Descent and its derivation
Let[ (3)ofFor

be30
awhichadjoint

andegreeofinvariant

U + 1.

G-gauge potential AICFA),

isa2n + 2 form

is··gauge

invariantclosedd[(FA)

Moreoverthere,aredeT(FA)
0=,,I(FA)=Ge[in (A)+=+,...

,

obeyingd[inCE A),,--Tcn(CE) A),,=>·
called theand[A],de [in (E2 A] dec [in[Er Al,

odifferential forms
sequence of,d[ch=.

[in [A] [inCE A) In-(Etz A]+

polynomial function

descent equations-d[n/Er E A],,,

For 9=0. 1Jan-g(E2n +,..,,depending linearly.

-24-9+1 form

aequation forgenera 9 isaqt(t : Egy Al,I H)2n-g+&Jahg(t=, --.,((Ei tj),+Egy AS,.

.

.

,

found

the sameinway

weas(--Ei,A)

0x92n + 1.

,

[In (A) satisfying I(FA)The form
beis,

infinitesimal gauge transformationson

The descent

E , -, Eg

Eg A]

:

found=&Jinn(A] canChern-Simons form

from the Chern character :Write [(3) usingas[(3)=[(3dI(FA).

,.

.

,an3).

adjoint invariant multilinear function [(5-3nn)

By the adjointFa) , , Fa)= I(Fait,DaFa

Also,for an& T(FA)invariancearbitrary variation A+= o0.

by Bianchi

A + JA,= ICFA--- , FA)
DadA

=i

=dICFAV-,

GA ...

=>J(FA JA),For

afamily

I(FA)

At-A

=te (0 1],,,[(Fac) [(Fa)==-Sjd + [(Fat)d J (FAt A),J(Fat A)d)jd: In (A)+=,+In what follows
Notation:A(X)

G)(X)

For

we,

[Enge forshall findmanifold Xa=

the spaceof G-gauge potentials=>

R(X g)

of valued 1-forms

the space ofG-gauge transformations

Map(X 6)G-valued functions,==,

We shall consider

Y
Define projection= =agauge

potentials

Xxg(X)
evalutation

-X

X

941.

E G;

aon

onmaps,

X

X,

on

X.

gauge transformations.

(x 9)

XonYK

4 g()

on

·At A(X)

Anydetermines= PY A·encanev

A differential form
whenasG(X)+iton

isagauge potential

agaugeX

said tois

The exterior derivative

Note E:While=bea(19) form-

NT X ** T g(X)ofsection**(p) forms

the space ofdontransformation.

19 is+X

Ri( *)

Ri(X) == donA(X).

=regarded

bea;

on

Xis-

denotedon.
Xby

did : -

PEA = e( : 9) < & (X ; 9)=enerteider
inReiderin&!

In( ** ]+->&2 / * )

En(A]

=>

Ein(A)+-Asd[En CA]

wehave& Fin)=[(Far)(d + 5) ( Ein(A]dEEn CAT=+in[(FA) holds=+++...

+Min+&2n

any space,on.

Ein(A)

GEEntCA]-in+Ein-(A]E (A])*+... +dEcnCA]+--Mintz

Rinei-

E (A]*-Min=

components

Fin(A]+in+

LHS

all-

Mineinhas

Gin[A] dEn(A]++...

(A].

...

-Men
RHS

=

[(FE)

=

[(PFA)+222n=

P* I(FA)

=

fintz

The

equsplits

todEnnCA]

Gin(A] dEinCAT+PEICFA)=o=o&=o②G Ein(A] + dEEn (A)+Iii-~ 2n+dIo (A)·For

athe·Todifferential form

2Xon

pullback of E

byxeX

X

fromtheg(X)acts

on&EachE=[in (e A]: =,En [t [2 A],

Map(X , )fieldvector

Put

8-,

ion+

right

=

denotewe,,(x 1),

9:

Lieg(x)=-Ein(A] ( En) /n,.
X

inducesEnn(AS (Esle:=by Wl129')#.
X-

11 , 99).

aThen,,(l ,thedescent

equationsd[in/,

nothing but⑧[A]=[(FA),

de[inCA]=

dIinCE A],+are...

.

,D,de [in(E2 A] -denlinCE A] -InSenE] AJ,,,

d[in (t En A),=,-②,·This

icanbe shown

Obviously,G(any)/1AlsoThus:

enderly,=** / =* /,+(1])

By this,I,& [in + (A)

=,is

=0.

O

=follows.

as.
erlos 11 = 1

since=,A,hence

andZen ( ** ])

what

we

=,

[ind1+(A)already know

PRI(FA))

=,

I(FA)

,

=,dA.
(a).

:

whichisO.

To show the rest

somepreparation

A useful tooltheleft,

Lg
·is

*(g)

:by geG(X)action+(needed.

is,99EXonX,.

It satisfies

PxoLg
Le

Using theseL

*==>

==Pig
we,.

.

(+erX

andin

G(X)

.

find(PA)=(LPA)Pg.

9TPxOLg)"A

3.

=*(4/ A )*==PEA(PXA9)ev

e

L [C]
i e.

(5)(=::*xnge=[in (ES][A]=

Enqu (A9](b)·Forfunction faonX,+1=(f)(x 1),iEfhaf f(x=+,1.

e

Lef/=

(t=0ab(Le++f((x )(+,=(c), toXb( * ),ForExIE)(x 1),**=X(E( e,~=>)/tEl s)

LgxE( 1),=,= ox(Le+ E(x ())(t,-= 0(Lex)(E ( 1),:

EXCElly = E Let x)(E) /1 to,Similarly,

for&ECX),/Ei Gall * (Le * x)(E·,.

.

,Ea) 11 t
,

= 0.

(d)

0

·Also

shall

we,

do iz
ThisoftheA=0onEl g) isgett)attvariationofC.

(2)

underfi * ).

the tangent,

curve

invariant·1zod+becaseisuse,=(e)rectorand0

isside remark#Evia

thevector

field

the leftactionof

do ijid=

do is + igod

=is+onXG(X)=Map(X 9)

induced fromon

X,

,

we

also

have0.

Thes

Leg (the Lie derivative by E).

Together with the commutativity of the left and the rightactionsthis(which implies ( T o) andcan,

be usedderivation of thetogiveabove=

analternativeformula(d).

=andat( 1),,mechanicalNow·,

areweD(E)12

isready.

JEin(A)(E))+,

The first term
J + (AJ(E))

Ein [A] I

=+(c)

(b)

**

Le=>

de [in[A]The second term=,t= 0(a)(A)() / to =-def>=DE),0.

dEcn(A] (E) 1,

disEin(Ablnd[inSt A],:=

idEin(A]I y(e)
=.

,,

=

0

=,,

Let Enti(A) /=dEinCA](E)1

reads=-d(i Ein(As( ),.

GelintCA]-dIinCE A)
,

=0,

which

isD.

·& EEr))
Note,

is

GEanCAT (E En)) + dEnzlA](E E))
,,,

=-,

that

GEinCA](EE)=EEcn(A]lE)-EclanCAz(E )-,EnCAS (S Ent),-actionright
11-[E [i],andEEnCAb(e))(d)=,Lee IinCA])(2) Into(b)=def= Tn[En=Alt](t,,=,in(A] (E , ) 1,

de [inStrA] -denTinCG Al,

The second term

·Je [in( A].
,Thus the first term=,&Ein(A) (E Ell..

,-[in(CEET A],.

(e)ii dra) I ,=digi,=,=>def=d (tin-CAT(E)/ /,dIn-SE En A),,,(E , [ ) ):, E.

reads,Je [in (En A] -denInCE A][in[[E ES As-,,,,=·&In (EE A],

For 092n + 1Iq2n-Cas I ,9+ 1(E)if,we,. .

.. ....

&put

Eg,A)

:71)[Iv anat (A) (Ea)

.

,

reads
,-Egy Al,I H 2n-g ((Ei tj)+dF.

, ,mat(t :=>iswhich,

9 19-1⑪ (E Egn))+,q(t, -..,Est A)
,,.

.

,.

.

(i--Ei,A)

1,