Born-Infeld theory with higher derivatives

Wissam Chemissany,1 Renata Kallosh,2 and Tomas Ortin3

1Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G12Stanford Institute for Theoretical Physics and Department of Physics, Stanford University, Stanford, California 94305-4060, USA

3Instituto de Fisica Teorica UAM/CSIC, C/ Nicolas Cabrera, 13-15, C. U. Cantoblanco, E-28049-Madrid, Spain(Received 12 December 2011; published 2 February 2012)

We present new models of nonlinear electromagnetism which satisfy the Noether-Gaillard-Zumino

current conservation and are, therefore, self-dual. The new models differ from the Born-Infeld–type

models in that they deform the Maxwell theory starting with terms like �ð@FÞ4. We provide a recursive

algorithm to find all higher-order terms in the action of the form �n@4nF2nþ2, which are necessary for the

Uð1Þ duality current conservation. We use one of these models to find a self-dual completion of the �ð@FÞ4correction to the open string action. We discuss the implication of these findings for the issue of UV

finiteness of N ¼ 8 supergravity.

DOI: 10.1103/PhysRevD.85.046002 PACS numbers: 11.25.�w, 04.65.+e

I. INTRODUCTION

In this paper, we discuss a method for constructingeffective Lagrangians for nonlinear theories with dualitysymmetries. This work builds on earlier papers [1–3]. Thehope is that this procedure may shed further light oncounterterms in maximal supergravity theories. In particu-lar it may improve our understanding of the role of E7ð7Þelectromagnetic duality symmetry inN ¼ 8 supergravity.

Here we study a simplified class of models with only onevector field, no scalars, and duality group Uð1Þ. Althoughthe E7ð7Þ symmetry of N ¼ 8 supergravity is a global

continuous symmetry it has some unusual features whichwere uncovered for the first time in 1981 by Gaillard andZumino [4] in the construction of extended supergravities(for a recent review see [5]). The familiar global continu-ous symmetries are defined by the Noether current conser-vation and have been well known since 1918. However,duality symmetries have subtleties in the vector sector ofthe theory. Namely, the vector part of the action is notinvariant under duality symmetry, but transforms in aspecific way, so that the Bianchi identities and equationsof motion transform into each other by duality symmetry.This feature is guaranteed by the conservation of theNoether-Gaillard-Zumino (NGZ) current and the corre-sponding NGZ identity.

Several theories with Uð1Þ duality are known. At thefree, linear level, there is Maxwell’s electromagnetism andthe higher-derivative generalizations constructed in [2]. Atthe interacting, nonlinear level, there is the Born-Infeld(BI) theory [6–8] and its generalizations [9,10]. The factthat the original BI theory has electromagnetic duality wasfirst noticed by Schrodinger [7]. The action of this modeland of the generalizations constructed so far only containpowers of the Maxwell field strength F, and no higherderivatives. The BI Lagrangian had been derived byFradkin and Tseytlin [8] as the low-energy spacetimeeffective Lagrangian for the vector field with a constant

field strength, coupled to a string. The self-duality of Born-Infeld action and the relation to the D3-brane of type IIBsuperstring theory and its SLð2; zÞ symmetry was studied in[11]. For a review on BI action and open superstring theory,we refer to [12].The action of the BI model has a well-known closed

form det1=2ð��� þ F��Þ, while the actions of its general-

ization do not, so the Lagrangian has to be written as aninfinite power series. Gibbons and Rasheed [9] have shownthat there is a function of one variable’s worth ofLagrangians admitting duality rotations and gave an ex-plicit algorithm for their construction. These models weredeveloped in more detail in [10] and more recently in [3].The action of all these models is identical at the F2; F4; F6

level, but they differ at the F8 and higher levels.In this paper, we will construct two simple self-dual

models of nonlinear electrodynamics whose first deviationfrom the free Maxwell theory starts with a ð@FÞ4 term andcontain terms of higher order in F and derivatives. We willpresent a recursive procedure to construct all of them.A term of this kind [ð@FÞ4] is known to arise in the

4-point amplitude of the open string1 [14]. It was shown in[15] that, with this term (and other F4 with higher deriva-tives present in the 4-point amplitudes), the theory satisfiesthe NGZ identity, and is consistent with electromagneticself-duality. Here we will show that a combination of thetwo simple self-dual models gives precisely the ð@FÞ4 termstudied in [15] as well as higher-order terms required tosatisfy the NGZ current conservation at the n-point level.We will also describe a more general class of models

where there are terms with Fn, without derivatives, aswell as terms with derivatives @2mF2n. In all cases, the

1The same types of terms have been considered in [13] as partof the effective action of a single D3-brane. They have beenshown to fit elegantly into an SLð2; zÞ-invariant function thatencodes both the perturbative and nonperturbative contributionsto the amplitude.

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algorithm for a construction of such actions satisfying theNGZ identity will be given.

II. Uð1Þ DUALITY, NO SCALARS

Our goal is to construct actions SðFÞ, where F�� �@�A� � @�A� is the Maxwell field strength, which have

a nonlinear Uð1Þ duality. Two classes of such actions areknown in the literature: that of the Born-Infeld theory andits generalizations [9,10], that depend only on F and not onits derivatives, and the action constructed in [2], which hashigher derivatives but is quadratic in F.

As usual, we define the dual field strength GðFÞ by

~G�� � 1

2�����G�� � 2

�SðFÞ�F��

: (2.1)

The infinitesimal Uð1Þ duality transformations that inter-

change the equations of motion @� ~G�� ¼ 0 and Bianchi

identities @� ~F�� ¼ 0 are given by

�F

G

!¼ 0 B

�B 0

!F

G

!: (2.2)

The necessary condition for the theory to be self-dual isconservation of the NGZ current [4], which inUð1Þmodelswithout scalars requires that

Zd4xðF ~FþG ~GÞ ¼ 0: (2.3)

TheUð1Þ case is a special case of a more general Spð2n;RÞduality group

�F

G

!¼ A B

C D

!F

G

!; (2.4)

which also acts on scalars, � ¼ �ðA; B; C;DÞ.In the general case, the NGZ identity requires the action

to be of the form [4]

S ¼ 1

4

Zd4xF ~Gþ Sinv; (2.5)

where Sinv is exactly invariant under the duality group�Sinv ¼ 0. This is a reconstructive identity, since, in prin-ciple, it may be used to find the action from the knowledgeof Sinv and GðFÞ. On the other hand,

�S ¼ 1

2

Zd4x�ðF ~GÞ ¼ 1

4

Zd4xð ~GBGþ ~FCFÞ: (2.6)

Equations (2.5) and (2.6) are equivalent to NGZ currentconservation, whereas Eq. (2.3) is a particular form of thecurrent conservation, valid only for Uð1Þ models withoutscalars. Indeed, only for Uð1Þ in absence of scalars

�S ¼ 12~GBG, with B ¼ �C and A ¼ 0 and therefore

1

2~GBG ¼ 1

4ð ~GBGþ ~FCFÞ )

Zd4xðF ~FþG ~GÞ ¼ 0:

(2.7)

A. New reconstructive identity in Uð1Þ modelswithout scalars

As mentioned before, to use the generic reconstructiveidentity (2.5) one needs, in addition to GðFÞ in eachparticular model, additional information on Sinv. In theUð1Þ models without scalars that we are consideringhere, this additional information comes form the followinggeneral observation: if � is the coupling constant of themodel (so the linear Maxwell term is independent of it),then, Sinv is related to the full action by

Sinv ¼ ��@S

@�: (2.8)

This relation follows from the uniqueness of Sinv for agiven nonlinear theory, with given nonlinear duality trans-formations and from the invariance of � @S

@� . The precise

coefficient relating these two objects follows from thestudy of the linear and next-to-linear terms of a genericaction. For example, it is well known [4] that in the BImodel one has Sinv ¼ �g2 @S

@g2.

Using this general observation, one can derive a new,more useful, reconstructive identity:

SðFÞ ¼ 1

4�

Zd4xd�F ~G: (2.9)

To prove this, we will first prove that � @S@� is duality

invariant,2 using the NGZ identity and the definition(2.1) and (2.3)

�@

@�

Zd4xðF ~FþG ~GÞ ¼ 2�

Zd4x

@ ~G

@�G

¼ �Z

d4x@

@�

��S

�F

�G ¼ 0:

(2.10)

Then, since the functional variation and the partialderivative with respect to � commute, and using (2.2),we find that

2This is just a particular case of the general theorem proven inAppendix B of Ref. [4]. Note that the general proof in Ref. [4] isbased on a condition that the duality transformation of scalarsdoes not depend on a coupling associated with the deformation;meanwhile, the transformation law of vectors does depend onsuch a coupling. This raises the issue of whether in extendedsupersymmetric theories, where scalars and vectors are in thesame multiplet, the construction of this type is available.

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0 ¼Z

d4x�

�F

��@S

@�

�G ¼ B�1

Zd4x

�

�F

��@S

@�

��F

¼ B�1�

��@S

@�

�: (2.11)

Now, using the observation (2.8) in (2.5) that

Sþ �@S

@�¼ 1

4

Zd4xF ~G; (2.12)

which can integrated immediately, leading to (2.9).The new reconstructive identity (2.9) is particularly well

suited to find the action as a series expansion in � when thedual field strength G is also available as a series expansionin �: defining3

~GðFÞ ¼ �Fþ 2X1n¼1

�n ~GðnÞðFÞ; (2.13)

S ¼ � 1

2

Zd4xF2 þ 2

X1n¼1

�nSðnÞ; (2.14)

so that the � ¼ 0 free limit of the theory is the Maxwelltheory; we find that each term in the expansion of theaction is given by

SðnÞ ¼ 1

4ðnþ 1ÞZ

d4xF ~GðnÞðFÞ: (2.15)

In the models that we are going to consider ~GðFÞ is givenby a series expansion of the above form with all the termsof higher order in � given by a simple recursion relationand these results can be checked explicitly order by orderin �.

B. NGZ identity with graviphoton convention

To proceed, we introduce the standard supergravitygraviphoton conventions [16], employed in [3] in the co-variant procedures for perturbative nonlinear deformationof duality-invariant theories. In the complex basis wedefine

T ¼ F� iG; T� ¼ Fþ iG; (2.16)

which transform under finite Uð1Þ duality transformationswith a phase, so, under (2.2)

�T ¼ iBT: (2.17)

We also introduce the self-dual notation,

T� ¼ 12ðT � i ~TÞ; (2.18)

and form 4 different combinations of the components ofthe graviphoton field, see Table I. Observe that T�þ ¼ T��.In this notation, the NGZ identity (2.3) takes the form

Zd4x½T�þTþ � T��T�� ¼ 0: (2.19)

In the linear Maxwell theory Tþ ¼ 0, so

Tþ ¼ Fþ � iGþ ¼ 0; (2.20)

which implies ~G ¼ �F. In more general theories in whichthe dual field G is treated as independent of F, this con-straint is used to eliminate the nonphysical degrees offreedom and express G as a function of F and scalars, ifany, and it is known as a linear twisted self-dualityconstraint.In [2], Bossard and Nicolai proposed to use a nonlinear

deformation of the twisted self-duality constraint based ona manifestly duality invariant source of deformation

I ð1ÞðTÞ to construct a self-dual theory. We will followhere the generalized procedure used in [3]. Let us assume

that a manifestly duality invariant I ð1ÞðTÞ is given. It wasshown in[3] that if, instead of vanishing as required by thelinear twisted self-duality condition, Tþ is given by thenonlinear twisted self-duality condition

Tþ�� ¼ �I ð1ÞðT�; T�þÞ

�T�þ��

;

ðTþ��Þ� ¼ T��

�� ¼ �I ð1ÞðT�; T�þÞ�T�

��

;(2.21)

it follows that the NGZ identity is satisfied automatically.One computes T�þTþ � T��T�, using (2.21) and findsthat it vanishes since it is proportional to the variation of

I ð1ÞðTÞ under duality, which vanishes since �I ð1Þ ¼ 0:

Zd4x½T�þTþ � T��T��

¼Z

d4x

�T�þ �I ð1ÞðT�; T�þÞ

�T�þ � T� �I ð1ÞðT�; T�þÞ�T�

�

¼ 1

B�I ð1Þ ¼ 0: (2.22)

Thus, once Eqs. (2.21) are solved for GðFÞ there is no needto check the NGZ identity; it is satisfied and we have theGðFÞ of a self-dual theory.In the models that we are going to study, the nonlinear,

twisted, self-duality constraint can be solved as a powerseries in a parameter �:

TABLE I. The 4 combinations of the graviphoton componentshave � chirality and � duality charge.

Graviphoton components Chirality Charge

Tþ ¼ Fþ � iGþ þ þT�þ ¼ Fþ þ iGþ þ �T� ¼ F� � iG� � þT�� ¼ F� þ iG� � �

3The global factors of 2 have been introduced for later con-venience, since they lead to simpler expressions for the coef-ficients TðnÞ� to be introduced later.

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Tþ ¼ �2X1n¼1

�nTðnÞþ; (2.23)

so

iGþ ¼ Fþ þ 2X1n¼1

�nTðnÞþ; (2.24)

from which we can get the coefficients of the series (2.13)for n > 0,

~G ðnÞ ¼ �ðTðnÞþ þ c:c:Þ; ðn > 0Þ; (2.25)

and of (2.14) for n > 0 (for n ¼ 0 they have been chosen tocorrespond to Maxwell’s theory),

2SðnÞ ¼ � 1

2ðnþ 1ÞZ

d4x½FþTðnÞþ þ c:c:�; ðn > 0Þ:(2.26)

III. BORN-INFELD WITH HIGHER DERIVATIVESAND DUALITY CURRENT CONSERVATION

In this section we are going to construct two deforma-tions of the Maxwell theory using two particularly simple

manifestly duality-invariant sources of deformation I ð1ÞA ðTÞ

and I ð1ÞB ðTÞ given, respectively, by

I ð1ÞA ðTÞ � �

23tð8Þ�1�1�2�2�3�3�4�4

@T�þ�1�1@

� T��2�2@�T�þ�3�3@�T��4�4 ; (3.1)

I ð1ÞB ðTÞ � �

23tð8Þ�1�1�2�2�3�3�4�4

@T�þ�1�1@�

� T��2�2@T�þ�3�3@�T��4�4 ; (3.2)

where the tensor tð8Þ is defined in the Appendix, or, usingthe shorthand notation introduced in the Appendix,

I ð1ÞA ðTÞ � �

23tð8Þabcd@T

�þa@T�b@�T�þc@�T�d; (3.3)

I ð1ÞB ðTÞ � �

23tð8Þabcd@T

�þa@�T�b@T�þc@�T�d: (3.4)

At first order in � the models that one obtains using theprocedure described in the previous section are associatedwith the following deformations of the action:

Sð1ÞA ¼ 1

4

Zd4xtð8Þabcd@F

þa@F�b@�Fþc@�F�d; (3.5)

Sð1ÞB ¼ 1

4

Zd4xtð8Þabcd@F

þa@�F�b@Fþc@�F�d: (3.6)

Alternative forms of these corrections which do not use

the tð8Þ tensor are Eqs. (A9) and (A10).

In what follows we are going to construct explicitly the

model A, using I ð1ÞA ðTÞ in the nonlinear twisted self-dual

condition.

A. Model A

The simplest way to solve the nonlinear twisted self-dual

condition with I ð1ÞA ðTÞ is to plug the series expansion (2.23)

into both sides of it and identify the terms with the samepowers of �. First, observe that the expansion (2.23) for Tþimplies for T�þ and T�,

T�þ ¼ 2Fþ þ 2X1n¼1

�nTðnÞþ; (3.7)

T� ¼ 2F� þ 2X1n¼1

�nTðnÞþ�: (3.8)

It is, then, convenient, to define4

Tð0Þþ ¼ Fþ; (3.9)

so

T�þ ¼ 2X1n¼0

�nTðnÞþ; (3.10)

T� ¼ 2X1n¼0

�nTðnÞþ�: (3.11)

With these definitions, the nonlinear twisted self-dualcondition for this model, which is

Tþa ¼ � �

22tð8Þabcd@ð@T�b@�T

�þc@�T�dÞ; (3.12)

takes the form

X1n¼1

�nTðnÞþa ¼ tð8Þabcd

Xp;q;r¼0

�pþqþrþ1@ð@TðpÞ�b@�

� TðqÞ�þc@�TðrÞ�dÞ; (3.13)

from which it follows that

TðnÞþa ¼ tð8Þabcd

Xp;q;r¼0

�pþqþrþ1;n@ð@TðpÞ�b@�

� TðqÞ�þc@�TðrÞ�dÞ; (3.14)

which can be solved recursively, given that Tð0Þþa ¼ Fþ

a .Thus,

Tð1Þþa ¼ tð8Þabcd@ð@F�b@�F

þc@�F�dÞ; (3.15)

4Notice, however, the expansion of Tþ is still given by (2.23)and has no term of zero order in �.

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Tð2Þþa ¼ tð8Þabcd½@ð@F�b@�F

þc@�Tð1Þ�dÞþ @ð@F�b@�T

ð1Þþc@�F�dÞþ @ð@Tð1Þ�b@�F

þc@�F�dÞ�; (3.16)

Tð3Þþa ¼ tð8Þabcd½@ð@F�b@�F

þc@�Tð2Þ�dÞþ @ð@F�b@�T

ð1Þþc@�Tð1Þ�dÞ þ permutations�;(3.17)

etc. The action can be obtained immediately by using thepower series expansion of reconstructive identity (2.26).Explicitly, we get

2Sð0Þ ¼ � 1

4

Zd4xF2; (3.18)

2Sð1Þ ¼ 1

2

Zd4xtð8Þabcd@F

þa@F�b@�Fþc@�F�d;

(3.19)

2Sð2Þ ¼ � 1

2

Zd4x½Tð1Þþ

a Tð1Þþa þ c:c:�

¼ � 1

2

Zd4xftð8Þabcdtð8Þdefg@ð@F�b@�F

þc@�F�dÞ� @�ð@�F�e@�F

þf@�F�gÞ þ c:c:g; (3.20)

etc., up to total derivatives. It can be checked order byorder that this action is related to the dual field strengthiGþ ¼ Fþ � Tþ by (2.1):

Gþ�� ¼ 2i

�S

�Fþ�� ; (3.21)

as required.

B. Model B

The recursive algorithm for generating a complete ac-tion above produces the �n term from the previous ones.The derivation of this model follows the exact stepswhich we outlined in case A. Each time, the sequence ofðþ �þ�Þ has to be replaced by ðþ þ��Þ, the rest is thesame. Therefore, we will not provide more details on thederivation of the B model.

IV. SUPERSYMMETRIZABLE BORN-INFELDDUALITY SYMMETRIC MODELWITH

HIGHER DERIVATIVES

The model with derivatives of F known from the opensuperstring effective action [14] was shown to satisfy theNGZ current conservation condition (2.3) in [15]. In thismodel, the first deformation of the Maxwell theory is givenby the quartic coupling term

Sð1Þ ¼ �

24

Zd4xtð8Þabcd@�F

a@�Fb@�Fc@�Fd; (4.1)

in the notation introduced in the Appendix. As shownthere, it can be rewritten in the form [Eq. (A11)]

Sð1Þ ¼ �

22

Zd4xtð8Þabcd

�@�F

þa@�F�b@�Fþc@�F�d

þ 1

2@�F

þa@�F�b@�Fþc@�F�d

�: (4.2)

It is clear that, to reproduce this quartic term in theaction, we must take a combination of models A and B,studied above, and use, as a manifestly self-dual source ofdeformation,

I ð1ÞstringðTÞ ¼ I ð1Þ

A ðTÞ þ 1

2I ð1ÞB ðTÞ

¼ �

23tð8Þabcd

�@�T

�þa@�T�b@�T�þc@�T�d

þ 1

2@�T

�þa@�T�þb@�T�c@�T�d

�: (4.3)

The resulting nonlinear, twisted self-duality constraint canbe solved by the same recursive procedure we employedfor model A above and the dual field strength GðFÞ andcorresponding action can be found by the use of the newreconstructive identity.It is interesting to observe that, after partial integration,

the above quartic term is very close to the ð@FÞ4 term foundin Ref. [17], although the latter, corresponding to an am-plitude calculation, is not real in its current form. It is likelythat for the effective action one can produce the realexpression dividing the one in Ref. [17] by two and addingthe Hermitian conjugate.

V. MORE GENERAL Uð1Þ DUALITY,NO SCALARS MODELS

Using the covariant procedures for perturbative nonlin-ear deformation of duality-invariant theories [3] we canconstruct more general models with NGZ current conser-vation. For example, we may consider more generalsources of deformation.

I ð1ÞfnðT�; T�þÞ ¼ X

n¼1

fnðI ð1ÞðT�; T�þÞÞn; (5.1)

where I ð1ÞðT�; T�þÞ is defined in (4.3) and fn are arbitraryconstants, and the model we described above in details hasf1 ¼ 1 and fn ¼ 0, n > 1. In addition, we may add termswhich depend only on F’s without derivatives, for ex-ample, the ones studied in [3].Any manifestly Uð1Þ duality invariant IðT�; T�þÞ with

the spacetime derivatives action of T’s, or without, has tohave the same number of T�’s as T�þ’s, and has to be

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Lorentz covariant. In such a case, one expects a recursiveequation, defining G��ðFÞ from the equation

Tþ�� ¼ �IðT�; T�þÞ

�T�þ��

; (5.2)

as shown in the simple models defined in [3] withoutderivatives, and in Sec. III in the case with derivatives.The solution is guaranteed to satisfy the NGZ Uð1Þ currentconservation [3].

Equation (5.2), in all cases, provides a recursive proce-dure determiningGðFÞ as a power series in �. The action inthese most general models of Uð1Þ duality without scalarsis given by the new reconstructive identity (2.9).

VI. DISCUSSION

In this paper we have constructed explicitly the firstcomplete model of a Born-Infeld type with higher deriva-tives, which has an electromagnetic Uð1Þ duality. Themodel is given by the power series expansion of theLagrangian and involves all powers of the Maxwell fieldstrength and their derivatives;

S ¼ SMaxwell �X1n¼1

�n

2ðnþ 1ÞZ

d4x½Fþ��TðnÞþ�� þ c:c:�:

(6.1)

Here TðnÞ � @4nF2nþ1 and the explicit expression is given

via a recursive algorithm in Eq. (3.14), which defines TðnÞ

in terms of TðmÞ with m< n and starts with Tð0Þþ ¼ Fþ.We have also outlined the procedure to produce morecomplicated models where terms with and without deriva-tives on F are mixed.

Apart from the intrinsic motivation to discover a non-linear model with higher derivatives and with dualitysymmetry, which was not known in the past, our goalhere was to test the Bossard-Nicolai proposal [2]. Theauthors conjectured that there is a straightforward algo-rithm which allows us to construct N ¼ 8 supergravitywith higher derivatives, consistent with E7ð7Þ duality.

This conjecture was used in [2] to counter the argumentof [1] suggesting that E7ð7Þ duality symmetry predicts the

finiteness of N ¼ 8 supergravity.However, there is no actual construction of N ¼ 8

supergravity with higher derivatives in [2], which wouldbe a formidable task. Therefore, we performed a detailedinvestigation of this issue in applications to much simplermodels, such as the Born-Infeld models and their general-izations. An investigation of this issue in [3] demonstratedthat the algorithm of construction of N ¼ 8 supergravitywith higher derivatives requires substantial modificationseven in application to the simplest Born-Infeld model.Moreover, it was observed in [3] that the presence of the4-point UV divergence F4fðs; t; uÞ term in the N ¼ 8supergravity would require us to produce a theory of theBorn-Infeld–type with derivatives leads to a nonstop pro-

liferation of the powers of the vector field strength withincreasing number of derivatives.A generalization of the results in [3] to the Born-Infeld

model with higher derivatives required additional efforts.In this paper, we were able to construct a toy model of aBorn-InfeldN ¼ 8 supergravity, withN ¼ 0 supersym-metry replacing N ¼ 8 and Uð1Þ duality replacing E7ð7Þ.The model indeed has a full nonlinearity in powers of�n@4nF2nþ2 with n ! 1, as predicted in [3].Thus, whereas we were able to construct the Born-Infeld

model with derivatives, and we are planning to develop asimilar construction for supersymmetric models, whichalso have a Uð1Þ duality symmetry, at present we do notsee any obvious way to extend this construction and de-velop the Born-Infeld version of N ¼ 8 supergravityalong the lines of [2]. Until the existence of the Born-Infeld version ofN ¼ 8 supergravity is demonstrated, theargument that E7ð7Þ duality symmetry predicts the finite-

ness of N ¼ 8 supergravity [1] seems to us still valid.Moreover, even if one manages to construct a consistenttwo-coupling model of N ¼ 8 supergravity with gravi-tational coupling 2 as well as Born-Infeld coupling �, itwill raise a new question of whether the conjecturedexistence of this new theory predicts anything for theUV behavior of the original one-coupling N ¼ 8 super-gravity, which depends only on gravitational coupling.

ACKNOWLEDGMENTS

We are grateful to G. Bossard, J. Broedel, J. J. Carrasco,S. Ferrara, D. Freedman, M. Green, A. Linde, H. Nicolai,R. Roiban, E. Silverstein, and A. Tseytlin for stimulatingdiscussions and especially to M. de Roo for his contribu-tion in the early stages of this project. This work is sup-ported by the Stanford Institute for Theoretical Physics andthe NSF Grant No. 0756174, the Spanish Ministry ofScience and Education Grant No. FPA2009-07692, theComunidad de Madrid grant HEPHACOS S2009ESP-1473, and the Spanish Consolider-Ingenio 2010 programCPAN CSD2007-00042. W.C. and T.O. wish to thank theStanford Institute for Theoretical Physics for its hospitalityand financial support.

APPENDIX: SOME USEFUL RELATIONS

Using the definition of the Hodge dual

~A�� � 12�����A

��; ~~A ¼ �A; (A1)

and of the self- and anti-self-dual parts

A� ¼ 12ðA� i ~AÞ; (A2)

for 2-forms, one can easily prove the following identitiesinvolving arbitrary 2-forms A and B:

~A ~B ¼ BA� 12 TrðABÞ1; (A3)

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~AB ¼ � ~BAþ 12 TrðA ~BÞ1; (A4)

A�B� ¼ �B�A� þ 12 TrðA�B�Þ1; (A5)

A�B� ¼ B�A�: (A6)

The tð8Þ tensor [18] is totally symmetric in four pairs ofantisymmetric indices. It is convenient to use only oneLatin index a; b; c . . . to denote each of these four pairs

and write tð8Þabcd instead of tð8Þ�1�1�2�2�3�3�4�4¼

tð8Þ½�1�1�½�2�2�½�3�3�½�4�4�. Then, in terms of these indices,

tð8Þ is completely symmetric tð8Þabcd ¼ tð8ÞðabcdÞ.tð8Þ can be defined by its contraction with 4 arbitrary 2-

forms A; B; C;D:

tð8ÞabcdAaBbCcDd ¼ 8½TrðABCDÞ þ TrðACBDÞ

þ TrðACDBÞ� � 2½TrðABÞTrðCDÞþ TrðACÞTrðBDÞ þ TrðADÞTrðBCÞ�:

(A7)

Then, using the above relations, one can write

tð8Þabcd@�Fa@�Fb@�F

c@�Fd

¼ 16

�Trð@�Fþ@�FþÞTrð@�F�@�F�Þ

þ 12 Trð@�Fþ@�FþÞTrð@�F�@�F�Þ

�; (A8)

and

tð8Þabcd@�Fþa@�F�b@�F

þc@�F�d

¼ 4Trð@�Fþ@�FþÞTrð@�F�@�F�Þ; (A9)

tð8Þabcd@�Fþa@�F

�b@�Fþc@�F�d

¼ 4Trð@�Fþ@�FþÞTrð@�F�@�F�Þ; (A10)

from which we find that

tð8Þabcd@�Fa@�Fb@�F

c@d

¼ 4tð8Þabcd½@�Fþa@�F�b@�Fþc@�F�d

þ 12@�F

þa@�F�b@�Fþc@�F�d�: (A11)

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