Global Anomalies in M-Theory

Home , Gravitational anomaly

CERN-TH/97-277

hep-th/9710126

Global Anomalies in M -theory

Mans Henningson

Theory Division, CERN

CH-1211 Geneva 23, Switzerland

[email protected]

Abstract

We rst consider M -theory formulated on an op en eleven-dimensional

spin-manifold. There is then a p otential anomaly under gauge transforma-

tions on the E bundle that is de ned over the b oundary and also under

di eomorphisms of the b oundary. We then consider M -theory con gura-

tions that include a ve-brane. In this case, di eomorphisms of the eleven-

manifold induce di eomorphisms of the ve-brane world-volume and gauge

transformations on its normal bundle. These transformations are also po-

tentially anomalous. In b oth of these cases, it has previously b een shown

that the p erturbative anomalies, i.e. the anomalies under transformations

that can be continuously connected to the identity, cancel. We extend this

analysis to global anomalies, i.e. anomalies under transformations in other

comp onents of the group of gauge transformations and di eomorphisms.

These anomalies are given by certain top ological invariants, that we explic-

itly construct.

CERN-TH/97-277

Octob er 1997

1. Intro duction

The consistency of a theory with gauge- elds or dynamical gravity requires that the

e ective action is invariant under gauge transformations and space-time di eomorphisms,

usually referred to as cancelation of gauge and gravitational anomalies. The rst step

towards establishing that a given theory is anomaly free is to consider transformations

that are continuously connected to the identity. The cancelation of the corresp onding

anomalies, often called p erturbative anomalies, imp oses some constraints on the chiral

eld content of the theory. Given that the p erturbative anomalies cancel, it makes sense

to investigate transformations in other comp onents of the group of gauge transformations

and di eomorphisms. An anomaly under such a transformation is usually referred to

as a global anomaly. A general formula for the quantum contribution of chiral elds to

global anomalies was given in [1]. Provided that the p erturbative anomalies cancel, the

global anomaly is a top ological invariant, i.e. it it invariant under smo oth deformations

of the data, und thus only dep ends on the top ological classes of for example the space-

time manifold and the gauge-bundle. The requirement that the anomaly vanishes for an

arbitrary transformation imp oses some restrictions on these ob jects.

In string theory, the requirements of sup ersymmetry and vanishing anomalies are par-

ticularly constraining, b ecause of the high dimensionality of space-time. For the purp ose

of computing anomalies, it is enough to know the low-energy e ective sup ergravity the-

ory. The non-chiral typ e IIA sup ergravity theory obviously has no anomalies. The chiral

typ e IIB sup ergravity theory has a p otential p erturbative gravitational anomaly, but the

contributions from the various chiral elds `miraculously' cancel against each other [2].

Typ e I sup ergravity coupled to some sup er-Yang-Mills theory always has a non-vanishing

p erturbative quantum anomaly. It can however be cancelled by a Green-Schwarz mech-

anism [3] involving an anomalous transformation law at tree-level for the two-form eld,

provided that the gauge group is SO(32) or E E . This discovery actually preceeded

8 8

the construction of the SO(32) and E E heterotic string theories. Global anomalies in

8 8

string theory was rst considered in [1]. The result is that the known string theories are

free from global anomalies when formulated in ten-dimensional Minkowski space. How-

ever, there are interesting non-trivial restrictions on consistent compacti ctions to lower

dimensions.

There is by now mounting evidence that the di erent string theories should be seen

as particular limits of a conjectured theory called M -theory. In the long wave-length limit, 1

this theory should reduce to eleven-dimensional sup ergravity. It should also admit certain

typ es of top ological defects, in particular space-time b oundaries and ve-brane solitons.

Much has b een learned ab out M -theory by studying the mechanism for cancellation of

p erturbative anomalies for various con gurations including such defects. In this way,itwas

discovered that when M -theory is de ned on an op en eleven-manifold, there is an E E

8 8

sup er-Yang-Mills multiplet propagating on the b oundary. The cancellation of p erturbative

gauge and gravitational anomalies involves a subtle interplay between contributions from

this multiplet, the bulk degrees of freedom and a generalized Green-Schwarz mechanism

[4][5]. The situation is even more interesting for M -theory con gurations involving a

ve-brane. The p erturbative gauge and gravitational anomalies from the ve-brane world-

volume theory can b e partially cancelled by an anomaly in ow from the surrounding eleven-

dimensional space [6][7]. There is however a remaining part, whose cancellation seems

to require additional world-volume interactions and also imp oses a certain top ological

restriction on the ve-brane con guration [7][8].

Given our present incomplete understanding of M -theory, it seems that any further

information ab out this theory would be valuable. The purp ose of the present pap er is to

carry the analysis describ ed ab ove one step further byinvestigating also global anomalies.

In section two, we consider the case of M -theory de ned on an op en eleven-manifold. In

section three, we instead consider M -theory con gurations including a ve-brane. In b oth

of these cases, we derive an explicit formula for the top ological invariants describing the

global anomalies.

2. Anomalies on op en eleven-manifolds

We consider M -theory on an eleven-dimensional op en spin-manifold Y . The massless

degrees of freedom propagating in the bulk of Y are those of the eleven-dimensional su-

p ergravity multiplet, i.e. a metric G , a three-form p otential C and a fermionic

MN MNP

Rarita-Schwinger eld . Atlow energies, the dynamics of these elds is governed by the

eleven-dimensional sup ergravity action [9]. The bulk action p ossesses a classical invariance

under di eomorphisms of Y , and since we are in an o dd numb er of space-time dimensions,

this symmetry is obviously not sp oiled by any chiral anomaly.

However, there is p otentially an anomaly, often called the parity anomaly, in the

bulk of Y , which is asso ciated with the Rarita-Schwinger eld [10]. The op erator in

the kinetic term of this eld is Hermitian in eleven dimensions, but has in nitely many 2

p ositive and negative eigenvalues, leading to a p otential sign problem in the de nition

of the fermionic path-integral measure [2]. This will show up as a change of the

bulk

e ective bulk action under a di eomorphism of Y . To describ e this anomaly, it

bulk

is convenient to intro duce a twelve-dimensional manifold (Y S ) , called the mapping

torus, as follows: We start with the cylinder Y I , where I is an interval, and equip it

with a metric that smo othly interp olates between the original metric on Y at one of the

b oundaries of I and the metric obtained from it by the transformation at the other

b oundary. Finally,we glue together the two b oundaries of I to form (Y S ) . The bulk

anomaly is then given by [11]

3 1

1 1

Index (RS ) Index (D ) ; (2:1) = i

0 bulk

(Y S ) (Y S )

2 2

1 1

where Index (RS ) and Index (D ) denote the indices of the Rarita-Schwinger

(Y S ) (Y S )

and Dirac op erators on (Y S ) . It follows from charge conjugation symmetry that these

indices are even in twelve dimensions, so is a multiple of i, corresp onding to the

bulk

sign ambiguity in the fermionic path integral measure. Precisely in twelve dimensions,

the combination of indices that app ears in (2.1) is related to the signature of

(Y S )

(Y S ) as

1 1 1

= Index (RS ) 3 Index (D ): (2:2)

(Y S ) (Y S ) (Y S )

It follows that is a multiple of 16 and that the bulk anomaly can be written as

(Y S )

: (2:3) =

bulk

(Y S )

The massless degrees of freedom on the b oundary M of Y include a left-handed Rarita-

Schwinger eld and a right-handed spinor eld originating from the Rarita-Schwinger

eld of the bulk theory. There is also a set of left-handed spinor elds in the

adjoint representation of E that together with a set of gauge elds A make up a sup er

Yang-Mills multiplet propagating on the b oundary. These elds give rise to an anomaly

under di eomorphisms of Y that induce di eomorphisms of M , and also under gauge

transformations of the E bundle V over M . To describ e such a transformation , we

consider the mapping torus (M S ) and the E bundle V over it. These ob jects are

constructed in analogy with (Y S ) by identifying the b oundaries of the cylinder M I

1 1

after a twist by . We note that (Y S ) is b ounded by(MS ) . The anomalous change

under the transformation of the e ective action obtained byintegrating out

ef f ef f

the fermionic elds is then given by a general formula derived in [1] as

= ; (2:4)

ef f

where denotes a certain -invariant on (M S ) .

The -invariant on a closed manifold C is de ned as

= lim sign( ) exp(j j); (2:5)

i i

where i indexes the eigenvalues of a certain op erator on C , and the sum runs over all

i such that 6= 0. In general, the expression (2.5) is prohibitively dicult to evaluate.

The situation is b etter if C b ounds some twelve-manifold B , and the gauge bundle can

be extended to a bundle over B . Whether this is actually p ossible or not is a problem in

cob ordism theory. In the situation at hand, where C = (M S ) , it is indeed p ossible

since we can for example cho ose B to be (V S ) . In any case, if C is the b oundary

of B , the -invariant on C can be expressed in terms of a certain op erator D on B . The

op erator D is in fact the one that arises in a calculation of the p erturbative anomaly, i.e.

the anomaly under a transformation that can b e continuously connected to the identity,

as we will now describ e. The Atiyah-Singer index theorem (see for example [12]) gives

the index of D on a closed twelve-manifold as the integral of some characteristic class

I . The p erturbative anomaly is then obtained through a descent pro cedure [13]: Since

I is closed, it can be written lo cally as I = d! , where ! is the asso ciated Chern-

12 12 11 11

Simons form. The latter form is not invariant under in nitesimal di eomorphisms and

gauge transformations, but its variation is a total derivative, i.e. ! = d , where the

sup erscript 1 indicates that the form is linear in the parameter of the transformation.

The p erturbative anomaly is now given by the integral of over the space-time manifold

M . Returning to the case of an op en manifold B with b oundary C , the Atiyah-Pato di-

Singer index theorem (see for example [12]) now states that the -invariant on C is given

Z Z

= Index (D ) I + ! : (2:6)

B 12 11

B C

In our case, C =(M S ) . Recalling the de nition of this manifold as the cylinder

M I with the two b oundaries identi ed after a twist by , we see that the last term

do es not really make sense, since the integrand ! is in general in the expression for

2 4

not invariant under such a transformation and therefore is not well-de ned on (M S ) .

This term should therefore more prop erly be written as an integral over M I , so that

the anomalous change of the e ective action is

Z Z

: (2:7) = i Index (D ) I + !

ef f B 12 11

B M I

As explained in [4], the anomaly from the and is given by half the standard

anomaly I (R) from these elds in typ e I sup ergravity, whereas the anomaly from

Sugra

is the standard anomaly I (R; F ) from this eld in sup er Yang-Mills theory with E

SY M 8

gauge group. The standard anomaly formulas [14] give

1 1 7 31

2 3 2 4 6

I (R)= (trR ) + trR trR trR

Sugra

(2 ) 1296 1080 2835

1 1 5 1 1

2 3 2 2 2 2 2 2 2 4

(trF ) + (trF ) trR trF (trR ) trF trR I (R; F )=

SY M

(2 ) 24 16 192 48

31 31 31

2 3 2 4 6

; + (trR ) + trR trR + trR

10368 4320 5670

(2:8)

where R and F are the Riemann curvature and eld-strength two-forms resp ectively and

tr for a power of F denotes 1=30 of the trace in the adjoint representation of E . Here

3 1

4 2 3 6 2 3

we have used the E identities trF = (trF ) and trF = (trF ) . The integrals of

10 8

I (R) and I (R; F ) on a closed twelve-manifold equal Index (RS ) 3 Index(D )

Sugra SY M 0

and Index(D ) resp ectively, where RS , D and D are the Rarita-Schwinger op erator,

V 0 V

the Dirac-op erator and the Dirac op erator for fermions in the adjoint of E resp ectively.

Again, Index (RS ) 3 Index(D )= , where denotes the signature in twelve dimensions.

Indeed, I (R) equals the Hirzebruch L-p olynomial in this dimension.

Sugra

The characteristic class I = I (R)+ I (R; F ) describing the anomaly from

12 Sugra SY M

the elds , and do es not vanish in general, so the p erturbative anomalies from the

fermionic elds do not cancel. However, it factorizes as I = I ^ I , where

12 4 8

1 1 1

2 2

I = trF trR

(2 ) 4 8

(2:9)

1 1 1 1 1

2 2 2 2 2 2 4

(trF ) + trF trR (trR ) trR I = :

(2 ) 6 6 48 12

We can thus write the anomalous change of the e ective action as

Z Z

= i + Index (D ) I ^ I + ! ^ I ; (2:10)

ef f B B V 4 8 3 8

B M I 5

where ! is the Chern-Simons form of I , i.e. d! = I .

3 4 3 4

The quantum anomaly (2.10) can b e cancelled by a generalized Green-Schwarz mech-

anism involving the three-form p otential C of eleven-dimensional sup ergravity [4][5]. With

some changes, the following discussion can also be adapted to the case of M -theory on a

Z orbifold, where the role of the b oundary is taken over by the orbifold xed p oints. We

b egin by decomp osing I as

I ^ I + I ; (2:11) I =

4 4 8

where

1 1 1

2 2 4 0

(trR ) trR : (2:12) I =

(2 ) 48 12

The Green-Schwarz counterterms are now

Z Z

= i C ^ G ^ G + C ^ I ; (2:13)

Y Y

where G = dC is the invariant four-form eld strength. Note that these terms are bulk

interactions, although the anomaly to b e cancelled is supp orted on the b oundary. The rst

of these terms is the familiar `Chern-Simons' interaction of eleven-dimensional sup ergravity.

The existence of the second term can be inferred from a one-lo op calculation for typ e IIA

strings [15] lifted to eleven dimensions, or from the requirement of p erturbative anomaly

cancelation on the M -theory ve-brane world-volume [6]. To b e able to write this term, it

is crucial that I dep ends only on R and not on F , since the latter eld only propagates

on the b oundary M and not in the bulk of Y .

The change in the Green-Schwarz terms under the transformation is given by

= i C ^ G ^ G + I ; (2:14)

Y @I

where the integral over Y @I means the di erence of the integrals over Y at the two

b oundary p oints of the interval I . By using Stokes' theorem and the fact that @ (Y I )=

M I +Y @I ,we can rewrite this as

Z Z

= i C ^ I + i G ^ G ^ G + I ; (2:15)

GS 8

M I Y I

where in the rst term we have used (2.11) and the condition that

G = I (2:16)

4 6

on M I . This condition follows from the requirement that the b oundary interactions

preserve half of the sup ersymmetry of the bulk theory [4]. In particular, the pullback of

I to M I is trivial in cohomology. We now assign an anomalous transformation law to

C such that the quantity

H = ! C (2:17)

is invarianton MI. This is consistent with the b oundary condition (2.16), which amounts

dH =0 (2:18)

on M I . The anomalous change of the total action = + + can now be

bulk ef f GS

written as

1 1

+ + Index (D ) = i

B B V

(Y S )

16 16

(2:19)

Z Z Z

: I ^ I + H ^ I + G ^ ( G ^ G + I )

4 8 8

1 1

B (M S ) (Y S )

Note that the since H and G are invariant, the integrands in the last two terms are indeed

1 1

well-de ned on (M S ) and (Y S ) resp ectively.

Before we continue, we will rst verify that the expression (2.19) do es not dep end

on the choice of B . We can replace B by some other twelve-manifold B with the same

b oundary (M S ) . The expression for then changes by

1 1

i + Index (D ) Index (D )

~ ~

V B B V

B B

16 16

(2:20)

Index (D ) :

~ ~

B (B ) B (B )

~ ~

Here B (B ) denotes the closed twelve-manifold constructed by gluing together B and

B with opp osite orientation along their b oundaries, and the last two terms originate from

R R

the integrals I ^ I + I ^ I . We can now use the Novikov formula (see for example

4 8 4 8

B B

[16])

= (2:21)

~ ~

B (B ) B

to cancel the signature terms. Furthermore, it follows from charge conjugation symmetry

and the reality of the adjoint representation of E that Index(D ) is always even in twelve

8 V

dimensions, so the expression (2.20) vanishes mo dulo 2i. Since the action always

app ears as exp , such an ambiguity in is harmless. To simplify the expression for the 7

anomaly, we can now cho ose B = (Y S ) and use the prop erties that the signature

is a multiple of 16 and Index (D ) is even. It then follows from (2.19) that, mo dulo 2i,

Z Z Z

= i I ^ I + H ^ I + G ^ ( G ^ G + I ) : (2:22)

4 8 8

1 1

B (M S ) (Y S )

We should also check that do es not dep end on the precise form of H . For this to

be true, we must actually require I to be trivial in cohomology on (M S ) . (This is

of course automatic if the eight-dimensional cohomology group of this space is trivial, as

would be the case for example in compacti cations to d 4 space-time dimensions.) It

then follows from (2.18) that the term in (2.22) involving H actually vanishes so that

Z Z

G ^ G + I ) ; (2:23) = i I ^ I + G ^ (

4 8

B (Y S )

again mo dulo 2i.

The total anomaly (2.23) is in fact a top ological invariant. In particular, vanishes

for a transformation that can b e continuously connected to the identity, i.e. the p ertur-

bative anomalies cancel. To see this, we consider a smo oth deformation of the data. The

variation of the characteristic classes I , I and I are total derivatives, and the same is

4 8

0 0 0

true for the eld strength G, i.e. I = d , I = d , I = d and G = d . To

4 3 8 7

8 7 3

preserve the conditions (2.11) and (2.16), the relations

0 0

= I ^ +

7 4

3 7

(2:24)

must hold mo dulo closed forms. It is then easy to see that the expression (2.23) is invariant,

again provided that I is trivial in cohomology on the b oundary (M S ) of B and

(Y S ) .

The total anomaly is thus determined by the top ological classes of the various

ob jects. We will not address the dicult problem of determining the conditions for it to

vanish. A particularly interesting case is of course M -theory on an eleven-manifold of the

form Y = X J for some closed ten-manifold X and an interval J , which is b elieved to

describ e the strong-coupling limit of the E E heterotic string on X [4]. In this case,

8 8

the considerations in [17][18] concerning global anomalies for the E E heterotic string

8 8

can be carried over to the M -theory setting. 8

3. Anomalies on the ve-brane world-volume

We consider an M -theory con guration including a ve-brane. The world-volume of

the ve-brane de nes a six-manifold W emb edded in the eleven-dimensional spin-manifold

Y on which the theory is de ned. (For simplicity, in this section we will only consider

the case when Y is closed and orientable.) The normal bundle N of W in Y is then an

SO(5) bundle over W . The massless elds of the world-volume theory are those of a six-

dimensional N = 4 tensor multiplet [19], i.e. ve scalars , i = 1;:::;5, a two-form

with anti-selfdual eld strength T = d and fermionic spinors that take their values in

the bundle S constructed from the normal bundle N by using the spinor representation of

the SO(5) structure group.

The classical theory is invariant under di eomorphisms of Y that map the ve-brane

world-volume W to itself. (The invariance under other di eomorphisms is explicitly broken

by the ve-brane.) As in the previous section, such a transformation is describ ed by

the mapping torus (Y S ) . The transformation induces a di eomorphism of W

and a gauge transformation on the bundle S , which we describ e by the mapping torus

1 1

(W S ) and an SO(5) bundle S over it. Obviously,(W S ) is a seven-dimensional

submanifold of the twelve-manifold (Y S ) . For the purp ose of computing anomalies,

we can regard the world-volume theory as an SO(5) gauge theory with fermions in the

spinor representation. This essentially amounts to replacing the eleven-manifold Y by the

total space of the normal bundle N . As stated ab ove, the theory also contains a chiral

two-form and is coupled to non-dynamical gravity induced from the emb edding of W

in Y . Because of the anti-selfduality constraint on the eld strength T = d , there is

no description in terms of a covariant action, but this can be remedied by adding further

anomaly-free elds [2]. The anomalous change of the e ective action under the

ef f ef f

transformation again follows from the general formula in [1]. Wethus get = ,

ef f

1 1

where now is an -invariant on (W S ) . Assuming that (W S ) b ounds some

eight-manifold E , this can be expressed as

Z Z

= i Index (D ) J + ! : (3:1)

ef f E 8 7

E W I

The anomaly p olynomial J is here given by

1 1 1 1 1 1

2 2 4 2 2 2 2 4

(trF ) trF trF trR (trR ) + trR ; (3:2) J =

(2 ) 256 192 384 768 192 9

where tr for a power of F denotes the trace in the fundamental representation of SO(5),

2 4 2

trF and TrF = which is related to the trace Tr for the spinor representation as TrF =

3 1

2 2 4

(trF ) trF . The integral of J over a closed eight-manifold yields Index(D ) =

16 4

1 1

Index(D ) , where D is the Dirac op erator for chiral fermions with values in the

S S

2 8

bundle S and is the signature. Finally, ! is the Chern-Simons form of J , i.e. d! = J .

7 8 7 8

The total anomaly on W also receives a contribution from the bulk theory on Y . The

anomalous interaction is in fact the second term in the Green-Schwarz interaction (2.13).

In the present context, this term is b etter written as

= i G ^ ! ; (3:3)

bulk

0 0 0 0

where the Chern-Simons form ! ob eys d! = I and I was de ned in (2.12). The reason

7 7 8 8

is that the three-form C is not globally well-de ned in the presence of the magnetically

charged ve-brane. The eld strength G makes sense, though, and ob eys

; (3:4) dG =

where is a representative of the Poincare dual of W supp orted in an in nitesimal

neighborhood of W . (The factor of is due to our normalization of G.) The anomalous

change of under the transformation is thus

bulk

Z Z Z

0 0 0

! ; (3:5) = i G ^ ! = i G ^ I +

bulk

7 7 8

W I Y @I Y I

where we have used Stokes' theorem and (3.4). The last term involves the Chern-Simons

0 0

form ! of I restricted to W . To evaluate this term in the present context, one should

7 8

note that there is an imp ortant change in notation between this section and the previous

one: In the formula (2.12) for I , R denotes the curvature on Y , whereas in this section we

take R to denote the induced curvature on W . We should therefore rewrite (2.12) using

the decomp osition of the tangent bundle of Y restricted to W as the direct sum of the

tangent bundle of W and the normal bundle N . The latter bundle is regarded as an SO(5)

bundle with eld strength F . In this way we get

1 1 1 1 1 1

0 2 2 4 2 2 2 2 4

I = (trF ) trF + trF trR + (trR ) trR (3:6)

(2 ) 48 12 24 48 12

on W . The combined anomaly of + is thus

ef f bulk

Z Z Z

1 1

0 00

+ = i Index (D ) J + G ^ I + ! ; (3:7)

ef f bulk E S E 8

8 7

2 8

E Y I W I 10

00 00 0 00 00

where ! is the Chern-Simons form of I = J + I , i.e. d! = I . We see that

7 8 8 7 8

1 1 1

00 2 2 4

I = (trF ) trF ; (3:8)

(2 ) 192 96

which in fact equals 1=24 times the second Pontrjagin class p (N ) of the normal bundle

N .

We will now describ e a mechanism, outlined in [7] and further elab orated in [8], to

cancel the remaining p erturbative anomaly in (3.7). In the following, we will use a vector

sign over a di erential form to denote that it takes its values in the normal bundle N .

Bilinears in such forms are understo o d to b e multiplied via the b er-metric on N . In this

way,we can write the characteristic class I as

~ ^ ;~ (3:9) I =

where the N -valued four-form ~ is a bilinear in the eld strength F contracted with the

invariant rank ve tensor of SO(5). (The eld strength F takes its values in the adjoint

representation of SO(5), i.e. in the antisymmetric pro duct of two copies of N .) We also

intro duce the N -valued Chern-Simons three-form ~! corresp onding to ~ so that D~! = ~ ,

where D denotes the SO(5) covariant exterior derivative. Although the square of D do es

not vanish, it follows from the Bianchi identity for F that D~ = 0. Furthermore, we

intro duce an N -valued three-form H as follows: When restricted to W , the tangent bundle

of Y decomp oses as a direct sum of the tangent bundle of W and the normal bundle N .

The eld strength G, which is a section of the fourth exterior p ower of the tangent bundle

of Y , can b e decomp osed accordingly. The form H is then prop ortional to the comp onent

which is a three-form on W with values in N . We must also require that

D H = ~ (3:10)

when restricted to W , i.e. ~ must be covariantly exact. This is a top ological restriction

for the anomaly cancelation mechanism to work. It also xes the normalization of H . The

requisite counterterm is now

= ~! ^ H: (3:11)

Its change under the transformation is

Z Z

1 1

~ ~

~! ^ H = ~ ^ H ~! ^ ~ ; (3:12) =

24 24

W @I W I 11

where we have used Stokes' theorem, the relationship of !~ to ~ , and the condition (3.10).

The anomalous change of the total action = + + is thus

total ef f bulk ct

Z Z Z

1 1 1

= i Index (D ) J + G ^ I + ~ ^ H ;

total E S E 8

2 8 24

1 1

E (Y S ) (W S )

(3:13)

where wehave used the relationship b etween the Chern-Simons forms ! and !~ that follows

from (3.9). Note that the integrands of the last two terms are well-de ned on (Y S ) and

(W S ) resp ectively, since the eld strength G (and thus also H ) transforms covariantly

under .

We will now discuss some prop erties of the expression (3.13) for the total anomaly.

First of all, should b e indep endent mo dulo 2i of the choice of the eight-manifold

total

E as long as it is b ounded by(W S ) . For this to b e true, we must, in addition to the

eight-dimensional analogue of the Novikov formula (2.21), assume that

Index (D ) = Index (D ) Index (D ) (3:14)

~ ~

S S E S

E (E ) E

mo dulo 4 for any eight-manifolds E and E with common b oundary. Furthermore,

total

is indep endent of the exact form of G and H as long as (3.4) and (3.10) are ful lled and I

and ~ are (covariantly) exact. (The exactness of I is again automatic in compacti cations

to d 4 dimensions, whereas the covariant exactness of ~ is assured by (3.10).) Finally,

is a top ological invariant. Indeed, the variations of the characteristic classes ~ , J

total 8

and I under a smo oth deformation of the data must b e (covariantly) exact, i.e. ~ = D,

0 0

J = d and I = d . To preserve the relationships (3.9) and (3.10), we must have

8 7

H = and

1 1

(3:15) ^ ~ = +

12 16

mo dulo closed forms. It is then easy to see that the expression (3.13) is invariant.

The requirement that the anomaly vanish for any di eomorphism of the eleven-

manifold Y that leaves the world-volume W invariant is a necessary restriction on a con-

sistent M -theory con guration. Obviously, a rst question to settle is the correct interpre-

tation of the condition (3.10), which entered already at the p erturbative level. Provided

that this equation is ful lled, it makes sense to consider the expression (3.13) for the global

anomaly. It is not clear to what extent its vanishing follows from already known restric-

tions on M -theory con gurations. Here we just remark that there is no anomaly for pure

gauge transformations, i.e. transformations induced by di eomorphisms of Y that act 12

trivially on W . The reason is that since the homotopy group (SO(5)) is trivial, all pure

gauge transformations can be continuously connected to the identity. One should there-

fore consider di eomorphisms of W that are not continuously connected to the identity,

p ossibly combined with gauge transformations. A basic case is when W is top ologically

6 1

a six-sphere S , in which case (W S ) is actually the connected sum of one of the 27

6 1

exotic seven-spheres and S S [1]. In fact, if the global anomaly do es not vanish in this

situation, it will not vanish for any W . This follows from the fact that a di eomorphism

of S always has an analogue in the di eomorphism group of an arbitrary six-manifold.

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