Killings, duality and characteristic polynomials

5 March 1998

Ž .Physics Letters B 421 1998 162–168

Killings, duality and characteristic polynomials

´ 1 2 3Enrique Alvarez , Javier Borlaf , Jose H. Leon´ ´´ ´Instituto de Fisica Teorica, CXVI and Departamento de Fisica Teorica, CXI, UniÕersidad Autonoma, 28049 Madrid, Spain´ ´ ´

Received 3 December 1997Editor: L. Alvarez-Gaume

Abstract

Ž .In this paper the complete geometrical setting of lowest order abelian T-duality is explored with the help of some newŽ . Žgeometrical tools the reduced formalism . In particular, all invariant polynomials the integrands of the characteristic

.classes can be explicitly computed for the dual model in terms of quantities pertaining to the original one and with the helpof the canonical connection whose intrinsic characterization is given. Using our formalism the physically, and T-dualityinÕariant, relevant result that top forms are zero when there is an isometry without fixed points is easily proved. q 1998Published by Elsevier Science B.V.

1. Introduction

Ž w x.T-duality cf. 1,2 has become a standard tool inthe general exploration of the space of vacua in

ŽString Theory apparently best thought of as coming.from an 11-dimensional M-theory , albeit almost

always in the somewhat trivial setting of toroidalcompactifications.

Although Buscher’s formulas are known not to beŽexact except in the simplest cases i.e., generically

X .there are a corrections to them , it is neverthelessvery interesting to get as much information as possi-ble on the ‘‘classical’’ geometry of the dual targetspace.

ŽIn the present paper we generalize to the case inwhich there is a n-dimensional abelian group of

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected].

.isometries and develop further a formalism firstw xdiscussed by one of us in 3 , thus allowing to

compute all interesting geometrical quantities of thedual space. Ariadne’s thread will consist in exploit-ing the residual gauge invariance in adapted coordi-nates.

As a subproduct all invariant polynomials in thedual space are explicitly determined, yielding somegeneral conclusions on the vanishing of certain topo-logical invariants when they are zero. Some new

Ž .T-invariants i.e., scalars under T-duality also stemfrom the analysis.

Our results are strongest when all points havetrivial isotropy group; that is, when the isometrydoes not have any fixed points, which in the eu-clidean signature is the same thing as to say that theKilling vector never has zero modulus.

It is a well known fact than when performing aT-duality transformation, the geometry of the dual

Ž .space insofar as this concept makes sense can bewildly different from the one of the original space.

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII S0370-2693 97 01585-2

´ ( )E. AlÕarez et al.rPhysics Letters B 421 1998 162–168 163

There are two qualifications. First, Buscher’s formu-las are expected to receive corrections in all but the

w xsimplest cases 5 . Second, the probes to be used toexplore the dual geometry are not necessarily the

Žsame as the ones to be used in the original one cf.w x .4 for some comments on the operator mapping .

It is nevertheless of great interest to determine the‘‘classical’’ geometry of the dual space in as precise

Ž w xa manner as possible. cf. 6 for previous attempts in.this direction .

2. The reduced formalism

In this section we will give a very brief summaryw xabout the Reduced Formalism introduced in 3 . We

shall assume that the Target Space manifold M isinvariant under a n-dimensional abelian group ofisometries, and the corresponding Killing vectorswill be denoted by k m where m,ns0,1, . . . , Dy1,a

a,bs0, . . . ,ny1 and D is the dimension of M. Westart with two conditions. The first one, we have aset S of G-invariant tensors, V, characterized by

LL Vs0 1Ž .k a

i a Žwhich in adapted coordinates x , x isn, . . . , Dy.1 reduces to E Vs0.a

The second condition consist on having a connec-tion compatible with the first one, which implies

s s

LL = s LL ,=Ž .k k mmna a n

sK aRs q= = K s q2= K aT s '0Ž .a amn m n a m a a n

2Ž .

Rs is the curvature of the connection G d and T ramn lb mn

Ž . sis the corresponding torsion. 2 is nothing but E Ga mn

s0 in adapted coordinates.ŽGiven a set of commuting vector fields in our

. mcase, the Killings , k , there always exists a systemaŽ . mof coordinates adapted coordinates such that k sa

Em Ž w x.d , i. e., k ' . cf. 7 . These conditions do notaa a E x

determine completely the system of coordinates; theresidual gauge group actually consists in arbitrary

Ž i Xcompositions of transverse diffeomorphisms x siŽ j..f x with redefinitions of the ignorable coordi-

nates themselves:

x X i sx i , x X a sx a qLa x j 3Ž . Ž .

Ž .Tensors in V transform linearly under 3 :

V X sJ EL V 4Ž . Ž .If we had at our disposal some transÕerse gauge

aŽ j. Ž .fields A x , i.e., fields transforming under 3 as:i

AX a x j sAa x j yE La x j 5Ž . Ž . Ž . Ž .i i i

Then we could associate the reduced tensor Õ toeach tensor V by:

Õ'J A V 6Ž . Ž .Ž . XReduced tensors are invariant under 3 , i.e., Õ s

Ž . Ž .J AyEL J EL VsÕ, because J provides a repre-sentation of the abelian group G in the space of

Ž .tensors characterized by 1 .Ž . Ž .It follows simply from 2 and 6 that there exists

a reduced covariant derivative

\ Õ'J A = V 7Ž . Ž .corresponding to a reduced connection given by:

a rb dr dg sJ yA J yA J A G yE J AŽ . Ž . Ž . Ž .m bn d ž /mn a b a

8Ž .

Ž . Ž .With the definition giving above, 6 and 7 , theoperation that gives the reduced tensors commuteswith the basic operations of the tensor calculus:linear combination, tensor product, contraction, per-mutation of indices and covariant derivation. That

Ž w x.feature together with its simplicity see 3 is thereason to call the whole setting the Reduced Geome-try.

In a Riemannian manifold with abelian Killingvectors

G Aab aiG smn cdˆA G qA A Gž /b j i j ic jd

with

LL G s0 9Ž .k mna

Ž .there is a natural gauge field 5 namely,

Aa x j sGabA x j 10Ž . Ž . Ž .i b i

where G Gbc sd c.ab a

The reduced LeÕi-CiÕita connection, g has al - c

non-zero torsion which is the responsible for thereduced curÕature is not simply the curvature of the

´ ( )E. AlÕarez et al.rPhysics Letters B 421 1998 162–168164

w xreduced connection, as can be seen in 3 . In thegeneral case, GsG qH, the resulting reduced cur-lc

vature ish a hhr sR g qh y2T g g qhŽ . Ž . Ž .ldp l - c l - c l - cldp ld ap

11Ž .

where h r is the reduced tensor corresponding tomnr Ž .H and T g is the Levi-Civita reduced torsion.mn l - c

2.1. Buscher’s formulas for n commuting Killings

The context of most applications of the formalismstarts from a two-dimensional non-linear sigma modelwith target space MM, whose bosonic part is given by:

Ss G qB E X mE X n 12Ž .Ž .H mn mn q y

The generalization of Buscher’s transformationsto the case where there are n commuting Killings

Ž .present LL Gs0 and LL BsdW follows easilyk ka a

w xfrom the gauging procedure 2 . The resulting dualmodel has the following backgrounds:

˜" ab ˜" ab "Q sQ , Q s"Q Q ,ab " ai " b i

˜ cd y qQ sQ yQ Q Q 13Ž .i j i j q ci d j

with the conventions Q" 'G "B , Q" Qbcsd c,ab ab ab ab " a

Q" 'G "B and Q 'G qB and QabQ" sai ai ai i j i j i j " bc

d a.c

The quotient metric is invariant under T-duality:

˜ ˆG sG 14Ž .i j i j

The three form H'dB is defined in tensorial termsas

1H ' = B q= B q= B 15Ž .Ž .mnr m nr n rm r mn2

where the Levi-Civita connection is used to defineŽ . Ž .covariant derivatives. Then, from 6 and 7 , the

explicit computation yields for its reduced partner1h s0; h s E B ;abc iab i ab2

1 Ab Abi jah sy F B y E B q E B ;Ž .ai j i j j ab i ab2 2 2

ˆh sh 16Ž .i jk i jk

aŽ .where F B 'E B yE B . The reduced three-i j i a j j ai

form h is actually T-invariant.i jk

3. The canonical map

The classical string dynamics is governed by thepullback of the generalized connection with torsionG "sG "H, and at one-loop level the beta-func-lc

tions of the bosonic and Ns1 supersymmetric stringmodels are proportional to the Ricci tensor of thatgeneralized connection. Therefore our fist interest is

" Žto determine the T-dual of the connection G de-.noted as usual by g when reduced in the reduced"

setting:

G " r sG r "H r 17Ž .mn Ž lc.mn mn

g " r sg r "h r 18Ž .mn Ž lc.mn mn

r Ž .where the torsion H sH G is given by 15 .mns mn rs

To be specific, the starting point is:

g " c s0ab

1" i i .ˆg sy E Qab ab2

1" b bc " . bg s G E Q sgi a i ac ai2

1" j jk " . jˆg s G C sgai i k a ia2

1" a ab "g sy G Ci j i jb2

" k ˆ " k ˆ k ˆ kg sG "h 'G 19Ž .i j i j i j i j

" Ž ". d " d "where C sF Q qA E Q yA E Q .i jb i jb i j b d j i b d

The T-duals are 4:

g " s s t . a t " b t s g " l 20Ž .mn m n " l a b

for all components except g " b sg . b, with˜ ˜i a ai

t " b s"Qab; t " j s t j sd ja " i " i i

t a s"Q. otherss0 21Ž ." b ab

The simplicity of the reduced transformationsŽ .20 , allow us to built a map between the original

˜Ž . Ž .S and dual S geometries, the canonical map,˜Ž .transforming tensors V™V with the property of

mapping the corresponding covariant derivatives lin-˜ " ˜ "Ž .early = VA= V :

˜ " m1 , . . . ,m lVn , . . . ,n1 m

l mm " a " b , . . . ,br s 1 ls T T V 22Ž .Ł Ł" b n a , . . . ,ar s 1 mž /ž /rs1 ss1

4 It is exceedingly convenient to take advantage of the transfor-" Ž " . bmation properties of the combination s sE Q y E Q Ai ja i ja i ab j

1namely, s s" s with C s s .i j i j i ja w i j xaQ "


where the T " and T can be viewed as a sort of"

vierbeins relating indices of the initial and dualgeometries. The covariant derivatives map linearlybut not canonically,

˜ " ˜ " m1 , . . . ,m l= Vr n , . . . ,n1 m

l m. l m " a " b , . . . ,br s 1 lsT T T = VŁ Łr " b n l a , . . . ,ar s 1 mž /ž /rs1 ss1

23Ž .

because the anomaly in the derivative index. The" w xmatrices T and T , first used by Hassan 8 , are:"

"Qab 0"" nT sm ab . iyQ Q dž /. b i j

"Q. "Q.ab ai

mT s" n i0 dž /j

n being the column index and m the row index. Theabove mentioned anomaly in fact implies that thecovariant derivative does not commute with thecanonical map, and as a simple corollary the curva-tures do not transform simply by trading indices ofT " as we shall see momentarily.

Ž ".Covariantly constant tensors with respect to =m

Ž .transform necessarily as 22 . This is the case for the" ˜ " ˜Ž .metric = Gs= Gs0 , for the holomorphicm m

complex structures underlying extended supersym-w x w xmetries 8 , p-forms associated to W-algebras 11 ,

and whatever holomorphic covariantly constant ten-sor we found in our geometry.

There are other tensors with non-canonical trans-formations, such as the 2-form B and the 3-formmn

H . Torsion is best studied as forming part of theabg

Ž .generalized connection. Actually, both Eqs. 22 andŽ .23 together easily yield:

˜ " r . l " b r " a " b rG sT T T G q E T TŽ .mn m n " a lb m n " b

24Ž .

The target-space connection transforms as a T-du-ality one except for the anomaly in the m-index.

4. Transformation of the curvature and canonicalconnection

Let us now consider the generalized curvature," Ž ". lR sR G G which obviously satisfies themnsr mns l r

symmetry relationships R" syR" syR" ,mnsr nmsr mnrs

R" sR. . Working with the transformations ofmnrs rsmn

Ž .the connection in 20 and using the expression forŽ . 5the reduced curÕature 11 we get :

˜" . a . b " l " dR sT T T Tmnsr m n s r

= R" y2Qab= "k a= .k b 25Ž .Ž .abld . a b l d

The fact that there is an inhomogeneous partŽ ab " a . b.y2Q = k = k in the transformation of the. a b l d

curvature turns out to be a useful clue.Actually, when an object does transform inhomo-

DŽ . 6geneously, such as: rs"Ł t rqc , the involu-˜tive property, T 2 s1, completely determines thetransformation of the inhomogeneous part, c ,namely˜ Dcs.Ł t c . This fact immediately suggests the

1definition w'rq c , which does transform homo-2

geneously.There is then a quantity, the corrected general-

ized curÕature W " which transforms linearly:abdh

W " 'R" yQab= "k a= .k b 26Ž .mnsr mnsr . m n s r

˜ " . a . b " l " d "W sT T T T W 27Ž .mnsr m n s r mnsr

Ž .This correction 26 is minimal because it includesfirst-derivative terms and therefore it cannot be ab-sorbed in a background’s redefinition.

We have just seen that the transformations of theŽ .ordinary connection is not canonical 22 . It is possi-

ble, however, to define a new connection withŽ .canonical transformation properties 24 We shall

refer to it as canonical connection:

" r " r ab a . b rG sG yG k = kmn mn m n

" r " l " b r " a " b rG sT T T G q E T TŽ .mn m n " a lb m n " b

28Ž .

5 k a ' knG .m a nm6 We denote as Ł t D the product of matrices t with D " or .

as appropriate.


"Therefore the canonical covariant derivative, =m

does now commute with the canonical mapping. Thismeans that a canonical transformation for tensors

˜ " m1 , . . . ,m lVn , . . . ,n1 m

m l" b m a , . . . ,aA B 1 ls T T V 29Ž .Ł Łn " a b , . . . ,bA B 1 mž / ž /As1 Bs1

corresponds with a canonical one for the covariantderivatives:

" " m , . . . ,m˜ 1 l˜= Vr n , . . . ,n1 m

m l" l " b m " a , . . . ,aA B 1 lsT T T = VŁ Łr n " a l b , . . . bA B 1 mž / ž /As1 Bs1

30Ž .

The connection is compatible with the metric,"= G s0 provided that the Killing conditionm nr

LL G s0 is satisfied. Moreover, = s0 acting onk mn 0aa "S, implying k R s0. At the end of this sectiona ansr

Ž .we will give an intrinsic T-duality independentcharacterization of this canonical connection forwhich the above properties will appear natural.

Also following simply from the commutativity isthe canonical transformation of the curvature:

" " a " b " d " h "R sT T T T R 31Ž .mnsr m n s r a bdh

With the canonical connection, the canonical T-du-ality map commutes with the basic operations of thetensor calculus, i.e., linear combinations, tensorproducts, permutation and contraction of indices andcovariant derivation. In particular it implies that

" "every tensor built from R , G , = , and anymnsr a b h

other tensor transforming canonically, transformscanonically. As a corollary, target-space canonical

" " " " 2˜ ˜ ˜Ž Ž .scalars are T-duality scalars R sR , = R" " 2 " " m n " " m n˜ ˜Ž .s = R , R R s R R ,m n m n

" " mnsr " " mnsr˜ ˜ .R R sR R . . . .mnsr mnsr

In complex manifolds, the presence of additionalcanonical tensors, i.e., the holomorphic complexstructures J " m, allows the construction of anothern

" mn " ms nrŽnew set of T-duality scalars R J , R J J ,mn " mnsr " "" ra " mns .R J R , . . . .mnsr " a

Ž .The definition of canonical connection in 28was motivated in its very convenient transformationproperties under T-duality. Nevertheless, an intrinsic

Ž .T-duality independent characterization can be givenfor it.

Ž .Let us start with our set of abelian Killing� m 4 7vectors k and the vector space KK spanned byŽa.

them. The presence of a metric G on our manifoldmn

induces the natural projector on KK, P n 'm

k kn Gab 8, with P 2 sP and Pk sk .Ža.m Žb. Ža. Ža.Now, let us take the quotient of the whole space

of connections CC by the projection P,i.e., CCrP.Then, two connections G 1 and G 2 belong to the

Ž 1 2 . nsame class on CCrP if G yG sP L for somem m n

matrix valued one-form L.In every class there is an unique covariant deriva-

tion,= , with the property

= sLL 32Ž .k kŽa. Ža.

9 HWriting =sP =qP= , the orthogonal com-Ž H .ponent P s1yP is common to every element

on the same class, say = H , and the KK projection isn ab abŽ .P= s k k G = s k G LL . Therefore,m Ža.m Žb. n Ža.m k Žb.

the barred connection is

H ab= s= qk G LL 33Ž .m m Ža.m k Žb.

in every class of CCrP.If we start in a class having a connection compati-

Žble with the metric as it is the case in T-dualityŽ ..17 , the compatibility of the barred connectiontrivially implies the Killing condition LL G s0.k mnŽa.

In adapted coordinates, the = sLL conditionk kŽa. Ža.rmeans = sE and then G s0. Taking an arbitrarya a a n

Ž .reference connection in the class, G = the aboveconditions imply

/r r ab rG sG yk G = k 34Ž . Ž .nmn mn Ža.m Žb.

where the / operation on connections simply flipsthe sign of the torsion, /:G r ™G r and /:G r

Ž mn . Ž mn . w mn x™yG r . Note that the / operation transforms ourw mn xstringy connections G " one into the other. In that

Ž . Ž .way, the barred connection 33 34 agree with theŽ .T-duality canonical one 28 if we choose the classes

which G "belong to.

7 k m ' k m.Ža. a8 We remember G s kn kn and GabG sd a.ab Žb. Ža. b c c9 m= ' k = .k Ža. mŽa.


With respect to the reference connection G , thebarred connection loses the information about theparallel transport in the Killing’s direction. This

Ž .orthogonal projection is easily seen rewriting 34 ina non-covariant form

r H a r ab rG sP G yk G E k 35Ž .mn m an Ža.m n Žb.

As a consequence of the defining properties,a m mk = k s LL k s 0, implying the barredŽa. a Žb. k Žb.Ža.

geodesics

m m a b¨ ˙ ˙X qG X X s0 36Ž .ab

have the free motion on the Killing’s direction

˙m a m ˙aX sC k X t ; C s0 37Ž . Ž .Ž .Ža.

as a consistent solution. Let us note this motion isallowed for eÕery barred connection provided that itprojects out the parallel transport in the KK direction.

/Ž .Finally let us remark that = ks0. Of course,/Ž .the = is in general non-compatible with the met-

Žric. As a corollary, we get the useful see next.section condition

sa sk R s LL = 38Ž .ž /Ža. amn k Ža. mn

which for the stringy connections means

"R s0 39Ž .amnr

5. Invariant polynomials

Ž .Invariant polynomials P V are characterized inŽ w x. Ž . Ž y1 . . Žgeneral cf. 9 by P V sP g V g , where

r Ž . r m nV 'R G dx ndx is the matrix-valued cur-s mns

. Ž .vature two-form , which implies that dP V s0;Ž .and moreover, that P V has topologically invariant

Ž .integrals on manifolds without boundary . Chernand Simons have proven the specific result thatgiven two different connections, v and v

X

P VX yP V sdQ v

X ,v 40Ž . Ž . Ž . Ž .where Q, the Chern-Simons term, is given byŽ X . 1 Ž X .Q v ,v 'rH P v yv,V , . . . ,V , r being the0 t t

degree of the polynomial, and V 'dv qv nv ,t t t tX Ž .with v ' tv q 1y t v.t

As a consequence, all those polynomials can beŽdetermined up to a total differential, that is in a

.cohomological sense , using any convenient connec-"tion; in our case the canonical connection G

imposes itself naturally, because as we have seen in" rdetail, the corresponding curvature, R transformsmns

canonically under T-duality.It is plain that the only non-zero components of

any canonical inÕariant polynomial would be theones with all indices transverse. This is clear,because

" d= s0 actually implies R s0.a aa b

In the particular case in which we are consideringa Pontryagin characteristic polynomial, appropriate

Ž .when the curvature lies in the Lie algebra of O k ;4 j a. m Ž4 j.Ž . Ž .p V gL M we can write k P sj mn . . . n1 4 jy1

0 ;a,n . . . n or, in adapted coordinates,1 4 jy1

Ž4 j.P s0 41Ž .an . . . n1 4 jy1

Now, P transforms canonically:

4 jŽ4 j. " b Ž4 j.˜ AP s T P 42Ž .Łn . . . n m b . . . b1 4 j A 1 4 jž /As1

And for the non-vanishing components:

4 jŽ4 j. k Ž4 j. Ž4 j.˜ AP s d P sP 43Ž .Łi . . . i i k . . . k i . . . i1 4 j A 1 4 j 1 4 j

As1

This means that the components of the canonicalPontryagin forms are actually invariant. A glance atŽ . Ž41 implies that Top Forms vanish because they

"necessarily include Killing indices, for which R s.0 .

This in turn means that if the canonical connec-Ž ab .tion has a global meaning det G /0 the topologi-

cal invariants obtained by integrating top forms arenecessarily zero, both in the original and in the dualmodel.

Chern classes are defined for complex manifoldsŽ . 2 jŽ .with Vggl k,C ; c gL M , but are otherwisej

similar to the Pontryagin ones from the point of viewof the present work.

A well known fact is that for even dimensionalŽ .manifolds a further SO 2 r invariant polynomial can

Ž .be defined the Pfaffian . The corresponding Eulerclass is essentially the square root of the highestPontryagin class. The mother of all index theorems isprecisely the Gauss-Bonnet theorem, which states

Ž .that the Euler characteristic x M is integral of the"Ž . ŽEuler class e R . Now we see that the integrand a

.top form is necessarily zero if the group G acts


Ž . Ž .freely without fixed points . From 43 , this asser-tion is T-duality invariant

When there are fixed points, we could define aone parameter family of connections compatible with

10 "the metric interpolating from G at ts1 toG " en ts0.

tkm" s " s . sG t sG y = k 44Ž . Ž .mn mn n22k q 1y tŽ .This is well defined ; t)0 whereas in the limitt™0 the top forms vanish everywhere except per-haps at the fixed points. This means that all thetopological information is stored in the fixed points.

In the case at hand this is contained in the well-known theorem by ‘‘Poincare-Hopf’’ asserting that´in a compact manifold M endowed with a differen-

Ž .tiable vector field, w Killings in our case withisolated zeroes, the sum of the corresponding indicesŽ Ž .i that is, the Brouwer degree of the mapping k x 'Ž .k x . w x10 is precisely Euler’s characteristic:< < < <Ž .k x

mi

x M s is y1 rank H m . 45Ž . Ž . Ž . Ž .Ý Ý iis0

Ž Ž .where H m stands for the i-th homology group ofi.M.As a trivial corollary, only in manifolds with zero

Euler Characteristic it is possible to have free Killingactions.

Owing to the fact that we do not have enoughcontrol on the topology of the dual manifold, we cannot say anything about the topological numbers,

10 Because of the Killing condition.

which are integrals of the invariant polynomials,except, of course, when they vanish. It would also bequite complicated to keep track of all boundaryterms in manifolds with non-trivial boundary in or-der to compute, say, the h invariant.

Acknowledgements

We are grateful to Luis Alvarez-Consul, Luis´´Alvarez-Gaume, Cesar Gomez, Tomas Ortin and´ ´ ´ ´

Kostas Sfetsos. This work was supported by theŽ . Ž .UAM J.H.L. , CAM J.B. ; CICYT AENr96r-1655

and AENr96r1664 and an EU TMR contractERBFMRXCT960012.

References

w x1 A. Giveon, M. Porrati, E. Rabinovici, Target Space DualityŽ .in String Theory, Physics Rept. 244 1994 77, hep-

thr9401139.w x2 E. Alvarez, L. Alvarez-Gaume, Y. Lozano, Nucl. Phys.´

Ž . Ž .Proc. Pub. 41 1995 1.w x3 J. Borlaf, Geometry of T-duality, hep-thr9707007.w x4 E. Alvarez, L. Alvarez-Gaume, J.L.F. Barbon, Y. Lozano,´ ´

Ž .Nucl. Phys. B. 415 1994 71.w x Ž .5 E. Alvarez, J. Borlaf, J.H. Leon, Phys. Lett. B 400 1997 97.´w x6 O. Alvarez, Classical geometry and Target Space Duality,

hep-thr9511024.w x7 A. Lichnerowicz, Theories relativistes de la gravitation et de´

l’electromagnetisme, Masson, Paris, 1955.´ ´w x Ž .8 S.F. Hassan, Nucl. Phys. B 454 1995 86; Nucl. Phys. B460

Ž .1996 362.w x Ž .9 T. Eguchi, P. Gilkey, A. Hanson, Phys. Rep. 66 1980 213.

w x10 J. Milnor, Topology from a differential viewpoint, The Uni-versity Press of Virginia, 1981.

w x Ž .11 B. Kim, Phys. Lett. B 368 1996 71.

Killings, duality and characteristic polynomials

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