[math-ph]
Axial Symmetry, Anti-BRST Invariance, and Modified Anomalies
arXiv:1011.1095v3 [hep-th] 27 Aug 2014
Amir Abbass Varshovi∗
School of Mathematics,
Institute for Research in Fundamental Sciences (IPM).
School of Physics,
Institute for Research in Fundamental Sciences (IPM).
Tehran-IRAN
Abstract: It is shown that anti-BRST symmetry is the quantized counterpart of local
axial symmetry in gauge theories. An extended form of descent equations is worked out
which yields a set of modified consistent anomalies.
I.
INTRODUCTION
There are intimate correlations between physical objects in the quantum theory of gauge fields,
such as anomalies, (anti-) BRST symmetry, (anti-) ghosts, ..., and geometric-topological concepts in
differential geometry, including Chern characters, index theorem, principal bundles, and topological
invariants, which have been extensively studied in last few decades with spectacular results and vast
amount of applications [1]. Specially, anomaly of axial symmetry at quantum levels was one of the
first significant gates toward illustration of such topological features inside a physical gauge theory,
the quantum electro-dynamics, as an Abelian model for gauge theories.
Historically, soon after the pioneering articles of Adler, Bell and Jackiw [2, 3] on the anomalous
axial Ward identity for U (1)-gauge theory, Bardeen [4] showed that for more general theories a generalized version of anomalous term exists. More precisely, Bardeen proved that the root of the anomalous
behavior is not entirely confined in axial currents, but in fact, it is basically hidden in the renormalization of the theory. In the other words, the anomalous terms come from those Feynman diagrams
with few enough vertices which diverge linearly at the least. But, however, despite of this primitive
topological dependence, the critical topological meaning of anomalous Ward identity was still not
clear. Bardeen also showed that the scalar and the pseudo-scalar fields do not appear in the anomalous terms after renormalizing the theory, and therefore, the anomaly merely depends upon the gauge
fields, or more precisely upon the curvature tensor, provided by gauge invariance of observables in
gauge theories. It was also known that the whole anomaly can never completely be removed by using
different forms of counter terms, the fact of which could be reported as an evidence for topological
structure of anomalous Ward identities.
The consistency condition discovered by Wess and Zumino [5] was essentially an alternative criterion
for gauge invariance of anomalous terms. The Bardeen’s results were also confirmed by the consistency
∗
Electronic address:
[email protected]
condition [5], which was then translated to topological structure for consistent anomalies as a nontrivial cohomology class of gauge transformation group.
Fujikawa was essentially the first one who shed the light on relationship between anomalies and
topology [6-9]. He precisely proved that the anomalies are basically rooted in the topology of principal
bundle of which the gauge theory is defined over. Specially, Fujikawa showed that the reason of axial
anomaly lies in non-equality of dimensions of kernels of left- and right-handed Dirac operators as
two mutually adjoint differential operators. This non-equality could be reported as breaking the
classical axial symmetry at quantum levels, appearing in variation of the path integral measure of
fermionic partition function against local axial transformations [6]. Indeed, Fujikawa rigorously proved
that anomalies are intimately related to the celebrated Atiyah-Singer index theorem, which basically
counts the analytic index of an elliptic pseudo-differential operator over a vector bundle (such as
the Dirac operator on the spinor bundle) with integrating a specific topological differential form,
a characteristic class, over the base manifold, substantiating an equivalence between analytic and
topological structures [10, 11]. Particularly, Fujikawa demonstrated that the nth Chern class of 2ndimensional base manifold, the space-time, describes the topological structure of axial anomalous
Ward identity through the index theorem. By this a fascinating feature for anomalous behaviors of
gauge theories was worked out via spectacular achievements of modern differential geometry.
On the other hand, the Faddeev-Popov path integral quantization method [12] for non-Abelian
gauge theories led to an alternative formulation of classical gauge symmetry, the BRST symmetry,
which was basically reflected from differential structures of the connection space over the principal
bundle of the gauge theory [13, 14]. Indeed, the Faddeev-Popov quantization method replaced the
classical gauge symmetry by its quantized version, the BRST invariance. In this formulation two
kind of unphysical fields, ghost and anti-ghost, emerged in the quantum Lagrangian within the path
integral setting together with Fermi-Dirac statistics and scalar behaviors (e.g.; spinless fermions).
Although there was no expectation to see more symmetries after quantizing the gauge theories,
but anti-BRST, another quantized symmetry of quantum Lagrangian [15, 16], emerged through the
Faddeev-Popov quantization approach without any classical counterpart. It was also shown that a deep
duality exists between BRST and anti-BRST symmetries and respectively for ghost and anti-ghost
fields [17], but the precise correlation was still unclear geometrically.
After Stora and Zumino [18-20] discovered a sequence of descent equations which could relate the
consistent anomalies and consistent Schwinger terms [21] through a set of deRham-BRST differential
equations, Faddeev showed that this new point of view could be explained in the setting of cohomology algebras [22], and in precise, of the cohomology of gauge group, which was proven with details
by Zumino [23]. Then, some rigorous geometric models for Faddeev-Popov quantization method,
ghosts, BRST transformations and BRST cohomology were constructed to explain the path integral
quantization of gauge theories in the setting of differential geometry [24-32] (for more collected and
recent works see [33-36]). These geometric formulations led to elegant topological descriptions of consistent anomalies and consistent Schwinger terms [37-39]. In this geometric point of view, ghosts are
left invariant operator valued 1-forms over the gauge transformation group and can be described by
principal connections over the principal bundle of the moduli space of connections.
The geometric procedure produces an alternative setting for Faddeev-Popov path integral quantization method. Indeed, this formulation can be considered as the geometric version of BRST quan-
tization of gauge theories [40-44]. More precisely, although the structure of BRST invariance made
its first appearance as an accidental symmetry of the quantum action, but it became apparent that
it takes its most sophisticated setting in the Hamiltonian formulation of gauge theories, where it is
used in the homological reduction of Poisson algebra of smooth functions on a symplectic manifold on
which a set of first class constraints is defined. This constructive formulation lead to more drastic and
deeper understanding of ghost fields, gauge fixing, and eventually BRST transformations in terms of
symplectic geometry. Such geometric method of quantization of gauge theories based on symplectic
data is conventionally referred to as geometric BRST quantization [45-47].
In this article the main geometric ideas of BRST quantization is applied for studying some general
version of gauge theories which admit the local axial symmetry. It is shown that because of enlargement
of the gauge transformation group, the algebra of ghosts and BRST transformations get larger. Then,
it is seen that the behaviors of new geometric objects are in complete agreement with those of antighosts and anti-BRST transformations in the standard fashions [15, 16]. Indeed, a generalized version
of BRST and anti-BRST transformations is found by considering the local axial symmetry. Strictly
speaking, this consideration leads to an enlargement of the algebra generated by gauge fields, ghosts
and anti-ghosts, by introducing the axial gauge fields to Yang-Mills gauge theories. It is shown that
despite of the standard BRST algebra generated by gauge, ghost and anti-ghost fields, this enlarged
algebra is closed under BRST and anti-BRST transformations [17]. The necessity of Nakanishi-Lautrup
fields [48, 49] is removed here and these auxiliary fields are replaced by a set of functionals of ghosts
and anti-ghosts. This functional can be considered as a colored scalar field with null ghost-number.
On the other hand, the extra degrees of freedom carried by auxiliary fields are compensated by axial
gauge fields in the extended Lagrangian. In fact, this development gives a conceptual framework and
a geometric formalism to algebraic anti-BRST transformations and anti-ghosts. This also leads to a
geometric description of extended BRST quantization of gauge theories [50, 51].
Using these extended BRST transformations via the approach of [18-20], an extended sequence
of descent equations and consequently a modified version of consistent anomaly and of consistent
Schwinger term is found. According to [4], none of vector and axial currents is the only responsible for
axial anomaly. Particularly, the anomalous behavior of the theory should be studied by considering
the contribution of vector and axial currents simultaneously. Thus, it seems more natural to consider
modified consistent anomalies and modified consistent Schwinger terms as anomalous behaviors of
gauge theories.
In section 1, Axial Extension of a Gauge Theory, the idea of axially extended gauge theories is
studied in the setting of differential geometry. In second 2, Extended BRST Transformations, a
summarized review over geometric ghosts and BRST transformations is given and then the geometrical
aspects of anti-ghosts and anti-BRST transformations are worked out. Finally, in section 3, Extended
Descent Equations, the modified version of anomalous terms is derived and studied properly.
In fact, in this article it is shown that the following diagram commutes for gauge theories;
Quantization
−→
Classical Gauge Symmetry
Quantum BRST Symmetry
Axial Extension ↓
↓ Axial Extension
Classical Local Axial Symmetry
(I.1)
Quantum Anti-BRST Symmetry
−→
Quantization
Although it was thought that quantization causes the classical gauge symmetry to produce two
different quantum symmetries [17], due to BRST and anti-BRST transformations, in this article it
is shown the anti-BRST invariance is the quantized version of classical local axial symmetry, which
is broken in standard Yang-Mills theories. This broken symmetry revives through quantization and
produces the anti-BRST invariance.
II.
AXIAL EXTENSION OF A GAUGE THEORY
Extending a renormalizable field theory is the process of adding a number of new renormalizable
terms to the Lagrangian density in order to enlarge its symmetry group. This process may or may not
add a number of new fields to the theory. It can be seen that this can replace a global symmetry by a
local one. From the perturbation theory point of view this may result in appearance of a number of
new Feynman diagrams which will affect the renormalization process and consequently the anomalous
behaviors of the theory. In this section, we study the axial extension of gauge theories in a geometric
framework. By axially extension we mean enlarging a Yang-Mills gauge theory to produce an axial
gauge theory which admits local axial symmetry.
A.
Axially Extended Gauge Theory
The main goal of axial extension of a gauge theory is to add a number of new renormalizable
terms to the Lagrangian density in order to make the axial symmetry appear in the local form. Thus,
generally consider a Yang-Mills gauge theory over R4 by;
L = LY ang−M ills + LDirac + Aaµ ψγ µ T a ψ ,
(II.1)
with gauge group G and the gauge transformation group G. Under the action of an element e−iα(x) ∈ G
with;
ψ ′ (x) = e−iα(x) ψ(x)
−iα(x)
A′µ (x) = e−iα(x) i∂µ eiα(x) + Ade
(Aµ (x))
(II.2)
L remains invariant. But the action of axial transformation e−iα(x)γ5 on L yields a nontrivial variation; δL = e−iα(x)γ5 i∂µ eiα(x)γ5 , which cannot be compensated by variations of gauge fields Aaµ (x).
Essentially, this is the reason that axial transformations are global symmetries; there is no geometric
structure for such transformations. To overcome this difficulty one should introduce a number of new
axial gauge fields to capture the variations of L under axial transformations. To this end, a set of new
terms, ψγ µ γ5 T a ψ, each of which coupled to an axial gauge field, say Bµa , should be added to L. The
resulting theory takes the form of;
Lex = LGauge + LDirac + Aaµ ψγ µ T a ψ + Bµa ψγ µ γ5 T a ψ ,
(II.3)
where LGauge is a functional of Aaµ (x) and Bµa (x) which will be determined in the following. It can be
easily checked that the invariance of Lex − LGauge under gauge transformation e−iα(x) requires (II.2)
together with;
Bµ′ (x) = Ade−iα(x) (Bµ (x)) ,
(II.4)
where Bµ (x) = Bµa (x)γ5 T a . On the other hand, the invariance of Lex − LGauge under axial transformation e−iα(x)γ5 requires;
ψ ′ (x) = e−iα(x)γ5 ψ(x) ,
A′µ (x) + Bµ′ (x) = e−iα(x)γ5 i∂µ eiα(x)γ5 + Ade−iα(x)γ5 (Aµ (x) + Bµ (x)) .
(II.5)
Notice that the axial transformation mixes the vector and axial gauge fields. This is a crucial fact
which is discussed in the following. To find a gauge/axial invariant Lagrangian density Lex , one should
look for a gauge/axial invariant LGauge . The second part of (II.5) forces LGauge to be a functional of a
mixed form of Aµ (x) and Bµ (x). Moreover, according to (II.2), (II.4) and (II.5), the gauge and axial
transformations are respectively given by;
(Aµ + Bµ )′ (x) = e−iα(x) i∂µ eiα(x) + Ade−iα(x) (Aµ (x) + Bµ (x)) ,
(Aµ + Bµ )′ (x) = e−iα(x)γ5 i∂µ eiα(x)γ5 + Ade−iα(x)γ5 (Aµ (x) + Bµ (x)) .
(II.6)
Both of these transformations are completely similar to the gauge field part of (II.2) which keeps the
pure Yang-Mills Lagrangian density invariant. Thus, if one defines;
Fµν = ∂µ (Aν + Bν ) − ∂ν (Aµ + Bµ ) − i[(Aµ + Bµ ), (Aν + Bν )] ,
(II.7)
the Yang-Mills Lagrangian density LY ang−M ills = − 14 T r{Fµν F µν } would be a compatible candidate
for LGauge . Here we use the normalized trace with tr 1 = 1 for 1 = γ52 , the 4 × 4 identity matrix
acting on spinors. Note that although the trace of γ5 vanishes the Lagrangian density cannot be split
into two different parts, each of which a functional of Aµ or Bµ . We emphasize that in the pure
Yang-Mills Lagrangian density, LY ang−M ills , the vector and the axial gauge fields, Aµ and Bµ , are
both considered conventionally in the fundamental representation of the gauge group, independent to
their representations on spinors which we show simply with T a matrices.
Take the infinitesimal gauge transformation of e−tiα ∈ G, t ∈ R;
∆α Aµ :=
d
−itα(x) i∂ eitα(x) + Ad
µ
e−itα(x) (Aµ (x))) = −∂µ α
dt |t=0 e
d
∆α Bµ := dt |t=0 Ade−itα(x) (Bµ (x)) = i[Bµ , α] ,
+ i[Aµ , α] ,
(II.8)
and the infinitesimal axial transformation of e−tiαγ5 , t ∈ R;
∆′α (Aµ + Bµ ) :=
d
|t=0 e−itα(x)γ5 i∂µ eitα(x)γ5 + Ade−itα(x)γ5 (Aµ (x) + Bµ (x)) .
dt
(II.9)
Indeed, it can be seen that the infinitesimal axial transformations of gauge and axial gauge fields
are given by;
∆′α Aµ = i[Bµ , αγ5 ] ,
∆′α Bµ = −∂µ αγ5 + i[Aµ , αγ5 ] .
(II.10)
If one defines the vector and axial gauge fields as operator valued 1-forms, then the gauge and axial
transformations of A and B are respectively given by the following forms;
∆α A = −dα + i[A, α] ,
∆α B = i[B, α] ,
(II.11)
∆′α A = i[B, αγ5 ] ,
∆′α B = −dαγ5 + i[A, αγ5 ] .
(II.12)
and;
Therefore, an extended Lagrangian density, Lex , is found properly. Moreover, as we expected before, Lex not only contains L (the original theory) but it admits the axial symmetry in the local form.
Indeed, up to anomalous behaviors, Lex is a renormalizable theory by power counting. Note that from
the perturbation theory point of view the fact of which Lex contains L, results in the appearance of
all Feynman diagrams of the original theory in the perturbative calculations of the extended theory.
Thus L may be considered as a sub-theory for Lex . In the following, the next subsection, it is shown
that the extended Lagrangian, Lex , can be considered as a gauge theory with an enlarged gauge group.
B.
Extended Gauge Transformation Group
As it was stated above, an axially extended gauge theory admits the local axial symmetry, but it
has not shown yet that geometrically the extended Lagrangian density is also a gauge theory by itself.
To show this fact the geometric theory of gauge fields is revisited now. Let M be an even dimensional
spin manifold and G a semi-simple (or compact) Lie group with Lie algebra g. Consider a principal
G-bundle over M ;
G ֒→ P ։ M ,
(II.13)
and set a connection over P with the Cartan connection form π [52]. Now suppose that V is a
complex irreducible representing space of G. Also consider a Hermitian inner product for V , say h, i,
and assume that the action of G is unitary. The representation of G on V defines a complex vector
bundle over M with;
V ֒→ E := P ×G V ։ M ,
(II.14)
where;
P ×G V := {[(p, v)]|(p, v) ∈ P × V and (p, v) ∼ (p ⊳ g, g −1 ⊲ v)} .
(II.15)
It can easily be checked that E ։ M is a vector bundle and any one of its sections can be represented
by an equivalence class, say [(s, ξ)], for some s ∈ Γ(P ) and ξ ∈ C ∞ (M, V ). Clearly, by definition
(II.15), the choices of s and ξ are not unique. As it will be discussed in the following, these different
choices produce the gauge symmetry. On the other hand, the connection of P induces a connection
over E with covariant derivative;
∇X [(s, ξ)] := [(s, dξ(X))] + [(s, π(ds(X)) ⊲ ξ)] ,
(II.16)
where π(ds(X)) ⊲ ξ is the action of g, induced by the representation of G (for any T ∈ g and v ∈ V ,
d
|t=0 etT ⊲ v). The definition (II.16) is independent of the choices s and ξ. Moreover,
define; T ⊲ v := dt
the pull back of π through s, s∗ (π), defines an operator valued 1-form over M . Conventionally, s∗ (π)
is called the connection form. Basically these operator valued 1-forms play the roles of gauge fields in
the context of Yang-Mills gauge theories. To see this note that;
(s ⊲ g)∗ (π) = g−1 dg + Adg−1 (s∗ (π)) ,
(II.17)
which is thoroughly similar to the gauge field part of (II.2) with replacing s∗ (π) (resp. (s ⊲ g)∗ (π))
with; −iA = −iAµ dxµ (resp. −iA′ = −iA′µ dxµ ).
Now consider the tensor product bundle;
S ⊗ V ֒→ S(M ) ⊗ E ։ M
(II.18)
for the spin bundle S(M ) ։ M and its standard fiber S. The definition (II.16) and the connection of
the spin bundle naturally induce a covariant derivative, ∇⊗ , over the tensor product bundle (II.18).
Thus if D is the Dirac operator of ∇⊗ , then the Lagrangian density of matter is given by;
LM atter (ψ, D) = hψ, Dψi
(II.19)
for Γ(S(M ) ⊗ E). According to definitions (II.15) and (II.16), it can be seen that LM atter (ψ, D) is a
gauge invariant functional. The other gauge invariant functional on 4-dimensional M with Minkowsian
signature is the Yang-Mills lagrangian density;
1
LY ang−M ills (∇⊗ ) := T r{R ∧ ⋆R} ,
2
(II.20)
with R the curvature tensor of ∇⊗ and with ⋆ the Hodge star operator. When M = R4 , the spin
bundle is trivial and thus it is seen from (II.16) that; R = ds∗ (π) + s∗ (π)2 , where here, as mentioned
above, R is written in the fundamental representation. Usually, LY ang−M ills is defined by referring to
F = 2iR. Thus, one may conventionally define; LY ang−M ills (∇⊗ ) = − 18 T r{F ∧ ⋆F } 1 .
1
Here we recall that the Hodge dual
√ of a differential m-form over a pseudo-Riemannian (m + n)-dimensional manifold,
| det g|
ǫµ1 ,...,µm ,ν1 ,...,νn ω µ1 ,...,µm dxν1 ∧ ... ∧ dxνn , for ω a differential m-form locally
locally is defined with; ⋆ω :=
n!
µ1
µm
given by ωµ1 ,...,µm dx ∧ ... ∧ dx , over local coordinates {xµ }m+n
µ=1 , for totally anti-symmetric components ωµ1 ,...,µm ,
1 ≤ µ1 , ..., µm ≤ m + n. Therefore, − 81 T r{F ∧ ⋆F } is essentially equal to − 41 T r{Fµν F µν }
Now the axially extended gauge theories can be studied by these geometric structures. First, note
that for any given d-dimensional Lie algebra g there exists a natural way to define a 2d-dimensional
Lie algebra g̃ which contains g as a Lie sub-algebra. To see this, chose a basis for g, say {ta }da=1 , and
define a set of new elements {sa }da=1 together with the following commutation relations;
[sa , sb ] = [ta , tb ] = C abc tc
,
[sa , tb ] = −[tb , sa ] = C abc sc .
(II.21)
It can be checked easily that the Jacobi identity is satisfied by the relations of (II.21) and thus they
form a Lie algebra, g̃. Indeed g̃ can be considered as an extension of g. However, it is clear that this
extension is well-defined, unique, and natural; i.e. if g̃′ is constructed over g for another basis, say
{t′a }da=1 , then g̃′ ∼
= g̃.
It is clear that for a given representation of g, say φ, and for a nontrivial involutive matrix γ which
commutes with all φ(ta )s, one can produce a representation of g̃, say φ̃. It is enough to set φ̃(sa ) =
γφ(ta ). This can be done with γ = γ5 for any representation of the Lie algebra g on the spinors
over an even dimensional spin manifold. Thus, we refer to the extension procedure of (II.21) as axial
extension of a Lie algebra.
As it was stated above, g̃ is itself a Lie algebra and thus there exists a simply connected Lie group
G̃ with LieG̃ = g̃. Since g̃ contains g as a Lie sub-algebra, then there is a closed Lie subgroup of G̃,
with Lie algebra g. Actually, G̃ contains a simply connected Lie subgroup, say G′ , with LieG′ = g.
Indeed, G̃ ∼
= G′ × G′ . To see this note that {taR := 12 (ta + sa ), taL := 21 (ta − sa )}da=1 , forms a basis for g̃
with the following Lie brackets;
[taR , tbR ] = C abc tcR , [taL , tbL ] = C abc tcL , [taL , tbR ] = 0 .
(II.22)
This shows that g̃ can be decomposed into the direct sum of two copies of g, g̃ = g ⊕ g. On the other
hand, since π1 (G′ × G′ ) = π1 (G′ ) × π1 (G′ ), it follows that G̃ = G′ × G′ , and thus the axial extension of
any Lie algebra induces an extension of its simply connected Lie group. Therefore, the axial extension
of su(N ) will induce an extension of SU (N ). Hence, the axial extension of a SU (N )-gauge theory
leads to a SU (N )L × SU (N )R -gauge theory.
Generally, for simply connected G and G̃ the inclusion of i : g ֒→ g̃ defines a natural inclusion of G
into G̃ denoted by I : G ֒→ G̃, with dI = i [53]. Thus, if P ։ M and P̃ ։ M are principal bundles
over M for simply connected Lie groups G and G̃ respectively, and if ξ : P → P̃ is a principal bundle
homomorphism which by definition makes the following diagram commutative
P ×G
→
P ։
M
ξ ↓
↓ I
ξ ↓
= ↓ ,
P̃ × G̃
→
P̃ ։
M
(II.23)
then a natural reduction of G̃-gauge theories to G-gauge theories over M is defined through ξ. It is
enough to set the Cartan connection form over P ։ M , the pull back of that over P̃ ։ M , say π̃,
through ξ, i.e. π := ξ ∗ (π̃). Such a reduction procedure is always possible for simply connected M , e.g.
M = R4 . In this sense, the reduction procedure can be considered as the inverse project of extension
gauge theories.
As it was stated above, one can consider {T a , T a γ5 }da=1 as a representing basis of the extended
Lie algebra g over the spinors. With this notation, (II.21) also confirms the infinitesimal gauge and
axial transformations of (II.11) and (II.12). More precisely, (II.3) defines an ordinary G̃-gauge theory.
Consequently, the gauge and axial transformations for L are essentially the gauge transformations
for Lex , the extended Lagrangian density. These transformations are conventionally referred to as
extended gauge transformations.
The relations of (II.22) assert that one can consider a mixed form of A and B to define a number
of new gauge fields which do not mix under extended gauge transformations. To this end set;
AR = (
A+B
)PR
2
,
AL = (
A+B
)PL ,
2
(II.24)
5
5
where PR = 1+γ
and PL = 1−γ
are respectively the projections onto right and left handed Weyl
2
2
spinors due to chiral fermions. Thus the infinitesimal transformations take the following form;
1
∆R
α AR = − 2 dαR + i[AR , αR ] ,
1
∆L
α AL = − 2 dαL + i[AL , αL ] ,
L
∆R
α AL = ∆α AR = 0 ,
(II.25)
for ∆R (resp. ∆L ), the right- (resp. left-) handed infinitesimal chiral transformation. This splits the
gauge fields into right- and left-handed components.
III.
EXTENDED BRST TRANSFORMATIONS
It is well-known that the BRST invariance is the quantized version of the classical gauge symmetry
in the context of quantum Yang-Mills gauge theories [12-14]. Actually, the BRST invariance can
reproduce all the information of gauge symmetries within the quantized formalism. In this section
the geometric structure of anti-BRST transformations and consequently of anti-ghost is given by
using the geometric ideas of BRST quantization. In the first subsection, a review over the geometrical
theory of ghosts and BRST transformations is given.
A.
Ghosts and BRST transformations, a Geometric Approach
The geometric description of Fddeev-Popov quantization, not only illustrates the canonical structures of ghost fields and BRST transformations, but it results in simple proofs for their substantial
behaviors [37-39]. To see this, assume that M , the space-time, is an even dimensional spin manifold.
Consider the principal G-bundle of (II.13) and the Cartan connection form π over it. The space of all
connection forms is an Affine space, denoted by A. The gauge transformation defines a right action
of the infinite dimensional Lie group C ∞ (M, G) on A. For example the action of g ∈ C ∞ (M, G) on
A ∈ A is given by;
A ⊳ g = ig−1 dg + Adg−1 (A) .
(III.1)
Consider G ∈ C ∞ (M, G) to be the set of base point preserving elements [37]. Then, the free action
of G on A defines a principal G-bundle;
G ֒→ A ։ A/G .
(III.2)
Fix a connection over this principal bundle and denote its Cartan connection form by Π. For a fixed
A0 ∈ A define the fiber map;
iA0 : M × G → M × A ,
iA0 : (m, g) 7→ (m, A0 ⊳ g) .
(III.3)
Define A as a g-valued 1-form over M × A with;
A(vm , ηA ) = A(vm ) − iΠ(ηA )(m) ,
(III.4)
for (vm , ηA ) ∈ Tm M × TA A. Then set A + ω := i∗A0 (A), where A and ω are the pull backs of
the first and the second parts of A respectively. Conventionally ω is called ghost. Indeed, ω is a
C ∞ (M, g)-valued left invariant 1-form over G and then its color components anti-commute with
each other as simple 1-forms. Thus, its color components behave like Grassmannian numbers in
path-integral frameworks as group indexed ghosts do.
Denote the exterior derivative operator over A by dA and define δ to be its pull back through i∗A .
Then a direct calculation gives;
δA = dω − iAω − iωA ,
δω = −iω 2 .
(III.5)
Moreover, since (d + dA )2 = d2A = 0, we find that; δ2 = 0 and δd + dδ = 0, similar to ordinary BRST
derivation. This is the reason for ω and δ to be respectively considered as the standard ghost and
BRST derivative [37]. In fact, for better understanding (III.5), it should be compared with the vector
gauge field part of (II.11). Indeed, any 1-parameter group gauge transformation, say eitα , t ∈ R,
induces a fiber-wise vector field over A, shown by η α , given by
α
ηA
=
d,D Z
X
a,µ=1 x∈M
(∆α A)aµ (x)
δ
δAaµ (x)
∈ TA A ,
(III.6)
for {Aaµ (x)}d,D
a,µ=1,x∈M (d = dim g and D = dim M ) the coordinate functions of A. Moreover, δA in
(III.5), is a differential 2-form on M × G with;
iηα δA = iηα (dω − iAω − iωA) = −dα + i[A, α] = ∆α A ,
(III.7)
for iηα the internal multiplier operator [53]. More generally, infinitesimal gauge transformations are
infinitesimal moves through the fibers of A [37] and therefore, they are smooth sections of tangent
bundle T A. Basically, infinitesimal gauge transformations are fiber-wise vector fields on A. On
the other hand, ghosts are fiber-wise 1-forms over A and thus, they can evaluate the infinitesimal
gauge transformations as their dual objects. In fact, the equivalence of gauge symmetry and BRST
invariance can be illustrated by noting that the former is defined with the elements of T A but the
later one is given in terms of their dual objects in the cotangent bundle, T ∗ A. It is the same idea of
Legendre transformation in classical mechanics which translates the Lagrangian formalism into the
Hamiltonian formulation over a symplectic manifold. This intuition is one of the cornerstones for
geometric approach to BRST quantization.
B.
Anti-Ghosts and Anti-BRST Transformations
As stated above, the extension of gauge group enlarges the algebra of group indexed ghosts. In
fact, in this subsection it is shown that how the anti-ghost fields emerge through the axial extension
of a Yang-Mills gauge theory. Consider A and A5 as the set of all vector and axial connection forms
respectively, and take the space of extended connection forms as the Cartesian product space A × A5.
Then consider g̃, G̃ and G̃ as the axially extended versions of Lie algebra g, gauge group G, and gauge
transformation group G, respectively. Finally, define A and Π similar to (III.4) and let P : g̃ ։ g and
P5 := 1 − P be two projections. The fiber map for a fixed point (A0 , B0 ) ∈ A × A5 is given by;
i(A0 ,B0 ) : M × G̃ → M × A × A5 ,
i(A0 ,B0 ) : (m, g) 7→ (m, (A0 , B0 ) ⊳ g).
(III.8)
Now define A + B + ω + ω ∗ := i∗(A0 ,B0 ) (A), with ω := i∗(A0 ,B0 ) (P (−iΠ)) and ω ∗ := i∗(A0 ,B0 ) (P5 (−iΠ)).
Thus if ω and ω ∗ are the pull backs of the exterior derivative operators on A, dA , and on A5 , dA5 ,
respectively, then it can similarly be shown that2 ;
δA = dω − iAω − iωA ,
δB = −iBω − iωB ,
δω = −iω 2 ,
δω ∗ = −iω ∗ ω − iωω ∗ ,
(III.9)
which is in complete agreement with BRST derivative, replacing the Nakanishi-Lautrup field h [48,
49, 54] by −iω ∗ ω − iωω ∗ . Note that h is a colored scalar field with ghost number zero similar to
−iω ∗ ω − iωω ∗ . On the other hand, as it will be shown in the following δ2 vanishes and this implies
that δ annihilates both h and −iω ∗ ω − iωω ∗ similarly. Thus, −iω ∗ ω − iωω ∗ can be considered as
the Nakanishi-Lautrup field which appears in the path integral quantization formalism. Indeed, it is
seen that the ordinary BRST derivation is not closed for classical fields, and group indexed ghosts
and anti-ghosts [13, 14, 17], but this new fashion of BRST algebra is closed by itself. Actually, the
auxiliary field is replaced by −iω ∗ ω − iωω ∗ in (III.9) which closes the BRST algebra. Moreover,
the degrees of freedom carried by the auxiliary field h are compensated with those due to axial
gauge fields. However, δ is also called the BRST operator while ω and ω ∗ are referred to as ghost
and anti-ghost fields, respectively. This gives an elegant geometric description to group indexed
anti-ghosts in terms of 1-forms dual to infinitesimal axial transformations.
On the other hand, one can similarly show that;
δ∗ A = −iBω ∗ − iω ∗ B ,
δ∗ B = dω ∗ − iAω ∗ − iω ∗ A ,
δ∗ ω = 0 ,
δ∗ ω ∗ = −iω ∗2 .
(III.10)
Actually, since d2A = d2A5 = 0, then δ2 = δ∗2 = 0. According to (III.10), δ∗ gives hand a set of
transformations which are similar to BRST ones and keep the quantum Lagrangian invariant. Clearly
δ∗ is in complete agreement with the algebraic anti-BRST transformation [15-17]. Thus, conventionally
2
One can simply apply the notation of super-commutator for elements of non-trivial degrees such as (operator valued)
differential forms; i.e. [a, b] := ab − (−1)|a||b| ba. Therefore, for example we write; δA = dω − i[A, ω]. Here and in the
following, to have more clear formulas we refuse this super-algebraic notation.
δ∗ is called the anti-BRST derivative. Moreover, from (d + dA )2 = (d + dA5 )2 = (dA + dA5 )2 = 0, one
simply finds;
δd + dδ = δ∗ d + dδ∗ = δδ∗ + δ∗ δ = 0 .
(III.11)
These anti-commutation relations will result in a generalized form of descent equations [18-20] by
using d, δ and δ∗ alternatively. Note that, δ∗ ω is given in term of the auxiliary field in the context of
the standard anti-BRST transformation [15-17], thus, (III.10) equalizes automatically the NakanishiLautrup field and −iω ∗ ω − iωω ∗ , which confirms the last part of (III.9). We refer to the union of
(III.9) and (III.10) as extended BRST transformation or extended BRST derivation. Consequently, it
was shown by (III.9)-(III.11) that the anti-BRST invariance is the quantized counterpart of the local
axial symmetry.
IV.
EXTENDED DESCENT EQUATIONS
In this section the extended BRST transformation is used to generalize the descent equations [1820] in a concrete manner. Indeed, this process leads to a modification of consistent anomalous terms in
gauge theories. In the other words, the enlargement of gauge group yields a set of generalized form of
anomalous terms including consistent anomalies and consistent Schwinger terms via extended BRST
derivative. The result will be delivered in a lattice diagram of differential forms which commutes up
to deRham exact forms.
A.
Analytic BRST and Anti-BRST Transformations
The smooth action of a Lie group G on a given smooth manifold M will induce a set of vector
fields over M . Indeed, if X is a left invariant vector field over G and etX , t ∈ R, is its integral
curve, then for any m ∈ M , γm (t) := m ⊳ etX , is a smooth curve in M . Actually, the collection of
d
all vectors X̃m := dγm ( dt
|t=o ) defines a smooth vector field over M with their integral curves γm ,
m ∈ M . Thus, the action of G̃ on A × A5 can be considered as a collection of fiber-wise smooth vector
fields over A × A5 . To formulate this idea analytically, consider a coordinate system over A × A5 ,
itα
say {Aaµ (x), Bµa (x)}d,D
a,µ=1,x∈M , with d = dim g and D = dim M . Thus, the gauge transformation e
induces a vector field, X α , over A × A5 with;
X Z
δ
δ
α
{(−∂µ αa (x) − C abc αb (x)Acµ (x)) a
+ (−C abc αb (x)Bµc (x)) a
} . (IV.1)
X =
δAµ (x)
δBµ (x)
M
a,b,c,µ
A generalized form of η α in (III.6). Moreover, the axial transformation eitαγ5 defines another vector
field, Y α , over A × A5 with;
X Z
δ
δ
α
Y =
+ (−∂µ αa (x) − C abc αb (x)Acµ (x)) a
} , (IV.2)
{(−C abc αb (x)Bµc (x)) a
δAµ (x)
δBµ (x)
M
a,b,c,µ
where it is supposed that;
δ
Abν (y)
δAa
µ (x)
δ
b
a (x) Bν (y)
δBµ
δ
Bνb (y)
δAa
µ (x)
= δba δνµ δ(D) (x − y) ,
= δba δνµ δ(D) (x − y) ,
=
δ
b
a (x) Aν (y)
δBµ
=0,
(IV.3)
since { δAaδ (x) , δB aδ(x) }d,D
a,µ=1,x∈M form a basis for tangent spaces of A × A5 . Using these notations the
µ
µ
infinitesimal gauge transformations of A0 and B0 as vector and axial gauge fields with respect to eitα ,
i.e. (II.8), are given by;
(∆α A0 )aµ (x) = Aaµ (x)(∆α A0 ) = XAα0 (Aaµ (x)) ,
(∆α B0 )aµ (x) = Bµa (x)(∆α B0 ) = XBα0 (Bµa (x)) .
(IV.4)
provided ∆α A0 and ∆α B0 be considered as two elements of A and A5 respectively. While the infinitesimal axial transformations of A0 and B0 as vector and axial gauge fields with respect to eitαγ5 ,
i.e. (II.10), are given by;
α (Aa (x)) ,
(∆′α A0 )aµ (x) = Aaµ (x)(∆′α A0 ) = YA
µ
0
α
′
a
a
′
(∆α B0 )µ (x) = Bµ (x)(∆α B0 ) = YB0 (Bµa (x)) .
(IV.5)
Here ∆′α A0 and ∆′α B0 are also considered as two elements of A and A5 respectively. In order to
extract the analytic forms of BRST and of anti-BRST operators one should go from the tangent
bundle T (A × A5 ) to its dual T ∗ (A × A5 ) and use the differential form alternatives. In fact, the
Cartan connection form Π, makes it possible to translate the fiber-wise tangential formalisms to the
cotangential ones. According to (IV.1) and (IV.2) one finds;
Z(Γ) = (dA Γ + dA5 Γ)(Z) =
PR
a
a
(δΓ/δAµ (x)){i∂µ P (Π (Z))(x) − iC abc Abµ (x)P (Πc (Z))(x) − iC abc Bµb (x)P5 (Πc (Z))(x)} (IV.6)
P RM
a
a
abc Ab (x)P (Πc (Z))(x) − iC abc B b (x)P (Πc (Z))(x)} ,
+
5
µ
µ
M (δΓ/δBµ (x)){i∂µ P5 (Π (Z))(x) − iC
where the sum is over group and space indices, a, b, c and µ, while Z ∈ spanC {X α , Y β }α,β acts on
Γ ∈ C ∞ (A × A5 ). Consequently, taking the pull back of (IV.6) via the fiber map i(A0 ,B0 ) leads to;
Z(Γ) = (δΓ + δ∗ Γ)(Z) =
R
P
(IV.7)
(δΓ/δAaµ (x)){−∂µ ω a (Z)(x) + C abc Abµ (x)ω c (Z)(x) + C abc Bµb (x)ω ∗c (Z)(x)}
R
Pa,b,c,µ M
+ a,b,c,µ M (δΓ/δBµa (x)){−∂µ ω ∗a (Z)(x) + C abc Abµ (x)ω ∗c (Z)(x) + C abc Bµb (x)ω c (Z)(x)} ,
for Z any arbitrary left invariant vector field on G̃. But since each fiber of A × A5 is precisely
a copy of G̃, then any vertical vector of the tangent space T(A0 ,B0 ) (A × A5 ) is an element of
spanC {X α |(A0 ,B0 ) , Y β |(A0 ,B0 ) }α,β . Thus, it can be seen that,
R
P
a
abc Ab (x)ω c (x) + C abc B b (x)ω ∗c (x)}(δ/δAa (x))
µ
µ
µ
a,b,c,µ M {−∂µ ω (x) + C
R
P
+ a,b,c,µ M {−∂µ ω ∗a (x) + C abc Abµ (x)ω ∗c (x) + C abc Bµb (x)ω c (x)}(δ/δBµa (x)) ,
(IV.8)
is the fiber-wise exterior derivative operator over the total space A×A5 , and equivalently is the analytic
form for extended BRST derivative operator, δex := δ + δ∗ , acting on elements of C ∞ (A × A5 ).
Note that using the same procedure, it is seen that for the ordinary geometric feature of BRST
quantization, group indexed anti-ghosts and axial gauge fileds never appear in (IV.8) to form the
exterior derivative operator. Therefore, for such cases, (IV.8) is replaced by;
X Z
δ
{−∂µ ω a (x) + C abc Abµ (x)ω c (x)}( a
),
(IV.9)
δAµ (x)
M
a,b,c,µ
while the anti-ghost part is exclusively related to the axial extension;
X Z
δ
δ
) + {−∂µ ω ∗a (x) + C abc Abµ (x)ω ∗c (x)}( a
).
{C abc Bµb (x)ω ∗c (x)}( a
δAµ (x)
δBµ (x)
M
(IV.10)
a,b,c,µ
Since each fiber of A × A5 is canonically diffeomorphic with G̃, it is seen that the ghost part of (IV.7)
defines the analytic form of BRST derivative. Specially, for any Γ ∈ C ∞ (A × A5 ) one has,
X Z
δΓ
δΓ
) + {C abc Bµb (x)ω c (x)}( a
) . (IV.11)
δΓ =
{−∂µ ω a (x) + C abc Abµ (x)ω c (x)}( a
δAµ (x)
δBµ (x)
M
a,b,c,µ
Similarly it is seen that (IV.10) defines the analytic form of anti-BRST derivative operator. Thus, for
any Γ ∈ C ∞ (A × A5 ) we have;
X Z
δΓ
δΓ
) + {−∂µ ω ∗a (x) + C abc Abµ (x)ω ∗c (x)}( a
) .(IV.12)
δ∗ Γ =
{C abc Bµb (x)ω ∗c (x)}( a
δA
δB
(x)
M
µ
µ (x)
a,b,c,µ
B.
Modified Anomalies and Schwinger Terms
Obviously, the forms of consistent anomalies and consistent Schwinger terms completely depend
on the analytic form of BRST derivative operator via the descent equations. Thus, it is expected
that when the algebra of ghost fields and eventually the BRST transformations get larger, then the
consistent anomalies and Schwinger terms should be modified properly.
By definition, gauge anomaly is the deviation of a second quantized Yang-Mills theory from its
classical gauge symmetry. It is well-known that, anomalies cannot be canceled out properly by renormalization counter terms while in return, they ruin the renormalizability of the gauge theory [2, 34,
54]. In fact, classically the equations of motion of the Lagrangian density (II.3) show that the vector
and the axial currents, should obey the following conservation laws;
D a J(x) := ∂µ J aµ (x) + C abc Abµ (x)J cµ) (x) + C abc Bµb (x)J5cµ (x) = 0 ,
D5a J(x) := ∂µ J5aµ (x) + C abc Abµ (x)J5cµ (x) + C a bcBµb (x)J cµ (x) = 0 .
(IV.13)
On the other hand, the variation of the quantum action W with respect to Aaµ (x) (resp. Bµa (x)) is
J aµ (x) (resp. J5aµ (x)). Thus, from (IV.11) and (IV.12) we have;
P R
ω a (x)D a J(x) = δW ,
Pa RM ∗a
∗
a
a M ω (x)D5 J(x) = δ W .
(IV.14)
P R
δ( a M ω a (x)D a J(x)) = 0 ,
P R
δ∗ ( a M ω ∗a (x)D5a J(x)) = 0 ,
P R
P R
δ∗ ( a M ω a (x)D a J(x)) + δ( a M ω ∗a (x)D5a J(x)) = 0 ,
(IV.15)
Thus, the properties of δ and δ∗ asserts that;
where the first equation is called the consistency condition [5, 22, 23] (with B = 0). The second
equation is somehow unfamiliar despite of its relation to axial currents. It is known that if a theory is
anomalous, then both δW and δ∗ W participate in anomalous behavior simultaneously [4]. Thus, it is
more convenient to consider δex W = δW + δ∗ W as the anomalous term. Actually, the counter terms
affect this description of anomalies such that a suitable choice of counter terms may lead to δW = 0
as a desired result. But, cohomologically the extended BRST class of δex W is unaffected with respect
to counter terms. Indeed, the cohomology class of δex W is independent of renormalization methods;
in the other words, δex W reveals the pathology structure of renormalizing the theory [23, 54]. Thus,
extended BRST derivation produces a framework to study the anomalous behaviors of gauge theories.
Geometrically, using the Quillen supper connection [55] and the family index theory [56-57], it can
R
be seen that Λ := M δex W is the Cartan connection form of the principal U (1)-bundle due to
[ constructed from kernel of perturbed Dirac operators D(A,B)
the determinant line bundle DET,
over A × A5 , for extended gauge fields, (A, B) ∈ A × A5 , as compact operators [37, 56, 58]. Any
U (1)-gauge transformation causes the anomalous term Λ to be added by a local term over A × A5 ,
say λ. Moreover, this local term is an exact extended BRST form, i.e. λ = δex w, for w an element
of C ∞ (A × A5 ), which never changes the cohomology class of Λ. Specially, such λs produce exactly
the whole set of possible counter terms appearing in renormalization the theory. Actually, (IV.15)
shows that the connection of Quillen determinant bundle, restricted to the fibers of A × A5 , is flat,
i.e.; δex Λ = 0, which is a geometric description of consistency conditions3 [37].
In non-commutative geometry, it is known that this anomalous behavior arises because (A+B)D −1 ,
for free Dirac operator D, is not a trace class operator in general [59, 60]. Actually, (A+B)D −1 belongs
to the Schatten class ℓp (L2 (S(M ) ⊗ E)) for p > dim M , which causes the quantum action W not to
be well defined over A × A5 . To see this recall that;
R
iS(ψ,ψ̄,A,B)
∞
X
(−1)n+1
ψ,ψ̄ e
R
T r{((A + B)D −1 )n } ) .
(IV.16)
= exp(
iS(ψ,ψ̄)
n
e
ψ,ψ̄
n=1
But for small enough integers n, n ≤ dim M , ((A + B)D −1 )n s are not trace class operators. Then,
one needs to modify the trace. This can be considered as the regularization method in the setting of
operator theory. Such fascinating feature of regularization takes place by using the Dixmier trace and
zeta Riemann function [59-61]. This causes the quantum action W to vary through the fibers. Indeed,
W is not even a single valued function over A × A5 . Therefore, δex W belongs to a nontrivial deRham
(extended BRST) cohomology class of G̃ [23, 59, 60]. As an intuitive example for this situation one
can consider the angle function θ over S 1 . θ is not a continuous function but its exterior derivative
1 (S 1 , R). Thus, integrating the pull back of δ W
dθ, forms a nontrivial cohomology class of HdeR
ex
1
1
through a smooth map g : S → G̃ over S measures the anomalous behavior of the theory [59, 60,
62]. In this way, one computes an element of the holonomy group of the Quillen determinant bundle
[ over G̃. As it was stated above, this bundle is equipped with a flat connection which leads to
DET
[
a projective group homomorphism of φ : π1 (G̃) = π1+dim M (G̃) ։ Hol(DET).
Actually, since this
[ then Hol(DET)
[ ⊆ U (1) [56-58]. But since Z(A + B), the
connection preserves the metric of DET,
3
By vanishing the curvature, i.e.; δex Λ = δΛ + δ ∗ Λ = 0, we basically focus on the whole three equations of (IV.15)
as extended consistency conditions. In fact, in the extended BRST formalism, the standard consistency condition,
δW = 0, is simultaneously replaced with three equations of (IV.15)
partition function of an extended gauge theory with extended gauge field (A, B), is single valued over
R
[ = {1}. Therefore, one concludes that 1 g∗ (Λ) = 2πm for m the winding
A × A5 , then Hol(DET)
S
number of the phase of det D(A,B)⊳g around the loop g : S 1 → G̃ [62]. This can be computed in terms
of periodic cyclic cohomology, using the Chern-Connes character and the local index formula [59-61].
To extract the modified consistent anomaly according to the Stora-Zumino procedure [18, 19], one
should use d, δ and δ∗ alternatively to provide a generalized formulation of descent equations. Initially,
the Bianchi identity asserts that;
δR = −iωR + iRω ,
δ∗ R = −iω ∗ R + iRω ∗ ,
dR = i(A + B)R − iR(A + B) ,
(IV.17)
where R = −idA − A2 is the curvature4 . Thus, (IV.17) implies that T r{Rn+1 } is a deRham, BRST,
and anti-BRST closed form. Set M = R2n and consider T r{Rn+1 } as a 2n + 2-form over R2n+2 . Thus,
the Poincare lemma leads to;
1,0
δΩ0,0
2n+1 = dΩ2n ,
2,0
δΩ1,0
2n = dΩ2n−1 ,
0,0
T r{Rn+1 } = dΩ2n+1
,
1,1
δ∗ δΩ0,0
2n+1 = dΩ2n ,
0,0
δ∗ Ω2n+1
= dΩ0,1
2n ,
0,1
0,2
∗
δ Ω2n = dΩ2n−1 ,
(IV.18)
where Ωj,k
is a deRham differential i-form with ghost number of j − k, while i + j + k = 2n + 1.
i
j,k
Actually, Ωi is simultaneously a differential j-form over A and a differential k-form over A5 . On the
0,1
1,1
0,1
∗ 1,0
∗ 1,0
other hand, it can easily be shown that Ω1,1
2n = δΩ2n = −δ Ω2n + dΩ2n−1 . Indeed, δΩ2n and −δ Ω2n
differ in an exact differential form. This fact plays an important role in calculating the consistent
Schwinger term.
From (IV.18) it is seen that;
0,1
2,0
1,1
0,2
δex (Ω1,0
2n + Ω2n ) = d(Ω2n−1 + Ω2n−1 + Ω2n−1 ) ,
and hence;
δex
Z
R2n
0,1
(Ω1,0
2n + Ω2n ) = 0 .
(IV.19)
(IV.20)
0,1
Therefore, up to a factor (Ω1,0
2n + Ω2n )(x) can be considered as the modified non-integrated
R
0,1
∗
consistent anomaly; in the other words, R2n (Ω1,0
2n + Ω2n ) is a candidate for δex W = δW + δ W . Here
we should emphasize that despite of the effect of counter terms to anomalous descriptions, which
makes us apply the term of ”candidate” for consistent anomaly and consistent Schwinger term, the
extended BRST cohomology class of Ωi,j
2n+1−i−j is exactly unaffected by renormalization methods. In
R
1,0
+ Ω0,1
fact, in the following we precisely prove that the extended BRST cohomology class of R2n (Ω2n
2n )
coincides exactly with that of δex W .
4
We should emphasize again that the gauge fields and consequently the curvature tensors are here considered in
representation of which they act on the spinors, as they appear in the quantum action
A direct calculation shows that when n = 2 then;
1
0,1
3
∗
Ω1,0
4 + Ω4 = T r{d(ω + ω )( i(A + B)d(A + B) + (A + B) )}
2
(IV.21)
1
which is the modified consistent anomaly up to a factor of c2 = 24π
2 . Indeed, if one sets B = 0 the
5
resulting form is equal to the well-known consistent anomaly [23, 54]. On the other hand, ghost
number counting leads to;
c2 Ω1,0
4 =
1
1
T r{dω( i(A + B)d(A + B) + (A + B)3 )} ,
2
24π
2
(IV.22)
which we called ”ghost consistent anomaly”. Moreover, the remained term;
c2 Ω0,1
4 =
1
1
T r{dω ∗ ( i(A + B)d(A + B) + (A + B)3 )} ,
2
24π
2
(IV.23)
is also referred to as ”anti-ghost consistent anomaly”.
If W =
R
R2n
w, for a non-local form w, then δex W = cn
R
1,0
R2n (Ω2n
+ Ω0,1
2n ) implies that;
dw = −cn Ω0,0
2n+1 + K ,
0,0
dw = −cn Ω2n+1 + K ′ ,
(IV.24)
for a BRST (resp. anti-BRST) closed form K (resp. K ′ ). Thus, the added term K (or K ′ ) must
be simultaneously a BRST and an anti-BRST closed form. This undetermined BRST/anti-BRST
(extended BRST) closed form can be compared with the gauge fixing term in the Faddeev-Popov
quantization method [12, 54]. It asserts that the quantum Lagrangian density is simultaneously invariant under BRST and anti-BRST transformations, the fact of which was studied in [17] with details.
Moreover, continuing the sequence of equations (IV.18), results in;
3,0
δΩ2,0
2n−1 = dΩ2n−2
,
0,3
δ∗ Ω0,2
2n−1 = dΩ2n−2 .
(IV.25)
Indeed, fallowing the standard descent equations for d and δex = δ + δ∗ yields the following equality,
0,2
1,1
3,0
2,1
1,2
0,3
δex (Ω2,0
2n−1 + Ω2n−1 + Ω2n−1 ) = d(Ω2n−2 + Ω2n−2 + Ω2n−2 + Ω2n−2 ) .
(IV.26)
Ghost number counting implies that,
3,0
δΩ2,0
2n−1 = dΩ2n−2 ,
0,3
δ∗ Ω0,2
2n−1 = dΩ2n−2 ,
(IV.27)
1,1
2,1
δ∗ Ω2,0
2n−1 + δΩ2n−1 = dΩ2n−2 ,
1,2
∗ 1,1
δΩ0,2
2n−1 + δ Ω2n−1 = dΩ2n−2 .
(IV.28)
0,2
1,1
Ω2,0
2n−1 + Ω2n−1 + Ω2n−1 ,
(IV.29)
and,
Therefore,
5
Actually, by setting B = 0 anti-ghost fields will be killed automatically after taking the trace.
is a candidate for the modified consistent Schwinger term up to a factor cn . For n = 2, the modified
consistent Schwinger term is given by;
0,2
1,1
c2 (Ω2,0
3 + Ω3 + Ω3 ) =
1
T r{(d(ω + ω ∗ ))2 (A + B)} .
24π 2
(IV.30)
where;
1
2
c2 Ω2,0
3 = 24π 2 T r{(dω) A} ,
1
∗ 2
c2 Ω30,2 = 24π
2 T r{(dω ) A} ,
1
∗
∗
c2 Ω1,1
3 = 24π 2 T r{(dωdω + dω dω)B} .
(IV.31)
From now on we refer to these differential forms with ”ghost/ghost”, ”anti-ghost/anti-ghost” and
”ghost/anti-ghost” consistent Schwinger term, respectively. Note that the ghost consistent anomaly
and the ghost/ghost consistent Schwinger term are related by an ordinary descent equation of d
and δ. It is also the case for anti-ghost consistent anomaly and anti-ghost/anti-ghost consistent
Schwinger term with replacing δ by δ∗ . More generally, the extended descent equation will give rise
to a lattice diagram which commutes up to exact deRham forms. This lattice, produces a bi-complex
for horizontal maps of (δ, d), and vertical ones with (δ∗ , d) as;
Ω0,0
2n+1
l
0,1
Ω2n
l
0,2
Ω2n−1
l
0,3
Ω2n−2
l
..
.
↔ Ω1,0
2n
l
1,1
↔ Ω2n−1
l
1,2
↔ Ω2n−2
l
1,3
↔ Ω2n−3
l
..
.
↔ Ω2,0
2n−1
l
2,1
↔ Ω2n−2
l
2,2
↔ Ω2n−3
l
2,3
↔ Ω2n−4
l
..
.
↔ Ω3,0
2n−2
l
3,1
↔ Ω2n−3
l
3,2
↔ Ω2n−4
l
3,3
↔ Ω2n−5
l
..
.
↔ ...
↔ ...
↔ ...
(IV.32)
↔ ...
..
.
In the diagram of (IV.32) the differential forms Ωi,j
at the inner vertices are
2n+2−i−j/2n+1−i−j
i,j
considered as Ωi,j
2n+2−i−j (resp. Ω2n+1−i−j ) for the range (resp. domain) of the relevant incoming
(resp. outgoing) arrows. Finally, it can be seen that the co-diagonal elements of (IV.32) (the elements
of Ωi,j
2n+1−i−j with the same number of i + j) produce modified consistent anomalies, modified
consistent Schwinger terms and so on. More precisely, the extended formalism of descent equations
is given for the co-diagonals of (IV.32). Note that the top row and the left column of (IV.32)
show respectively the ordinary deRham/BRST and deRham/anti-BRST descent equations as was
mentioned above.
Here for the last description we should precisely prove that the extended BRST cohomology class
R
0,1
of R2n (Ω1,0
2n + Ω2n ) coincides exactly with that of δex W . To show this note that we are essentially
looking for a deRham differential 2n-form with (anti-) ghost number equal to one, as the non-integrated
consistent anomaly. Physically we need an observable with mass dimension 2n and (anti-) ghost
number equal to one. This differential form is basically a part of a differential 2n + 1-form over G̃ × M ,
which can be decomposed as;
Ξ2n+1 =
2n+1
X
(−1)i Ξi2n+1−i = Ξ02n+1 − Ξ12n + Ξ22n−1 − ... ,
(IV.33)
i
where here i is the extended ghost number, i.e. the sum of ghost and anti-ghost numbers. Moreover,
in (IV.33) we let the differential forms on M to get degrees higher than 2n, the dimension of M .
However, if Ξ2n+1 defines a deRham cohomology class then, (δex + d)Ξ2n+1 should vanish. Therefore,
by extended ghost number counting we have;
dΞ02n+1 = 0 ,
δex Ξ02n+1 = dΞ12n ,
δex Ξ12n = dΞ22n−1 ,
..
.
(IV.34)
which except the first equation produces properly the extended descent equations of (IV.18). As stated
R
above, topologically we demand the pull back of M Ξ12n through any smooth g : S 1 ֒→ G̃ defines an
element of H 1 (S 1 , Z). Applying the index theorem and the Poincare lemma for the case of M = R2n
we deduce that dΞ02n+1 doesn’t vanish, but in return, as the only choice, it coincides with cn T r{Rn+1 }
with null cohomology class, and with specific topological constant cn , as we discussed above. Now,
applying the dimension of M , we get a topological cohomology class with;
Ω2n+1 =
2n+1
X
(−1)i+1 Ωi2n+1−i = Ω12n − Ω22n−1 + ... ,
(IV.35)
i
deduced thoroughly from (n + 1)th Chern character cn T r{Rn+1 } which essentially vanishes. On the
other hand, it is clear that by adding any extended BRST exact form to Ω12n , the cohomology class of
Ω2n+1 doesn’t change. This basically rejects the effect of renormalization methods on the BRST class
of anomaly terms. This precisely proves our claim.
V.
CONCLUSIONS
In this article it was shown that in gauge theories, anti-BRST invariance is the quantized version
of local axial symmetry. Basically, the axially extension of a gauge theory enlarges the gauge group
and consequently the algebra of ghost fields. Initially, it was shown that the anti-BRST derivation
and infinitesimal axial transformations are mutually dual in the sense of differential geometric objects.
Moreover, an elaborate geometric description for anti-ghost and anti-BRST transformation was given
by means of differential objects over the space of connections of the gauge theory principal bundle.
Finally, a collection of extended descent equations was formulated by using BRST and anti-BRST
derivatives alternatively. This results led to a modified version for consistent anomalies and consistent
Schwinger terms.
VI.
ACKNOWLEDGMENTS
The author expresses his gratitude to F. Ardalan, R. Bertlmann, A. Davody, D. Freed, M. Khalkhali,
D. Perrot and A. Shafiei Deh Abad for their use-full comments. However my special thanks go to M.
M. Sheikh-Jabbari for his elegant remarks and pedagogical discussions. Also the author should confess
that this article owes most of its appearance to S. Ziaee for many reasons. Finally, I should recall that
most parts of this research was done during my PhD period in Sharif University of Technology.
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