gauge theories on a(ds) space and killing vectors

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Annals of Physics 323 (2008) 705–751

www.elsevier.com/locate/aop

Gauge theories on A(dS) space and Killing vectors

Rabin Banerjee *, Bibhas Ranjan Majhi

S.N. Bose National Centre for Basic Sciences, Sector-3, Block-JD, Salt Lake City, Kolkata 700 098, India

Received 30 March 2007; accepted 16 August 2007Available online 1 September 2007

Abstract

We provide a general technique for collectively analysing a manifestly covariant formulation ofnon-abelian gauge theories on both anti-de Sitter as well as de Sitter spaces. This is done by stereo-graphically projecting the corresponding theories, defined on a flat Minkowski space, onto the sur-face of the A(dS) hyperboloid. The gauge and matter fields in the two descriptions are mapped byconformal Killing vectors and conformal Killing spinors, respectively. A bilinear map connectingthe spinors with the vector is established. Different forms of gauge fixing conditions and their equiv-alence are discussed. The U(1) axial anomaly as well as the non-abelian covariant and consistent chi-ral anomalies on A(dS) space are obtained. Electric-magnetic duality is demonstrated. The zerocurvature limit is shown to yield consistent findings.� 2007 Elsevier Inc. All rights reserved.

Keywords: A(dS) space; Stereographic projection; Killing vectors

1. Introduction

Quantum field theories on anti-de Sitter and de Sitter, collectively denoted as A(dS),space-times have a long history originating from the pioneering paper by Dirac [1]. Thesespace-times are crucial in cosmological studies since they are the only maximally symmet-ric examples of a curved space-time manifold. Incidentally, the A(dS) space-time is a solu-tion of the negative (positive) cosmological Einstein’s equations having the same degree ofsymmetry as the flat Minkowski space-time solution. Moreover, recently a non-zero

0003-4916/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.aop.2007.08.010

* Corresponding author.E-mail addresses: [email protected] (R. Banerjee), [email protected] (B.R. Majhi).

mailto:[email protected]

mailto:[email protected]

706 R. Banerjee, B.R. Majhi / Annals of Physics 323 (2008) 705–751

cosmological constant has been proposed to explain the luminosity observations of thefarthest supernovae [2]. The A(dS) metric is therefore expected to play an important roleif this proposal is validated. These developments show that the study of field theories onA(dS) space is highly desirable, if not essential.

Most of the available treatments of quantum field theory in curved space-time employhigh powered mathematical tools [3]. While exploiting such techniques for the A(dS) caseare feasible, it is not particularly practical since it misses the special symmetry properties ofthis space-time. Examples of such approaches are the dimensional reduction scheme usingvierbein language [4,5] or those based on group theoretical notions [6,7]. Yet anothermethod is to use the coordinate independent approach, also called the ambient formalism.For scalar fields this was done in [8,9] which was later extended to include gauge theories[10,11]. While an advantage of this approach is its link (although not a complete one toone mapping) with the corresponding analysis on a flat Minkowski space-time, there isan unpleasant feature which also exists in other approaches [1,6–11]. The point is thatwhereas the electron wave equation involves the angular momentum operator, the gaugefield equation involves both this operator as well as the ordinary momentum operator.Since the A(dS) space is a hyperboloid (pseudosphere) the natural operator entering intothe equation of motion should be the relevant angular momentum operator, since trans-lations on the A(dS) space correspond to rotations on the pseudosphere. This is usuallycorrected by imposing subsidiary conditions to avoid going off the pseudosphere of con-stant length.

In this paper, we develop a manifestly covariant method of formulating interactinggauge theories on the A(dS) space-time. Some basic features of this method were alreadydiscussed by one of us [17] in the context of de Sitter space and its Wick rotated version,the hypersphere which is the n-dimensional sphere immersed in (n + 1)-dimensional flatspace [18,19]. The relevant wave operators always incur the angular momentum operatorsso that subsidiary conditions necessary in other approaches are avoided. The method isgeneral enough to collectively dicuss both the de Sitter as well as the anti-de Sitter exam-ples. Extention to arbitrary dimensions is straightforward. Our method is applicable forhigher rank tensor fields. An exact one to one correspondence with the theories on the flatspace has been established. Effectively, the theories on the flat space are projected on to theA(dS) space by a stereographic transformation which is basically a conformal transforma-tion. We show that variables in the gauge sector (like potentials, field strengths, etc.) in thetwo descriptions are related by rules similar to usual tensor analysis, with the conformalKilling vectors playing the role of the metric. Likewise, quantities in the matter (fermionic)sector are related by the conformal Killing spinors. A bilinear map connects these spinorswith the conformal Killing vector. Apart from formulating gauge theories we have alsocomputed the chiral anomalies and exhibited electric-magnetic duality rotations inA(dS) space.

The analysis presented in this paper is basically classical and the extension to quantumfield theory will be quite nontrivial. While certain related points are studied in Section 5,fully addressing this issue is beyond the scope of the present paper.

In Section 2, the connection between stereographic projection and conformal Killingvectors is shown including an explicit derivation of the latter using the Cartan-Killingequation. The use of these Killing vectors is also elaborated. The pure Yang–Mills theoryon A(dS) space is formulated in details in Section 3. The action is derived. Its equivalencewith the standard action defined on an arbitrary curved space is shown by using the

R. Banerjee, B.R. Majhi / Annals of Physics 323 (2008) 705–751 707

explicit form of the induced metric. The gauge symmetry is discussed and its connectionwith the gauge identity is shown. Different forms of the Lorentz gauge condition and theirequivalence are analysed. Finally, implications of subsidiary conditions used in the litera-ture [1,6–11] are mentioned. Section 4 discusses the stereographic projection of the Diraclagrangian by means of conformal Killing spinors. The bilinear map connecting these spi-nors with the conformal Killing vector is given. Section 5 provides a detailed calculation ofboth the U(1) (axial) anomaly as well as the non-abelian (covariant and consistent) chiralanomalies. The counterterm connecting the covariant and consistent anomaly is also com-puted. Electric-magnetic duality rotations in an abelian theory are discussed in Section 6.A second rank anti-symmetric tensor gauge theory is formulated in Section 7. The zerocurvature limit, analysed in Section 8, yields consistent results. The equations of motionon A(dS) space smoothly pass to corresponding equations on the flat Minkowski space.Finally, our conclusions are given in Section 9. An Appendix A discussing the role ofboundary conditions has also been included.

2. Stereographic projection and Killing vectors on A(dS) space

Amongst curved space-times, the de Sitter and anti-de Sitter spaces are the only possi-bilities that have maximal symmetry admitting the highest possible number of Killing vec-tors. The role of these vectors in suitably defining gauge theories on such spaces is crucialto this analysis. We shall do our discussions for de Sitter and anti-de Sitter spacescollectively.

The A(dS) universe is a pseudosphere in a five-dimensional flat space with Cartesiancoordinates ra = (r0, r1, r2, r3, r4) satisfying,

r2 ¼ rara ¼ glmrlrm þ sðr4Þ2 ¼ sl2 ð1Þ

where s = �1 for de Sitter space, s = +1 for anti-de Sitter space and l is the A(dS) lengthparameter. The metric of the de Sitter space dS(4,1) is induced from the pseudoeuclideanmetric g = diag(+1,�1,�1,�1,�1). It has the pseudoorthogonal group SO(4,1) as thegroup of motions. Anti-de Sitter comes from g = diag(+1,�1,�1,�1,+1). It has apseudoorthogonal group SO(3, 2)as the group motions.The mostly negative flat Minkow-ski metric is glm = diag(+1,�1,�1,�1) with l, m = 0,1,2,3.

It should be mentioned that A(dS) is only locally Weyl (conformally) equivalent tothe flat space. Although this is enough for our purposes, we remark that the corre-spondence is globally more intricate and not discussed here. For instance, Eq. (1)defines a geometry that can be considered to describe a patch covering only a finitetime interval in A(dS); it cannot be used to describe field configurations in the fullA(dS) geometry.

A useful parametrisation of these spaces is done by exploiting the stereographic projec-tion. The four-dimensional stereographic coordinates (xl) are obtained by projecting theA(dS) surface into a target Minkowski space. The relevant equations are [20],

rl ¼ XðxÞxl; XðxÞ ¼ 1þ sx2

4l2

� ��1

ð2Þ

and,


r04 ¼ �XðxÞ 1� sx2

4l2

� �ð3Þ

where x2 = glmxlxm and r04 ¼ r4

l ¼ s r4

l .The inverse transformation is given by

xl ¼ 2

1� r04rl ð4Þ

In order to define a gauge theory on the A(dS) space analogous stereographic pro-jections for gauge fields have to be obtained. This is done following the method devel-oped by us [18,19] in the example of the hypersphere which was later extended to thede Sitter hyperboloid [17]. The point is that there is a mapping of symmetries on theflat space and the pseudosphere (e.g.translations on the former are rotations on the lat-ter) that is captured by the relevant Killing vectors. Furthermore since stereographicprojection is known to be a conformal transformation, one expects that the cherishedmap among gauge fields would be provided by the conformal Killing vectors. We maywrite this relation as,bAa ¼ Kl

aAl þ ra/ ð5Þ

where the conformal Killing vectors Kla satisfy the transversality condition,

raKla ¼ 0 ð6Þ

and an additional scalar field /, which is just the normal component of bAa,1 is introduced,

/ ¼ s1

l2ra bAa ð7Þ

The five components of bA are expressed in terms of the four components of A plus ascalar degree of freedom. To simplify the analysis the scalar field is put to zero. It isstraightforward to resurrect it by using the above equations. With the scalar field gone,bA is now given bybAa ¼ Kl

aAl ð8Þ

and satisfies the transversality condition originally used by Dirac [1]

ra bAa ¼ 0 ð9Þ

The conformal Killing vectors Kla are now determined. These should satisfy the Cartan-

Killing equation which, specialised to a flat four-dimensional manifold, is given by

omKla þ olKm

a ¼2

4okKk

aglm ð10Þ

The most general solution for this equation is given by [21]

Kla ¼ tla þ �axl þ xlm

a xm þ klax2 � 2kr

axrxl ð11Þ

where xlm = �xml. The various transformations of the conformal group are characterisedby the parameters appearing in the above equation; translations by t, dilitations by �, rota-

1 Hat variables are defined on the A(dS) universe while the normal ones are on the flat space.


tions by x and inversions (or the special conformal transformations) by k. Imposing thecondition (6) and equating coefficients of terms with distinct powers of x, we find the basicstructures of the Killing vectors:

Klm ¼ 1þ s

x2

4l2

� �gl

m � sxmxl

2l2ð12Þ

Kl4 ¼ sK4l ¼ xl

lð13Þ

These Killing vectors establish the link between the A(dS) coordinates and the flat ones bythe relation,

Kal ¼ 1þ sx2

4l2

� �2ora

oxlð14Þ

With the above solution for the Killing vectors, the stereographic projection for the gaugefields (8) is completed leading to, in component form,

bAl ¼ 1þ sx2

4l2

� �Al � s

xmxl

2l2Am ð15Þ

bA4 ¼xl

lAl ð16Þ

The inverse relation is given by

1þ sx2

4l2

� �Al ¼ bAl þ s

xlbA4

2lð17Þ

Before proceeding to discuss gauge theories some properties of these Killing vectors aresummarised. There are two useful relations,

KlaKam ¼ 1þ s

x2

4l2

� �2

glm ð18Þ

and,

KlaKbl ¼ 1þ s

x2

4l2

� �2

hab ¼ 1þ sx2

4l2

� �2

gab � srarb

l2

� �ð19Þ

These relations are also valid for general D-dimensions. The transverse projector hab sat-isfies,

rahab ¼ rbhab ¼ 0; habhbc ¼ hc

a ð20Þ

and will be subsequently used in the construction of transverse entities like transversederivatives or angular momentum operators.

Relation (18) shows that the product of the Killing vectors with repeated ‘a’ indicesyields, up to the conformal factor, the induced metric. The other relation can be inter-preted as the transversality condition emanating from (6). For computing derivativesinvolving Killing vectors, a particularly useful identity is given by

KlaolKam ¼ � 1þ s

x2

4l2

� �s

xm

l2ð21Þ


A simple use of (18) yields the inverse of (8) as,

Al ¼ 1þ sx2

4l2

� ��2

KalbAa ¼ ora

oxlbAa ð22Þ

where the second equality follows from (14). The above relation is an illuminating rephras-ing of (17).

Note that the conformal (Jacobian) factor that relates the volume element on the A(dS)space with that in the four-dimensional flat manifold,

d4x ¼ dx0 dx1 dx2 dx3 ¼ 1þ sx2

4l2

� �4

dX ð23Þ

naturally emerges in (18) and (19). For D-dimensions this generalises to,

dDx ¼ dx0 dx1 dx2 . . . dxD�1 ¼ 1þ sx2

4l2

� �D

dX ð24Þ

The invariant measure is given by

dX ¼ lr4

d4r ¼ lr4

� �dr0 dr1 dr2 dr3 ð25Þ

To observe the use of these Killing vectors, let us analyse the generators of the infinitesimalA(dS) transformations. In terms of the host space Cartesian coordinates ra, these are writtenas,

Lab ¼ rao

orb� rb

o

orað26Þ

which satisfy the algebra,

½Lab; Lcd � ¼ gbcLad þ gadLbc � gbdLac � gacLbd ð27Þ

In terms of the stereographic coordinates the generator is expressed as,

Lab ¼ raKlb � rbKl

a

� �ol; ol ¼

o

oxlð28Þ

which can be put in a more illuminating form by contracting with ra,

raLab ¼ sl2Klbol ð29Þ

clearly showing how rotations on the A(dS) space are connected with the translations onthe flat space by the Killing vectors.

An object of related interest is the transverse (or tangent) derivative, expressed in termsof the transverse projector,

ra ¼ hab o

orbð30Þ

This derivative satisfies the properties ra$a = 0, $ara = 4 (or ‘D’ in D-dimensions) and

obeys the following commutation relations,

½ra;rb� ¼ s1

l2Lab; ½ra; rb� ¼ hab ð31Þ

Also, the angular momentum operator is directly written in terms of the transverse deriv-ative as,


Lab ¼ rarb � rbra; ra ¼ Kalol ¼ s1

l2rbLba ð32Þ

where we have used (29).

3. Formulation of Yang–Mills theory on A(dS) space

In this section we discuss the formulation of Yang–Mills theory on the A(dS) space. Thetheory is obtained by stereographically projecting the usual theory defined on the flat Min-kowski space. A comparison with other approaches will be done pointing out the advantagesof our formalism.

The pure Yang–Mills theory on the Minkowski space is governed by the standardlagrangian,

L ¼ � 1

4TrðF lmF lmÞ ð33Þ

where the field tensor is given by

F lm ¼ olAm � omAl � i½Al;Am� ð34ÞTo define the field tensor on the A(dS) space we proceed systematically by looking at

the gauge symmetries. If the ordinary potential transforms as,

A0l ¼ U�1ðAl þ iolÞU ð35Þ

then the projected potential transforms as,

bA0a ¼ KlaA0l ¼ U�1 bAa þ s

i

l2rbLba

� �U ð36Þ

obtained by using (8) and (29).The infinitesimal version of these transformations obtained by taking U = e�ik is then

found to be,

dAl ¼ Dlk ¼ olk� i½Al; k� ð37Þfor the flat space while for the A(dS) space it is given by

d bAa ¼ KladAl ¼ s

1

l2rbLbak� i½ bAa; k� ð38Þ

This is put in a more transparent form by introducing, in analogy with the flat space, a‘covariantised angular derivative’ [19,24] on the A(dS) space,bLab ¼ Lab � i½ra

bAb � rbbAa; � ¼ �bLba ð39Þ

so that,

d bAa ¼ s1

l2rb bLbak ð40Þ

Note that, this is consistent with rad bAa ¼ 0 which is a consequence of the transversalitycondition (9).


The covariantised angular derivative satisfies a relation that is the covariantised versionof (29),

rabLab ¼ sl2KlbDl ð41Þ

obtained by using the transversality condition on the gauge fields.It is feasible to extend the definition (30) of the transverse derivative to the ‘covarian-

tised transverse derivative’ as,

~ra ¼ ra � i½ bAa; � ð42ÞThis is related to the ‘covariantised angular momentum’ (39) by equations similar to (31)and (32). These are given by their covariantised versions,

~ra ¼ KalDl ¼ Kalðol � i½Al; �Þ ¼s

l2rbbLba ð43Þ

and,

½ ~ra; rb� ¼ hab

½ ~ra; ~rb� ¼ s

l2bLab

Also, ~ra satisfies the following properties that are exactly identical to $a;

ra~ra ¼ rara ¼ 0

~rara ¼ rara ¼ 4ð44Þ

The field tensor bF abc on the A(dS) space is now defined. It has to be a fully anti-sym-metric three index object that transforms covariantly. The covariantised angular derivativeis the natural choice for constructing it. We define,bF abc ¼ Lab

bAc � ira½ bAb; bAc��

þ c:p: ð45Þ

where c.p. stands for the other pair of terms involving cyclic permutations in a, b, c.To see that bF abc transforms covariantly it is convenient to recast this in a form involving

the Killing vectors, analogous to the relation (8). Indeed it is mapped to the field tensor onthe flat space by the following relation,bF abc ¼ raKl

bKmc þ rbKl

c Kma þ rcKl

aKmb

� �F lm ð46Þ

so that symmetry properties under exchange of the indices is correctly preserved. To showthe equivalence, (8) and (28) are used to simplify (45), yielding,bF abc ¼ raKl

b � rbKla

� �ol Km

cAm

� �� ira Km

bAm;Klc Al

� þ c:p: ð47Þ

The derivatives acting on the Killing vectors sum up to zero on account of the identity,

raKlb � rbKl

a

� �olKm

c þ c:p: ¼ 0 ð48Þ

The derivatives acting on the potentials, together with the other pieces, combine to repro-duce (46), thereby completing the proof of the equivalence.It is now trivial to see, using theabove relation (46), that bF abc transforms covariantly simply because Flm does. The exacttransformation is given by,


dbF abc ¼ �i½bF abc; k� ð49Þ

Of course this can also be derived by starting from (45) and using (38).The inverse relation following from (46), obtained by contracting with ra and the Kill-

ing vectors, is given by

F lmðxÞ ¼ s

1

l21þ s

x2

4l2

� ��4

KblKcmðrabF abcÞ ¼ s

1

l2

orb

oxl

orc

oxmðrabF abcðrÞÞ ð50Þ

where use has been made of (14) to get the final result.Incidentally, the generalisation of (22) and (50) to arbitrary rank tensors is easily done,

Al1l2...lnðxÞ ¼ ora1

oxl1

ora2

oxl2. . .

oran

oxln

bAa1a2...anðrÞ ð51Þ

and

F l1l2...lnlnþ1ðxÞ ¼ s

1

l2

orb1

oxl1

orb2

oxl2. . .

orbnþ1

oxlnþ1ðrabF ab1b2...bnþ1ðrÞÞ ð52Þ

where F ðbF Þ denotes the field strength corresponding to the potential Að bAÞ.It is possible to extend the above relations to establish a mapping between the covariant

derivatives on the A(dS) local coordinates and the transverse derivatives (30) or the angularmomentum operator (32). Let us first recall that the A(dS) space is immersed in a (4 + 1)-dimensional flat space which has the metric ds2 = gabdradrb and A(dS) is the subspacegabrarb = sl2 with metric ds2 = glmdxldxm. So, locally both metrics must agree on A(dS); i.e.

gab dra drb ¼ glm dxl dxm ð53Þ

But,

dra ¼ olra dxl ¼ ora

oxldxl ¼ 1þ s

x2

4l2

� ��2

Kal dxl ð54Þ

Hence,

1þ sx2

4l2

� ��4

gabKalKbm ¼ glm ð55Þ

Using the identity (18), we obtain the induced metric and its inverse,

glm ¼ 1þ sx2

4l2

� ��2

glm; glm ¼ 1þ sx2

4l2

� �2

glm: ð56Þ

Incidentally, the induced metric glm is related to the transverse projector hab (see (19)) by

glm ¼ora

oxl

orb

oxmhab ð57Þ

which is similar to (50). This follows from (55), the identification (14) and the transversal-ity condition (6).

The Affine connection for the vector field,

Cmlr ¼

1

2gmqðgql;r þ gqr;l � glr;qÞ ð58Þ


is then found to be,

Cmlr ¼ s

1

2l21þ s

x2

4l2

� ��1

xmglr � xlgmr � xrg

ml

� �ð59Þ

The covariant derivative acting on a scalar and a covariant vector yields,

rl/ ¼ ol/ ð60Þand

rlAm ¼ olAm � CrlmAr ¼ olAm � s

1

2l21þ s

x2

4l2

� ��1

ðxrArglm � xlAm � xmAlÞ ð61Þ

These results are reproduced by the maps,

rl/ðxÞ ¼ora

oxlra/ðrÞ ð62Þ

and

rlAmðxÞ ¼ora

oxl

orb

oxmra bAbðrÞ ð63Þ

To show this, consider the first of the above relations. Using (14) and the definition (32) ofthe transverse derivative $a, we obtain,

ora

oxlra/ ¼ 1þ s

x2

4l2

� ��2

KalKaror/ ¼ ol/ ð64Þ

that follows on using the identity (18). The cherished result $l/ = ol/ is thereby repro-duced.

Now using (14), (8), (34) and the identity (18), the R.H.S. of (63) becomes,

ora

oxl

orb

oxmra bAbðrÞ ¼ 1þ s

x2

4l2

� ��2

KbmolðKbrArÞ ð65Þ

Putting the structures of the Killing vectors (12), (13) above, we get the R.H.S. of (61),which proves (63).

Now, it is possible to generalise the relations (62) and (63) to include a chain of covar-iant derivatives. This leads to the following results,

rlrm/ðxÞ ¼ora

oxl

orb

oxmrarb/ðrÞ ð66Þ

and

rlrmArðxÞ ¼ora

oxl

orb

oxm

orc

oxrrarb bAcðrÞ ð67Þ

The generalisation to higher orders is straight forward. The d’Alembertian on the scalarfield is next calculated, using (66),

�/ðxÞ ¼ glmrlrm/ðxÞ ¼ 1þ sx2

4l2

� �2

glm ora

oxl

orb

oxmrarb/ðrÞ ¼ rara/ðrÞ

¼ r2/ðrÞ ð68Þ


Likewise, the d’Alembertian on the vector potential yields, using (67),

�ArðxÞ ¼ glmrlrmArðxÞ ¼orc

oxrr2 bAcðrÞ ð69Þ

The action for the Yang–Mills theory on the A(dS) space is now defined by first con-sidering the repeated product of the field tensors. Taking (46) and using the transversalityof the Killing vectors, we get,bF abc

bF abc ¼ 3sl2 KblKcmKkbKq

c

� �F lmF kq ð70Þ

Finally, using (18), we obtain,

bF abcbF abc ¼ 3sl2 1þ s

x2

4l2

� �4

F lmF lm ð71Þ

Using this identification as well as (23), the actions on the flat space and the A(dS) spaceare mapped as,

S ¼ � 1

4

Zd4xTr � ðF lmF lmÞ ¼ �s

1

12l2

ZdXTr � ðbF abc

bF abcÞ ð72Þ

The lagrangian following from this action is given by

LX ¼ �s1

12l2Tr � ðbF abc

bF abcÞ ð73Þ

This completes the construction of the Yang–Mills action which can be taken as thestarting point for calculations on the A(dS) space. This action is manifestly gaugeinvariant under (49). Later on we will discuss an alternative approach to understandthis invariance.

Now using the definition of the induced metric (56), it is possible to show that the aboveaction can also be obtained from the standard action defined on the curved space which istaken as,

S ¼ � 1

4

Zd4xðp � gÞglqgmr Tr � F D

lmFDqr

� �ð74Þ

where

F Dlm ¼ rlAm �rmAl � i½Al;Am� ð75Þ

and Al has Weyl weight zero for the above action to be conformally invariant. Hence

F DlmF

Dqr ¼ olAm � Cr

lmAr � omAl þ CrmlAr � i½Al;Am�

� �� oqAr � Ck

qrAk � orAq þ CkrqAk � i½Aq;Ar�

� �ð76Þ

Since Crlm is symmetric in two lower indices,

F DlmF

Dqr ¼ F lmF qr ð77Þ

where Flm is defined in (34). Taking the explicit form of the induced metric (56) on theA(dS) space, we obtain,


pð�gÞ ¼ pð� detðglmÞÞ ¼p

� 1þ sx2

4l2

� ��8 !

det glm

!¼ 1þ s

x2

4l2

� ��4

ð78Þ

Putting all these values in (74) and using (23), we get

S ¼ � 1

4

ZdX 1þ s

x2

4l2

� �4

1þ sx2

4l2

� ��4

1þ sx2

4l2

� �4

glqgmr Tr � ðF lmF qrÞ ð79Þ

Finally, using (71) we obtain

S ¼ �s1

12l2

ZdXTr � ðbF abc

bF abcÞ ð80Þ

which reproduces (72).This section is concluded by providing a brief discussion of the gauge fixing condition.

On the hypersphere, Adler [22] used the following analogue of the Lorentz condition,

LabbAb � bAa ¼ 0 ð81Þ

The same condition is also viable on the A(dS) pseudosphere. Since there is a free index in(81) its connection with the Lorentz gauge is not particularly transparent. A straightfor-ward algebra, however, yields,

LabbAb � bAa ¼ s

1

l2raðrbLbc bAcÞ ð82Þ

so that (81) may be equivalently replaced by

raLabbAb ¼ 0 ð83Þ

as the pseudospherical analogue of the Lorentz condition. In this form there is no free in-dex. Self-consistency between (81) and (83) is established by contracting the former with ra

and using the transversality condition (9).In order to prove the identity (82) we simplify its R.H.S. as follows,

s

l2rarbLbc bAc ¼ raKclol

bAc ¼ raKcl � rcKla

� �olbAc þ rcKl

aolbAc

¼ LcabAc þ rcKl

aol Krc Ar

� �ð84Þ

where (28) and (29) have been used. Now, exploiting the identity,

rcKlaolKr

c ¼ �Kra ð85Þ

and the transversality condition (9), we obtain,

s

l2rarbLbc bAc ¼ Lc

abAc � Kr

aAr ¼ LcabAc � bAa ð86Þ

thereby proving (82).Actually, it is possible to generalise the relation (82) to any vector bV a which is projected

to the flat space by

bVa ¼ 1þ sx2

4l2

� �n

KlaV l ð87Þ

One follows the same steps as before leading to the equivalence,


LabbV b � bV a ¼

s

l2raðrbLbc bV cÞ ð88Þ

This will be used later. Note that, the mapping for the vector potential (8) corresponds toputting n = 0 in (87).

It is simple to prove that the operator in (83) corresponds to the covariant derivative$lAl in the local coordinates. From (63), we obtain,

rlAl ¼ glmrlAm ¼ 1þ sx2

4l2

� �2

glm ora

oxl

orb

oxmra bAb ¼ 1þ s

x2

4l2

� ��2

KalKlbra bAb ð89Þ

Now using the identity (19) and (32) we have,

rlAl ¼ s1

l2rbLba

bAa ð90Þ

This further justifies (83) as the pseudospherical analogue of the Lorentz gauge.The gauge condition in the form (81) is particularly useful for simplifying the equation

of motion. To see this consider the non-abelian equation of motion obtained from (80), byemploying the variational principle,2bLab

bF abc ¼ 0 ð91Þwhere the covariantised angular momentum bLab is defined in (39). This equation is nextwritten in terms of the potential bAa. Using the definition of the field tensor (45), the aboveequation reduces to,

LabLab bAc � 2LabLac bAb � 2iLabfra½ bAb; bAc�g � iLabfrc½ bAa; bAb�g� i½ra

bAb � rbbAa; bF abc� ¼ 0 ð92Þ

Now, using (27) we obtain,

½Lab; Lac� ¼ �3Lcb ð93Þ

Exploiting this identity in (92) yields,

LabLab bAc þ 6Lbc bAb � 2LacLabbAb � 2iLabfra½ bAb; bAc�g � iLabfrc½ bAa; bAb�g

� i½rabAb � rb

bAa; bF abc� ¼ 0 ð94Þ

In terms of the ‘covariantised angular momentum’ (39) this is written as,

P acbAc ¼ 0 ð95Þ

where the wave operator Pac is defined by

P ac ¼ bLbdbLbdgac � 6bLac þ 2bLab

bLbc ð96Þ

Exploiting the angular momentum algebra (27) it is possible to check the following prop-erties of the wave operator,bLbcP ca ¼ P bc bLca ¼ P b

a;

rbP ba ¼ P bara ¼ 0ð97Þ

2 See the Appendix A for details.


It is worthwhile to point out the significance of the identities (97). These are related to thegauge invariance of the action (80) under the transformations (40). The action (80) is man-ifestly gauge invariant under (49). However, there is an alternative way of understandingthis invariance which illuminates its connection with the gauge identity. As is known, forgauge theories defined on a flat space, gauge invariance of an action is enforced by a gaugeidentity; the number of gauge parameters being equal to the number of gauge identities.To see this in the present context, we consider the variation of the action (80) under anarbitrary transformation of the potential, d bAa:

dS ¼Z

dXTr � ½ðd bAcÞðbLabbF abcÞ� ¼

ZdXTr � ½ðd bAcÞðP ca bAaÞ� ð98Þ

where for simplicity, we have omitted the prefactor ð s12l2Þ. Invariance of the action under an

arbitrary d bAa yields the equation of motion (91) or (95). If the gauge transformation (40) isnow considered, then,

dS ¼ s

l2

ZdXTr � ½rbðbLbckÞðP ca bAaÞ� ð99Þ

Using an integration by parts we find,

dS ¼ � s

l2

ZdXTr � ½kbLbcfrbP ca bAag� ð100Þ

Gauge invariance of the action (i.e. dS = 0) requires that the factor multiplying the gaugeparameter k should vanish identically, i.e. without recourse to any equations of motion.This indeed happens as a consequence of the properties (97),bLbcfrbP ca bAag ¼ ðbLbcrbÞP ca bAa þ rbbLbcP ca bAa ¼ ðLbcrbÞP ca bAa þ rbP a

bbAa

¼ �ðobrbÞrcP ca bAa ¼ 0 ð101Þ

where the definition (39) of the ‘covariantised angular momentum’ bLbc has been explicitlyused to simplify the first piece. The above identity is usually referred as a gauge identity.

Let us now consider the gauge condition (81). It is equivalently expressed as follows,bLabbAb � bAa ¼ Lab

bAb � i½rabAb � rb

bAa; bAb� � bAa ¼ LabbAb � bAa ¼ 0 ð102Þ

where the transversality condition (9) forces the commutator term to vanish. Now usingthe gauge condition (102) in (95) we obtain,

ðbLabbLab � 4Þ bAc ¼ 0 ð103Þ

Some comments concerning the equation of motion (103) are now in order. This equa-tion dose not involve the parameter ‘s’ and is identical for both AdS as well as dS spaces.Of course this parameter enters when interactions are introduced. For instance, as dis-cussed later, for Yang–Mills field coupled to fermionic matter, the equation of motion(91) is replaced by

s

2l2bLabbF abc þ j c ¼ 0 ð104Þ

where ja is the fermionic current. Imposing the gauge condition (81) yields the form anal-ogous to (103),


ðbLabbLab � 4Þ bAc ¼ �2sl2j c ð105Þ

Finally, note that all derivatives occur only through the angular momentum operatorwhich is the correct derivative operator on the A(dS) pseudosphere.

The last point mentioned above is usually violated in other formulations [1,6–11] wherethe wave operator has a rather complicated structure so that derivatives involve both Lab

as well as oa. Hence, to ensure that it dose not go off the pseudosphere, subsidiary condi-tions are imposed on the field variables. This is now elaborated further.

In the ambient space formulation, starting from the Casimir operator and using theinfinitesimal generators, the Casimir eigenvalue equation and some algebric properties,the field equation for a massless abelian vector field in A(dS) space is found to be [10],3

ðsl2r2 � 2Þ bAa þ 2s

l2rarbLbc

bAc � sl2raðob bAbÞ ¼ 0 ð106Þ

A similar equation is also given in [6] which exploits the irreducible representations of thekinamatical A(dS) groups or using the triplet formalism [7]. The subsidiary (or divergence-less) condition o

b bAb ¼ 0 is next imposed to eliminate the last term that explicitly involvesthe derivative ob. For comparing (106) with our result, it is now necessary to identify $2

with LabLab. To see this we use the expression for the angular momentum operator(32), to derive,

s

2l2LabLab ¼ r2 þ s

l2ðrbraÞðrarbÞ ð107Þ

where we have used the results, ra$a = 0 and $ara = 4. Then,

r2 � s

2l2LabLab ¼ � s

l2ðrbraÞðrarbÞ ¼ 1þ s

x2

4l2

� �2

raðolraÞol

¼ 1þ sx2

4l2

� �21

2olðraraÞol ¼ 0 ð108Þ

where the constancy of the A(dS) pseudosphere rara = sl2 is used to get the vanishingresult. Hence, we obtain the identification,

r2 ¼ s

2l2LabLab ð109Þ

The Lorentz gauge fixing condition in the form (83) is now used to eliminate the secondterm in (106). Finally, putting (109) in (106) we find,

ðLbcLbc � 4Þ bAa ¼ 0 ð110Þ

which exactly corresponds to the abelian version of (103). It might be recalled that (103)was also obtained by using the Lorentz gauge, albeit in the form (81). Incidentally it is alsopossible to rewrite (103) using the covariantised transverse derivative (42). First, weexpress (39) in terms of this derivative as,bLab ¼ ra

~rb � rb~ra ð111Þ

3 This paper discusses only the de Sitter example while Refs. [6,7] consider the anti-de Sitter case. There is nocollective discussion.


Taking its repeated product yields,

s

2l2bLabbLab ¼ ~ra

~ra þ s

l2rað ~rbraÞ ~rb ¼ ~ra

~ra þ s

l2raðrbraÞrb ¼ ~ra

~ra ¼ ~r2 ð112Þ

where use has been made of (108). Hence the following form of (103) is obtained,

ðsl2 ~r2 � 2Þ bAa ¼ 0 ð113Þ

which can be interpreted as the non-abelian extension of (106), subject to subsidiary andLorentz gauge conditions.

We conclude this section by discussing two issues; the possibilities of hyperbolic A(dS)space as infrared regulators and secondly, the A(dS)/CFT correspondence in ourapproach.

It was shown in [12] that a space of constant negative curvature like the AdS space pro-vided a good infrared regulator for euclidean quantum field theory. The method used therewas also based on stereographic projection. Hence it is simple to relate the findings of [12]with our analysis. In fact the AdS gauge field equation of motion, which is the startingpoint in [12], just corresponds to the s = 1 version of (105). There was an arbitrarinessin the general solution of the Green function which was appropriately tuned to improvethe asymptotic behaviour. Since the de Sitter case implies s = �1 in (105), these argumentsgo through and one expects that the infrared properties also improve here. This is presum-ably related to the fact that time slices of the de Sitter space can be chosen to make thespatial volume finite thereby strongly affecting the infrared behaviour. Indeed it is preciselythis property of the boundedness of volume that makes euclidean QED on the sphere man-ifestly infrared-finite [22].

Recently, there have been several studies [13,14] to extend the AdS/CFT correspon-dence [15] to include the de Sitter space. While it is unclear whether the hypothesiseddS/CFT correspondence is on the same footing as the AdS/CFT one, it is possible to showtheir equivalence at the algebric level. The point is that the isometry group SO(2,n) (forAdSn+1) or SO(1, n + 1) (for dSn+1) acts on the boundary as the conformal group actingon Minkowski (euclidean) space. The same geometry as defined by (1) is used. Light conecoordinates are then introduced to define the ‘projective boundary’ of A(dS) space. Thepurported results follow by considering the action of the group – say SO(1,n + 1) – onthe boundary points. We refer to [16] for relevant technical details.

4. Stereographic projections of the Dirac equation on A(dS) space

In order to discuss the stereographic projection of the Dirac equation it is first necessaryto introduce the various Dirac matrices. In the ordinary D = 4 Minkowski space, thesematrices satisfy,

fcl; cmg ¼ clcm þ cmcl ¼ 2glm; fcl; c4g ¼ 0 ð114Þcyl ¼ c0clc0; cy4 ¼ c4; c4 ¼ ic0c1c2c3; ðc4Þ

2 ¼ 1 ð115Þ

Now we define the gamma matrices on the de Sitter (dS) space as,

Cl ¼ clc4 ð116ÞC4 ¼ c4 ð117Þ


while, for the anti-de Sitter (AdS) space as,

Cl ¼ iclc4 ð118ÞC4 ¼ c4 ð119Þ

These gamma matrices are defined so that, in either case, they obey the following proper-ties,

fCa;Cbg ¼ 2sgab ð120Þ

and

ðr � CÞ2 ¼ l2 ð121Þ

In order that the current satisfies a transversality condition like (9), we take its form onA(dS) space as,

ja ¼ p1

2lbW½C � r;Ca� bW ð122Þ

so that raja ¼ 0. Here, p = 1 for dS space and p = �i for AdS space. The field, bW is theDirac spinor on the A(dS) space which is mapped to the Dirac spinor W on the Minkowskispace through the following relation,

bW ¼ 1� x2

4l2

� �1� xl

2lcl

� �W ð123Þ

for dS space. For AdS space the corresponding map is,

bW ¼ 1þ x2

4l2

� �1� i

xl

2lcl

� �W ð124Þ

From these relations, we define the adjoint spinor as,

bW ¼ bWyC0 ¼ 1� x2

4l2

� �W 1þ xl

2lcl

� �ð125Þ

for dS space and for AdS space as,

bW ¼ bWyC0C4 ¼ 1þ x2

4l2

� �W 1þ i

xl

2lcl

� �ð126Þ

This completes the stereographic mapping of the Dirac spinor. We now use these expres-sions to prove,

ja ¼ 1þ sx2

4l2

� �2

Klajl ð127Þ

which is the exact analog of (8), apart from the conformal factor.Here the explicit calculation for dS space is given. Calculations for AdS space are sim-

ilar. For a = l we have from (122),


jl ¼1

2l1� x2

4l2

� �2

ra½ca; cl� �raxk

2l½ca; cl�ck þ 2r4cl �

2r4xk

2lclck

�þ raxm

2lcm½ca; cl� �

raxkxm

4l2cm½ca; cl�ck þ

2r4xm

2lcmcl �

2r4xkxm

4l2cmclck

�ð128Þ

Using the identity,

clcmca ¼ glmca � glacm þ gmacl � i�lmakckc4 ð129Þ

and the properties of gamma matrices (114) and (115) we get,

jl ¼ 1� x2

4l2

� �2

Kmljm ð130Þ

Similarly for a = 4 one can show,

j4 ¼ 1� x2

4l2

� �2

Kl4jl ð131Þ

These two can be written as,

ja ¼ 1� x2

4l2

� �2

Klajl ð132Þ

For AdS space this will be,

ja ¼ 1þ x2

4l2

� �2

Klajl ð133Þ

This proves Eq. (127). The implication of the conformal factor in (127) becomes evidentwhen we introduce coupling with the gauge fields. This factor is necessary to ensure theform invariance of the interaction term in the action,Z

dXðjabAaÞ ¼

Zd4xðjlAlÞ ð134Þ

This is verified by an explicit use of the various maps. Indeed in D-dimensions, the map(127) reads as,

ja ¼ 1þ sx2

4l2

� �D�2

Klajl ð135Þ

so that form invariance of the interaction term in the action as illustrated in (134) is validin any dimensions. The inverse map, computed from (127) by contracting with the Killingvector and using the identity (18), is given by

jl ¼ 1þ sx2

4l2

� ��4

Kalja ¼ 1þ sx2

4l2

� ��2ora

oxlja ð136Þ


We have seen that the vector fields are mapped by the conformal Killing vectors. So, wecan expect that the Dirac spinors are mapped by the conformal Killing spinors. To showthis we define a transformation matrix W such that,

W ¼ 1þ xl

2lcl

� �ð137Þ

for dS space and for AdS space,

W ¼ 1þ ixl

2lcl

� �ð138Þ

The adjoint of ‘W’, following (125), is given by

W ¼ W yC0 ¼ C0 1� xl

2lcl

� �ð139Þ

for dS space. For AdS space the corresponding definition follows from (126),

W ¼ W yC0C4 ¼ C0C4 1� ixl

2lcl

� �ð140Þ

Therefore, for dS space we have,

W W ¼ C0 1� x2

4l2

� �ð141Þ

and the corresponding relation for AdS space is,

W W ¼ C0C4 1þ x2

4l2

� �ð142Þ

Using the above equations, the fermion maps (123) and (124) are written collectivelyas,

W ¼ 1þ sx2

4l2

� ��2

W bW ð143Þ

The maps for the adjoint spinors, on the other hand, are different. For dS space (125) be-comes,

W ¼ � 1� x2

4l2

� ��2 bWC0W ð144Þ

while for AdS space, (126) simplifies to,

W ¼ 1þ x2

4l2

� ��2 bWC0C4W ð145Þ

The mapping (143) of fermions from A(dS) space to flat space has now been expressed interms of ‘W’ which is the conformal Killing spinor satisfying the equation,

olW ¼ 1

4clðcrorW Þ ð146Þ


This is analogous to (10) that defines the conformal Killing vector. The above relationyields a general definition for a conformal Killing spinor. Similar relations have appearedpreviously in the literature [23].

It is known that [18] there is a bilinear map connecting the conformal Killing vectorswith the conformal Killing spinors. To see this we have to write (122) in a different formwhich is given below. Using (120) and (121) one can show that

½C � r;Ca� ¼ �2 gba � s

rarb

l2

� �CbðC � rÞ ð147Þ

Therefore, the current (122) on A(dS) space is also expressed as,

ja ¼ �p1

lbW gb

a � srarb

l2

� �CbðC � rÞ bW ¼ �p

lbWhb

aCbC � r bW ð148Þ

Now with the help of (19), the above equation is written as,

ja ¼ �p1

l1þ s

x2

4l2

� ��2 bWKalKblCbðC � rÞ bW ð149Þ

Therefore, by (127) we have,

�p1

l1þ s

x2

4l2

� ��4 bWKalKblCbðC � rÞ bW ¼ WKlaclW ð150Þ

Using the fermionic spinor maps (123) and (125) (for dS space), the above can be writtenas,

1

l1� x2

4l2

� ��2

WKblCbðC � rÞC0W ¼ cl ð151Þ

Multiplying ‘W ’ from left and then ‘W’ from right on both sides of the above equation andusing (141) we get,

W clW ¼ 1

lKblðC0C

bÞðC � rÞ ð152Þ

for dS space. For AdS space, the calculation is similar and yields,

W clW ¼ i

lKblðC0C4C

bÞðC � rÞ ð153Þ

These are the cherished bilinear relations between the Killing spinors and the Killing vec-tors.

Let us next consider the definition of the axial current. The analogoue of c4 (chiralityoperator on Minkowski space) is given here by 1

l C � r since it satisfies,

1

lC � r

� �2

¼ 1 ð154Þ

and the anti-commutator,

C � rl; ½C � r;Ca�

�¼ 0 ð155Þ


The projection operators are given by

P� ¼1� C � r

l

2; ; P 2

� ¼ P�; ; PþP� ¼ 0 ð156Þ

Hence the axial current is defined as,

ja5 ¼ �p1

2l2bW½C � r;Ca�ðC � rÞ bW ð157Þ

This also satisfies the transversality condition raja5 ¼ 0. As was shown for the vector cur-rent, one can prove,

ja5 ¼ 1þ sx2

4l2

� �2

Klajl5 ð158Þ

where jl5 ¼ bWclc4W is the axial current on the flat space. This can also be written (similarto (148)) as,

ja5 ¼ �p bW gba � s

rarb

l2

� �CbbW ¼ � bWhb

aCbbW ð159Þ

Now we will write the Dirac operator and the Dirac equation on A(dS) space. For thiswe define,

Sab ¼1

4½Ca;Cb� ð160Þ

Using the definition for Sab given above, one can show the algebric relation,

SabLab ¼ � 1

4½cl; cm�Llm þ clL4l ð161Þ

for dS space. The corresponding relation for AdS space is,

SabLab ¼ 1

4½cl; cm�Llm � iclL4l ð162Þ

These relations are collectively expressed as,

SabLab ¼ s4½cl; cm�Llm þ pclL4l ð163Þ

Again using (12), (13) and (28) we get the form of the angular momentum operator on theflat space as,

Llm ¼ xlom � xmol ð164Þ

L4l ¼ �sl 1� sx2

4l2

� �ol �

1

2lxlxmom ð165Þ

Using the fermion maps (123), (125) and (161) together with the expression for the angularmomentum operator given above we obtain,

bWðSabLab þ 2Þ bW ¼ l 1� x2

4l2

� �4

WclolW ð166Þ


for dS space. The corresponding calculations for AdS space are exactly the same. For thisone can show,

bWð�SabLab þ 2Þ bW ¼ l 1þ x2

4l2

� �4

WclolW ð167Þ

So, in general we can write,

bWð�sSabLab þ 2Þ bW ¼ l 1þ sx2

4l2

� �4

WclolW ð168Þ

This enables us to convert the Dirac operator into the pseudospherical operator by the map,

lclol ! �sSabLab þ 2 ð169Þ

The Dirac action on the flat Minkowski space is given by

S ¼ �i

Zd4xWclolW ð170Þ

Using the map (23) for the measure and (168), we obtain the following action for A(dS)space by a projection of the above flat action,

S ¼ � i

l

ZdX bWð�sSabLab þ 2Þ bW ð171Þ

It is possible to show that the above action, exactly in analogy to the case of the gauge field,can be obtained from the standard action defined on the curved space which is taken as,

S ¼Z

d4xðp � gÞWcelmcmrlWc ð172Þ

where ‘Wc’ is the Dirac spinor on the curved space and is connected to the flat space Dirac

spinor through the conformal factor 1þ s x2

4l2

� �32

, which is required for the above action to

be conformally invariant. The vielbein elm is related to the flat Minkowski metric by thefollowing relation,

elm ¼ 1þ sx2

4l2

� ��1

glm ð173Þ

The covariant derivative $l for a spinor is,

rl ¼ ol þ1

2xlabr

ab ð174Þ

where the spin connection is defined as,

xlab ¼1

2ðCl;ab � Ca;bl � Cb;laÞ ð175Þ

Cm;ab ¼ oaem

b � obema

� �ð176Þ

and

rab ¼ 1

4½ca; cb� ð177Þ


Now using (23), (56) and (173) and the transformation relation Wc ¼ ð1þ s x2

4l2 Þ32W, the first

part of the action (172) involving the ordinary derivative can be written as,

S1¼�i

ZdX 1þ s

x2

4l2

� �4

1þ sx2

4l2

� ��4

1þ sx2

4l2

� �32

W 1þ sx2

4l2

� �glmcmol 1þ s

x2

4l2

� �32

W

" #

¼�i

ZdX 1þ s

x2

4l2

� �4

WclolW�3is

4l2

ZdX 1þ s

x2

4l2

� �3

WclxlW

ð178Þ

For the second part we have to first calculate the form of the spin connection. Using (173)and (176) we have,

Cl;ab ¼ elmCm;ab ¼ elm½oaðgmkebkÞ � obðgmkeakÞ�

¼ s

2l21þ s

x2

4l2

� ��1

ðxagbl � xbgalÞ ð179Þ

Therefore, using the definition for the spin connection (175), we obtain,

xlab ¼s

2l21þ s

x2

4l2

� ��1

ðxagbl � xbgalÞ ð180Þ

Putting all the results in the second part of the action we get,

S2 ¼ �is

16l2

ZdX 1þ s

x2

4l2

� �3

Wðxagbl � xbgalÞcl½ca; cb�W ð181Þ

Now, using the previous identity (129), we find,

ðxagbl � xbgalÞcl½ca; cb� ¼ �12xlcl ð182Þ

Putting this result in (181) we obtain,

S2 ¼3is

4l2

ZdX 1þ s

x2

4l2

� �3

WclxlW ð183Þ

Therefore, adding these two parts (178) and (183) of the action (172) we get,

S ¼ S1 þ S2 ¼ �i

ZdX 1þ s

x2

4l2

� �4

WclolW ð184Þ

Finally using (168) we obtain the cherished form,

S ¼ � i

l

ZdX bWð�sSabLab þ 2Þ bW ð185Þ

which reproduces (171).So the Dirac equation on A(dS) space is given by

ð�sSabLab þ 2Þ bW ¼ 0 ð186Þ

For the hypersphere an analogous relation is given in [22,24].For general D-dimensions the Dirac equation on A(dS) space can be written as:


�sSabLab þ D2

� �bW ¼ 0 ð187Þ

For D = 4 we get back equation(186).In presence of fermionic matter the flat space Yang–Mills action has the form,

S ¼Z

d4x � 1

4Tr � ðF lmF lmÞ � iWclðol � ieAlÞW

� ð188Þ

Using appropriate expressions for each piece, the stereographically projected action on theA(dS) space becomes,

bS ¼ Z dX � s

12l2Tr � ðbF abc

bF abcÞ � i

lbWð�sSabLab þ 2Þ bW � e bAaja

� ð189Þ

where ja is defined in (122) and we have used (127) to project the interaction term. Hence,in the presence of gauge fields, Eq. (186) becomes,

islðSabÞijgm

n Lab � 2i

lgm

n dij þ2ep

lðSabÞijrbðcAk aÞðT kÞmn

� bWjm ¼ 0 ð190Þ

By looking at the Dirac Eq. (186) or (187) one might be tempted to interpret the numericalfactor as a mass term on A(dS) space. But this is not true. It is seen from (169) that thisDirac operator was obtained from a projection of the massless Dirac operator on the flatspace. Also, it is equivalent to the massless Dirac operator (172) defined on a curved back-ground.

Yet another way is to use chiral invariance. In the flat space the mass termbreaks this invariance in the sence that the anti-commuting property {clol,c5} = 0of the massless Dirac operator no longer holds. In A(dS) space the analogue ofc5 is C�r

l , as already discussed. We now compute the anticommutator of the Diracoperator appearing in (187) with C�r

l and show that it vanishes, thereby reconfermingthat the numerical factor should not be interpreted as a mass term. The de Sitterspace calculation is given below. The calculation for AdS space is exactly similar.Now,

S � Lþ D2;C � r

l

�/ ¼ 1

lfS � L;C � rg/þ D

C � rl

/ ð191Þ

The first term of the above equation yields,

1

lfS � LðC � r/Þ þ C � rS � L/g ¼ 1

lfSabðLabC � rÞ/þ SabC � rLab/þ C � rS � L/g ð192Þ

Now the last two terms of the above cancel each other, which is shown below.

SabC � rLab þ C � rS � L ¼ 1

4½Ca;Cb�C � rLab þ 1

4C � r½Ca;Cb�Lab

¼ 1

2CaCbC � rðraob � rboaÞ þ 1

2C � rCaCbðraob � rboaÞ

¼ 1

2½ðC � rÞCaðC � rÞoa � CaðC � rÞ2oa þ ðC � rÞ2Cao

a

� ðC � rÞCaðC � rÞoa� ¼ 0 ð193Þ


where we have used (154) in the last line.Now,

SabLabðC � rÞ ¼ SlmLlmðC � rÞ þ SDlLDlðC � rÞ ¼ 1

2½cmclLlmðC � rÞ � clLDlðC � rÞ�

¼ 1

2½cmclcacDLlmra þ cmclcDLlmrD þ clcmcDLDlrm þ clcDLDlrD� ð194Þ

Using the expressions for Llm(164), LDl(165), rl and rD(2) we have,

Llmra ¼ 1

1� x2

4l2

ðxlgma � xmglaÞ

LlmrD ¼ 0

LDlrm ¼ �l 1þ x2

4l2

� �1� x2

4l2

glm

LDlrD ¼ xl

1� x2

4l2

ð195Þ

Therefore,

SabLabðC � rÞ ¼ð1� DÞxlclcD

1� x2

4l2

þ Dl 1þ x2

4l2

� �1� x2

4l2

cD �xl

1� x2

4l2

clcD

¼ ð1� DÞrlCl � DrDCD � rlCl ¼ �DðC � rÞ ð196Þ

Using all these results, we obtain,

S � Lþ D2;C � r

l

�¼ 0 ð197Þ

For AdS space this will be f�S � Lþ D2; C�r

l g ¼ 0. So, in general we obtain,

�sS � Lþ D2;C � r

l

�¼ 0 ð198Þ

This shows that D2

cannot be the mass term and the free Dirac operator for masslessfermion on general D-dimensional A(dS) space is 1

l ð�sS � Lþ D2Þ.

Now if we include a mass term, then the massive Dirac action on A(dS) space for4-dimensions can be written as,

S ¼Z

dX bW � i

lð�sS � Lþ 2Þ þ m

� bW ð199Þ

where ‘m’ plays the role of mass of fermion on A(dS) space. ‘m’ can be determined by ourprojection method. The mass term of the flat space action is given by mWW. Using the pro-jection for the spinor field (143)–(145) we obtain,

mWW ¼ m 1þ sx2

4l2

� ��3 bW bW ð200Þ

So the Dirac action on A(dS) space for the mass sector is,


Sm ¼Z

dX 1þ sx2

4l2

� �4

� m 1þ sx2

4l2

� ��3 bW bW ¼ Z dXm 1þ sx2

4l2

� �bW bW ð201Þ

Therefore the mass of the fermion on A(dS) space is given by

m ¼ m 1þ sx2

4l2

� �ð202Þ

In the flat space limit l fi1 we get back the usual mass.It may be mentioned that so far the analysis has been basically classical. Exten-

sion to quantum field theory is quite nontrivial. In the next section we will take upthis issue in some detail where an explicit calculation of the quantum anomaly ispresented and its equivalence is established with our stereographic projectionapproach.

There is another point to consider. Any quantum effect on the A(dS) space need nothave a counterpart on the flat space. For instance, a quantum field theory on dS space willreveal the analogue of Hawking radiation because of the cosmological horizon. However,it appears highly unlikely that this physical effect would naturally connect to a correspond-ing feature of the projected theory on flat space.

As a simple illustration consider the factorisation of the wave operator for Bose parti-cles in terms of the fermionic wave operator that holds in flat space,

ðic � oÞðic � oÞ ¼ �� ð203Þ

This dose not have an analogue on the A(dS) space. As we have shown, here the Boseoperator (in four dimensions) is (LabLab � 4) while the Fermi operator is (�sSabLab + 2).It is now possible to show [12,22],

ð�sSabLab þ 2Þð�sSabLab þ 1Þ ¼ � 1

2ðLabLab � 4Þ ð204Þ

The simple factorisation in flat space therefore dose not hold in the A(dS) space.

5. Chiral anomalies on A(dS) space

In this section we discuss the structure of chiral anomalies on A(dS) space. First, anexplicit evaluation of the axial U(1) anomaly is computed in the path integral approach.This result is next reproduced by an appropriate stereographic map of the usual Adler-Bell-Jackiw [25] flat space expression. This shows that the methods developed here aremeaningful for considering quantum effects. Finally, we obtain the non-abelian chiralanomalies by our projection technique.

5.1. The axial anomaly in the path integral formalism

In a series of papers, Fujikawa [26] has shown how the chiral anomalies encountered inperturbation theory may be derived in a path integral framework. In this approach theanomalous behaviour of Ward-Takahashi identities is traced to the Jacobian factor arisingfrom the noninvariance of the path integral measure under chiral transformations. Herewe compute the chiral anomaly in the A(dS) space using this method. We follow theapproach of [27] where the analysis has been done on the sphere. In our case an analytic


continuation of the A(dS) pseudosphere is implied. The advantage of working on the com-pact space is that it admits a complete set of familiar basis functions, namely the sphericalharmonics. The generating functional is,

Zðg; �g; vaÞ ¼Z

dl expðZ

dX½Lþ �g bW þ bWgþ v � bA�Þ ð205Þ

where dl ¼ ½d bW�½d bW�½d bAa� is the functional measure including the Faddeev–Popov factorand dX is the volume element.

Now the action (in four dimensions) is given by

S ¼Z

dX bW 1

lð�sS � Lþ 2Þ bW þ ie bAaja

� ¼Z

dX bW 1

lð�sS � Lþ 2Þ bW � iep bAa

bWhabCbC � r

lbW�

¼Z

dX bW 1

lð�sS � Lþ 2Þ � ig bAah

abCbC � r

l

� bW¼Z

dX bW½Q� ighabCbC � r

lbAa� bW ð206Þ

where in the second line use has been made of (148) and Q ¼ 1l ð�sS � Lþ 2Þ, g = ep.

Here we will show the calculation for dS space (i.e. s = �1). AdS space calculation issimilar to it. Following Fujikawa, let /n be a complete set of eigenfunction for the Diracoperator

DA ¼ Q� ighabCbC � r

lbAa ð207Þ

i.e.

DA/n ¼ kn/nZdX/ynðrÞ/mðrÞ ¼ dnm

ð208Þ

Under the chirality transformation bW ! ei�ðrÞC�rl bW, bW ! bWei�ðrÞC�rl the functional measuretransforms as dl! dl exp½�2i

RdX�ðrÞAðrÞ�, where

AðrÞ ¼X

n

/ynC � r

l/n ð209Þ

and the lagrangian transforms as L! L� i�½Labð bWSabC � r

l2bWÞ � 2 bW C � r

l2bW�. Now the

requirement of invariance of the generating functional under chiral transformation givesthe correct anomaly equation, which turns out to be,

Lab bWSabC � r

l2bW� � 2 bW C � r

l2¼ �2AðrÞ ð210Þ

Multiplying the above by rc we obtain the final form of the anomaly equation as,

jc5 � Lcbjb5 ¼ 2rcAðrÞ ð211Þwhere ja5 is given by Eq. (159). A(r) is the anomaly factor which will now be explicitly com-puted.


The conditionally convergent sum in A(r) (209) is evaluated by regularizing large eigen-values and changing to the free spinor harmonic basis. The spinor harmonics WðlÞl0msðrÞ areconstructed to be orthonormalized eigenfunctions of the free massless Dirac operatorQ ¼ 1

l ðS � Lþ 2Þ; i.e.

QWðlÞl0ms ¼ lWðlÞl0ms; ; l ¼ �ðl0 þ 2Þðl0 þ 1Þ;WðlÞl0ms ¼ P ðlÞY l0mvs;

P�ðl0þ2Þ ¼ l0 þ 3� S � L

2l0 þ 3;

P ðl0þ1Þ ¼ 1þ S � L

2l0 þ 3

ð212Þ

In the above set Y l0mðrÞ are four-dimensional spherical harmonics which satisfy,ZdXY l01m1

ðrÞY l02m2ðrÞ ¼ dl01l02

dm1m2;X

m

Y l0mðrÞY l0mðr0Þ ¼2l0 þ 3

4p52

C3

2

� �C

32

l0r � r0

l2

� �;X

l0m

Y l0mðrÞY l0mðr0Þ ¼ dðr � r0Þ

ð213Þ

where the index ‘m’ actually stands for the three ‘magnetic’ quantum numbers and the ‘vs’

are constant orthonormal 22 componant spinors. The C32ð Þ

l0 are Gegenbauer polynomials.

Therefore,

AðrÞ ¼ limM!1

Xn

/ynðrÞC � r

le�

knMð Þ2/nðrÞ

¼ limM!1

Xl0lms

WðlÞl0msyðrÞC � r

le�

DAMð Þ

2

WðlÞl0msðrÞ ð214Þ

Now from (207),

D2A ¼ Q2 � igðS � Lþ 2ÞhabCb

C � rl2

bAa � ighabCbC � r

l2bAaðS � Lþ 2Þ � g2 habCb

C � rlbAa

� �2

ð215Þ

In the previous section we have seen that fS � Lþ 2; C � rl g ¼ 0 (Eq. (197)). Also, one can

show another anti-commuting relation,

habCb;C � r

l

�¼ 0: ð216Þ

Using these anti-commuting relations, (215) can be written as,

D2A ¼ Q2 � ig

C � rl2ðS � Lþ 2ÞhabCb

bAa þ igC � r

l2habCb

bAaðS � Lþ 2Þ þ g2ðhabCbbAaÞ2

¼ Q2 � igC � r

l2S � LhabCb

bAa þ g2ðhabCbbAaÞ2 þ ig

C � rl2

habCbbAaS � L ð217Þ


Also, after some simplifications one can show,

C � rhabCbbAaS � L ¼ bAarbCaCcLbc ð218Þ

Therefore,

AðrÞ ¼ limM!1

Xl0lms

WðlÞl0msðrÞC � r

lexp �M�2 Q2 � ig

C � rl2

S � LhabCbbAa

�

þ g2ðhabCbbAaÞ2 þ ig

C � rl2

habCbbAaS:L

�WðlÞl0msðrÞ

¼ limM!1;r!r0

Xl0

TrC � r

lexp �M�2 Q2 � ig

C � rl2

S:LhabCbbAa

��

þ g2ðhabCbbAaÞ2 þ

ig

l2bAarbCaCcLbc

��

C 32

� �4p

52

ð2l0 þ 3ÞC32ð Þ

l0r � r0

l2

� �ð219Þ

Now one can check the following trace relation,

TrðCaCbCcCdCeÞ ¼ �4i�abcde ð220Þ

for four dimensions. The behaviour of the summand in (219) for large l 0 is l03e�l02M2 . To-

gether with the above trace relation and (216) we can eliminate several terms in the expo-nentials. Then (219) simplifies to,

AðrÞ ¼ limM!1;r!r0

Xl0

e�l0Mð Þ

2

C32ð Þ

l0r � r0

l2

� �C 3

2

� �4p

52

ð2l0 þ 3Þ

� TrC � r

lexp � 1

M2�ig

C � rl2

S:LhabCbbAa þ g2ðhabCb

bAaÞ2� � ��

¼C 3

2

� �4p

52

limM!1

Xl0

e�l0Mð Þ

2

ð2l0 þ 3Þ Cðl0 þ 3ÞCðl0 þ 1ÞCð3Þ

� TrC � r

lexp � 1

M2�ig

C � rl2

S � LhabCbbAa þ g2ðhabCb

bAaÞ2� � ��

ð221Þ

where we have used Cbað1Þ ¼ ðaþ2b�1ÞCa in the last line. The sum in (221) gives,

C 32

� �2p

52Cð3Þ

Xl0

e�l0Mð Þ

2

l0 þ 3

2

� �ðl0 þ 2Þðl0 þ 1Þ

¼ 2

ð4pÞ2MZ

dl0e�l02 ½ðMl0Þ3 þ OðMl0Þ2� ¼ 1

ð4pÞ2M4 1þ O

1

M

� �� ð222Þ

Expansion of the exponential in (221) gives terms like,


1

t!

�� 1

M2

�tTr

C � rl

�igC � r

l2S:LhabCb

bAa þ g2ðhabCbbAaÞ2

�t� ¼ 1

t!� 1

M2

� �t

TrC � r

l�ig

C � rl2

S � LhabCbbAa þ g2ðC � bAÞ2 �t�

¼ 1

t!ig

M2

� �t

TrC � r

lC � r

l2S � LhabCb

bAa þ igðC � bAÞ2 ��C � r

l2S � LhabCb

bAa þ igðC � bAÞ2 �t�1#

ð223Þ

Using the value of hab and Sab the terms in the trace simplify to,

C � rl

C � rl2

S:LhabCbbAa þ igðC � bAÞ2 �

¼ 1

2lCaCbCcðLab bAc � 2igra bAb bAcÞ ð224Þ

and

C � rl2

S � LhabCbbAa þ igðC � bAÞ2 ¼ � rd

l2CeCf ðLde bAf � igrd bAe bAf Þ ð225Þ

where for the last relation we have used,

raCaCbCcCdLbc bAd ¼ rað�2gab � CbCaÞCcCdLbc bAd

¼ �2rbCcCdLbc bAd � CbC � rCcCdðrboc � rco

bÞ bAd

¼ �2rbCcCdLbc bAd ð226Þ

since (C Æ r)2 = l2. So, R.H.S of (223) becomes,

1

t!ð�1Þt�1 ig

M2

� �t

Tr1

2lCaCbCcðLab bAc � 2igra bAb bAcÞ rd

l2CeCf ðLde bAf � igrd bAe bAf Þ

�t�1" #

ð227Þ

Now looking at (222) and (227) and considering the limit M fi1 it is seen that only thet = 2 term contributes in (227). Therefore,

AðrÞ ¼ 1

4ð4pÞ2g2Tr CaCbCcðLab bAc � 2ig

ra

lbAb bAcÞ rd

l3CeCf ðLde bAf � ig

rd

lbAe bAf Þ

� ¼ �i

g2

16p2l3�abcef rdðLab bAcÞðLde bAf Þ þ iglðLab bAcÞ bAe bAf

h�2ig

ra

lbAb bAcrdLde bAf þ 2ra bAb bAc bAe bAf

¼ �i

g2

16p2l3�abcef rdðLab bAcÞðLde bAf Þ ð228Þ

where in the last line anti-symmetricity of �abcef has been used. Now from (45) we obtain(for abelian case),bF abcrd

bF def ¼ 6�abcef Lab bAcrdLde bAf þ 3�abcef Lab bAcrdLef bAd ð229Þ


Again, multiplying (48) from left by �abcef and from right by rdLefbAd we obtain,

�abcef ðLabKclÞrdLef bAd ¼ 0 ð230Þ

Therefore the last term in (229) reduces to,

�abcef ðLab bAcÞrdðLef bAdÞ ¼ �abcef ðLabKcmAmÞrdðLef bAdÞ¼ �abcef ðLabKcmÞAmrdLef bAd þ �abcef KcmðLabAmÞrdLef bAd

¼ �abcef KcmðLabAmÞrdðLef KdlÞAl

þ �abcef KcmðLabAmÞrdKdlðLef AlÞ ¼ 0 ð231Þ

where we have used (230) and raKal = 0. So (229) yields,bF abcrdbF def ¼ 6�abcef Lab bAcrdLde bAf ð232Þ

Substituting this in (228) we obtain,

AðrÞ ¼ �ig2

96p2l3�abcef

bF abcrdbF def ¼ �i

g2

96p2l3ra�bcdef

bF abcbF def ð233Þ

Putting everything in (211), we get the anomaly equation,

jg5 � Lgbjb5 ¼ � ig2

48p2l3rgra�bcdef

bF abcbF def : ð234Þ

We now reproduce the above relation by stereographically projecting the familiarAdler-Bell-Jackiw anomaly [25] on the A(dS) pseudosphere. For the axial current, employ-ing a gauge invariant regularisation, the familiar result on the flat space is known to be[25],

oljl5 ¼ ig2

16p2�lmkqF lmF kq ð235Þ

Using (29) and the definition of the current (127) (appropriately interpreted for the axialvector currents), it is possible to obtain the identification,

raLabjb5 ¼ sl2 1þ sx2

4l2

� �4

oljl5 ð236Þ

In getting at the final result, use was made of the identity (21). This provides a map for oneside of (235). To obtain an analogous form for the other side, it is necessary to consider thecompletely anti-symmetric tensor �lmkq whose value is the same in all systems.

In order to provide a mapping among the �-tensors in the two spaces, we adopt thesame rule (46) used for defining the antisymmetric field tensor. However there is a slightsubtlety. Strictly speaking, this Levi–Civita epsilon is a tensor density. Hence its transfor-mation law is modified by an appropriate conformal (weight) factor,

�abcde ¼1

l1þ s

x2

4l2

� ��4

raKlbKm

cKkdKq

e þ cyclic permutations inða; b; c; d; eÞ� �

�lmkq

ð237Þ


It is possible to verify the above relation by an explicit calculation, taking the conventionthat both the epsilons are +1(�1) for any even (odd) permutation of distinct entries (0, 1,2, 3, 4) in that order.

The inverse relation is obtained from (237) by appropriate contractions and exploitingthe identity (18),

�lmkq ¼ s1

l1þ s

x2

4l2

� ��4

raKblKcmKdkKeq�abcde

¼ sl

1þ sx2

4l2

� �4orb

oxl

orc

oxm

ord

oxk

ore

oxqðra�

abcdeÞ ð238Þ

Now the explicit expressions for the anomaly are identified with the minimum of effort.Indeed, using (50) and (238), the ABJ anomaly is projected as,

1

16p2�lmkqF lmF kq ¼ s

1

16p2l51þ s

x2

4l2

� ��4

rarf ri�abcdebF fbcbF ide ð239Þ

The weight factors cancel out from both sides of the projected anomaly Eq. (235) and weobtain, using (236) and (239),

raLabjb5 ¼ig2

16p2l3rarf ri�abcde

bF fbcbF ide ð240Þ

It is also possible to rewrite the anomaly expression in a form that resembles the expres-sion on the hypersphere [18,27]. To do this we have to exploit the identity,

�abcdebF abc ¼ s

3

l2rarf �abcde

bF fbc ð241Þ

Then the A(dS) anomaly equation reduces to,

raLabjb5 ¼ sig2

48p2lra�bcdef


This is the desired anomalous current divergence equation in the A(dS) space which has aclose resemblance with the corresponding equation on the hypersphere. It is the exact ana-logue of the ABJ-anomaly equation on the flat space.

There is another way in which the anomaly equation can be expressed. To see this,observe that the projection (158) for the current corresponds to n = 2 in the general for-mula (87). Hence the identity (88) holds and we obtain,

rcraLabjb5 ¼ sl2ðLcbjb5 � jc5Þ ð243Þ

Thus the anomaly Eq. (242) takes the form given in (234), thereby completing our proof ofequivalence. Compatibility between the two forms (242) and (234) is easily established bycontracting the latter with rg and using the transversality of the current ðraja5 ¼ 0Þ.

The normal Ward identity for the vector current is obtained by setting the right handside of either (242) or (234) equal to zero.

It is known that on the flat space, it is feasible to redefine the current so that the anom-aly vanishes. In that case, however, the current is no longer gauge invariant. This compen-sating term is given by


X l ¼ ig2

8p2�lmkqAmF kq ð244Þ

such that olJl5 = 0, where,

Jl5 ¼ jl5 � X l: ð245Þ

Observe that Jl5 is not gauge invariant due to the presence of Xl.The same phenomenon also occurs on the A(dS) space. Here the compensating piece is

obtained from a projection of (244),

bX a ¼ 1þ sx2

4l2

� �2

KlaX l ð246Þ

Since,

raLab bX b ¼ sl2 1þ sx2

4l2

� �4

olX l ¼ sig2

48p2lra�bcdef


we observe that the modified current,

bJ a5 ¼ ja5 � bX a ¼ 1þ sx2

4l2

� �2

KlaJl5 ð248Þ

is anomaly free, i.e. raLabbJ b5 ¼ 0. Also note that the anomaly free currents are mapped in

the same way as the anomalous ones. The transversality condition rabJ a5 ¼ 0 is obviouslysatisfied by (248). Expectedly, the current bJ a5 is not gauge invariant.

It is straightforward to extend this calculation for arbitrary D = 2n dimensions.4 Theflat space expression (235) is known to be generalised as [28,29],

oljl5 ¼ 2i

ð4pÞnn!�l1l2...l2n

F l1l2 F l3l4 . . . F l2n�1l2n ð249Þ

The map for the Levi–Civita tensor is the generalised version of (238),

�l1l2...l2n¼ s

l1þ s

x2

4l2

� ��2n

raKa1l1Ka2l2

. . . Ka2nl2n�aa1a2...a2n ð250Þ

while the map for the field tensor is given by (50). Using these mappings the projectedexpression for the anomaly is,

2

ð4pÞnn!�l1l2...l2n

F l1l2 . . . F l2n�1l2n ¼ 2

ð4pÞnn!

s

l2nþ11þ s

x2

4l2

� ��2n

� rara1ra2

. . . ran�ab1b2...b2nbF a1b1b2 bF a2b3b4 . . . bF anb2n�1b2n

ð251Þ

Next, the projection of the L.H.S. of (235) has to be found. Using (29) and (135) we ob-tain,

4 For simplicity, the coupling factor g is not included.


raLabj5b ¼ sl2 1þ s

x2

4l2

� �2n

olj5

l ð252Þ

Finally, exploiting (251) and (252) the projected form of (249) is derived,

raLabj5b ¼

2i

ð4pÞnn!

1

l2n�1rara1

ra2. . . ran�ab1b2...b2n

bF a1b1b2 bF a2b3b4 . . . bF anb2n�1b2n ð253Þ

Following the same steps employed for obtaining (234) from (240), the above anomalyequation is expressed as,

jg5 � Lgbjb5 ¼ � 2i

3ð4pÞnn!l2n�1rgra�a1a2...a2nþ1

bF aa1a2 bF a3a4a5 . . . bF a2n�1a2na2nþ1 ð254Þ

5.2. Non-abelian chiral anomalies

Now we will discuss about the non-abelian chiral anomaly of spin one-half fermions. Asis well known [28–31], there are two types of anomaly on the flat Minkowski space: thecovariant anomaly and the consistent anomaly. The covariant anomaly, as its nameimplies, transforms covariantly under the gauge transformation. The consistent anomaly,on the other hand, is the one that satisfies the Wess-Zumino consistency condition. Just asthe covariant anomaly does not satisfy this condition, the consistent anomaly does nottransform covariantly.

The covariant anomaly is given by

ðDljlÞðaÞðcovariantÞ ¼ ðoljlÞðaÞ � i½Al; jl�ðaÞ ¼ i

32p2�lmqrTr � fkaF lmF qrg ð255Þ

where ka are the symmetry matrices and (jl)(a) is the chiral current,

ðjlÞðaÞ ¼ Wkacl

1þ c5

2W ð256Þ

The result (255) has been obtained by various methods [28–31], all of which basically relyon regularising the current (256) in a covariant manner.

We shall now stereographically project (255) to obtain the covariant anomaly on theA(dS) space. The map for the current (256) is similar to (127) and is given by

ðjaÞðaÞ ¼ 1þ sx2

4l2

� �2

KlaðjlÞ

ðaÞ ð257Þ

Now, using (39) and following steps similar to the abelian case, we find,

ðra bLabjbÞðaÞ ¼ sl2 1þ sx2

4l2

� �4

ðDljlÞðaÞ ð258Þ

which is the non-abelian version of (236).The R.H.S. of (255) is obtained by a straightforward generalisation of (239),

1

32p2�lmqrTr � fkaF lmF qrg ¼

s

32p2l51þ s

x2

4l2

� ��4

rarf ri�abcdeTr � fkabF fbcbF ideg ð259Þ

Hence, the covariant anomaly on A(dS) space is given by


ðrf bLfgjgÞðaÞðcovariantÞ ¼is

96p2lra�bcdef Tr � fkabF abcbF def g ð260Þ

which follows on exploiting (255), (258), (259) and the identity (241).Next, the consistent anomaly is considered. On the flat Minkowski space this is given by

ðDljlÞðaÞðconsistentÞ ¼i

24p2�lmqrTr � kaolðAmoqAr �

i

2AmAqArÞ

�ð261Þ

In order to project this equation it is convenient to recast it in the following form,

ðDljlÞðconsistentÞ ¼AðaÞðconsistentÞ

¼ i

96p2�lmqrTr � fkaðF lmF qrþ iF lmAqArþ iAlAmF qr� iAmF lqArÞg ð262Þ

where the definition (34) of field tensor Flm has been used. Using the inverse maps for �lmqr

(238), field tensor (50) and the vector field (22), the stereographic projection of the consis-tent anomaly (262) on A(dS) space is,

ðAÞðaÞðconsistentÞ ¼i

96p2l31þ s

x2

4l2

� ��4

�abcderarfs

l2riTr � fkabF fbcbF ideg

�þ iTr � fkaðbF fbc bAd bAe þ bAc bAd bF fbc � bAcbF fbd bAeÞg

ið263Þ

Now using the identity (241) one can show,

ðAÞðaÞðconsistentÞ ¼i

288p2lð1þ s

x2

4l2Þ�4 1

l2ra�bcdef Tr � fkabF abcbF def g

�þis�abcdeTr � fkaðbF abc bAd bAe þ bAc bAd bF abc � bAcbF abd bAeÞg

ið264Þ

Therefore, the final equation for the consistent anomaly on the A(dS) space is,

ðrf bLfgjbÞðaÞðconsistentÞ ¼ sil

288p2

1

l2ra�bcdef Tr � fkabF abcbF def g

�þis�abcdeTr � fkaðbF abc bAd bAe þ bAd bAebF abc þ bAd bF abc bAeÞg

ið265Þ

It is known that, on the flat Minkowski space, the covariant and consistent currents arerelated by a local counterterm given by

ðX lÞðaÞ ¼ ðjlÞðaÞðconsistentÞ � ðjlÞðaÞðcovariantÞ

¼ � i

48p2�lmqrTr � fkaðAmF qr þ F qrAm þ iAmAqArÞg ð266Þ

As was shown in [29], this ambiguity in the current is such that it does not affect the effec-tive action. This is a consequence of the fact that,

ðAlÞðaÞðX lÞðaÞ ¼ 0 ð267Þ

From (266) we observe that the covariant and consistent anomalies on flat space are re-lated by

ðDljlÞðaÞðcovariantÞ ¼ ðDljlÞðaÞðconsistentÞ � ðDlX lÞðaÞ ð268Þ


The above analysis is now carried out on the A(dS) space. The projection of (266) yields,

ðbX bÞðaÞ ¼ 1þ sx2

4l2

� �2

KblðX lÞðaÞ

¼ � i

48p2lra�abcde

1

l2riTr � fkað bAcbF ide þ bF ide bAcÞg þ isTr � fka bAc bAd bAeg

� ð269Þ

Note that the condition analogous to (267) is,

ð bAaÞðaÞðbX aÞðaÞ ¼ 0 ð270Þwhich is obviously satisfied by (269). Now, using the definition (39) of ‘covariantised angu-lar momentum’ and (269), we have,

ðrf bLfgbX gÞðaÞ ¼ � s

il288p2

2

l2ra�bcdef Tr � fkabF abcbF def g � is�abcdeTr

�� fkaðbF abc bAd bAe þ bAd bAebF abc þ bAd bF abc bAeÞg

ið271Þ

Exploiting the expressions for the consistent anomaly, covariant anomaly and the aboverelation we have,

ðrf bLfgjgÞðaÞðcovariantÞ ¼ ðrf bLfgjgÞðaÞðconsistentÞ � ðr

f bLfgbX gÞðaÞ ð272Þ

which is the A(dS) space analogue of (268). Here, ðrf bLfgbX gÞðaÞ plays the role of the local

counterterm for the anomaly on the A(dS) space.

6. Duality symmetry

The well known electric-magnetic duality symmetry swapping field equations with theBianchi identity in flat space has an exact counterpart on the A(dS) hyperboloid. To seethis it is essential to introduce the dual field tensor that enters the Bianchi identity. Thedual tensor is defined by

~F ab ¼ �1

6�abcde

bF cde ð273Þ

Using (46) and (237) together with the properties of the Killing vectors the dual on theA(dS) space is expressed in terms of the dual on the flat space as,

~F ab ¼ �slKkaKq

b~F kq ð274Þ

where the flat space dual is given by

~F kq ¼1

2�kqlmF lm ð275Þ

Inversion of (274) yields,

~F lr ¼ �s1

l1þ s

x2

4l2

� ��4

KalKbr~F ab ¼ �s

1

lora

oxl

orb

oxr~F ab ð276Þ

where we have used (14) to obtain the final result. Apart from a dimensional scale themapping is exactly identical to that of a second rank tensor given in (51).


The Bianchi identity on the A(dS) space is then given by

raLab ~F bc ¼ 0 ð277ÞThis is confirmed by a direct calculation. Alternatively, it becomes transparent by projectingit on the flat space by means of Killing vectors. Using the basic definitions and the identity,

KqboqðKblKm

cÞ~F lm ¼ 0 ð278Þ

we obtain,

raLab ~F bc ¼ �l3KblKkbKq

col~F kq ð279Þ

Finally, exploiting (18) we get the desired projection,

raLab ~F bc ¼ �l3 1þ sx2

4l2

� �2

Kqco

k ~F kq ð280Þ

which vanishes since ok ~F kq ¼ 0.Now the abelian equation of motion following from a variation of the action (72) is

given by

LabbF abc ¼ 0 ð281Þ

The duality transformation is next discussed. Analogous to the flat space rule,~F ! F ; F ! �~F the duality map here is provided by

~F ab !rc

lbF abc;

rc

lbF abc ! �~F ab ð282Þ

It is easy to check the consistency of this map. The inverse of (273) yields,

bF abc ¼ s1

2�abcde

~F de ð283Þ

while,

bF abc ¼ � 1

2�abcde ~F de ð284Þ

Under the first of the maps in (282), the above relation is transformed as,

bF abc ! �s1

4lrf �abcde�

defgh ~F gh ¼ �s1

lra

~F bc þ rb~F ca þ rc

~F ab

� �ð285Þ

where use was made of (284) at an intermediate step. Contracting the above map by rc

immediately leads to the second relation in (282).Likewise, under the map (285), the dual field (273) transforms as,

~F ab ! s1

2l�abcderc ~F de ð286Þ

Substituting the expression (273) for ~F de we reproduce the first of the maps given in (282).Now the effect of the duality map on the equation of motion (281) is considered. Using

(285) and the correspondence (274) along with the identity (278) we find,

1

2LabbF abc ! sl2 1þ s

x2

4l2

� �2

Kcqol ~F lq ð287Þ


Finally, using (280) we obtain the cherished mapping,

1

2LabbF abc ! �s

1

lraLab ~F bc ð288Þ

showing how the equation of motion passes over to the Bianchi identity. Likewise theother map swaps the Bianchi identity to the equation of motion,

s1

lraLab ~F bc !

1

2LabbF abc ð289Þ

It is feasible to perform a continuous SO(2) duality rotations through an angle h. The rel-evant transformations are then given by,

rc

lbF 0abc ¼ cos h

rc

lbF abc � sin h~F ab ð290Þ

~F 0ab ¼ sin hrc

lbF abc þ cos h~F ab ð291Þ

This mixes the equation of motion and the Bianchi identity in the following way,

1

2LabbF 0abc ¼ cos h

1

2LabbF abc � sin hs

1


s1

lraLab ~F 0bc ¼ sin h

1

2LabbF abc þ cos hs

1


The discrete duality transformation corresponds to h ¼ p2.

7. Formulation of anti-symmetric tensor gauge theory

Our analysis can be extended to include higher rank tensor gauge theories. Some typicalexamples are the linearised version of gravity which uses a symmetric second rank tensoror the p-form gauge theories employing anti-symmetric tensor fields.

In this section we discuss our formulation for the second rank antisymmetric tensorgauge theory. Also, there are some features which distinguish it from the analysis forthe vector gauge theory. The extension for higher forms is obvious. Both abelian andnon-abelian theories will be considered. To set up the formulation it is convenient to beginwith the abelian case which can be subsequently generalised to the non-abelian version.The action for a free 2-form gauge theory in flat four-dimensional Minkowski space isgiven by [32],

S ¼ � 1

12

Zd4xF lmqF lmq ð294Þ

where the field strength is defined in terms of the basic field as,

F lmq ¼ olBmq þ omBql þ oqBlm ð295Þ

The infinitesimal gauge symmetry is given by the transformation,

dBlm ¼ olKm � omKl ð296Þ

which is reducible since it trivialises for the choice Kl = olk.It is sometimes useful to express the action (or the lagrangian) in a first order form by

introducing an extra field,


L ¼ � 1

8�lmqrF lmBqr þ 1

8AlAl ð297Þ

where the B ^ F term involves the field tensor,

F lm ¼ olAm � omAl ð298Þ

Eliminating the auxiliary Al field by using its equation of motion, the previous form (294)is reproduced. The gauge symmetry is given by (296) together with dAl = 0. The first orderform is ideal for analysing the non-abelian theory.

To express the theory on the A(dS) pseudosphere, the mapping of the tensor field is firstgiven. From the previous analysis, it is simply given bybBab ¼ Kl

aKmbBlm ð299Þ

and satisfies the transversality condition,

rabBab ¼ rbbBab ¼ 0: ð300Þ

The tensor field with the latin indices is defined on the pseudosphere while those with thegreek symbols are the usual one on the flat space. This is written in component notation byusing the explicit form for the Killing vectors given in (12) and (13),

bBlm ¼ 1þ sx2

4l2

� �1þ s

x2

4l2

� �Blm � s

xqxm

2l2Blq � s

xqxl

2l2Bqm

� �ð301Þ

and,

bBl4 ¼1

l1þ s

x2

4l2

� �xqBlq ð302Þ

These are the analogues of (16). The inverse relation is given by,

1þ sx2

4l2

� �4

Blm ¼ KlaKm

bbBab ð303Þ

which may also be put in the form,

1þ sx2

4l2

� �2

Blm ¼ bBlm � sxlbBm4

2lþ s

xmbBl4

2lð304Þ

which is the direct analogue of (17). Moreover using (14), the map (303) may also be ex-pressed in a more transparant form as,

Blm ¼ora

oxl

orb

oxmbBab ð305Þ

which is just the result (51) for a second rank tensor.Next, the gauge transformations are discussed. From (296), the defining relation (299)

and the angular momentum operator (28), infinitesimal transformations are given by

dbBab ¼ src

l2Kl

bLca � KlaLcb

� �Kl ð306Þ

This is consistent with radbBab ¼ 0 which is imposed by (300). In this form the expression isnot manifestly covariant. This may be contrasted with (38) which has this desirable fea-


ture. The point is that an appropriate map of the gauge parameter is necessary. In the pre-vious example the gauge parameter was a scalar which retained its form. Here, since it is avector, the required map is provided by a relation like (8), so that,bKa ¼ Kl

aKl ð307Þ

Pushing the Killing vectors through the angular momentum operator and using the abovemap yields, after some simplifications,

dbBab ¼ s1

l2rc Lca

bKb � LcbbKa

� �� ra

bKb þ rbbKa

h ið308Þ

It is also reassuring to note that (308) manifests the reducibility of the gauge transforma-tions. Since Kl = olk leads to a trivial gauge transformation in flat space, it follows from(307) that the corresponding feature should be present in the pseudospherical formulationwhen,

bKa ¼ s1

l2rcLcak ð309Þ

It is easy to check that with this choice, the gauge transformation (308) trivialises; i.e.dbBab ¼ 0.

The field tensor on the pseudosphere is constructed from the usual one given in (295).Since the Killing vectors play the role of the metric in connecting the two surfaces, thisexpression is given by a natural extension of (46),bF abcd ¼ raKl

bKmcK

qd þ rbKl

c KmaKq

d þ rcKldKm

aKqb þ rdKl

aKmcK

qb

� �F lmq ð310Þ

Note that cyclic permutations have to taken carefully since there is an even number ofindices. The inverse mapping is provided by

F lmq ¼ s1

l2ð1þ s

x2

4l2Þ�6KblKcmKdqðra

bF abcdÞ ¼ s1

l2

orb

oxl

orc

oxm

ord

oxqðrabF abcdÞ ð311Þ

This is a particular case of the general result (52).In terms of the basic variables, the field tensor is known to be expressed as,

bF abcd ¼ LabbBcd þ Lbc

bBad þ LbdbBca þ Lca

bBbd þ LdabBcb þ Lcd

bBab

� �ð312Þ

To show that (310) is equivalent to (312), the same strategy as before, is adopted. Usingthe definition of the angular momentum (28), (312) is simplified as,bF abcd ¼ raKl

b � rbKla

� �ol Km

cKrdBmr

� �þ � � � ð313Þ

where the carets denote the inclusion of other similar (cyclically permuted) terms. Nowthere are two types of contributions. Those where the derivatives act on the Killingvectors and those where they act on the fields. The first class of terms cancel out asa consequence of an identity that is an extension of (48). The other class combinesto reproduce (310).

The action on the A(dS) pseudosphere is now obtained by first taking a repeated prod-uct of the field tensor (310). Using the properties of the Killing vectors, this yields,


bF abcdbF abcd ¼ 4sl2 1þ s

x2

4l2

� �6

F lmqF lmq ð314Þ

From the definition of the flat space action (294) and the volume element (23), it followsthat the above identification leads to the pseudospherical action,

SX ¼ �s1

48l2

ZdX 1þ s

x2

4l2

� ��2bF abcdbF abcd ð315Þ

Thus, up to a conformal factor, the corresponding lagrangian is given by,

LX ¼ �s1

48l2bF abcd

bF abcd ð316Þ

By its very construction this lagrangian would be invariant under the gauge transforma-tion (308). There is however another type of gauge symmetry which does not seem to haveany analogue in the flat space. To envisage such a possibility, consider a transformation ofthe type,5

dbBab ¼ Labk ð317Þ

which could be a meaningful gauge symmetry operation on the A(dS) space. However, inflat space, it leads to a trivial gauge transformation. To see this explicitly, consider the ef-fect of (317) on (303),

1þ sx2

4l2

� �4

dBlm ¼ KlaKm

bLabk ð318Þ

Inserting the expression for the angular momentum from (28) and exploiting the transver-sality (6) of the Killing vectors, it follows that,

dBlm ¼ 0 ð319Þ

thereby proving the statement. To reveal that (317) indeed leaves the lagrangian (316)invariant, it is desirable to recast it in the form,

LX ¼ �s1

32l2bRabRa ð320Þ

wherebRa ¼ �abcdeLbcbBde ð321Þ

Under the gauge transformation (317), a simple algebra shows that dbRa ¼ 0 and hence thelagrangian remains invariant.

The inclusion of a non-abelian gauge group is feasible. Results follow logically from theabelian theory with suitable insertion of the non-abelian indices. As remarked earlier it isuseful to consider the first order form (297). The lagrangian is given by its straightforwardgeneralisation, where the non-abelian field strength has already been defined in (34). It isgauge invariant under the non-abelian generalisation of (296) with the ordinary derivativesreplaced by the covariant derivatives with respect to the potential Al, and dAl = 0. By the

5 Recently such a transformation was considered on the hypersphere [33,18].


help of our equations it is possible to project this lagrangian on the A(dS) space. Forinstance, the corresponding gauge transformations look like,

dbBab ¼ s1

l2rc Lca

bKb � LcbbKa

� �� ra

bKb þ rbbKa

h i� i½ bAa; bKb� þ i½ bAb; bKa�

¼ s

l2½rcðbLca

bKb � bLcbbKaÞ � ra

bKb þ rbbKa� ð322Þ

and so on. Expectedly, the ordinary angular momentum operator gets replaced by itscovariantised version.

Matter fields may be likewise defined. The fermion current jlm will be defined just as thetwo form field,

jab ¼ KalKbmjlm ð323Þwhile the inverse relation is given by

jlm ¼ 1þ sx2

4l2

� ��4

KalKbmjab ¼ ora

oxl

orb

oxmjab ð324Þ

It is simple to verify that this map preserves the form invariance of the interaction,Zd4xðjlmB

lmÞ ¼Z

dXðjabbBabÞ ð325Þ

quite akin to (134).

8. Zero curvature limit

The null curvature limit (which is also equivalent to a vanishing cosmological constant)is obtained by setting l fi1. Then the A(dS) group contracts to the Poincare group sothat the field theory on the A(dS) space should contract to the corresponding theory onthe flat Minkwski space. This is shown very conveniently in the present formalism usingKilling vectors. The example of Yang–Mills theory with sources will be considered.

The equation of motion in the A(dS) space obtained by varying the action composed ofthe pieces (72) and (134) is found to be,

s1

2l2bLabbF abc þ jc ¼ 0 ð326Þ

The operator appearing in the above equation is now mapped to the flat space. The map-ping for the usual angular momentum part is first derived,

LabbF abc ¼ 2raKl

bol ½raKbmKcq þ c:p:�F mq

� �ð327Þ

Using the transversality condition and the identities among the Killing vectors it is seenthat the only nonvanishing contribution comes from the action of the derivative on thefield tensor yielding,

LabbF abc ¼ 2sl2 1þ s

x2

4l2

� �2

KcqolF lq ð328Þ

It is straightforward to generalise this for the covariantised angular momentum and onefinds,


bLabbF abc ¼ 2sl2 1þ s

x2

4l2

� �2

KcqDlF lq ð329Þ

Using the map (127) for the currents, the equation of motion on the A(dS) space finallygets projected on the flat space as,

1þ sx2

4l2

� �2

KcqðDlF lq þ jqÞ ¼ 0 ð330Þ

This equation is now multiplied by the Killing vector Kkc . Using the identity among the

Killing vectors yields,

1þ sx2

4l2

� �4

ðDlF lk þ jkÞ ¼ 0 ð331Þ

The zero curvature limit (l fi1) is now taken. The prefactor simplifies to unity and thestandard flat space Yang–Mills equation with sources is reproduced.

9. Conclusions

We have provided a manifestly covariant formulation of non-abelian interacting gaugetheories defined on the A(dS) hyperboloid. The various expressions, at each step of the anal-ysis, preserved this covariance under the appropriate kinematic groups associated with theA(dS) space. A distinctive feature was to bypass the general formulation of field theoriesdefined on a curved space [3] in favour of exploiting the symmetry properties peculiar tothe A(dS) hyperboloid. This enabled us to set up a formulation that was general enoughto include both de Sitter as well as anti-de Sitter space-times, arbitrary dimensions, non-abe-lian gauge groups and higher rank tensor fields. Also, a complete one to one mapping with thecorresponding results on a flat Minkowski space-time was established. Using this correspon-dence it was possible to show that in the zero cuvature limit, the A(dS) field equation passedon to the flat space field equation. This was reassuring since it is known that the groups of theA(dS) space are deformations of the Poincare group which is the kinematical group of flatMinkowiki space.

Our method consists in embedding the d-dimensional A(dS) space in a flat (d + 1)-dimen-sional space, sometimes called the ambient space. While this is a time honoured approach[34,35], we have deviated on two important issues. First, instead of working with arbitrarycoordinate transformations that provide the map between the A(dS) space and the flat space,a particular type - the stereographic projection - has been used. An advantage of this is thatthis projection being conformal, the coordinate transformations were expressed in terms ofthe conformal Killing vectors. These vectors were explicitly computed by solving the Cartan-Killing equation. All expressions were thereby written in terms of the Killing vectors. Variousproperties of these vectors derived here were used to obtain the results compactly and trans-parently. The second distinctive feature was to avoid group theoretical techniques based onCasimir operators to construct the lagrangian or the action. We gave maps, involving theKilling vectors, for projecting the gauge fields on the flat space to the A(dS) space. Usingthese maps the action on the A(dS) space was constructed from a knowledge of the flat spaceaction. The action so obtained was manifestly covariant under the kinematical symmetrygroup of the A(dS) hyperboloid. All derivatives appeared only through the angular momen-


tum operator Lab. In other approaches, apart from Lab, the usual derivative oa also appears.This has to be removed by choosing subsidiary conditions which are obviously not requiredhere. Our results are in general valid for arbitrary dimensions. In particular, the expressionfor the axial anomaly in A(dS) space was given for any D = 2n dimensions.

We analysed the Yang–Mills theory and the two form gauge theory in details. Exten-sion to higher rank tensor fields is straightforward. Also, a discussion of the matter (ferm-ionic) sector was provided. The Lorentz gauge fixing condition was analysed. It appearedin two equivalent versions, one of which did not have any free index (reminiscent of theusual Lorentz gauge on a flat space) while the other had a single free index. The utilityof both forms was revealed. Specifically, the equivalence of our abelian gauge field equa-tion with that obtained in other (say ambient) formalisms (after imposing appropriate sub-sidiary conditions) was established in the Lorentz gauge fixed sector, where both versionsof the gauge fixing had to be employed.

We have computed the singlet (axial) anomaly as well as the non-abelian covariant andconsistent chiral anomalies. This was done by projecting the relevant expressions from theflat to the A(dS) space. For the singlet case, we also computed the redefined expression forthe axial current such that it was anomaly free. However, the current was no longer gaugeinvariant. This revealed the interplay between the anomaly and gauge invariance, exactlyas happens for the flat example. In the non-abelian context the counterterm connecting thecovariant and consistent anomalies has been calculated.

The dual field tensor was introduced from which a form of the Bianchi identity wasgiven. Electric-magnetic duality rotations swapping this identity with the equations ofmotion were found.

We feel our approach gives an intuitive understanding of the closeness of formulatinggauge field theories on flat and A(dS) space-time. Apart from the issues dealt here, a furtherapplication would be to develop the complete BRST formulation. This was earlier done byone of us [19] in a collaborative work for the case of the hypersphere (an n-dimansionalsphere embedded in (n + 1)-dimensional flat space). Also, a possible connection betweenmassless and massive higher rank tensor theories with superstring theory could be envisaged.

Appendix A. Variation principle and boundary condition

The equation of motion of any system can be derived by using the variation principleaccording to which the action of the system is extremised. Here we will explicitly show howthe equation of motion (91) comes by extremising the corresponding action (73) with suit-able boundary conditions. The compatibility of these conditions on the A(dS) space andthe flat space is also shown.

Varying the action (73) and using the definition for field tensor (45) we obtain,

dS ¼ � s

6l2

ZdXTr½dbF abc

bF abc� ¼ � s

2l2

ZdXTr½dðLab

bAc � ira½ bAb; bAc�ÞbF abc�

¼ � s

2l2

ZdXTr½Labd bAc � bF abc � 2ira½cdAb; bAc�bF abc� ð332Þ

where in the last line we have used the anti-symmetric property of bF abc. Now,ZdXðLabd bAcÞ � bF abc ¼

ZdX½ðraob � rboaÞd bAc� � bF abc ¼ 2

ZdXðobd bAcÞra

bF abc ð333Þ


Using the expression for invariant measure (25) we obtain,ZdXðLabd bAcÞ � bF abc ¼ 2l

Zd4rr4

½ðobd bAcÞrabF abc�

¼ 2lZ

d4rr4

ðglbold bAcÞra

bF abc

¼ 2lZ

d4rðold bAcÞ1

r4

rabF alc

¼ 2lZ

d4r ol d bAc1

r4

rabF alc

� �� 2l

Zd4r d bAcol

1

r4

rabF alc

� �� ð334Þ

Here ol ¼ oorl. By Gauss’ divergence theorem the first term of the above gives the surface

term which vanishes at the boundary if we consider d bAc ¼ 0 at the boundary. Using thisboundary condition the above expression simplifies to,Z

dXðLabd bAcÞ � bF abc ¼ �2lZ

d4rd bAcol1

r4

rabF alc

� ð335Þ

Now,

ol1

r4

rabF alc

� ¼ � 1

ðr4Þ2or4

orlrabF alc þ 1

r4

ol½rmbF mlc þ r4

bF 4lc�

¼ � 1

ðr4Þ2or4

orlrabF alc þ 1

r4

½bF llc þ rmolbF mlc þ olr4 � bF 4lc þ r4ol

bF 4lc�

ð336Þ

Since bF llc ¼ 0 and or4

orl ¼ � srl

r4the above expression reduces to,

ol1

r4

rabF alc

� ¼ s

ðr4Þ3rlðrm

bF mlc þ r4bF 4lcÞ þ 1

r4

ðraolbF alc � srl

r4

bF 4lcÞ

¼ s

ðr4Þ3rlrm

bF mlc þ s

ðr4Þ2rlbF 4lc þ 1

r4

raolbF alc � s

ðr4Þ2rlbF 4lc ð337Þ

The first term vanishes as bF mlc is anti-symmetric in l and m and second and last terms can-cel each other. Hence we get,

ol1

r4

rabF alc

� ¼ 1

r4

raolbF alc ¼ 1

r4

raobbF abc ¼ 1

2r4

LabbF abc ð338Þ

Substituting this in (335) we obtain,ZdXðLabd bAcÞbF abc ¼ �

ZdXd bAcLab

bF abc ð339Þ

Likewise the second term of (332) can be simplified as,

ZdXTr½�2ra½d bAb; bAc�bF abc� ¼
ZdXTr½�2raðd bAb

bAc � bAcd bAbÞbF abc�

¼Z

dXTrð�2rad bAb½ bAc; bF abc�Þ

¼Z

dXTrðd bAc � 2ra½ bAb; bF abc�Þ

¼Z

dXTrfd bAcðra½ bAb; bF abc� � rb½ bAa; bF abc�Þg ð340Þ

Substituting (339) and (340) in (332) we obtain,

dS ¼ s

2l2

ZdXTr½d bAcfLab

bF abc � ira½ bAb; bF abc� þ irb½ bAa; bF abc�g� ð341Þ

Hence dS = 0 (subjected to the boundary condition d bAa ¼ 0) yields,

fLab � i½rabAb � rb

bAa; �gbF abc ¼ 0 ð342Þi.e bLab

bF abc ¼ 0 ð343Þwhich is the equation of motion (91).

The equivalence of this equation of motion with the stereographically projected YMequation of motion on flat space was shown in Section 8. Now the equation of motionon flat space is derived by the variation principle where we use the boundary condition,dAl = 0. Again, the vector field on the A(dS) space is the projection of the flat space fieldthrough the Killing vector which we have shown in (8). Taking the variation of (8) we find,d bAa ¼ Kl

adAl ¼ 0 on the boundary which is precisely the boundary condition used in theobtention of the equation of motion on A(dS) space. Therefore we can conclude that theboundary condition on the flat space is really compatible with the boundary condition onthe A(dS) space as the fields and their variations are connected by a Killing vector.

References

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gauge theories on a(ds) space and killing vectors

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