Extension of Chern-Simons forms

0.2 MB·

by  Spyros Konitopoulos

NRCPS-HE-2-2014
Extension of Chern-Simons Forms
4 Spyros Konitopoulos and George Savvidy
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2 + Demokritos National Research Center, Ag. Paraskevi, Athens, Greece

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Abstract

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- Weinvestigatemetricindependent, gaugeinvariantandclosedformsinthegeneralizedYMtheory.
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These forms are polynomial on the corresponding fields strength tensors - curvature forms and are
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analogous to the Pontryagin-Chern densities in the YM gauge theory. The corresponding secondary

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v characteristic classes have been expressed in integral form in analogy with the Chern-Simons form.
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1 Because they are not unique, the secondary forms can be dramatically simplified by the addition of
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4 properly chosen differentials of one-step-lower-order forms. Their gauge variation can also be found
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yielding the potential anomalies in the gauge field theory.
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Introduction
The chiral anomalies, Abelian and non-Abelian [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18], can be
derivedbyadifferentialgeometricmethodwithouthavingtoevaluateFeynmandiagrams. Indeed, the
non-Abelian anomaly in (2n−2)-dimensional space-time may be obtained from the Abelian anomaly
in 2n dimensions by a series of reduction (transgression) steps [6, 7, 8, 9, 10, 11, 12, 13, 18]. The
U (1) gauge anomaly is given by the Pontryagin-Chern-Simons 2n-form [6, 7, 8, 9, 10, 11, 12, 13, 18]:
A
d∗JA ∝ P2n = Tr(Gn) = d ω2n−1, (1.1)
where ω2n−1 is the Chern-Simons form in 2n−1 dimensions [6, 7, 12]:
1
ω2n−1(A) = n dt Str(A,Gtn−1), (1.2)
Z
0
G = dA+ A2 is the 2-form Yang-Mills (YM) field-strength tensor of the 1-form vector field1 A =
−igAaLadxµ and G = tG+(t2 −t)A2. The non-Abelian anomaly [1, 2, 3, 4, 5] can be obtained by
µ t
the gauge variation of ω2n−1 [6, 7, 8, 9, 10, 12, 13, 16, 17, 18]:
δω2n−1 = dω21n−2 , (1.3)
where the (2n−2)-form has the following integral representation [6, 7]:
1
ω1 (ξ,A) = n(n−1) dt(1−t) Str dξ,A, Gn−2 . (1.4)
2n−2 t
Z
0 (cid:0) (cid:1)
Here ξ = ξaL is a scalar gauge parameter and Str denotes a symmetrized trace 2. The covariant
a
divergence of the non-Abelian left and right handed currents is given by this (2n−2)-form.
In recent articles [19, 20, 21] the authors found closed invariant forms similar to the Pontryagin-
Chern-Simons forms in non-Abelian tensor gauge field theory [22, 23, 24]. The first series of closed
invariant forms are defined in D = 2n+4 dimensions and are given by the expression
Φ = tr(G Gn) = Str(G ,Gn) = dψ , (1.6)
2n+4 4 4 2n+3
where the corresponding secondary (2n+3)-form ψ is in D = 2n+3 dimensions
2n+3
ψ = Str(A ,Gn) (1.7)
2n+3 3
1La are the generators of the Lie algebra.
2In this article we shall use the symmetrized trace
1
Str(A ,A ,...,A )≡ (A A ...A ), (1.5)
1 2 n n! i1 i2 in
(i1X,...,in)
where the sum is over all permutations. Its properties are described in the Appendix B of the article [10].
and G = dA +{A,A }3. It turns out that the introduction of Str in the above equations leads to
4 3 3
very crucial simplifications in all our subsequent derivations. For compact notation, when some of
the entries of Str are the same, we write them in power form. The second series of forms is defined
in D = 2n+6 dimensions [21]:
Ξ = Str(G ,Gn)+nStr(G2,Gn−1) = dφ . (1.8)
2n+6 6 4 2n+5
The general expression for the secondary (2n+5)-form φ will be constructed in this article. The
2n+5
third series of invariant closed forms found in this article Υ in D = 2n+8 dimensions is
2n+8
Υ = Str(G ,Gn)+3nStr(G ,G ,Gn−1)+n(n−1)Str(G3,Gn−2) = dρ . (1.9)
2n+8 8 4 6 4 2n+7
Its secondary form ρ will be presented in the next sections.
2n+7
All forms Φ , Ξ and Υ are analogous to the Pontryagin-Chern-Simons densities P in
2n+4 2n+6 2n+8 2n
the YM gauge theory (1.1) in the sense that they are gauge invariant, closed and metric independent.
Our aim is to investigate this rich class of topological invariants of extended gauge theory as well as
to find out potential gauge anomalies performing transgressions analogous to (1.1) and (1.3):
P2n ⇒ ω2n−1 ⇒ ω21n−2. (1.10)
Therefore we shall perform the following transgressions:
Φ ⇒ ψ ⇒ ψ1 ,
2n+4 2n+3 2n+2
Ξ ⇒ φ ⇒ φ1 , (1.11)
2n+6 2n+5 2n+4
Υ ⇒ ρ ⇒ ρ1 .
2n+8 2n+7 2n+6
We shall find explicit expressions for these primary invariants in terms of higher order polynomials
of the curvature forms on a vector bundle. The most difficult challenge will be the evaluation and
differentiation of the very complicated noncommutative polynomial expressions as well as the search
of the most simple expressions for the secondary forms. The secondary forms are not uniquely
defined. Indeed, the secondary form ψ is defined modulo the exterior derivative of an arbitrary
2n+3
(2n+2)-form ψ ∽ ψ +dα , the form φ modulo the exterior derivative of a (2n+4)-
2n+3 2n+3 2n+2 2n+5
form φ ∽ φ +dβ and the form ρ modulo the exterior derivative of a (2n+6)-form
2n+5 2n+5 2n+4 2n+7
ρ ∽ ρ +dγ . When the difference of two closed forms is an exact form, they are said to
2n+7 2n+7 2n+6
be cohomologous to each other. Therefore the problem is to find out the most simple representatives
in the set of equivalence classes. Conveniently chosen exact forms will dramatically simplify the
3In the Appendix one can find the definition of tensor gauge fields and the corresponding curvature forms.
expressions. These problems will be solved by using properties of symmetrized traces (1.5) defined
in [10].
In Section 2 we shall present a general construction and analysis of the primary forms Φ and
2n+4
Ξ , their secondary forms ψ and φ and the corresponding anomalies represented by ψ1
2n+6 2n+3 2n+5 2n+2
and φ1 . The material of this section is not completely new, but the alternative derivation in
2n+4
terms of symmetrized traces will allow to extend the results to the higher-dimensional forms Υ .
2n+8
In Section 3 we shall derive the explicit expressions for the primary form Υ , written in terms of
2n+8
the symmetrized traces (1.5), starting from the low-dimensional forms listed in [21]. Next, we shall
find the secondary forms ρ and the corresponding gauge anomalies ρ1 associated with each of
2n+7 2n+6
the independent gauge transformations δ ,δ ,δ ,δ . In the conclusion we summarize the primary
ξ ζ2 ζ4 ζ6
and secondary invariant forms constructed in the article. In the Appendix we present useful formulas
for the gauge transformations of the fields, the corresponding Bianchi identities and a one-parameter
deformation of fields generalizing deformation of [6, 7].
2
Gauge and Metric Independent Forms
We shall start by deriving the already known results for the form Φ using the properties of the
2n+4
symmetrized traces. This approach will allow to extend the derivation to more complicated cases.
Indeed, the form can be represented in terms of a symmetrized trace as [20, 21]
Φ = Str G ,Gn (2.1)
2n+4 4
(cid:16) (cid:17)
so that its gauge invariance with respect to the standard-scalar gauge transformations δ and the
ξ
tensor gauge transformations δ can be easily checked:
ζ2
δ Φ = Str δ G ,Gn +nStr G ,δ G,Gn−1 =
ξ 2n+4 ξ 4 4 ξ
(cid:16) (cid:17) (cid:16) (cid:17)
= Str [G ,ξ],Gn +nStr G ,[G,ξ],Gn−1 = 0,
4 4
(cid:16) (cid:17) (cid:16) (cid:17)
δ Φ = Str δ G ,Gn = Str [G,ζ ],Gn =
ζ2 2n+4 ζ2 4 2
(cid:16) (cid:17) (cid:16) (cid:17)
1
= Str [G,ζ ],Gn +...+Str Gn,[G,ζ ] = 0. (2.2)
n+1(cid:20) 2 2 (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17)
On the last steps we used the identity (B.10) of the Appendix B of the article [10]. Our next step is
to check that Φ is a closed form. Indeed,
2n+4
dΦ (A,A ) = Str DG ,Gn +nStr G ,DG,Gn−1 = Str [G,A ],Gn =
2n+4 3 4 4 3
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
1
= Str [G,A ],Gn +Str G,[G,A ],Gn−1 +...+Str Gn,[G,A ] = 0,
n+1(cid:20) 3 3 3 (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
where we used the Bianchi identity DG = 0. To find out the secondary form we shall use a one-
parameter deformation of the gauge potentials: [6, 20, 21]
A = tA, A = tA , A = tA , A = tA , A = tA ,....,
t 3t 3 5t 5 7t 7 9t 9
defined in the Appendix (5.6), (5.7), take the derivative and employ (B.13) of [10]:
d dG dG
Φ (A ,A ) = Str 4t,Gn +nStr G , t,Gn−1 =
dt 2n+4 t 3t dt t 4t dt t
(cid:16) (cid:17) (cid:16) (cid:17)
= Str D A +t{A,A },Gn +nStr G ,D A,Gn−1 =
t 3 3 t 4t t t
(cid:16) (cid:17) (cid:16) (cid:17)
= dStr A ,Gn +ndStr G ,A,Gn−1 +
3 t 4t t
(cid:16) (cid:17) (cid:16) (cid:17)
+Str {A,A },Gn −nStr [G ,A ],A,Gn−1 =
3t t t 3t t
(cid:16) (cid:17) (cid:16) (cid:17)
= d Str A ,Gn +nStr G ,A,Gn−1 . (2.3)
(cid:26) 3 t 4t t (cid:27)
(cid:16) (cid:17) (cid:16) (cid:17)
Integrating the above equation over the parameter t in the interval [0,1] we get the integral repre-
sentation of the secondary form
1
ψ = dt Str A ,Gn +nStr G ,A,Gn−1 . (2.4)
2n+3 Z0 (cid:20) (cid:16) 3 t(cid:17) (cid:16) 4t t (cid:17)(cid:21)
This secondary formis not unique. It can bemodified by the additionof thedifferential of a one-step-
lower-order formdα . Two closed formswhich differ by anexact formaresaidto becohomologous
2n+2
to each other. The integral on the right hand side of the equation looks complicated, but if we add
a properly chosen exact form dα , then it will be dramatically simplified. Let us take it in the
2n+2
following form:
1
α = −n dt Str A ,A,Gn−1 .
2n+2 3t t
Z0 (cid:16) (cid:17)
Then we get:
1
ψ ∽ ψ +dα = ψ −n dt Str D A ,A,Gn−1 +
2n+3 2n+3 2n+2 2n+3 t 3t t
Z0 (cid:16) (cid:17)
1 1
+ n dt Str A ,D A,Gn−1 −n(n−1) dt Str A ,A,D G ,Gn−2 ,
3t t t 3t t t t
Z0 (cid:16) (cid:17) Z0 (cid:16) (cid:17)
where D G = 0 4 . Using the relations (5.2), (5.7) and the integral representation (2.4), we have:
t t
1 1 ∂G
ψ ∽ ψ −n dt Str G ,A,Gn−1 +n dt Str A , t,Gn−1 =
2n+3 2n+3 Z0 (cid:16) 4t t (cid:17) Z0 (cid:16) 3t ∂t t (cid:17)
1 ∂G 1 ∂
= dt Str A ,Gn +nStr A , t,Gn−1 = dt Str A ,Gn = Str A ,Gn .
Z0 (cid:26) (cid:16) 3 t(cid:17) (cid:16) 3t ∂t t (cid:17)(cid:27) Z0 ∂t (cid:16) 3t t(cid:17) (cid:16) 3 (cid:17)
(2.5)
4The symbol ”∽” denotes the cohomology relation between the two forms.
Thus, the secondary form gets the following compact form
ψ = Str A ,Gn . (2.6)
2n+3 3
(cid:16) (cid:17)
The secondary forms (2.4) and (2.6) are representatives of the same cohomology class, because their
difference is an exact form dα , but, as one can see (2.6), has a much more simple expression.
2n+2
Using (5.2), (5.5) we can verify that dψ = Φ :
2n+3 2n+4
dStr A ,Gn = Str DA ,Gn −nStr A ,DG,Gn−1 = Str G ,Gn .
3 3 3 4
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
The form (2.6) allows to find the potential anomalies of the theory, by the following transgression
steps. The gauge invariance of the primary form Φ means that δΦ = d(δψ ) = 0. By
2n+4 2n+4 2n+3
employing the Poincare’s lemma it follows that δψ = dψ1 , where ψ1 is the potential
2n+3 2n+2 2n+2
anomaly. Thus, in order to proceed we have to calculate the gauge variation of the secondary form
with respect to the gauge transformations δ and δ . We have5
ξ ζ2
δ ψ = Str δ A ,Gn +nStr A ,δ G,Gn−1 =
ξ 2n+3 ξ 3 3 ξ
(cid:16) (cid:17) (cid:16) (cid:17)
= Str [A ,ξ],Gn +nStr A ,[G,ξ],Gn−1 = 0, (2.7)
3 3
(cid:16) (cid:17) (cid:16) (cid:17)
that is, the secondary form is gauge invariant with respect to standard-scalar gauge transformations
δ and therefore there are no gauge anomalies associated with the scalar gauge transformations. But
ξ
with respect to the tensor gauge transformations δ there are anomalies
ζ2
δ ψ = Str δ A ,Gn−1 = Str Dζ ,Gn = dStr ζ ,Gn . (2.8)
ζ2 2n+3 ζ2 3 2 2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Therefore the anomaly is
ψ(1) (ζ ,A) = Str ζ ,Gn . (2.9)
2n+2 2 2
(cid:16) (cid:17)
In summary, we have the expressions (2.1) for primary form Φ , the expression (2.6) for the
2n+4
secondary form ψ and (2.9) for the anomaly.
2n+3
We shall now move to the next primary formΞ , which canbe written in terms of symmetrized
2n+6
traces as [21]:
Ξ = Str G ,Gn +nStr G2,Gn−1 . (2.10)
2n+6 (cid:18) 6 (cid:19) (cid:18) 4 (cid:19)
Each term of this expression is independently gauge invariant. Indeed,
δ Str G ,Gn = Str δ G ,Gn +nStr G ,δ G,Gn−1 =
ξ 6 ξ 6 6 ξ
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
= Str [G ,ξ],Gn +nStr G ,[G,ξ],Gn−1 = 0, (2.11)
6 6
(cid:16) (cid:17) (cid:16) (cid:17)
5The identity (B.10) of the Appendix B [10] should be used.
and for the second term we shall get
δ Str G2,Gn−1 = 2Str δ G ,G ,Gn−1 +(n−1)Str G2,δ G,Gn−2
ξ 4 ξ 4 4 4 ξ
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
= 2Str [G ,ξ],G ,Gn−1 +(n−1)Str G2,[G,ξ],Gn−2 = 0,
4 4 4
(cid:16) (cid:17) (cid:16) (cid:17)
where in the last two equations we again used (B.10) of [10]. However only the sum of these terms
is a closed form. We can check the closeness of the form Ξ by taking the exterior derivative:
2n+6
dΞ = Str DG ,Gn +nStr G ,DG,Gn−1 +
2n+6 6 6
(cid:16) (cid:17) (cid:16) (cid:17)
+ 2nStr DG ,G ,Gn−1 +n(n−1)Str G2,DG,Gn−2 =
4 4 4
(cid:16) (cid:17) (cid:16) (cid:17)
= 2Str [G ,A ],Gn +Str [G,A ],Gn +2nStr [G,A ],G ,Gn−1 = (2.12)
4 3 5 3 4
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
= 2 Str [G ,A ],Gn +nStr G ,[G,A ],Gn−1 +Str [G,A ],Gn = 0.
4 3 4 3 5
(cid:26) (cid:27)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
On the first step we used (B.13) of [10] and on the second - the Bianchi identities DG = 0. On the
last step, the terms in the big brace as well as the last term are zero because of (B.10) of [10]. Again,
according to Poincar´e’s lemma, this equation implies that Ξ can be locally written as an exterior
2n+6
derivative of a certain (2n +5)-form. In order to find that form we need to differentiate Ξ over
2n+6
the deformation parameter t. We have:
d dG dG dG
Ξ (A ,A ,A ) = Str 6t,Gn +nStr G , t,Gn−1 +2nStr 4t,G ,Gn−1 +
dt 2n+6 t 3t 5t dt t 6t dt t dt 4t t
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
dG
+n(n−1)Str G2 , t,Gn−2 = Str D A ,Gn +Str {A,A },Gn +2Str {A ,A },Gn +
4t dt t t 5 t 5t t 3 3t t
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
(cid:1)
+nStr G ,D A,Gn−1 +2nStr D A ,G ,Gn−1 +2nStr {A,A },G ,Gn−1 +
6t t t t 3 4t t 3t 4t t
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
+n(n−1)Str G2 ,D A,Gn−2 = dStr A ,Gn +Str {A,A },Gn +2Str {A,A },Gn +
4t t t 5 t 5t t 3t t
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
+ndStr G ,A,Gn−1 −nStr D G ,A,Gn−1 +
6t t t 6t t
(cid:16) (cid:17) (cid:16) (cid:17)
+2ndStr A ,G ,Gn−1 +2nStr A ,D G ,Gn−1 +2nStr {A,A },G ,Gn−1
3 4t t 3 t 4t t 3t 4t t
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
n(n−1)dStr G2 ,A,Gn−2 −2n(n−1)Str D G ,G ,A,Gn−2 =
4t t t 4t 4t t
(cid:16) (cid:17) (cid:16) (cid:17)
= d nStr G ,A,Gn−1 +n(n−1)Str G2 ,A,Gn−2 +2nStr G ,A ,Gn−1 +Str A ,Gn +
(cid:26) 6t t 4t t 4t 3 t 5 t (cid:27)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
+2 Str {A ,A },Gn +nStr A ,[G ,A ],Gn−1 +
(cid:20) 3 3t t 3 t 3t t (cid:21)
(cid:16) (cid:17) (cid:16)
+2n Str {A,A },G ,Gn−1 −Str [G ,A ],A,Gn−1 +
(cid:20) 3t 4t t 4t 3t t (cid:21)
(cid:16) (cid:17) (cid:16)
+2n Str {A,A },G ,Gn−1 +Str A,[G ,A ],Gn−1 −(n−1)Str [G ,A ],G ,A,Gn−2 +
(cid:20) 3t 4t t 4t 3t t t 3t 4t t (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
+Str {A,A },Gn −nStr [G ,A ],A,Gn−1 = dφ . (2.13)
5t t t 5t t 2n+5
(cid:16) (cid:17) (cid:16) (cid:17)
On the second, third and last steps we used the relations (5.7) and (B.13), (B.10) of [10] respectively.
Integrating the above equation over the parameter t in the interval [0,1] we shall get the following
integral representation of the secondary form:
1
φ = dt nStr G ,A,Gn−1 +n(n−1)Str G2 ,A,Gn−2 + (2.14)
2n+5 Z0 (cid:26) (cid:16) 6t t (cid:17) (cid:16) 4t t (cid:17)
+2nStr G ,A ,Gn−1 +Str A ,Gn .
4t 3 t 5 t (cid:27)
(cid:16) (cid:17) (cid:16) (cid:17)
As we already mentioned above, the secondary form is not unique and it can be modified by the
addition of the differential of a one-step-lower-order form dβ (1.11). As in the case of ψ , we
2n+4 2n+3
will use this freedom in order to simplify our result. Adding dβ , where
2n+4
1
β = − dt nStr A ,A,Gn−1 +n(n−1)Str G ,A ,A,Gn−2 ,
2n+4 Z (cid:20) 5t t 4t 3t t (cid:21)
0 (cid:16) (cid:17) (cid:16) (cid:17)
we get:
φ ∽ φ +dβ =
2n+5 2n+5 2n+4
1
= φ + dt −nStr D A ,A,Gn−1 +nStr A ,D A,Gn−1 −
2n+5 Z (cid:20) t 5t t 5t t t
0 (cid:16) (cid:17) (cid:16) (cid:17)
−n(n−1)Str A ,A,D G ,Gn−2 −n(n−1)Str D G ,A ,A,Gn−2 −
5t t t t t 4t 3t t
(cid:16) (cid:17) (cid:16) (cid:17)
−n(n−1)Str G ,D A ,A,Gn−2 +n(n−1)Str G ,A ,D A,Gn−2 −
4t t 3t t 4t 3t t t
(cid:16) (cid:17) (cid:16) (cid:17)
−n(n−1)(n−2)Str G ,A ,A,D G ,Gn−3 ,
4t 3t t t t (cid:21)
(cid:16) (cid:17)
where one should use the Bianchi identities D G = 0 and (B.13) of [10]. Next, with the aid of (5.2)
t t
and (5.7) we get,
1
φ ∽ φ + dt −nStr G ,A,Gn−1 +nStr {A ,A },A,Gn−1 +
2n+5 2n+5 Z0 (cid:20) (cid:16) 6t t (cid:17) (cid:16) 3t 3t t (cid:17)
∂G
+nStr A , t,Gn−1 −n(n−1)Str [G ,A ],A ,A,Gn−2 −
5t ∂t t t 3t 3t t
(cid:16) (cid:17) (cid:16) (cid:17)
∂G
−n(n−1)Str G2 ,A,Gn−2 +n(n−1)Str G ,A , t,Gn−2 .
4t t 4t 3t ∂t t (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17)
One can see that the first, the third and fifth terms cancel with the first two terms of φ (2.14).
2n+5
Withtheaidof(B.9)of[10], thesecond andtheforthtermscombinetogiveStr A ,{A,A },Gn−1 .
3 3t t
(cid:16) (cid:17)
Finally using the equation tD A = G and (5.2) we shall get the following expression for φ :
t 3 4t 2n+5
1
dt 2nStr G ,A ,Gn−1 +Str A ,Gn +nStr {A,A },A ,Gn−1 +
Z (cid:20) 4t 3 t 5 t 3t 3t t
0 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
∂G ∂G
+nStr A , t,Gn−1 +n(n−1)Str G ,A , t,Gn−2 =
5t ∂t t 4t 3t ∂t t (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17)
1 ∂G
= dt Str A ,Gn +nStr A , t,Gn−1 +
Z0 (cid:20) (cid:16) 5 t(cid:17) (cid:16) 5t ∂t t (cid:17)
+nStr D A +t{A,A },A ,Gn−1 +nStr G ,A ,Gn−1 +
t 3 3 3t t 4t 3 t
(cid:16) (cid:17) (cid:16) (cid:17)
∂G
+n(n−1)Str G ,A , t,Gn−2 =
4t 3t ∂t t (cid:21)
(cid:16) (cid:17)
1 ∂
= dt Str A ,Gn +nStr G ,A ,Gn−1 .
Z ∂t(cid:20) (cid:18) 5t t(cid:19) (cid:18) 4t 3t t (cid:19)(cid:21)
0
Hence, after the integration we get
φ = Str A ,Gn +nStr A ,G ,Gn−1 . (2.15)
2n+5 5 3 4
(cid:18) (cid:19) (cid:18) (cid:19)
By comparing the representations of the secondary form φ in (2.14) and in (2.15) it becomes
2n+5
clear that the last expression is much more simple and transparent. Let us verify that the exterior
derivative of the above form leads us back to Ξ .
2n+6
dφ = Str DA ,Gn −nStr A ,DG,Gn−1 +nStr DG ,A ,Gn−1 +
2n+5 5 5 4 3
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
+ nStr G ,DA ,Gn−1 −n(n−1)Str G ,A ,DG,Gn−2 =
4 3 4 3
(cid:18) (cid:19) (cid:18) (cid:19)
= Str G ,Gn −Str {A ,A },Gn +nStr [G,A ],A ,Gn−1 +
6 3 3 3 3
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
+nStr G2,Gn−1 = Str G ,Gn +nStr G2,Gn−1 = Ξ ,
(cid:18) 4 (cid:19) (cid:18) 6 (cid:19) (cid:18) 4 (cid:19) 2n+6
where on the second step the second and third terms cancel because of (B.10) of [10].
In order to find out the potential anomalies we have to calculate the gauge variation of the
secondary form φ with respect to the scalar, rank-2 and rank-4 gauge parameters. We have
2n+5
δ φ = δ Str A ,Gn +nStr G ,A ,Gn−1 =
ξ 2n+5 ξ 5 4 3
(cid:20) (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17)
= Str δ A ,Gn +nStr A ,δ G,Gn−1 +nStr δ G ,A ,Gn−1 +
ξ 5 5 ξ ξ 4 3
(cid:19) (cid:19)
(cid:16) (cid:16) (cid:16) (cid:17)
+nStr G ,δ A ,Gn−1 +n(n−1)Str G ,A ,δ G,Gn−2 =
4 ξ 3 4 3 ξ
(cid:16) (cid:17) (cid:16) (cid:17)
= Str [A ,ξ],Gn +nStr A ,[G,ξ],Gn−1 +nStr [G ,ξ],A ,Gn−1 +
5 5 4 3
(cid:19) (cid:19)
(cid:16) (cid:16) (cid:16) (cid:17)
+nStr G ,[A ,ξ],Gn−1 +n(n−1)Str G ,A ,[G,ξ],Gn−2 = 0.
4 3 4 3
(cid:16) (cid:17) (cid:16) (cid:17)
Thus, there are no anomalies in the standard gauge symmetry. But there are potential anomalies in
the higher-rank gauge symmetries. Indeed,
δ φ = Str δ A ,Gn +nStr δ G ,A ,Gn−1 +nStr G ,δ A ,Gn−1 =
ζ2 2n+5 ζ2 5 ζ2 4 3 4 ζ2 3
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
= 2Str [A ,ζ ],Gn +nStr [G,ζ ],A ,Gn−1 +nStr G ,Dζ ,Gn−1 =
3 2 2 3 4 2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
= Str [A ,ζ ],Gn +nStr G ,Dζ ,Gn−1 =
3 2 4 2
(cid:16) (cid:17) (cid:16) (cid:17)
= Str [A ,ζ ],Gn +ndStr G ,ζ ,Gn−1 −nStr [G,A ],ζ ,Gn−1 =
3 2 4 2 3 2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
= ndStr ζ ,G ,Gn−1 (2.16)
2 4
(cid:16) (cid:17)
and
δ φ = Str Dζ ,Gn = dStr ζ ,Gn . (2.17)
ζ4 2n+5 4 4
(cid:16) (cid:17) (cid:16) (cid:17)
Hence the anomalies are:
φ(1) (ζ ,A) = Str ζ ,Gn
2n+4 4 4
(cid:16) (cid:17)
φ(1) (ζ ,A,A ) = nStr ζ ,G ,Gn−1 (2.18)
2n+4 2 3 2 4
(cid:16) (cid:17)
In summary, we have the expressions (2.10) for primary form Ξ , the expression (2.15) for the
2n+6
secondary form φ and (2.18) for the anomalies.
2n+5