A bound on Yang-Mills action

Se(A)(a Fru Fr=-Rt-=Fa=Fudd

Hodge * operation,W:"Standard Trace"?Oof-valued I-form

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12 Tr(FprFino)

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11 * W=

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9

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self dual.

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2nd order differentialequA- (ppTrIFE)

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1st order differential equr(A]Ea

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r(A]=SppCharCAS.

Recall from Lecture 12··SECA) <D

Tr(FA)u(A)=:gidg

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11 + 0dTr(AdA =A3)+=-&Tr(AdA+-A-STAA)===:Fact]g Tr(gdg(3)the winding number of g Son/g]

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⑦ vacuaConsider the Yang-Mills

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and

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conclude

EolAs]=ein(s)0 FoCAS.

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localized state

H(j)
T(j)

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ini
-of

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properties :ForaForg=Thus,variation ofEL equ:·,Y.

on

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FA

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0

=- Tr(AFA)(flatness (Tr(AdAY +* A33),

Tr(AdA + FA)

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+=:n+(9).

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Recall·forThuswithX.

.

: =.

4-dimensional manifold X possibly

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of-valued I-form A on X

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1 x (Ti [+ ] (Enclidean (=&X

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boundary &X

rx(A)
eg

an

=,

=&(3x(ti Tf),/Ox (ti [ ].

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CSp (Al :7>+-CSp(Alet]In Minkowski space

So(A)=TFFOCSRs (Alti]@CSir (Alt + ]=->.

CS(A)

CS(AS)
So,if

wewriteiOCS(Al)=Ho(Al),CS(A) + n(g).

=FolA)
thenobeys

CSp(A]:=iseKolA]

invariantunder"Block-type
We may consider
same

If

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Let
Then

write

is,

theproperty

as

Fo(A)

ein(sIOE/A]=EIA)

invariant

E : 0 Ef

functional"

general states[A] with themoretransformation

FLAS]

wave

allo=under

amplitudeie.

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f&athen,.

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(Ef,eviltisHEi)-1iS(A]S=2*

Ef(Acty] E (Altis]-iOCS(ACtyS + iOCS (Actil]em-*Ue(AStA)] I (Al(t : ]:iSp(A]1L-We

DIemayiSlA] + iSoLA)

O-sector

consider theAll states transform

forge Ofor&Uf(A)(ts)]All
butstates

andas

*

U : /Altis]

whichin&(AS)the action=isein(S10FLAl)
SIAT,

equivalently
arethe action

invariantisSLA)

under+So(A]9)

Note thattothe state(orMo) belongs to. That's
ground state
why

theAll states obtained from Fo

gauge

invariant

call

weit

the

thissector as

O-vacuum.

lov 20) by operating

local operators

inare.
the same sector,
This implies that sectors of different values of O

02-sectoreg.

O-sectoranddonot mix

with

with

each other0 . 02 (mod 22),.

We shall consider different sectors to be different &FTs
In other words

specify#↑thethe states

avoid,

value

areinfinity

A/gwhereSatisfyn(ght)

specify

toofinas,

G=theOf

theoryR/2X,

weneedafixed OcI/2TTY

path-integral must beof gto.

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0 :a·which doesovernot

have to/(Ef,eil-tis HE ):==ExCACe] FiCAltis]iS[A] + iSo(A)

e

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D

Uf(ACtH] H /AllI] in: