An invitation to Anomalies -...

Paris, October 16th 2006 Luis Alvarez-Gaume, CERN-TH

An invitation to

Anomalies

Consistency conditions on Supergravity theories in diverse dimensions

Paris October 16th 2006 Luis Alvarez-Gaume CERN-TH

3. Chirality, mass and regulators. Heuristics

2. The simplest possible example with most of the ingredients

1. Symmetries in Physics. Different realizations

Outline

4. Feynman graphs versus index theory

6. Sample computations: The SM and type IIB string theory

5. A lightning review of index theory and characteristic polynomials

7. Canceling ingenuity: Green and Schwarz; the Heterotic string

8. Local anomalies is not the end ...

9. Conclusions

E8 ! E8 and SO(32)


Symmetries in Physics

Wigner-Weyl [Q, H] = 0 Q |0! = 0

Nambu-Goldstone [Q, H] = 0 Q |0! "= 0

Anomalous symmetries

They hold at the classical level, but not quantum mechanically

Anomalies in global currents. Often they are welcome:

Neutral pion decay

Scale invariance: beta function in field theory

Anomalies in gauge currents: They are fatal


Gauge (and diffeomorphism) symmetries are redundancies in our description of the physical degrees of freedom

In four dimensions the photon and the graviton have only two degrees of freedom

Gauss’s law is required to set the number of physical degrees of freedom straight

In the gravitational case we need the diffeomorphism and the hamiltonian constraints

With local currents, conservation is not a luxury, it is needed for the physical consistency of the theory

Unitarity and/or renormalizability of the theory break down. You are

not describing the physics you want to describe if anomalies appear

What we verify with anomaly computations is that the theory contains the required degrees of freedom, which istantamount to preserving some basic symmetries.


Computing an anomaly

This is the only example where you will see the explicit details. In other cases details will be scarce, but the physics is essentially the same. So, pay attention!

We consider a two-dimensional problem to start with

L = !i "µ(#µ ! eAµ)! ! =

!

u+

u!

"

!0= "1 !1

= i "2 !5 = !!0!1=

!

1 0

0 !1

"

We have a left and a right moving fermion and we the action is formally invariant under independent change of the phase (if we compensate with the appropriate gauge transformation)

We have a left moving current and a right moving current. Both of them are anomalousas soon as a gauge field is turned on. If these currents are associated to true gauge symmetries

then the theory is clearly inconsistent. The procedures we follow in the simple example are generalized to higher dimensions later on

(!0 ! !1)u+ = 0, (!0 + !1)u!= 0


p

E E

p

!"# !"!

!"#$ %%& '()**+ ,- ./0 ./0,12$

+,+03.*+ p < 04 5/010(6 ./0 30#(."70 0301#2 6,8*.",36 (10 98(30 5(706 5"./ p > 0$ !,1 ./080-.:/(3;0; 69"3,1 u! ./0 6".*(.",3 "6 1070160;$ <06";064 6"3)0 ./0 69(."(8 ;"10).",3 "6 ),+9(). 5"./

803#./ L ./0 +,+03.*+ p "6 =*(3.">0; ()),1;"3# .,

p =2!n

L, n ! Z ?@$%@A

B/0 690).1*+ ,- ./0 ./0,12 10910603.0; "3 !"#$ %C$

D3)0 50 #,. ./0 690).1*+ ,- ./0 ./0,12 ./0 30E. 6.09 "6 ., ,F.("3 ./0 7()**+$ G6 5"./ ./0

H"1() 0=*(.",3 "3 -,*1 ;"+036",36 50 !88 (88 ./0 6.(.06 5"./ E " 0 ?!"#$ %%A$ G3 0E)".(.",3 ,- (9(1.")80 ,- ./0 H"1() 600 91,;*)06 ./03 ( 9,6"."70 0301#2 -01+",3 98*6 ( /,80 ./(. "6 "3.01910.0; (6 (3

(3."9(1.")80$ B/"6 #"706 *6 ./0 )8*0 ., 91,)00; ., ./0 =*(3.">(.",3 ,- ./0 ./0,12$ I3 ./0 0E9(36",3 ,-

./0 ,901(.,1 u± "3 .01+6 ,- ./0 +,;06 ?@$%JA 50 (66,)"(.0 9,6"."70 0301#2 6.(.06 5"./ (33"/"8(.",3

,901(.,16 5/010(6 ./0 6.(.06 5"./ 30#(."70 0301#2 (10 (66,)"(.0; 5"./ )10(.",3 ,901(.,16 -,1 ./0

),11069,3;"3# (3."9(1.")80

u±(x) =!

E>0

"a±(E)v(E)

± (x) + b†±(E)v(E)± (x)"

#?@$%KA

B/0 ,901(.,1 a±(E) ()."3# ,3 ./0 7()**+ |0,±# (33"/"8(.06 ( 9(1.")80 5"./ 9,6"."70 0301#2 E (3;

+,+03.*+ $E$ I3 ./0 6(+0 5(2 b†±(E) )10(.06 ,*. ,- ./0 7()**+ (3 (3."9(1.")80 5"./ 9,6"."70

0301#2 E (3; 69(."(8 +,+03.*+ $p$ I3 ./0 H"1() 60( 9").*10 ./0 ,901(.,1 b±(E)† "6 ,1"#"3(882(3 (33"/"8(.",3 ,901(.,1 ( 6.(.0 ,- ./0 60( 5"./ 30#(."70 0301#2 %E$ G6 "3 ./0 -,*1:;"+036",3(8)(60 ./0 91,F80+ ,- ./0 30#(."70 0301#2 6.(.06 "6 6,870; F2 "3.01910."3# (33"/"8(.",3 ,901(.,16 -,1

30#(."70 0301#2 6.(.06 (6 )10(.",3 ,306 -,1 ./0 ),11069,3;"3# (3."9(1.")80 5"./ 9,6"."70 0301#2 ?(3;

7")0 7016(A$ B/0 ,901(.,16 (990(1"3# "3 ./0 0E9(36",3 ,- u± "3 L=$ ?@$%KA 6(."6-2 ./0 *6*(8 (8#0F1(

{a!(E), a†!!(E #)} = {b!(E), b†!!(E #)} = "E,E!"!!! , ?@$%MA

J%

! !

p p

E E

v v

!"#$ %&' ()*+,-./ 01 ,2* /3445*44 ,6078"/*94"0935 :"-3+ !*58$

;-",,"9# ,2* ,607+0/)09*9, 4)"90- ! 34

! =

!u+

u!

"<=$%>?

398 8*!9"9# 34 .4.35 ,2* )-0@*+,0-4 P± = 12(1 ± "5) 6* !98 ,23, ,2* +0/)09*9,4 u± 01 ! 3-*

-*4)*+,"A*5B 3 -"#2,7 398 5*1,72398*8 ;*B5 4)"90- "9 ,60 8"/*94"094$

C9+* 6* 23A* 3 -*)-*4*9,3,"09 01 ,2* "7/3,-"+*4 6* +39 6-",* ,2* :"-3+ *D.3,"09$ EF)-*44"9#", "9 ,*-/4 01 ,2* +0/)09*9,4 u± 01 ,2* :"-3+ 4)"90- 6* !98

(#0 ! #1)u+ = 0, (#0 + #1)u! = 0 <=$%G?

H2* #*9*-35 405.,"09 ,0 ,2*4* *D.3,"094 +39 I* "9/*8"3,*5B 6-",,*9 34

u+ = u+(x0 + x1), u! = u!(x0 ! x1) <=$%J?

K*9+* u± 3-* ,60 63A* )3+L*,4 /0A"9# 3509# ,2* 4)3,"35 8"/*9,"09 -*4)*+,"A*5B ,0 5*1, (u+) 398 ,0,2* -"#2, (u!)$ M0,"+* ,23, 3++0-8"9# ,0 0.- +09A*9,"09 ,2* 5*1,7/0A"9# u+ "4 3 -"#2,72398*8 4)"90-

<)04","A* 2*5"+",B? 62*-*34 ,2* -"#2,7/0A"9# u! "4 3 5*1,72398*8 4)"90- <9*#3,"A* 2*5"+",B?$

N1 6* 639, ,0 "9,*-)-*, <=$%G? 34 ,2* 63A* *D.3,"09 10- ,6078"/*94"0935 ;*B5 4)"90-4 6* 23A*

,2* 105506"9# 63A* 1.9+,"094 10- 1-** )3-,"+5*4 6",2 6*55 8*!9*8 /0/*9,./ pµ = (E, p)$

v(E)± (x0 ± x1) =

1"L

e!iE(x0±x1) 6",2 p = #E <=$%O?

P4 ", "4 3563B4 ,2* +34* 6",2 ,2* :"-3+ *D.3,"09 6* 23A* I0,2 )04","A* 398 9*#3,"A* *9*-#B 405.,"094$

!0- u+Q 4"9+* E = !pQ 6* 4** ,23, ,2* 405.,"094 6",2 )04","A* *9*-#B 3-* ,204* 6",2 9*#3,"A*

O&

p

E

!"#$ %&' ())*+, -) ,.* */*+,0"+ !*/1$

2.*0* e "3 ,.* +.40#* -) ,.* )*05"-63$ 7"6+* 2* 43385*1 ,.4, ,.* 9*+,-0 :-,*6,"4/ 940"*3 41"4;4,"<+4//=> 2* +46 43385* ", ,- ;* 4::0-?"54,*/= +-63,46, 4, *4+. ,"5*$

@.*6> 2* -6/= .49* ,- 3** 2.4, "3 ,.* *))*+, -) AB$&CD -6 ,.* 94+885 1*:"+,*1 "6 !"#$ A%%D$

E.4, 2* !61 "3 ,.4, ,.* ,2- ;046+.*3 5-9* 43 3.-26 "6 !"#$ A%&D 0*38/,"6# "6 3-5* -) ,.* 6*#4,"9*

*6*0#= 3,4,*3 -) ,.* v+ ;046+. 41F8"0"6# :-3","9* *6*0#= 2."/* 4 345* 685;*0 -) ,.* *5:,= :-3","9*

*6*0#= 3,4,*3 -) ,.* -,.*0 ;046+. v! 2"// ;*+-5* *5:,= 6*#4,"9* *6*0#= 3,4,*3$ G.=3"+4//= ,."35*463 ,.4, ,.* *?,*064/ */*+,0"+ !*/1 E +0*4,*3 4 685;*0 -) :40,"+/*<46,":40,"+/* :4"03 -8, -) ,.*94+885$ H*6-,"6# ;= N ! eE ,.* 685;*0 -) 38+. :4"03 +0*4,*1 ;= ,.* */*+,0"+ !*/1 :*0 86", ,"5*>,.* !64/ 94/8*3 -) ,.* +.40#*3 QV 461 QA 40*

QA(!0) = (N " 0) + (0 " N) = 0

QV (!0) = (N " 0) " (0 " N) = 2N AB$&BD

@.*0*)-0* 2* +-6+/81* ,.4, ,.* +-8:/"6# ,- ,.* */*+,0"+ !*/1 :0-18+*3 4 9"-/4,"-6 "6 ,.* +-63*094,"-6

-) ,.* 4?"4/ +.40#* :*0 86", ,"5* #"9*6 ;= !QA ! eE $ I6 ,806 ,."3 "5:/"*3 ,.4,

"µJµA ! e!E AB$&JD

2.*0* 2* .49* 0*3,-0*1 ! ,- 54K* +/*40 .-2 ,.* 9"-/4,"-6 "6 ,.* +-63*094,"-6 -) ,.* 4?"4/ +800*6,"3 4 :80*/= F846,85 *))*+,$ L, ,.* 345* ,"5* !QV = 0 #84046,**3 ,.4, ,.* 9*+,-0 +800*6, 0*54"63+-63*09*1 4/3- F846,85 5*+.46"+4//=> "µJµ

V = 0$

E* .49* M83, 3,81"*1 4 ,2-<1"5*63"-64/ *?45:/* -) ,.* L1/*0<N*//<O4+K"2 4?"4/ 46-54/= P&%Q$

CR

First we compute the spectrum

Then we construct the second quantized vacuum

It is like defining a determinant, they represent fermionicdegrees of freedom

When the gauge field is turned on, the left and right handed currents are not conserved, but the vector current is. The

anomaly is related to the chiral structure of the theory. Vector like currents represent no problem

These lessons are easy to generalize, but we need more sophisticated technology


Chirality, mass and regulators. Heuristics

exp(!!eff [A]) = “DetD(A)”

We are evaluating the effective action for the fermions in the presence of external fields

!"# #$%&'#'()*+ ,-%%.#,,)&' &/ ("# %*). %.&0-1()&' 0),*%%#*., 2"#' ("# #+#1(.)1 !#+0 .#*1"#, ("#

1.)()1*+ 3*+-# Ecrit *( 2")1" ("# #$%&'#'( ), &/ &.0#. &'#

Ecrit =m2c3

!e! 1.3 " 1016 V cm!1 456789

!"), ), )'0##0 * 3#.: ,(.&'; !#+0 2")1" ), #$(.#<#+: 0)/!1-+( (& %.&0-1#6 = ,)<)+*. #//#1(> "&2#3#.>

(*?#, %+*1# *+,& )' * ()<#@3*.:)'; #+#1(.)1 !#+0 ABCD *'0 ("#.# ), ("# "&%# ("*( %*). %.&0-1()&' 1&-+0

E# &E,#.3#0 )' ("# %.#,#'1# &/ ("# *+(#.'*()'; #+#1(.)1 !#+0 %.&0-1#0 E: * +*,#.6

!"# "#-.),()1 0#.)3*()&' ("*( 2# /&++&2#0 "#.# 1*' E# <*0# <&.# %.#1),# )' FGH6 !"#.# ("#

0#1*: &/ ("# 3*1--< )'(& #+#1(.&'@%&,)(.&' %*)., 1*' E# 1&<%-(#0 /.&< ("# )<*;)'*.: %*.( &/ ("#

#//#1()3# *1()&' i![Aµ] )' ("# %.#,#'1# &/ * 1+*,,)1*+ ;*-;# %&(#'()*+ Aµ

i![Aµ] # + + + . . .

= log det

!1 $ ie/A

1

i!/ $ m

"4567I9

!"), 0#(#.<)'*'( 1*' E# 1&<%-(#0 -,)'; ("# ,(*'0*.0 "#*( ?#.'#+ (#1"')J-#,6 !"# %.&E*E)+)(: &/

%*). %.&0-1()&' ), %.&%&.()&'*+ (& ("# )<*;)'*.: %*.( &/ i![Aµ] *'0 ;)3#,

W =e2E2

4"3

"#

n=1

1

n2e!n ! m2

eE 456779

K-. ,)<%+# *.;-<#'( E*,#0 &' (-''#+)'; )' ("# H).*1 ,#* ;*3# &'+: ("# +#*0)'; (#.< &/ L1"2)';#.M,

.#,-+( 4567796 !"# .#<*)')'; (#.<, 1*' E# *+,& 1*%(-.#0 )' ("# NOP *%%.&$)<*()&' E: (*?)'; )'(&

*11&-'( ("# %.&E*E)+)(: &/ %.&0-1()&' &/ ,#3#.*+ %*).,> )6#6 ("# (-''#+)'; &/ <&.# ("*' &'# #+#1(.&'

(".&-;" ("# E*..)#.6

Q#.# 2# "*3# )++-,(.*(#0 ("# 1.#*()&' &/ %*.()1+#, E: ,#<)1+*,,)1*+ ,&-.1#, )' F-*'(-< R)#+0

!"#&.: -,)'; ,)<%+# #$*<%+#,6 S#3#.("#+#,,> 2"*( 2# +#*.'#0 "*, )<%&.(*'( *%%+)1*()&', (& ("#

,(-0: &/ J-*'(-< !#+0, )' 1-.3#0 E*1?;.&-'0,6 T' F-*'(-< R)#+0 !"#&.: )' U)'?&2,?) ,%*1#@()<#

("# 3*1--< ,(*(# ), )'3*.)*'( -'0#. ("# V&)'1*.W# ;.&-% *'0 ("),> (&;#("#. 2)(" ("# 1&3*.)*'1# &/

("# ("#&.: -'0#. X&.#'(Y (.*',/&.<*()&',> )<%+)#, ("*( *++ )'#.()*+ &E,#.3#., *;.## &' ("# '-<E#. &/

%*.()1+#, 1&'(*)'#0 )' * J-*'(-< ,(*(#6 !"# E.#*?)'; &/ ,-1" )'3*.)*'1#> *, "*%%#'#0 )' ("# 1*,# &/

1&-%+)'; (& * ()<#@3*.:)'; ,&-.1# *'*+:Y#0 *E&3#> )<%+)#, ("*( )( ), '&( %&,,)E+# *':<&.# (& 0#!'#

* ,(*(# 2")1" 2&-+0 E# .#1&;')Y#0 *, ("# 3*1--< E: *++ &E,#.3#.,6

!"), ), %.#1),#+: ("# ,)(-*()&' 2"#' !#+0, *.# J-*'()Y#0 &' 1-.3#0 E*1?;.&-'0,6 T' %*.()1-+*.>

)/ ("# E*1?;.&-'0 ), ()<#@0#%#'0#'( 4*, )( "*%%#', )' * 1&,<&+&;)1*+ ,#(-% &. /&. * 1&++*%,)'; ,(*.9

0)//#.#'( &E,#.3#., 2)++ )0#'()/: 0)//#.#'( 3*1--< ,(*(#,6 =, * 1&',#J-#'1# 2"*( &'# &E,#.3#. 1*++

("# 3*1--< 2)++ E# /-++ &/ %*.()1+#, /&. * 0)//#.#'( &E,#.3#.6 !"), ), %.#1),#+: 2"*( ), E#")'0 ("#

%"#'&<#'&' &/ Q*2?)'; .*0)*()&' ABZD6 !"# #<),,)&' &/ %*.()1+#, E: * %":,)1*+ E+*1? "&+# /&.<#0

58

!+[A] = parity invariant = Re!eff [A]

!!

[A] = parity violating = Im!eff [A]

In the first case we can always find a mass term and regulate the theory preserving gauge invariance

When parity invariant does not hold, the effective action is complex, in general there is no a priori regulartor, and anomalies may appear

Gauge invariance and current conservation are intimately connected


exp(!!eff [A]) = “DetD(A)” =

!d!d! exp(

!! D(A) !)

![Aµ ! Dµ!] =

!Tr ! Dµ

" !

" Aµ

"

!Dµjµ

Similarly, with diffeomorphisms we test energy-momentum conservation! !

! gµ!! Tµ!

Gauge anomalies can appear in any even number of dimensions. This depends of course on the fermion representation of the gauge group

In dimension 2n, the first anomalous graph is an n+1 -gon !µ1µ2...µ2n

In one of the vertices we test current conservation and in the other we have the polarization tensors,there are n independent momenta and n independent polarization vectors

!gµ! = !µ"! + !!"µ


Pure gravity anomalies only occur in 4n+2 dimensions

[C,!] = 0 in d = 4n + 2

{C,!} = 0 in d = 4n

CPT transforms particles of given helicity into particles of the same helicityWe can have chirally asymmetric gravitational couplings

Precisely the opposite of the previous case

! = !0!

1. . . !

d!1

Is the equivalent of the chirality operator in d=4! !5


Feynman graphs vs Index Theory

The anomaly computes the change of the fermion determinant under infinitesimal gauge transformations.

The result of testing current conservation will be a polynomial in the external gauge and gravitational fields

We can use Feynman graphs, or use a deep result in mathematics: THE ATIYAH-SINGER INDEX THEOREM

The AS index theorem also applies in more general settings, and it is necessary for global anomalies,however one should not track of the fact that the result obtain is purely perturbative

detDL(A, e) detDR(A, e) D =

!

0 DL

DR 0

"

IndexD = dimKerDL ! dimKerDR

=

!M

ch(F )A(TM)

We now explain the meaning of the symbols under the integral sign


F = dA + A2

=1

2Fµ!dx

µdx

!

We have an antisymmetric matrix of two-forms that can be skew-diagonalized formally

F is an antihermitian matrix of two-forms

!

2!= d" + "

2 =1

4!Rabµ! dx

µdx

!

!

2!

!

!

"

0 x1 0 . . .

"x1 0 0 . . .

.

.

....

. . .

#

$ In 2n dimensions there are n skew eigenvalue two-forms

ch(F ) = Tr eiF/2! A(TM) =!

i

xi/2

sinhxi/2

Chern character, and A-roof genus respectively. The second is a symmetric polynomial in the eigenvalues x_i, hence it can be written always in terms of products of traces of

the curvature two-forms


An Introduction to Anomalies 27

D± = i! µ!

!µ +12

"µab !ab + Aµ

"P± . (2.99)

The index of D+ is defined to be the dimension of the kernel of D+ minusthe kernel of D+

† = D! .

indD+ = dimker D+ ! dimker D! . (2.100)

In principle this is a problem in analysis. What Atiyah and Singer showed [15]is that ind D+ is a number which only depends on the topological set up (2.98)and it is given by the integral over M of a particular characteristic class. Inparticular, for the operator (2.99) the answer is:

indD+ =#

M

[ch(F)A(M)]vo1

A(M) "$

a

xa/2

sinh xa/2

ch(F) = Tr eiF/2# . (2.101)

A(M), the A or Dirac genus of M, is a polynomial in the two forms xa whichcan be rewritten in terms of the Pm’s. Since M is finite dimensional and thexa’s are the 2-forms (2.96), A(M) is a finite polynomial (same for ch(F)).The subscript ‘vol’ in (2.101) means that one has to extract the form whosedegree is equal to the dimension of M. The expansion of A(M) is rathercumbersome, and we will write it in terms of the Pm’s and as a polynomialin the Pontrjagin classes. A careful expansion yields: [20]

A(M) = 1 +1

(4#)2112

Tr R2 +1

(4#)4

%1

288(Tr R2)2 +

1360

Tr R4

&

+1

(4#)6

%1

10368(Tr R2)3 +

14320

Tr R2 Tr R4 +1

5670Tr R6

&

+1

(4#)8

%1

497664(Tr R2)4 +

1103680

(Tr R2)2 Tr R4

+1

68040Tr R2 Tr R6 +

1259200

(Tr R4)2 +1

75600Tr R8

&+ . . .

= 1 +122

!!1

6P1

"+

124

!7

360P2

1 !190

P2

"

+126

!! 31

15120P3

1 +11

3780P1 P2 !

1945

P3

"

+128

'127

604800P4

1 !113

226800P2

1P2 + . . .

+4

14175P1 P3 +

13113400

P22 !

19450

P4

(+ . . . (2.102)

28 L. Alvarez-Gaume

If dim V = r, we have

ch(F) = r +i

2!Tr F +

i2

2(2!)2Tr F2 + . . . +

in

n!(2!)nTr Fn + . . . (2.103)

Combining (2.102) and (2.103) we can compute the form of the index theoremin any dimension. For example, in d = 4 and d = 8:

indD+ =1

(2!)2

!

M

"i2

2Tr F2 +

r48

Tr R2

#d = 4

=1

(2!)4

!

M

"i4

24Tr F4 +

i2

96Tr F2 Tr R2

+r

4608(Tr R2)2 +

r5760

Tr R4

#d = 8

(2.104)

The other operators whose indices are of interest in the computation of anom-alies are obtained by replacing V by some particular vector bundle. For in-stance for the graviton field V = TM, the tangent bundle over M. In this caseA is simply the spin connection taking values on the Lie algebra of SO(2n)in the vector representation (Tab)cd = "a

c"bd ! "a

d"bc . Thus

Tr eRabTab/4! =$

a

2 cosh xa (2.105)

In the standard quantization of a spin 3/2 field [45], one has to add ghostfields to remove unphysical degrees of freedom. The constraints ka #a = 0 and#a " #a + ka! remove two spin 1/2 degrees of freedom of the same chiralityas #a, and the constraint $ a#a = 0 removes one spin 1/2 degree of freedomof opposite chirality. Including the ghost field in the index theorem for a spin3/2 field we get:

ind iD/ 3/2 =!

A(M)(Tr eR/2! ! 1)ch(F) (2.106)

where the last factor accounts for the possibility that the spin 3/2 fieldcarries some extra gauge index. Because of the dimensional dependence ofTr exp R/2!, we exhibit the quantity A(M)Tr(eR/2! – 1). To order 16, thispolynomial has the following form:

A(M)Tr(eR/2! ! 1) = ! 1(4!)2

2 Tr R2

+1

(4!)4

%!1

6(Tr R2)2 +

23Tr R4

&

+1

(4!)6

%! 1

144(Tr R2)3 +

120

Tr R2 Tr R4 ! 445

Tr R6

&

We expand formally in terms of forms of different degree both the gravity part and ...

...the gauge part

28 L. Alvarez-Gaume

If dim V = r, we have

ch(F) = r +i

2!Tr F +

i2

2(2!)2Tr F2 + . . . +

in

n!(2!)nTr Fn + . . . (2.103)

Combining (2.102) and (2.103) we can compute the form of the index theoremin any dimension. For example, in d = 4 and d = 8:

indD+ =1

(2!)2

!

M

"i2

2Tr F2 +

r48

Tr R2

#d = 4

=1

(2!)4

!

M

"i4

24Tr F4 +

i2

96Tr F2 Tr R2

+r

4608(Tr R2)2 +

r5760

Tr R4

#d = 8

(2.104)

The other operators whose indices are of interest in the computation of anom-alies are obtained by replacing V by some particular vector bundle. For in-stance for the graviton field V = TM, the tangent bundle over M. In this caseA is simply the spin connection taking values on the Lie algebra of SO(2n)in the vector representation (Tab)cd = "a

c"bd ! "a

d"bc . Thus

Tr eRabTab/4! =$

a

2 cosh xa (2.105)

In the standard quantization of a spin 3/2 field [45], one has to add ghostfields to remove unphysical degrees of freedom. The constraints ka #a = 0 and#a " #a + ka! remove two spin 1/2 degrees of freedom of the same chiralityas #a, and the constraint $ a#a = 0 removes one spin 1/2 degree of freedomof opposite chirality. Including the ghost field in the index theorem for a spin3/2 field we get:

ind iD/ 3/2 =!

A(M)(Tr eR/2! ! 1)ch(F) (2.106)

where the last factor accounts for the possibility that the spin 3/2 fieldcarries some extra gauge index. Because of the dimensional dependence ofTr exp R/2!, we exhibit the quantity A(M)Tr(eR/2! – 1). To order 16, thispolynomial has the following form:

A(M)Tr(eR/2! ! 1) = ! 1(4!)2

2 Tr R2

+1

(4!)4

%!1

6(Tr R2)2 +

23Tr R4

&

+1

(4!)6

%! 1

144(Tr R2)3 +

120

Tr R2 Tr R4 ! 445

Tr R6

&

Multiply them together, and select the 2n-th form


The characteristic polynomials have important properties:

They are closed, as a consequence of the Bianchi identities

F = dA + A2

dF + AF ! FA = 0 = DF

dTrF 2 = 2 TrFdF = 2 TrFDF ! 2 Tr[A, F ] = 0

The difference between two characteristic polynomials computed with two continuouly connected connections differ

by a total differential: A Chern-Simons form

Pm(F1) ! Pm(F0) = dQ0

2m!1(A, F )

A(t) A(0) = A0 A(1) = A1

Q0

3 = Tr(AdA +2

3A3)

Basic result 1 : !Q0

2m!1 = dQ1

2m!2

Stora-Zumino descent equations


Basic result II :

For each chiral field involved in d=2n dimensions, we determine its Index polynomial, extract the form of degree 2n+2, and then apply

the Stora-Zumino descent procedure: This is the ANOMALY

Pure gauge in d=2n :

I1/2(F ) ! ±TrFn+1± StrT a1T

a2 . . . Tan+1

!"#$# Q %& '"# #(#)'$%) )"*$+# ,- '"# ),$$#&.,/0%/+ .*$'%)(# */0 T3 %& '"# #%+#/1*(2# !%'" $#&.#)'

', '"# '"%$0 +#/#$*',$ ,- '"# 34567 +$,2. %/ '"# ),$$#&.,/0%/+ $#.$#&#/'*'%,/8 T3 = 12!

3 -,$ '"#

0,29(#'& */0 T3 = 0 -,$ '"# &%/+(#'&: ;,$ '"# !$&' -*<%(= ,- >2*$?& 5u@ d7 */0 (#.',/& 5e@ "e7 !# "*1#

'"# -,((,!%/+ !#(0 ),/'#/'

>2*$?&8

!u!

d!

"

L, 16

u!R, 23

d!R, 23

(#.',/&8

!"e

e

"

L,! 12

eR,!1 5A:BC7

!"#$# # = 1, 2, 3 (*9#(& '"# ),(,$ >2*/'2< /2<9#$ */0 '"# &29&)$%.' %/0%)*'#& '"# 1*(2# ,- '"# !#*?"=.#$)"*$+# Y : D#/,'%/+ '"# $#.$#&#/'*'%,/& ,- 345E7!34567!45F7 9= (nc, nw)Y @ !%'" nc */0 nw

'"# $#.$#&#/'*'%,/& ,- 345E7 */0 34567 $#&.#)'%1#(= */0 Y '"# "=.#$)"*$+#@ '"# <*''#$ ),/'#/' ,-

'"# 3'*/0*$0 G,0#( ),/&%&'& ,- * '"$## -*<%(= $#.(%)*'%,/ ,- '"# $#.$#&#/'*'%,/&8

(#-'H"*/0#0 -#$<%,/&8 (3, 2)L16

(1, 2)L! 1

2

5A:AI7

$%+"'H"*/0#0 -#$<%,/&8 (3, 1)R23

(3, 1)R! 1

3(1, 1)R

!1.

J/ ),<.2'%/+ '"# '$%*/+(# 0%*+$*< !# "*1# FI .,&&%9%(%'%#& 0#.#/0%/+ ,/ !"%)" -*)',$ ,- '"# +*2+#

+$,2. 345E7!34567!45F7 ),2.(#& ', #*)" 1#$'#K8

345E73 345673 45F73

345E72 34567 34567 45F7

345E7245F7 34567 45F72

345E7 345672

345E7 34567 45F7

345E7 45F72

J' %& #*&= ', )"#)? '"*' &,<# ,- '"#< 0, /,' +%1# $%&# ', */,<*(%#&: ;,$ #K*<.(# '"# */,<*(= -,$

'"# 345E73 )*&# )*/)#(& 9#)*2&# (#-' */0 $%+"'H"*/0#0 >2*$?& '$*/&-,$< %/ '"# &*<# $#.$#&#/'*'%,/:

J/ '"# )*&# ,- 345673 '"# )*/)#((*'%,/ "*..#/& '#$< 9= '#$< 9#)*2&# ,- '"# L*2(% <*'$%)#& %0#/'%'=

!a!b = $ab + i%abc!c '"*' (#*0& ',

tr#!a{!b, !c}

$= 2 (tr!a) $bc = 0. 5A:AF7

M,!#1#$ '"# "*$0#&' */,<*(= )*/)#((*'%,/ ),/0%'%,/ ', &*'%&-= %& '"# ,/# !%'" '"$## 45F7N&: J/ '"%&

AB

Possible anomalies in the Standard Model


In supergravity theories we have to identify the fields that will lead to anomalies, compute the corresponding anomaly polynomials, and verify whether they cancel by themselves or

with the help of counterterms

Chiral fermions Chiral gravitini Self-dual RR fields

! !µ !ab ! Fµ1µ2...µn" F = ±F

I1/2 =!

i

xi/2

sinhxi/2I3/2 =

!

i

xi/2

sinhxi/2(!1 +

"

j

2 cosh xj) IA = !

1

8

!

i

xi

tanhxi

If there are gauge quantum numbers, we have to multiply each of these polynomials by the corresponding Chern characters

We are ready to compute some interesting examples


Type IIB d=10

(!I1/2 + I3/2 + IA)12 = 0

A related cancellation in d=6 21 I1/2 ! I3/2 + 8 IA = 0

Type I d=10


tensor gauge field. Although the cancellation of anomalies is rather strik-ing, the possible phenomenological applications of this theory are not verypromising (in all known Kaluza-Klein compactifications it leads to a vector-like four-dimensional theory, and the gauge group cannot be very big). Thesecond non-trivial anomaly cancellation was recently discovered by Greenand Schwarz for type I superstring theories with gauge group SO(32) [12,13].Green and Schwarz also noticed that in the field theory limit of the stringtheory (and N = 1 super Yang–Mills coupled to N = 1 supergravity in d =10) the anomalies cancel also for E8 ! E8. Soon after the discovery of thisanomaly cancellation, the Princeton group [58] constructed the so-called “het-erotic” string with gauge group E8 ! E8. This theory is now being consideredrather intensely and it may have non-trivial phenomenological implications.In order to compute the anomalies in this theory we have to first know thefield content. The N = 1 supergravity multiplet contains the graviton ea

µ,a left-handed Weyl–Majorana gravitino !µ, a right-handed Weyl–Majoranafield !, a 2-index antisymmetric tensor Bµ" and the dilaton #. The N = 1super–Yang–Mills multiplet contains simply the gauge field Aa

µ and a left-handed Weyl–Majorana gluino !a where a = 1, . . . ,dim G = N (G = gaugegroup). Let us shift all the gravitational anomalies to the local Lorentz trans-formations for simplicity. From the formulae in Sect. 3 we can write downimmediately the 12-form which determines the anomaly through the descentequations. The characteristic polynomial we have to expand is in dimension d:

A!

R2$

"Tr eiF/2$ (gluinos)

A!

R2$

" #Tr(eR/2$ " I) + d " 1

$(gravitino)

"A!

R2$

"(right-handed Weyl–Majorana field)

(4.75)

For d = 10 we have to extract the 12-form piece of (4.75). If we factor outvarious common numerical factors, we get after some algebra:

I12 = " 115

Tr F6 +124

Tr F4Tr R2 " Tr F2

960(5(Tr R2)2 + 4Tr R4)

+N " 496

7560Tr R6 +

%N " 496

5760+

18

&Tr R4TrR2

+%

N " 49613824

+132

&(Tr R2)3 (4.76)

where the trace over the gauge field F is taken in the adjoint representation.By computing the Chern–Simons form associated with I12, and going throughthe descent equations, we will get the anomaly under combined gauge andlocal Lorentz transformations. If we only included the fermion field contribu-tion to the anomaly, it is clear that (4.76) does not vanish for any gauge groupand we would have to throw away the theory. The non-trivial step taken by


Cancelling ingenuity

Green and Schwarz proved a remarkable result

68 L. Alvarez-Gaume

Green and Schwarz is to analyze under what circumstances the anomaly in-duced by (4.76) can be cancelled by counterterms. Let us now repeat theirargument. If we consider the leading terms in I12, i.e. the term Tr F6 andTr R6, the anomalies they will induce are respectively Q1

10(!) and Q110 (A),

and we know from previous arguments that neither can be obtained as thevariation of a local functional. Thus if we require that the Tr R6 term vanish,we obtain N = 496, i.e. the gauge group must have 496 generators. Next werequire that the group in the adjoint representation should not have a 6thorder irreducible Casimir, i.e. we require that

Tr F6 = " Tr F2 Tr F4 + #(Tr F2)3 . (4.77)

In this case the anomalous variation of the e!ective action will be proportionalto Q1

2 Tr F4 and Q16 Tr F2, and there may be hope of finding a local countert-

erm cancelling this variation (for instance terms like Q03 Q0

7 will under gaugevariations generate Q1

2 Tr F4 and Tr F2 Q16). A crucial role in constructing

the appropriate counterterms is played by the two form B = Bµ$dxµ ! dx$/2appearing in the supergravity multiplet. A three-form field strength is formedfrom this potential:

H = dB + Q3L + k Q3y

d Q3L = Tr R2d Q3y = Tr F2 . (4.78)

The introduction of the Lorentz Chern–Simons form is an important modi-fication of the definition of H in the N = 1 d = 10 supergravity Lagrangian[60].

Since

%Q3y = "d Q12y

%Q3L = "d Q12L . (4.79)

H is gauge and local Lorentz invariant only if

%B = Q12L + k Q1

2y . (4.80)

What Green and Schwarz [12] proved (see also [13] for the presentation we arefollowing here) is that a local counterterm exists which cancels the anomalieswhenever I12 can be factorized in the form:

I12 = (Tr R2 + k Tr F2)X8 (4.81)

where X8 is an invariant 8-form constructed in terms of F and R. Straight-forward, but somewhat tedious algebra shows that the factorization (4.81)occurs only if

Tr F6 =148

Tr F2 Tr F4 " 114400

(Tr F2)3

k ="130

(4.82)

68 L. Alvarez-Gaume

Green and Schwarz is to analyze under what circumstances the anomaly in-duced by (4.76) can be cancelled by counterterms. Let us now repeat theirargument. If we consider the leading terms in I12, i.e. the term Tr F6 andTr R6, the anomalies they will induce are respectively Q1

10(!) and Q110 (A),

and we know from previous arguments that neither can be obtained as thevariation of a local functional. Thus if we require that the Tr R6 term vanish,we obtain N = 496, i.e. the gauge group must have 496 generators. Next werequire that the group in the adjoint representation should not have a 6thorder irreducible Casimir, i.e. we require that

Tr F6 = " Tr F2 Tr F4 + #(Tr F2)3 . (4.77)

In this case the anomalous variation of the e!ective action will be proportionalto Q1

2 Tr F4 and Q16 Tr F2, and there may be hope of finding a local countert-

erm cancelling this variation (for instance terms like Q03 Q0

7 will under gaugevariations generate Q1

2 Tr F4 and Tr F2 Q16). A crucial role in constructing

the appropriate counterterms is played by the two form B = Bµ$dxµ ! dx$/2appearing in the supergravity multiplet. A three-form field strength is formedfrom this potential:

H = dB + Q3L + k Q3y

d Q3L = Tr R2d Q3y = Tr F2 . (4.78)

The introduction of the Lorentz Chern–Simons form is an important modi-fication of the definition of H in the N = 1 d = 10 supergravity Lagrangian[60].

Since

%Q3y = "d Q12y

%Q3L = "d Q12L . (4.79)

H is gauge and local Lorentz invariant only if

%B = Q12L + k Q1

2y . (4.80)

What Green and Schwarz [12] proved (see also [13] for the presentation we arefollowing here) is that a local counterterm exists which cancels the anomalieswhenever I12 can be factorized in the form:

I12 = (Tr R2 + kTr F2)X8 (4.81)

where X8 is an invariant 8-form constructed in terms of F and R. Straight-forward, but somewhat tedious algebra shows that the factorization (4.81)occurs only if

Tr F6 =148

Tr F2 Tr F4 " 114400

(Tr F2)3

k ="130

(4.82)An Introduction to Anomalies 69

and

X8 =124

Tr F4 ! 17200

(Tr F2)2 ! 1240

Tr F2 Tr R2 +18Tr R4 +

132

(Tr R2)2 .

(4.83)Using (2.35) it is easy to show that (4.79) is satisfied by SO(32). Moreover,if we use (2.39) and the fact that the gluinos belong to the representation(248,1) + (1,248) of E8 " E8 (so that Tr(F1 + F2)6 = Tr F6

1 + Tr F62, etc.).

It can be checked that (4.79) also holds for E8 " E8, and that these twogroups have 496 generators. Once the factorization (4.81) takes place, thenthe counterterm that cancels the anomaly induced by I12 is simply:

Sc =! "

4#

Q3L ! 130

Q3y

$X7 ! 6BX8

%

dX7 = X8 . (4.84)

(X8 is a closed form because it is an invariant polynomial.) For the consci-entious reader who will check this computation, we should add one moretechnical detail: given an invariant polynomial of the form Tr Fn, using thedescent equations, it follows that Q1

2n!2 will be the anomaly. If, however, wehave terms like Tr F2 Tr F4, when we compute its “Chern–Simons” form, wecould have in general Q(!, ") = ! Q0

3 Tr F4 + " Tr F2 Q07 with !+" = 1. Vari-

ous choices of Q(!, ") di!er only by a total derivative, i.e. Q(!, ")!Q(!", "") =d"2n!2, so that one can go from the form of the anomaly given by Q(!, ")to that induced by Q(!", "") by simply varying a local counterterm. Thuswhich choice one makes of ! and " is essentially irrelevant. There is howevera canonical choice selected by Bose symmetry (of the external gluon or gravi-ton lines in the Feynman diagrams contributing to the anomaly). Using Bosesymmetry it is easy to show that the correct Chern–Simons form one wouldget for a term of the form Tr Fp Tr Fq is ! Q0

2p!1 Tr Fq + " Tr Fp Q02q!1 with

! = p/(p + q), " = q/(p + q). It is quite remarkable that in this context theanomaly cancellation pins down the gauge group to just two choices. This to-gether with the possibility that superstring theories with these gauge groupsare very likely finite in the ultraviolet (and therefore provide a consistent the-ory including quantum gravity non-trivially), make it very worthwhile tryingto understand what makes these theories so special. (See [59] for some of therecent attempts to make contact between superstring theory and low-energyphenomenology.)

5 Anomalies in #-Models

There is an extension of the treatment of anomalies presented in Sect. 3 whichapplies to the coupling of fermions to arbitrary #-model fields. These typeof anomalies where introduced in [8] where a thorough topological analysisis presented in terms of K-theory. The general conclusions concerning the

Then by modifying the transformation rules and adding some specific counterterms, the anomaly cancels


and

X8 =124

Tr F4 ! 17200

(Tr F2)2 ! 1240

Tr F2 Tr R2 +18Tr R4 +

132

(Tr R2)2 .

(4.83)Using (2.35) it is easy to show that (4.79) is satisfied by SO(32). Moreover,if we use (2.39) and the fact that the gluinos belong to the representation(248,1) + (1,248) of E8 " E8 (so that Tr(F1 + F2)6 = Tr F6

1 + Tr F62, etc.).

It can be checked that (4.79) also holds for E8 " E8, and that these twogroups have 496 generators. Once the factorization (4.81) takes place, thenthe counterterm that cancels the anomaly induced by I12 is simply:

Sc =! "

4#

Q3L ! 130

Q3y

$X7 ! 6BX8

%

dX7 = X8 . (4.84)

(X8 is a closed form because it is an invariant polynomial.) For the consci-entious reader who will check this computation, we should add one moretechnical detail: given an invariant polynomial of the form Tr Fn, using thedescent equations, it follows that Q1

2n!2 will be the anomaly. If, however, wehave terms like Tr F2 Tr F4, when we compute its “Chern–Simons” form, wecould have in general Q(!, ") = ! Q0

3 Tr F4 + " Tr F2 Q07 with !+" = 1. Vari-

ous choices of Q(!, ") di!er only by a total derivative, i.e. Q(!, ")!Q(!", "") =d"2n!2, so that one can go from the form of the anomaly given by Q(!, ")to that induced by Q(!", "") by simply varying a local counterterm. Thuswhich choice one makes of ! and " is essentially irrelevant. There is howevera canonical choice selected by Bose symmetry (of the external gluon or gravi-ton lines in the Feynman diagrams contributing to the anomaly). Using Bosesymmetry it is easy to show that the correct Chern–Simons form one wouldget for a term of the form Tr Fp Tr Fq is ! Q0

2p!1 Tr Fq + " Tr Fp Q02q!1 with

! = p/(p + q), " = q/(p + q). It is quite remarkable that in this context theanomaly cancellation pins down the gauge group to just two choices. This to-gether with the possibility that superstring theories with these gauge groupsare very likely finite in the ultraviolet (and therefore provide a consistent the-ory including quantum gravity non-trivially), make it very worthwhile tryingto understand what makes these theories so special. (See [59] for some of therecent attempts to make contact between superstring theory and low-energyphenomenology.)

5 Anomalies in #-Models

There is an extension of the treatment of anomalies presented in Sect. 3 whichapplies to the coupling of fermions to arbitrary #-model fields. These typeof anomalies where introduced in [8] where a thorough topological analysisis presented in terms of K-theory. The general conclusions concerning the

This is the case for:

Spin(32)/Z2

E8 ! E8

HeteroticStrings


Local anomalies is not the end...

Once the local anomalies have been cancelled, we may start worrying about global anomalies. This is a rather subtle subject, where the index theorem,the spectral flow and a good deal of differential topology are needed in order to test the theory

under these transformations. This is Witten’s landfor the most part.

The group of gauge transformations corresponds to transformations that are continuously related to the identity. These are the gauge or diff that are dealt

with by Gauss’s law. This space however need not be simply connected. If:

S2n!1

! R G ! {g : S2n!1 " G}

!1(G) != "

Gauss’s law can be integrated locally, but not globally, it may turn out that the partition function of the theory vanishes identically. This is the case the

famous Witten anomaly in d=4 G=SU(2) !4(SU(2)) = Z2


The computations in the gravitational case are rather involved, and it isremarkable that all theories that are free of local gauge and gravity

anomalies, do not suffer from global diff anomalies (Witten)

We have not exhausted the fauna of possible anomalies in string and supergravity theories, specially if we consider the contributions of branes, however the same technology applies to them, although the form of the relevant anomaly polynomials includes contributions also from Chern-

Simons forms and RR-fields. K-theory is quite useful

Similar arguments apply to orbifold field theories. They imply Horava-Witten, and Green-Schwarz anomaly analysis. Useful in the treatment of

theories with large extra dimensions


Conclusions

The cancellation of anomalies in supergravity theories puts very stringent contraints on the theories that are consistent

It provides an excellent arena to learn the subtle properties of gauge and gravitational theories with chiral matter

Apart from the original applications, they are needed still when tryingto obtain low-dimensional realistic models in theories with flux

compactifications, large extra dimensions, etc where the contributionof branes wrapped around nontrivial cycles is important.

An invitation to Anomalies -...

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