformations around the homology cycle generators. The
requirement of gauge invariance of all Wilson loops hence constrains the parameters of large gauge
transformations to belong to the dual of the free abelian homology group. As a consequence, finally
the physical classical phase space in the TFT sector is the quotient of the de Rham cohomology
group Hp(Σ,R)⊕Hd−p(Σ,R) by the additive lattice group defined by the transformations,
a′γ = aγ + 2π ℓ
, b′γ = bγ + 2π ℓ
(d−p)
namely a finite dimensional compact space having the topology of a torus of dimension 2Np.
3 Canonical Quantisation and Physical States
3.1 Physical Hilbert space factorisation
The BRST formalism offers a powerful and elegant quantisation procedure for TMGT but requires
the introduction of ghosts. In some respects, this formalism has also been used for the definition
and characterisation of topological quantum field theories [1]. In a related manner, the path integral
quantisation of these theories also brings to the fore the characterisation of topological invariants
through concepts of quantum field theory. For example, the two-point correlation function of B∧F
(and A∧F ) theories provides a quantum field theoretic realisation of the linking number of two sur-
faces of dimensions p and (d− p) embedded in M and its path integral representation through the
Ray-Singer analytic torsion of the underlying manifold. Notwithstanding these achievements, this
paper will not rely on such methods which necessarily require some gauge fixing procedure. Rather,
ordinary Dirac canonical quantisation methods will be implemented to unravel the physical content
of TMGT. First, this quantisation procedure is best adapted to a condensed matter interpretation.
It also enables to deal with large gauge transformations on homologically non trivial manifolds. Sec-
ond, the new Physical-Topological (PT) factorisation identified within the Hamiltonian formulation
independently of any gauge fixing procedure makes canonical quantisation especially attractive.
Canonical quantisation readily proceeds from the correspondence principle, according to
which classical Poisson brackets are mapped onto equal time quantum commutation relations for
the classical variables which are promoted to linear self-adjoint operators acting on the Hilbert
space of quantum states in the Schrödinger picture at the reference time t = t0,
Âi1···ip(t0, ~x ), B̂j1···jd−p(t0, ~y )
ǫi1···ipj1···jd−p δ
(d)(~x− ~y ),
Êi1···ip(t0, ~x ), Ĝ
j1···jd−p(t0, ~y )
ǫi1···ipj1···jd−p δ(d)(~x− ~y ).
A possible representation of the associated Hilbert space is in terms of functionals Ψ[A, E] with
their canonical hermitean inner product defined in terms of the field degrees of freedom A(~x ) and
E(~x ).
It should be clear that the PT factorisation identified at the classical level extends to the
quantum system. The full Hilbert space of the system factorises into the tensor product of two
separate and independent Hilbert spaces, each of which is the representation space of the operator
algebra of either the gauge invariant dynamical sector or the TFT sector. As a consequence of
the complete decoupling of these two sectors, one of which contributes to the physical Hamiltonian
only, the other to the first-class constraint operators only, a basis of the space of quantum states
may be constructed in terms of a likewise factorisation of wave functionals. Symbolically one has
Ψ[A, E] = Φ[E] Ψ[A].
The component Φ[E] associated to the dynamical sector is manifestly gauge invariant and is the
only one which contributes to the energy spectrum. The physical Hilbert subspace associated to the
TFT sector consists of those states wave functionals ΨP [A] which are invariant both under small
gauge transformations, namely which belong to the kernel of the first-class constraint operators
generating these transformations, and under the large gauge transformations4 characterised in the
previous Section in terms of their lattice action on harmonic p- and (d− p)-forms.
When the space manifold Σ is topologically trivial, for instance in the case of the hyperplane,
quantisation of TMGT does not offer much interest per se besides the free dynamics of the dynam-
ical sector, since the TFT sector then possesses a single gauge invariant quantum state. However
in the presence of external sources, or when the space manifold Σ does have non trivial topology,
new and interesting features arise. In the latter situation, to be addressed hereafter, the finite
though multi-dimensional gauge invariant content of the TFT sector, ΨP [A], does not contribute
to the energy spectrum. Hence it induces a degeneracy of the energy eigenstates of the complete
system. As demonstrated later, this degeneracy is restricted by gauge invariance under large gauge
transformations (LGT). Since the physical wave functional ΨP [A] in the TFT sector coincides with
that of a pure topological quantum field theory, one recovers the results of R. J. Szabo [10] who
solved in the Schrödinger picture the pure topological B ∧ F theory (as well as in the presence of
sources) in any dimension.
3.2 The topological sector: Gauss’ constraints and LGT
3.2.1 Hilbert space and holomorphic polarisation
At the classical level, phase space has been separated into two decoupled sectors: the TFT and the
dynamical sectors. According to the Hodge decomposition theorem (22), each of the corresponding
fields may in turn be decomposed into three further subsectors in terms of their longitudinal,
transverse and global components. The Gauss law constraints in conjunction with invariance under
small gauge transformations reduce the TFT sector to its global variables only, characterised by
the vector space P of the de Rham cohomology group in (32), which is to be restricted further
into a compact torus by the lattice action of the appropriate discrete large gauge transformations.
Likewise in the dynamical sector, the global degrees of freedom of phase space are also purely
topological and are again isomorphic to the 2Np-dimensional symplectic vector space P in (32).
In each case, these spaces are spanned by the global variables defined as in (27) and (28), namely
(aγ , bγ) and (Eγ , Gγ), respectively. It is quite natural to introduce for these even dimensional vector
spaces a complex structure parametrised by a Np ×Np complex symmetric matrix, τ = ℜ(τ) + i ρ,
such that (−τ) takes its values in the Siegel upper half-space. Such a complex structure introduced
over the phase space of global degrees of freedom enables the definition of a holomorphic phase
space polarisation, hence quantisation of these sectors.
The same decomposition in terms of longitudinal, transverse and global degrees of freedom
applies at the quantum level. Through the correspondence principle, these three subsectors of
4Otherwise, the physical wave functional carries a projective representation of the group of LGT [10].
quantum operators obey the Heisenberg algebra, whether for the TFT or dynamical sector. Let us
presently restrict to the TFT sector. For what concerns the local operators, one has
ϕ̂i1···ip−1(t0, ~x ), P̂
j1···jp−1
ϕ (t0, ~y )
)j1···jp−1
i1···ip−1
δ(d)(~x− ~y ),
ϑ̂i1···id−p−1(t0, ~x ), Q̂
j1···jd−p−1
(t0, ~y )
= − i~
)j1···jd−p−1
i1···id−p−1
δ(d)(~x− ~y ), (35)
while for the global operators,
âγ(t0), b̂
γ′(t0)
Introducing now the holomorphic combinations of the latter operators5,
ĉγ =
Iγδ â
δ + τγδ b̂
, ĉ†γ =
Iγδ â
δ + τγδ b̂
, (36)
where Iγδ is the inverse of the intersection matrix,
Iγδ I
δγ′ = δγ
one finds the Fock type algebra
ĉγ , ĉ
= ℑ(τ)γγ′ = ργγ′ , (37)
all other possible commutators vanishing identically. Note that this result implies that the inner
product in this sector of Hilbert space is to be defined in terms of the imaginary part (ρ−1)γγ
, in a
manner totally independent from the Riemannian metric structure of the compact space submani-
fold Σ. A priori , physical observables in pure topological quantum field theories ought nevertheless
to be independent from any extraneous ad hoc structure introduced through the quantisation pro-
cess such as the present complex structure.
Gauss law constraints and large gauge transformations are to be considered in the wave
functional representation of Hilbert space. The latter is spanned by the direct product of basis
vectors for the representation spaces of the algebras (35) and (37). These consist of functionals
Ψ[ϕ, ϑ, c] of the infinite dimensional space of field configurations in the TFT sector. Accordingly,
the inner product of such states is defined by
〈Ψ1 |Ψ2〉 =
[Dϕ] [Dϑ]
Ψ∗1[ϕ, ϑ, c] Ψ2[ϕ, ϑ, c],
which requires the specification of a functional integration measure. This measure is taken to be
the gaussian measure for fluctuations in the corresponding fields, which is induced by the complex
structure τ in the global sector or else by the Riemannian metric on Σ for fluctuations in ϕ and ϑ,
δϕ2 =
d~xhi1k1(~x ) . . . hip−1kp−1(~x ) δϕi1···ip−1(~x ) δϕk1···kp−1(~x ),
δϑ2 =
d~xhj1l1(~x ) . . . hjd−p−1ld−p−1(~x ) δϑj1···jd−p−1(~x ) δϑl1···ld−p−1(~x ),
δc2 =
γ,γ′=1
(ρ−1)γγ
δcγ δcγ′ . (38)
5It is implicitly assumed here that the parameter κ is strictly positive. If κ is negative, the roles of the operators
γ and b̂γ are simply exchanged in the discussion hereafter.
In contradistinction to an ordinary pure topological quantum field theory, such a space metric is
readily available within the context of TMGT, being necessary for the specification of the dynamical
fields. Independently from the complex structure introduced in the global sector, independence of
the physical Hilbert space measure in the (ϕ, ϑ) sector on the metric on Σ will be established
hereafter. Consequently the canonical commutation relations (35) and (37) in the TFT sector are
represented by the following functional operators acting on the Hilbert space wave functionals,
ϕ̂i1···ip−1(~x ) ≡ ϕi1···ip−1(~x ), P̂
i1···ip−1
ϕ (~x ) ≡ −
)i1···ip−1
j1···jp−1
δϕj1···jp−1(~x )
, (39)
ϑ̂i1···id−p−1(~x ) ≡ ϑi1···id−p−1(~x ), Q̂
i1···id−p−1
ϑ (~x ) ≡
)i1···id−p−1
j1···jd−p−1
δϑj1···jd−p−1(~x )
, (40)
ĉγ ≡ cγ , ĉ†γ ≡ −
. (41)
3.2.2 Gauss law constraints
The physical Hilbert space is invariant under all gauge transformations. A first restriction arises by
requiring the physical quantum states to be invariant under small gauge transformations generated
by the first class constraints. This set is the kernel of the Gauss law constraint operators (21) which
remain defined as in the classical theory since no operator ordering ambiguity is encountered,
Ĝ(1)
= 0 ⇒ δ
δϑi1···id−p−1(~x)
ΨP [ϕ, ϑ, c] = 0,
Ĝ(2)
= 0 ⇒
δϕi1···ip−1(~x)
ΨP [ϕ, ϑ, c] = 0.
Hence physical quantum states necessarily consist of wave functionals which are totally independent
of the longitudinal variables (ϕ, ϑ). When restricted to such states and properly renormalised, the
inner product integration measure is constructed from the definition (38) of the gaussian metric on
the space of fluctuations in the global coordinates,
〈Ψ1 |Ψ2〉 =
dcγ (det ρ)
Ψ∗1(c)Ψ2(c).
This measure on the physical Hilbert space is thus indeed independent of the Riemannian metric
on Σ, and involves only the ad hoc complex structure τ introduced towards the quantisation of the
global TFT sector.
3.2.3 LGT and global variables
The structure of the physical Hilbert space dramatically depends on the way one deals with
LGT. Given the holomorphic parametrisation (36), under the lattice action of LGT of periods
(d−p)
) ≡ (ℓ(p), ℓ(d−p)) as defined in (34) the new global operators should transform as,
c′γ = cγ +
2π2 κ
Iγγ′ ℓ
+ τγγ′ ℓ
(d−p)
= c†γ +
2π2 κ
Iγγ′ ℓ
+ τγγ′ ℓ
(d−p)
. (42)
Using the Baker-Campbell-Hausdorff (BCH) formulae for any two operators Â and B̂ commuting
with their own commutator,
eÂ B̂ e−Â = B̂ +
Â, B̂
, eÂ+B̂ = e−
2 [Â,B̂] eÂ eB̂ , (43)
it may be seen that the quantum operator generating the LGT of periods (k(p), k(d−p)) is
k(p), k(d−p)
k(p), k(d−p)
γ,γ′,ǫ
×(ρ−1)γγ′
Iγǫ k
(p) + τγǫ k
(d−p)
ĉγ′ −
Iγǫ k
(p) + τγǫ k
(d−p)
. (44)
The 1-cocycle C
k(p), k(d−p)
will be determined presently. This operator (44) defines the action
of LGT on the Hilbert space in the global TFT sector,
k(p), k(d−p)
Ψ(cγ) =
γ,γ′,δ
Iγδ k
+τγδ k
(d−p)
(ρ−1)γγ
Iγ′ǫ k
+τγ′ǫ k
(d−p)
k(p), k(d−p)
cγ + π
Iγγ′ k
+ τγγ′ k
(d−p)
 , (45)
where the BCH formula (43) has been used. However, a U(1)×U(1) 2-cocycle ω2(k; ℓ) appears in
the composition law of this quantum representation,
k(p) + ℓ(p), k(d−p) + ℓ(d−p)
= e2 i π ω2(k;ℓ) Û
k(p), k(d−p)
ℓ(p), ℓ(d−p)
ω2(k; ℓ) ≡ ω2
k(p), k(d−p); ℓ(p), ℓ(d−p)
γ,γ′=1
(d−p)
(d−p)
The 1-cocycle C
k(p), k(d−p)
appearing in (44) may be determined by requiring that the abelian
group composition law for LGT is recovered. This implies that ω2(k; ℓ) is a coboundary,
ω2(k; ℓ) = C1
k(p) + ℓ(p), k(d−p) + ℓ(d−p)
k(p), k(d−p)
ℓ(p), ℓ(d−p)
(mod Z),
C(k) ≡ C
k(p), k(d−p)
2 i π C1(k(p),k(d−p)).
A careful analysis, analogous to the one in [20], finds that the unique solution to this coboundary
condition is
I k, C
k(p), k(d−p)
γ,γ′=1
i π k I k
(d−p)
Iγγ′ k
(p) , (46)
where k ∈ Z and6 I = det
∈ N. It is noteworthy to recall that although Iγγ′ is a rational
valued matrix, I Iγγ′ is integer valued. Note also the quantisation condition arising for the coefficient
κ multiplying the topological terms in the original action of TMGT.
If k is rational, namely if k = k1/k2 with k1, k2 strictly positive natural numbers, invariance
of physical states under LGT cannot be achieved. However in this case the LGT group has a finite
6Recall that κ, hence k is assumed to be strictly positive in the present discussion, while the situation for a
negative κ or k is obtained through the exchange of the sectors aγ and bγ .
dimensional projective representation which may be constructed by finding a normal subgroup gen-
erated by the LGT operators. As demonstrated in [10], the TFT part of the physical wave functions
carries a projective representation of the group of LGT while the above discussion establishes that
the dimension of Hilbert space is
δ=1 k1 k2 IMin(Iδδ′). Any state of a given irreducible repre-
sentation gives the same matrix element for a physical observable. The characterisation of Hilbert
space changes qualitatively for integer or rational values of k, but the theory remains well-defined.
If we take k to be an integer, see (46), wave functions of the physical Hilbert space may be
classified in terms of irreducible representations of the group of LGT (45),
η1; cγ +
π I k
Iγγ′ k
+ τγγ′ k
(d−p)
γ,γ′,δ=1
Iγδ k
(p) + τγδ k
(d−p)
(ρ−1)γγ
cγ′ +
Iγ′ǫ k
(p) + τγ′ǫ k
(d−p)
γ,γ′=1
2 i π η1(k(p), k(d−p))− i π k I k
(d−p)
Iγγ′ k
Ψ(η1; cγ) , (47)
where the 1-cocycle η1
k(p), k(d−p)
characterises the irreducible representation. Since for an
abelian group each of its irreducible representations is one-dimensional, physical states correspond-
ing to a given irreducible representation are singlet under LGT.
As is well-known, functions obeying such a double periodicity condition are nothing other than
the generalised Riemann theta functions defined in any dimension on the complex Np-torus [10],
with the compact reduced phase space resulting from the requirement of invariance under LGT,
(cγ) =
γ,γ′=1
cγ (ρ
−1)γγ
(aδ′ + rδ′)
− I k τ
, (48)
where rδ ∈ [0, k IMin(Iδδ′)− 1] ⊂ N. Each physical subspace, characterised by the 1-cocycle
1 (k(p), k(d−p)) = aγ k
+ bγ k
(d−p)
where aγ , bγ ∈ [0, 1[⊂ R, is invariant under a particular irreducible representation of LGT. The
TFT component of each physical Hilbert space is of dimension
k IMin(Iδδ′). In general, the
choice of physical Hilbert space which is invariant under all LGT is the representation space with
η1(k(p), k(d−p)) ∈ Z, namely corresponding to aγ , bγ = 0.
3.3 The dynamical sector: Hamiltonian diagonalisation
Based on Hodge’s theorem, (22) and (24) define the decomposition of the dynamical sector into
three decoupled subsectors of canonically conjugate variables: the global harmonic sector and the
local (EL, PE) and (GL, QG) sectors. In turn the classical Hamiltonian (20) decomposes into three
separate contributions, one for each subsector. When quantising the system in each subsector, the
total quantum Hamiltonian follows from the classical one without any operator ordering ambiguity,
Ĥ[Ê, Ĝ] = Ĥh[Êh, Ĝh] + Ĥ1[ÊL, P̂E ] + Ĥ2[ĜL, Q̂E ].
The physical spectrum is thus identified by diagonalising each of these contributions separately.
3.3.1 Global degrees of freedom
The choice of normalisation used previously in the harmonic sector relies on the Poincaré duality
between the basis elements [Xγ ] and [Y γ ] of the relevant cohomology groups and their associated
homology generators Σ
and Σ
(d−p)
, respectively, see (26). This choice is of a purely topological
character. However in the dynamical sector, there is a remaining freedom as far as the normalisation
of the choice of the harmonic representative of the cohomology group is concerned, depending on
the metric structure, and thus fixing the basis elements Xγ of ker△(p) and Y γ of ker△(d−p). This
choice involves the inner product in (13) on which the Hodge decomposition relies. Hence one sets
Xγ ∧ ∗Xγ′ =
Ωγγ′ ,
Yγ ∧ ∗Yγ′ =
Ω̃γγ′ , (49)
where Ωγγ′ and Ω̃γγ′ are Np × Np real symmetric matrices. Given this normalisation, the global
part of the metric dependent quantum Hamiltonian operator constructed from (20) is expressed as
Ĥh[Ê
γ , Ĝγ
γ,γ′=1
Êγ Êγ
Ωγγ′ + Ĝ
Ω̃γγ′
, (50)
while the non vanishing commutation relations between the global phase space operators read
Êγ , Ĝγ
= −i~κ Iγγ′ . (51)
As in the TFT sector, see (36), the following holomorphic polarisation of the global dynamical
sector is used,
Iγα Ê
α − υγα Ĝα
, d†γ =
Iγα Ê
α − υγα Ĝα
where υ = ℜ(υ)+i σ is the Np×Np complex symmetric matrix characterising the complex structure
introduced in the global dynamical phase space sector, of which the imaginary part determines the
non vanishing commutation relations of the Fock like algebra
dγ , d
= σγγ′ . (52)
In order to readily diagonalise the Hamiltonian in the global sector which is of the harmonic
oscillator form, it is convenient to make the following choice for the complex structure matrix v as
well as for the normalisation quantities specified in (49),
ℜ(υ) = 0, σγγ′ = Ω̃γγ′ = δγγ′ , Ωγγ′ =
α,β=1
Iαγ Iβγ′ δ
αβ , (53)
where δγγ
is the Np × Np Kronecker symbol. With these choices, the contribution of the global
variables to the Hamiltonian is indeed diagonal,
~µNp + ~µ
γ,γ′=1
d†γ dγ′ δ
γγ′ , µ = e g κ.
One recognizes the Hamiltonian of a collection of Np independent harmonic oscillators of angular
frequency7 µ = e g κ, which turns out to be the mass gap of the quantum field theory. The operators
7Recall that under the assumptions of the analysis, this combination of parameters is indeed positive.
dγ and d
γ are, respectively, annihilation and creation operators obeying the Fock algebra (52) now
with σγγ′ = δγγ′ . The energy spectrum in the global dynamical sector of the system is readily
identified. The normalised fundamental state is the kernel of all annihilation operators,
dα |0〉 = 0, εh(0) =
Np ~µ, 〈0|0〉 = 1,
where εh
is the vacuum energy. Excited states, |nγ〉, are obtained through the action of the Np
creation operators d
γ on the fundamental state. This leads to the energy eigenvalue for any of these
states,
|nγ〉 =
|0〉, ε(nγ ) = ε
(0) + ~µ
nγ , (54)
γ=1 being the eigenvalues of each of the number operators d
γdγ , hence positive integers.
3.3.2 Local degrees of freedom on the torus
The canonical treatment of the global degrees of freedom in both the TFT and dynamical sectors
does not require the explicit specification of the space manifold Σ with its topology and Riemannian
metric, yet allowing the general discussion of the previous Sections. However, in order to identify the
full spectrum of dynamical physical states, the space manifold Σ including its geometry has now to
be completely specified. The explicit choice to be made for the purpose of the present discussion is
that of the d-dimensional Euclidean torus, Σ = Td, enabling straightforward Fourier mode analysis
of the then infinite discrete, thus countable set of degrees of freedom, and diagonalisation of the
harmonic oscillator structure of the Hamiltonian. This particular choice of the d-torus is motivated
by the fact that this manifold is the simplest flat yet homologically non trivial manifold. The
notations used are those of [21] where pure quantum electrodynamics is solved on the torus, of
which the conventions are extended to any p-form.
Accordingly, the variables E and G of the dynamical sector are periodic around the torus p-
and (d− p)-cycles, respectively. Their Fourier mode expansions read
i1...ip
(~x ) = δi1j1 . . . δipjp
k 6=0
αp−1=1
α1···αp
j1··· jp
(k)Eα1···αp(k) e2 i π k(~x) ,
where Eα1···αp(k) is a complex valued antisymmetric tensor and k are discrete vectors of the torus
dual lattice of which the components are measured in units of L−1. Their norm is expressed as
ω(k) =
ki kj δ
ij . Note that the zero modes of the fields are not included in these expressions, as
emphasized by the subscript ⊥. In fact, these zero modes are the global degrees of freedom which
have already been dealt with in the previous Section. The real valued tensors ε
α1···αp
i1···ip
(k) define a
basis of orthonormalised polarisation tensors for each k 6= 0. In our conventions, these tensors are
constructed from a orthonormalised basis of polarisation vectors εαi (k) for a vector field such that
εαi (k) ε
j (k) δ
ij = δαβ , (55)
where δαβ is the Kronecker symbol in polarisation space. This basis is chosen in such a way that,
for each k 6= 0, the dual lattice vector εd(k) is longitudinal whereas the vectors εα(k) are transverse
for α = 1, · · · , d− 1. Finally, it is convenient to choose for the longitudinal vector
εd(k) =
, k 6= 0.
Given the recursion relation induced by the Hodge decomposition theorem, the general polarisation
tensor of any p-tensor field may be expressed as
α1···αp
i1···ip
(k) =
(k) . . . ε
which may likewise be decomposed into transverse and longitudinal components,
Longitudinal :
α1···αp−1 d
i1··· ip−1 ip
α1,...,αp−1=1
; Transverse :
α1···αp
i1···ip
α1,...,αp=1
. (56)
Given any mode, the C
d degrees of freedom of a phase space field then separate into C
d−1 longi-
tudinal and C
d−1 transverse degrees of freedom. These notations having been specified, and using
the decompositions defined in (24), the relevant quantum operators are Fourier expanded as
i1···ip
(~x ) =
k 6=0
δi1j1 . . . δipjp p
αp−1=1
α1···αp−1 d
j1··· jp−1 jp
(k) Ê
α1···αp−1
L (k)
ǫi1···ip j1···jd−p
(d− p− 1)!
αd−p−1=1
α1···αd−p−1 d
j1··· jd−p−1 jd−p
(k) Q̂
α1···αd−p−1
G (k)
e2 i π k(~x),
ip···id−p
(~x ) =
k 6=0
δi1j1 . . . δid−pjd−p (d− p)
αd−p−1=1
α1···αd−p−1 d
j1··· jd−p−1 jd−p
(k) Ĝ
α1···αd−p−1
L (k)
(p − 1)!
ǫj1···jp i1···id−p
αp−1=1
α1···αp−1 d
j1··· jp−1 jp
(k) P̂
α1···αp−1
E (k)
e2 i π k(~x) .
The self-adjoint property of the operator Êi1···ip(~x ) translates into the following relations between
the associated mode operators and their adjoint,
αp−1=1
α1···αp−1 d
j1··· jp−1 jp
(k) Ê
α1···αp−1
L (k) =
αp−1=1
α1···αp−1 d
j1··· jp−1 jp
(−k) Ê† α1···αp−1L (−k),
αd−p−1=1
α1···αd−p−1 d
i1··· id−p−1 id−p
(k) Q̂
α1···αd−p−1
G (k) =
αd−p−1=1
α1···αd−p−1 d
i1··· id−p−1 id−p
(−k) Q̂† α1···αd−p−1G (−k).
Similar relations apply for the modes of the self-adjoint operator Ĝip···id−p(~x ).
Consequently, this decomposition of the non zero modes of the field operators in the dynamical
sector leads to two decoupled subsectors, each of which is comprised of a countable set of mode
operators with k 6= 0. In the first subsector one has the operators ÊL(k) and P̂E(k) with the
following non vanishing commutation relations,
† α1···αp−1
L (k), P̂
β1···βp−1
δα1[β1 . . . δαp−1 βp−1]δkk′ , (57)
while in the second subsector the operators ĜL(k) and Q̂G(k) possess the commutator algebra,
† α1···αd−p−1
L (k), Q̂
β1···βd−p−1
δα1[β1 . . . δαd−p−1 βd−p−1]δkk′ , (58)
V being the volume of the space torus Σ = Td.
This Fourier mode decomposition reduces the problem of diagonalising the Hamiltonian to a
simple exercise in decoupled quantum oscillators, with
Ĥ1[ÊL, P̂E ] =
2 (p − 1)!
k 6=0
κ2 g2
α1···αp−1
E (k)
ω̃2(k)
κ2 g2
α1···αp−1
L (k)
, (59)
Ĥ2[ĜL, Q̂E ] =
V κ2 e2
2 (d − p− 1)!
k 6=0
α1···αd−p−1
G (k)
ω̃2(k)
κ4 e4
α1···αd−p−1
L (k)
. (60)
In these expressions the following notation is being used,
α1···αp−1
L (k)
α1,...,αp−1=1
β1,...,βp−1=1
α1···αp−1
L (k) Ê
† β1···βp−1
L (k) δ
α1β1 . . . δαp−1βp−1 .
The operators (59) and (60) are nothing other than the Hamiltonians of a collection of C
d−1 and
d−1 independent harmonic oscillators, respectively, all of angular frequency
ω̃(k) =
4π2 ω2(k) + µ2, µ = e g κ.
The physical spectrummay easily be constructed by introducing annihilation and creation operators
associated to the algebras (57) and (58). The annihilation operators are defined by
α1···αp−1(k) =
V ω̃(k)
α1···αp−1
L (k) + i
g2 κ2
ω̃(k)
α1···αp−1
E (k)
α1···αd−p−1(k) =
V ω̃(k)
α1···αd−p−1
L (k) + i
κ2 e2
ω̃(k)
α1···αd−p−1
G (k)
whereas the creation operators a† α1···αp−1(k) and b† α1···αd−p−1(k) are merely the adjoint operators
of aα1···αp−1(k) and bα1···αd−p−1(k), respectively. One then establishes the Fock algebras,
α1···αp−1(k), a† β1···βp−1(k′)
= δα1 [β1 . . . δαp−1 βp−1] δkk′ ,
α1···αd−p−1(k), b† β1···βd−p−1(k′)
= δα1 [β1 . . . δαd−p−1 βd−p−1] δkk′ , (61)
whereas (59) and (60) then reduce to the simple expressions,
Ĥ1[a, a
†] = ~
k 6=0
ω̃(k)
d−1 +
α1<···<αp−1
† α1···αp−1(k) aα1···αp−1(k)
Ĥ2[b, b
†] = ~
k 6=0
ω̃(k)
d−1 +
α1<···<αd−p−1
† α1···αd−p−1(k) bα1···αd−p−1(k)
The Fock space representation is based on the normalised Fock vacuum |0〉, 〈0|0〉 = 1, which is the
kernel of all annihilation operators
α1···αp−1(k) |0〉 = 0, bα1···αd−p−1(k) |0〉 = 0, ε1+2
k 6=0
ω̃(k),
where ε1+2
is the divergent total vacuum energy. Excited states are obtained through the action
onto the Fock vacuum of all C
creation operators, see (61). This leads to states
|nγ(k)〉 with energy eigenvalues
ε(nγ(k)) = ε
k 6=0
nγ(k) ω̃(k), (62)
where {nγ(k)}
γ=1 are positive integers corresponding to number operator eigenvalues. A shorthand
notation is used in (62) with the index γ labelling the C
d possible combinations of a set of p distinct
integers in the range [1, d], {α1, . . . , αi, . . . , αp}dαi=1, which will be referred to as Γ
4 Spectrum and Projection onto the TFT Sector
4.1 Physical spectrum on the torus
Combining all the results of the previous Sections for what concerns the diagonalisation of the
physical TMGT Hamiltonian on the spatial d-torus Σ = Td, the complete energy spectrum of
states is given as
ε(nγ(k)) = ε(0) + ~
nγ(k) ω̃(k) , (63)
which is the sum of the contributions (54) and (62). Note that on the d-torus, the pth Betti number,
Np, equals C
. The components of the vector k of the dual lattice may take any integer values since
it is implicit in (63) that {nγ(0)}
γ=1 = {nγ}
γ=1. However, the index γ has a different meaning
whether k 6= 0 or k = 0. In the first case it refers to a value in the set Γpd and denotes one of the
possible C
d polarisations, while in the second case it is a (co)homology index, γ = 1, · · · , C
d . The
total vacuum energy ε(0) in (63) is divergent,
ε(0) =
ω̃(k) ,
and must be subtracted from the energy spectrum.
The positive integer valued functions nγ(k) count, for each k 6= 0, the number of massive
quanta of a p- or (d− p)-tensor field of momentum 2π~ k, of polarisation (56), namely
Transverse : ε
i1···ip
(k), γ ∈ Γp
; Longitudinal : ε
i1···ip
(k), γ ∈ Γp−1
and of rest mass8
M = ~µ = ~κ e g. (64)
There are also the contributions of the global quanta of the p- and (d − p)-tensor fields, where
{nγ(0)}
γ=1 count the numbers of excitations along the homology cycle generators Σ
and Σ
(d−p)
8A quantity indeed positive under the assumptions made.
In the particular case when p = 1, the integers {nγ(k)}dγ=1 count, for each k 6= 0, the number of
massive photons of momentum 2π~ k, of rest mass M and of polarisation
Transverse : {εγi (k)}
, Longitudinal : εdi (k) .
Depending on how one deals with large gauge transformations in the TFT sector, each energy
state is either infinitely degenerate for a real valued k, see (46), or (
δ=1 k1 k2 IMin(Iδδ′ )) times
degenerate if k is a rational number of the form k = k1/k2. If k is an integer, each energy state is
k IMin(Iδδ′)) times degenerate and the mass gap is then quantised,
I k e g .
In the Maxwell-Chern-Simons (MCS) case in 2 + 1 dimensions, we recover in the global sector a
quantum mechanical system corresponding to the Landau problem of condensed matter physics on
the 2-torus.
4.2 Projection onto the topological field theory sector
At least formally, the naive limit of infinite coupling constants, e → ∞ and g → ∞, in the classical
Lagrangian of topologically massive gauge theories, see (4) and (5), must lead to a pure topological
field theory (TFT) of the B ∧ F or A ∧ F type, see (1) or (3). However, as pointed out by several
authors (see for example [19, 22]), a paradox seems to arise at the quantum level (as well as within
the classical Hamiltonian formulation) when the pure Chern-Simons (CS) theory is viewed as the
limit e → ∞ of the Maxwell-Chern-Simons (MCS) theory. The Hilbert space of the CS theory is
constructed from the algebra of the non commuting configuration space operators which are in fact
canonically conjugate phase space operators. As far as the MCS theory is concerned, its Hilbert
space is constructed from the Heisenberg algebras of twice as many phase space operators. This
problem is generic whenever a pure quantum TFT (TQFT) is considered as the limit of its associated
TMGT because two distinct Hilbert spaces are being compared. Actually, due to the second-class
constraints appearing in the Hamiltonian analysis of a TFT which is already in Hamiltonian form,
non vanishing commutation relations apply to the configuration space operators. Furthermore, the
Gauss law constraints of pure TFT are not the limit of the Gauss law constraints of TMGT. The
former operators tend to restrict too drastically the physical Hilbert space in comparison to the
limit of the TMGT physical Hilbert space.
This problem of an ill-defined limit is usually handled by projecting from the Hilbert space of
the TMGT onto its degenerate ground state. This projection actually acts in a manner similar to
second-class constraints which then lead to a reduced phase space and non vanishing configuration
space commutation relations determined from the associated Dirac brackets. The global sector of
the MCS theory is analogous to the classical Landau problem of a charged point particle of mass
m moving in a two dimensional surface in the presence of an uniform external magnetic field B
perpendicular to that surface. Within the latter context the mass gap (64) corresponds to the
cyclotron frequency ωc [22, 23]. The spectrum of the quantised model is organised into Landau
levels (with a degeneracy dependent on the underlying manifold), of which the energy separation
ωc is proportional to the ratio B/m. The limit B → ∞ or m → 0 effectively projects onto the
lowest Landau level (LLL) in which one obtains a non commuting algebra for the space coordinates.
By analogy, projection onto the ground state reduces the phase space of the MCS theory to the
canonically conjugate configuration space operators of a pure CS theory. In the global sector, the
projection from a general TMGT, (4), to a pure TQFT offers in some sense a generalisation of
the LLL projection in any dimension. Considering that the mass gap (64) of the TMGT becomes
infinite for coupling constants running to infinity, all excited states decouple from the physical
spectrum, leaving over only the degenerate ground states. Projection onto these ground states
restricts the Hilbert space to that of a TQFT.
Interestingly, the PT factorisation established in this work enables the usual projection from
TMGT to TQFT to be defined in a natural way. Already in the classical Hamiltonian formulation
phase space is separated into two decoupled sectors, the first being dynamical and manifestly gauge
invariant, and the second being equivalent to a pure TFT with identical Gauss law constraints and
commutation relations. As a matter of fact in the present approach which does not require any
gauge fixing procedure whatsoever, the non commuting sector of a CS theory or, more generally,
the reduced phase space of a TQFT appears no longer after the projection onto the ground state
at the quantum level (or after the introduction of Dirac brackets) but is manifest already at the
classical Hamiltonian level. By letting e or g grow infinite, the mass gap (64) becomes infinite,
hence dynamical massive excitations decouple whereas the TFT sector, which is independent of the
coupling constants, remains unaffected. In this limit, the system looses any dynamics, the latter
being intimately related to the Riemannian metric structure of the spacetime manifold, while all
that is then left is a wave function depending on global variables only, namely the quantum states
of a TQFT.
5 Conclusion and Outlook
The main result of this paper is the identification of a Physical-Topological (PT) factorisation of
the classical phase space of abelian topologically massive gauge theories (TMGT) in any dimension,
into a manifestly gauge invariant and dynamical sector of non commuting “electric fields” and a
gauge variant purely topological sector of the B ∧ F or A ∧ F type. This factorisation is achieved
through a canonical transformation in the phase space of TMGT. The discussion considers the most
general action for abelian TMGT in any dimension and for any p-form fields, including the two
possible types of topological terms related through an integration by parts. The clue to this PT
factorisation relies on the identification of a topological field theory embedded in the full TMGT,
which is not manifest in the original Hamiltonian formulation. Let us emphasize that the procedure
does not require any gauge fixing choice whatsoever, with its cortege of second-class constraints or
ghost degrees of freedom. Rather, the PT classical factorisation readily allows for a straightforward
quantisation of these systems and the identification of their spectrum of gauge invariant physical
states, accounting also for all the topological features inherent to such dynamics.
In the early 1990’s, A. P. Balachandran and P. Teotonio-Sobrinho [24] established that it is
possible to identify among the phase space variables of a TMGT combinations corresponding to
those of a TFT. They noticed that, in a particular case of an underlying manifold with boundary,
edge states may be understood in terms of a TFT, already at the classical level. Nevertheless, this
paper did not realise the powerful gauge fixing free factorisation of the theory into two decoupled
sectors as described in the present work. Incidentally, it should be of interest to analyse how this
new approach may shed new light onto this paper, in a manner akin to that in which it makes most
transparent and natural the projection onto a topological field theory through the limits e, g → ∞,
generalising the concept of projection onto the lowest Landau level of the Landau problem. In the
present approach, the TFT sector with its reduced phase space is actually made manifest already at
the classical level, independently of any projection onto the quantum ground state, or any classical
projection onto physical edge states in the case of a manifold with boundary.
The formalism of TMGT defined by the actions in (4) or (5) offers a possible description
of some phenomenological phenomena such as effective superconductivity [24, 25], Josephson ar-
rays [26], etc. Furthermore, it is certainly of interest to investigate the perspectives offered by the
PT factorisation when a B ∧F or A∧F field theory is coupled to (non)relativistic matter fields as
an effective description of phenomena related, for example, to QCD confinement [27]. It is also of
interest to extend this approach to Yang-Mills-Chern-Simons theories [15, 23] or to the nonabelian
generalisation of the Cremmer-Scherk theory which requires the introduction of extra fields or to
allow for non renormalisable couplings since the generalisation to a local, power counting renor-
malisable action while preserving the same field content and the same number of local symmetries
as the abelian theory (4) is not possible (see [28] and references therein). However the long term
goal is to gain a deeper understanding of the influence of topological terms, such as the topological
mass gap, and of topological sectors in field configuration space on the nonperturbative dynamics of
gauge theories, beginning with Yang-Mills theories coupled to whether fermionic or bosonic matter
fields.
Acknowledgments
The work of B. B. is supported by a Ph.D. Fellowship of the “Fonds pour la formation à la Recherche
dans l’Industrie et dans l’Agriculture” (FRIA), of the Associated Funds of the National Fund for
Scientific Research (F.N.R.S., Belgium).
J. G. acknowledges the Institute of Theoretical Physics for an Invited Research Staff position
at the University of Stellenbosch (Republic of South Africa). He is most grateful to Profs. Hendrik
Geyer and Frederik Scholtz, and the School of Physics for their warm and generous hospitality
during his sabbatical leave, and for financial support. His stay in South Africa is also supported in
part by the Belgian National Fund for Scientific Research (F.N.R.S.) through a travel grant.
J. G. acknowledges the Abdus Salam International Centre for Theoretical Physics (ICTP,
Trieste, Italy) Visiting Scholar Programme in support of a Visiting Professorship at the ICMPA.
This work is also supported by the Institut Interuniversitaire des Sciences Nucléaires and by
the Belgian Federal Office for Scientific, Technical and Cultural Affairs through the Interuniversity
Attraction Poles (IAP) P5/27 and P6/11.
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	Introduction
	Gauge Invariant Factorisation of the Classical Theory
	Hamiltonian formulation
	The Physical-Topological (PT) factorisation
	Hodge decomposition
	Large and small gauge transformations
	Canonical Quantisation and Physical States
	Physical Hilbert space factorisation
	The topological sector: Gauss' constraints and LGT
	Hilbert space and holomorphic polarisation
	Gauss law constraints
	LGT and global variables
	The dynamical sector: Hamiltonian diagonalisation
	Global degrees of freedom
	Local degrees of freedom on the torus
	Spectrum and Projection onto the TFT Sector
	Physical spectrum on the torus
	Projection onto the topological field theory sector
	Conclusion and Outlook
