Polynomial Duality-Symmetric Lagrangians for Free p-Forms
aa r X i v : . [ h e p - t h ] J a n Polynomial Duality-SymmetricLagrangians for Free p -Forms Sukruti Bansal a , Oleg Evnin a,b , Karapet Mkrtchyan c,d a Department of Physics, Faculty of Science, Chulalongkorn University,Phayathai Rd., Bangkok 10330, Thailand b Theoretische Natuurkunde, Vrije Universiteit Brussel andThe International Solvay Institutes, Pleinlaan 2, Brussels 1050, Belgium c Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, 56126 Pisa, Italy d Blackett Laboratory, Imperial College London SW7 2AZ, U.K.
Abstract:
We explore the properties of polynomial Lagrangians for chiral p -forms pre-viously proposed by the last named author, and in particular, provide a self-containedtreatment of the symmetries and equations of motion that shows a great economy andsimplicity of this formalism. We further use analogous techniques to construct polyno-mial democratic Lagrangians for general p -forms where electric and magnetic potentialsappear on equal footing as explicit dynamical variables. Due to our reliance on the dif-ferential form notation, the construction is compact and universally valid for forms ofall ranks, in any number of dimensions. [email protected] [email protected] [email protected] ontents p -forms 3 D = 3 13 da ∧ dB = 0
17D PST-like form of the democratic Lagrangian 18
D.1 Derivation 18D.2 Dynamics 19– i –
Introduction
Duality between electric and magnetic degrees of freedom has been a source of in-spiration to theoretical physics since the formulation of Maxwell’s theory of electro-magnetism. In particular, it has prompted Dirac’s influential treatment of magneticcharges. More contemporary related developments include the monopole condensationpicture of confinement, Montonen-Olive duality, and various aspects of differential formfields ubiquitous in supersymmetric field theories.While the electric-magnetic symmetry is apparent, say, in the vacuum Maxwellequations, creating a Lagrangian description of such duality-symmetric theories thatwould feature electric and magnetic degrees of freedom on an equal footing is knownto be involved. A closely related question is giving a Lagrangian description to chiralforms, invariant under electric-magnetic dualities and often arising in supersymmetricfield theories. We provide a selection of relevant literature [1–53] that represents someof the history of the subject.To set the stage, one can visualize the familiar example of the Maxwell equationsin vacuum, which can be given using two-form field strength F as dF = 0 , d ⋆ F = 0 . (1.1)Solving the first equation using the Poincar´e lemma, we introduce the vector potential: F = dA . Then, the second equation is a gauge-invariant wave equation for the vector A . We could also solve the second equation using Poincar´e lemma, thus introducing adual potential: F = ⋆dB (in d dimensions, B is a ( d − B . In both cases, it isstraightforward to write a standard action in terms of potentials A or B , which producethe corresponding equations. The choice of one or another potential for the descriptionof the theory breaks, however, its manifest electric-magnetic duality symmetry. Instead,one could aim at maintaining the duality manifest and keeping both potentials A and B . In this case, (1.1) is equivalently re-encoded into the twisted selfduality relation dA = ⋆ dB . (1.2)(The term ‘twist’ denotes here the interchange of the two potentials that accompaniesthe Hodge dualization.) Lagrangian formulations giving rise to equation (1.2) are muchless obvious to construct than the corresponding formulations with only one of the twopotentials. Analogously and more generally, in d space-time dimensions, equations ofmotion for a free p -form can be written in terms of the twisted selfduality condition(1.2) employing both the curvature of the p -form potential, and its dual ( d − − p )-formpotential. – 1 –imilarly, in the special cases of (anti)selfdual (chiral) fields, the equations can begiven as duality relations dA = ± ⋆ dA . (1.3)While in the cases of twisted selfduality equations (1.2) it is possible to describe thecorresponding degrees of freedom using two-derivative equations and straighforward(Maxwell) Lagrangians, the chiral fields force us to look for Lagrangians that give riseto first derivative equations (1.3). There are several reasons why such Lagrangiansare hard to construct, one of which is that the ( p + 1)-form equation (1.2) cannot begenerated as a variation of a scalar Lagrangian with respect to the p -form field A . Thisimplies a necessity to break manifest Lorentz symmetry or introduce auxiliary fields.A number of approaches to constructing Lagrangians for chiral forms and demo-cratic formulations with both electric and magnetic potentials have appeared in theliterature over the years, and it is beyond the limits of this introduction to criticallyreview all of them; instead, we refer the reader to the original literature in our selectionof references [1–53] and beyond. Our focus here will be on analyzing and developinga particular proposal put forth in [43]. This proposal is, in turn, rooted in the attrac-tive approach originally developed in [16, 17, 22] by Pasti, Sorokin and Tonin (PST).While the formalism of [43] arises from the PST theory through introduction of anextra auxiliary form field via a particular version of the Hubbard-Stratonovich trick, itruns counter to the long perceived tension between keeping Lagrangian descriptions ofchiral forms local, manifestly Lorentz-invariant, ghost-free and polynomial with a finitenumber of terms and fields in the Lagrangian. Considerations of [43] have produced atheory that simultaneously possesses all of these appealing features.The analysis of [43] has focused on designing a polynomial Lagrangian for chiralforms satisfying (1.3), and showing its equivalence to PST theory. In our presenttreatment, we start by taking up the result of the considerations of [43] and showinghow to analyze the symmetries and equations of motion in this formalism. The resultis, of course, the same as what one would get in PST theory (to which the formulationof [43] can be related), but it is visible from our treatment that the derivations areconsiderably more compact and transparent than what one would have had to undertakeby reverting to PST theory first. This shows the efficiency of the polynomial formulationof [43]. To take the formulation of [43] further, we then turn to the topic of democraticelectric-magnetic theories that give a Lagrangian description to the duality relation(1.2) rather than the selfduality relation (1.3). A formalism analogous to [43] can bedeveloped for this objective, with a similarly high level of efficiency. This generates acompact treatment of democtratic polynomial Lagrangians for form fields of all ranks,in any number of dimensions. – 2 – Chiral p -forms We start by revisiting the polynomial formulations for a free chiral form proposed in [43].Our purpose is twofold. First, we would like to give a demonstration of the algebraicefficiency of this formalism, which allows for an economical handling of the equationsof motion and gauge symmetries. Second, we would like to thoroughly reformulate theformalism in the differential form notation, which makes it possible to treat all numbersof dimensions and form ranks in a uniform fashion. Much of the classic literature onrelated subjects [16, 17, 22] relies on the index notation, where numerical coefficientsof combinatorial nature appear in the formulas. The form notation eliminates suchcoefficients and makes the formulas more compact. (Some earlier treatments of relatedsubjects relying on the form notation can be found in [34, 37].) We give conversionformulas connecting the form and index notation in appendix A, and a collection ofidentities for differential forms in appendix B (this material is completely standard, butit is convenient to have it handy while going through our subsequent derivations). Ofthese, the projection rejection identity v ∧ ι v A + ι v ( v ∧ A ) = A, (2.1)valid for any unit vector v and any form A , and the action of the Hodge star on theinterior and exterior product, ⋆ ι v ( v ∧ A ) = v ∧ ι v ( ⋆A ) , ⋆ ( v ∧ i v A ) = ι v ( v ∧ ⋆A ) , (2.2)will be encountered especially frequently in our derivations (normally, with da/ p ( ∂a ) playing the role of v for a scalar field a ). In a slight abuse of notation, we shall employthe same letter to refer to vectors and their dual 1-forms, but there should be nosituations where this could cause confusion. We shall also use the notation A ≡ A ∧ ⋆A. (2.3)The formulation of [43] comes in two versions, related by a simple field redefinition,but endowed with rather different flavors. In the first formulation introduced in [43],the field content is a gauge p -form, an auxiliary p -form satisfying an algebraic equationof motion, and an auxiliary scalar field playing the same role as in PST formalism.The advantages of this formalism are its immediate relation to PST theory (which isrecovered upon integrating out the auxiliary p -form) and the simpler form of someof its gauge symmetries. The second formulation appears, at the first sight, to havea doubled field content consisting of two p -form gauge fields and the same auxiliaryscalar as before. The truth is that, because of the rich gauge symmetry, the actual– 3 –ynamical content of the theory is halved rather than doubled, leaving a single chiral p -form. The advantage of this formalism is its very conventional appearance in termsof two ordinary p -forms with Maxwellian gauge symmetries.In what follows, we shall first review the ‘Maxwellian’ formalism and provide acomplete self-contained analysis of the symmetries and the equations of motion, showingthe algebraic appeal of this formulation. Then for the convenience of the reader we willsummarize the structure of the formalism with an algebraic auxiliary field. Following [43], we consider the Lagrangian L = ( F + a Q ) + 2 aF ∧ Q , (2.4)where F = dA and Q = dR . Here, A and R are p -forms and a is a scalar field. A p -form may have a selfdual field strength in d = 2 p + 2 dimensions, and furthermore,in Minkowski signature, p must be even, which is what we shall assume until the endof section 2. (Note that, in this situation, ⋆⋆ = +1 when acting on any ( p + 1)-form,which we shall apply automatically throughout until the end of section 2.) Note thevery conventional field content with two Maxwellian p -form gauge fields A and R , onlyentering the Lagrangian through their respective field strengths F and Q .Lagrangian (2.4) possesses the following four gauge symmetries: δa = 0 , δA = dU , δR = 0 ; (2.5) δa = 0 , δA = 0 , δR = dU ; (2.6) δa = 0 , δA = − a da ∧ U , δR = da ∧ U ; (2.7) δa = ϕ , δA = − a ϕ ( ∂a ) ι da ( Q + ⋆Q ) , δR = ϕ ( ∂a ) ι da ( Q + ⋆Q ) , (2.8)where ϕ is an arbitrary scalar function and U is an arbitrary ( p − A and R , obvious as the Lagrangian onlydepends on the corresponding field strengths. The third symmetry leaves the first termof (2.4) unchanged and modifies the second one by a total derivative, which vanishesupon integration. This symmetry will be crucial below for establishing the selfdualityof the physical degrees of freedom. Validating the last transformation requires slightlymore work. The variation of Lagrangian (2.4) under arbitrary shifts of a , A and R , upto total derivative terms, is δ L = 2 (cid:2) δa { ( F + a Q ) ∧ ⋆Q + F ∧ Q } − δA ∧ [ d { ⋆ ( F + aQ ) } + da ∧ Q ]– 4 – δR ∧ [ d { a ⋆ ( F + aQ ) } − da ∧ F ] (cid:3) (2.9)Note that the content of the square brackets in the last line can be recast as a d [ ⋆ ( F + aQ )] + da ∧ [ ⋆ ( F + aQ ) − F ]. Thereafter, assuming δA = − a δR , which holds for (2.8),one gets δ L = 2 [ δa { ( F + aQ ) ∧ ⋆Q + F ∧ Q } + δR ∧ da ∧ { ( F + aQ ) − ⋆ ( F + aQ ) } ] , (2.10)or δ L = 2 h ϕ ( H ∧ ⋆Q + F ∧ Q ) + ϕ ( ∂a ) da ∧ ι da ( Q + ⋆Q ) ∧ ( H − ⋆H ) i , (2.11)where we have introduced H ≡ F + aQ for brevity. The second ϕ -term above getssimplified as follows: ϕ ( ∂a ) da ∧ ι da ( Q + ⋆Q ) ∧ ( H − ⋆H )= ϕ ( ∂a ) [( ∂a ) ( Q + ⋆Q ) − ι da { da ∧ ( Q + ⋆Q ) } ] ∧ ( H − ⋆H ) (by (2.1))= (cid:2) ϕ ( Q + ⋆Q ) − ϕ ( ∂a ) ⋆ { da ∧ ι da ( Q + ⋆Q ) } (cid:3) ∧ ( H − ⋆H ) (by (2.2))= − ϕ ( H ∧ ⋆Q + F ∧ Q ) − ϕ ( ∂a ) ⋆ ( H − ⋆H ) ∧ { da ∧ ι da ( Q + ⋆Q ) } (by (B.5))= − ϕ ( H ∧ ⋆Q + F ∧ Q ) − ϕ ( ∂a ) da ∧ ι da ( Q + ⋆Q ) ∧ ( H − ⋆H ) . Taking the second term to the left-hand side, one concludes that ϕ ( ∂a ) da ∧ ι da ( Q + ⋆Q ) ∧ ( H − ⋆H ) = − ϕ ( H ∧ ⋆Q + F ∧ Q ) . (2.12)Substituting this formula back into (2.11) results in δ L = 0, and hence (2.8) is a validsymmetry.Note that transformations (2.8) imply that the field a can be shifted arbitrarily, andis thus a pure gauge degree of freedom. Admissible gauges, as in PST formalism [22],are those that provide a as a good global coordinate function on the Minkowski space.In particular, gradients of a should not vanish anywhere, and the level hypersurfaces a = const must be of topology R d − and furnish a globally nondegenerate foliation of R d . This, obviously, implies that one cannot choose a to be a constant function (in thisdegenerate case, the formalism produces a single nonchiral propagating p -form, insteadof a chiral p -form). – 5 –mong all the commutators of the symmetry transformations (2.5) to (2.8), theonly non-zero commutators are those of transformation (2.8) with itself and (2.7). Thecommutator of (2.7) with (2.8) is δ U δ ϕ − δ ϕ δ U = δ V + δ W + δ X , (2.13)where δ V is transformation (2.5) with V = aϕU substituted for U , δ W is transfor-mation (2.6) with W = − ϕU substituted for U , and δ X is transformation (2.7) with X = ( ϕ ι da dU ) / ( ∂a ) substituted for U . The commutator of tranformation (2.8) withparameters ϕ and ϕ is δ ϕ δ ϕ − δ ϕ δ ϕ = δ V , (2.14)where δ V is transformation (2.7) with V = ( ϕ ι dϕ − ϕ ι dϕ ) ι da ( Q + ⋆Q ) substitutedfor U .From (2.9), the equations of motion are E a ≡ δ L δa ≡ ( F + aQ ) ∧ ⋆Q + F ∧ Q = 0 , (2.15) E A ≡ δ L δA ≡ d [ ⋆ ( F + aQ )] + da ∧ Q = 0 , (2.16) E R ≡ δ L δR ≡ d [ a ⋆ ( F + aQ )] − da ∧ F = 0 . (2.17)We would like to examine the combination E R − aE A , which can be simply read offthe δR -term in (2.10), or recovered directly from (2.16-2.17) in the following form byoperations similar to the derivation of (2.10): da ∧ [ F + aQ − ⋆ ( F + aQ )] = 0 . (2.18)Applying a Hodge star to the above gives ι da [ F + aQ − ⋆ ( F + aQ )] = 0. Acting on thislast equation with da ∧ , acting on (2.18) with ι da , adding the results and applying theprojection-rejection identity (2.1), one gets F + aQ = ⋆ ( F + aQ ) . (2.19)In other words, F + aQ is selfdual on any solutions of the equations of motion.While (2.19) has been derived from (2.16-2.17), it ensures that (2.15) is automati-cally satisfied. Indeed, for any two forms A and B of the same rank, A ∧ ⋆B = B ∧ ⋆A (this combination is often called simply A ⋆ B ). Hence, the left-hand side of (2.15) isrewritten as Q ∧ ⋆ ( F + aQ ) + F ∧ Q = Q ∧ ( F + aQ ) + F ∧ Q , which is identically– 6 –ero by (B.3) since the ranks of both F and Q are odd. This lack of a constrainingdynamical equation for a is another manifestation of the gauge symmetry (2.8).It remains to reveal that the full dynamical content of our theory is a single chiral p -form. To this end, we substitute (2.19) into (2.16) to obtain da ∧ dR = 0 . (2.20)Equations of this form are ubiquitous within PST theory and its relatives, and appear,in particular, among the PST equations of motion [22, 37]. We give a careful derivationof the general solution in appendix C. In brief, this equation is solved by restrictingit to hypersurfaces a = const, where it says simply that a tangential restriction of R to each such hypersurface is a closed form within the hypersurface. In this form, theequations are immediately integrated to yield R = dU + da ∧ V, (2.21)with arbitrary U and V . The first term can be completely gauged away by (2.6), whilethe second term can be completely gauged away by (2.7). As a result, the most generalsolution can be gauged to R = 0 , (2.22)whereupon (2.19) gives F = ⋆F. (2.23)We have thus proved that for any solution to (2.15-2.17), a can be changed arbitrarilyand R can be gauged to 0 by the symmetries of the theory, and then the field strengthof A becomes selfdual. Hence, the full dynamical content of (2.4) is a single propagatingchiral p -form. We now recast Lagrangian (2.4) into an alternative form such that no derivatives of R appear in the Lagrangian. Redefining the field A as A old = A new − aR , we get L = ( F − da ∧ R ) + 2 da ∧ F ∧ R. (2.24)The above Lagrangian has the following four gauge symmetries, which are direct coun-terparts of (2.5-2.8): δa = 0 , δA = dU , δR = 0; (2.25) By a ( x ) = const, we of course mean hypersurfaces defined by a specific value of a non-constantfunction a ( x ), and not that a ( x ) is a constant function, which would have been an inadmissible gauge. – 7 – a = 0 , δA = 0 , δR = da ∧ U ; (2.26) δa = 0 , δA = − da ∧ U , δR = dU ; (2.27) δa = ϕ , δA = ϕR , δR = ϕ ( ∂a ) ι da ( dR + ⋆dR ) . (2.28)While these symmetries are guaranteed to work due to the straightforward relationbetween (2.24) and (2.4), we also demonstrate how to validate them directly for theconvenience of the reader. Verification of the first three symmetries is straightforward:(2.25) does not change (2.24) as the latter only depends on F ; (2.26) does not change(2.24) as R only enters through da ∧ R ; (2.27) leaves the first term in (2.24) invari-ant, while the variation of the second term is a total derivative that vanishes uponintegration. For (2.28), the first term of (2.24) transforms as δ [( F − da ∧ R ) ]= 2 ( F − da ∧ R ) ∧ ⋆ h ϕ dR − ϕ ( ∂a ) da ∧ ι da ( dR + ⋆dR ) i = 2 ϕ ( ∂a ) ( F − da ∧ R ) ∧ [ ⋆ι da ( da ∧ dR ) − ι da ( da ∧ dR )]= 2 ϕ ( ∂a ) h F ∧ { ⋆ι da ( da ∧ dR ) − ι da ( da ∧ dR ) } + da ∧ R ∧ ι da ( da ∧ dR ) i = 2 ϕ ( ∂a ) F ∧ [ ⋆ι da ( da ∧ dR ) − ι da ( da ∧ dR )] + 2 ϕ da ∧ R ∧ dR, and second term transforms as2 δ ( F ∧ da ∧ R ) = 2 dϕ ∧ R ∧ da ∧ R + 2 ϕ dR ∧ da ∧ R + 2 F ∧ dϕ ∧ R + 2 F ∧ da ∧ ϕ ( ∂a ) ι da ( dR + ⋆dR )= 2 d ( ϕ da ∧ R ∧ R ) + 2 ϕ da ∧ R ∧ dR − d ( ϕ F ∧ R ) − ϕ F ∧ dR + 2 F ∧ da ∧ ϕ ( ∂a ) ι da ( dR + ⋆dR )= 2 d ( ϕ da ∧ R ∧ R − ϕ F ∧ R ) + 2 ϕ da ∧ R ∧ dR + 2 ϕ ( ∂a ) F ∧ [( − ι da ( da ∧ dR ) + ⋆ι da ( da ∧ dR )] . Subtracting the second variation from the first leaves a total derivative, which yieldszero upon integration.As mentioned above, and discussed in detail in [43], solving the (algebraic) equationof motion for R and substituting the result into (2.24) recovers the (nonpolynomial)PST Lagrangian. As a consequence, the standard analysis of PST formalism [22, 37]– 8 –uarantees that the dynamical content of (2.24) amounts to a single chiral p -form, andno other propagating degrees of freedom. That is, of course, in agreement with what wehave already shown for Lagrangian (2.4) entirely within the context of the polynomialtheory. We now turn to an ordinary nonchiral Maxwellian p -form, which could of course bedescribed with the Lagrangian F . However, we would like to represent it in a formalismwhere the electric potential A and the magnetic potential B both appear as explicitdynamical variables, while the corresponding field strengths F = dA and G = dB satisfy the electric-magnetic duality relation F = ⋆G. (3.1)This gives the problem a structure rather similar to the chiral form story of the previoussection.Democratic formulations of this type, where electric and magnetic degrees of free-dom appear on equal footing are of interest in the context of supersymmetric theories, aswell as in cases where it is necessary to couple the p -form to both electric and magneticcharges, see [30]. Such theories have previously been considered in [16, 17, 34] usingthe nonpolynomial PST formalism, and in specific numbers of dimensions. Our goalis to construct a polynomial formulation akin to the chiral forms of section 2, relyingon the differential form notation and keeping the treatment applicable to differentialforms of any rank in any number of dimensions.As in section 2, there are two closely related versions of our formalism. One caneither have a doubled set of Maxwellian forms and an extra auxiliary scalar that getsreduced to a single p -form dynamically due to the gauge symmetries, or instead ofthe doubled set of Maxwellian forms, one can have electric and magnetic potentialsand two additional auxiliary form fields satisfying algebraic equations of motion. Thislatter formalism is more directly related to PST-like nonpolynomial formulation, whichis obtained by integrating out the two algebraic auxiliary fields. We propose the following Lagrangian, which is overall a direct generalization of (2.4)adapted to condition (3.1) rather than (2.23), though it is essential to fix some signsand the ordering of forms in wedge-products: L = ( F + aP ) + ( G + a Q ) − a Q ∧ F + 2 a G ∧ P. (3.2)– 9 –e are in d spacetime dimensions; a is a scalar field; F = dA where A is a p -form gaugepotential (which can be thought of as electric); G = dB where B is a ( d − p − P = dR where R is a p -formfield; Q = dS where S is a ( d − p − p and d are completely arbitrary in this section. Note that ⋆ ⋆ F = ( − ( p + d + pd ) F according to (B.4), while F ∧ Q = − ( − ( p + d + pd ) Q ∧ F accordingto (B.3).The variation of the Lagrangian is given by: δ L = − h δa [ { ( − ( p + d + pd ) ⋆ ( F + aP ) − G } ∧ P − Q ∧ { ⋆ ( G + aQ ) − F } ]+ ( − p δA ∧ d [ ⋆ ( F + aP ) + ( − ( p + d + pd ) aQ ]+ ( − ( d − p ) δB ∧ d [ ⋆ ( G + a Q ) + aP ]+ ( − p δR ∧ d [ a ⋆ ( F + aP ) − ( − ( p + d + pd ) a G ]+ ( − ( p + d ) δS ∧ d [ a ⋆ ( G + a Q ) − aF ] i (3.3)The theory is invariant under the following gauge transformations: δa = 0 , δA = dU , δB = 0 , δR = 0 , δS = 0 ; (3.4) δa = 0 , δA = 0 , δB = dV , δR = 0 , δS = 0 ; (3.5) δa = 0 , δA = 0 , δB = 0 , δR = dU , δS = 0 ; (3.6) δa = 0 , δA = 0 , δB = 0 , δR = 0 , δS = dV ; (3.7) δa = 0 , δA = − a da ∧ U , δB = 0 , δR = da ∧ U , δS = 0 ; (3.8) δa = 0 , δA = 0 , δB = − a da ∧ V , δR = 0 , δS = da ∧ V ; (3.9) δa = ϕ , δA = − a ϕ ( ∂a ) ι da [ P + ( − ( p + d + pd ) ⋆ Q ] , δB = − a ϕ ( ∂a ) ι da ( ⋆P + Q ) ,δR = ϕ ( ∂a ) ι da [ P + ( − ( p + d + pd ) ⋆ Q ] , δS = ϕ ( ∂a ) ι da ( ⋆P + Q ) ; (3.10)where ϕ is an arbitrary scalar, U is an arbitrary ( p − V is an arbitrary( d − p − δ U δ ϕ − δ ϕ δ U = δ V + δ W + δ X , (3.11)where δ V is transformation (3.4) with V = aϕU substituted for U , δ W is transformation(3.6) with W = − ϕU substituted for U , and δ X is transformation (3.8) with X =– 10 – ϕ ι da dU ) / ( ∂a ) substituted for U . There is a directly analogous expression for thecommutator of (3.9) with (3.10). The commutator of two transformations given by(3.10) with parameters ϕ and ϕ is δ ϕ δ ϕ − δ ϕ δ ϕ = δ X + δ Y , (3.12)where δ X is transformation (3.8) with X = ( ϕ ι dϕ − ϕ ι dϕ ) ι da [ P + ( − ( p + d + pd ) ⋆ Q ]substituted for U and δ Y is transformation (3.9) with Y = ( ϕ ι dϕ − ϕ ι dϕ ) ι da ( ⋆P + Q )substituted for V .The equations of motion can be extracted from (3.3) as E a ≡ [( − ( p + d + pd ) ⋆ ( F + aP ) − G ] ∧ P − Q ∧ [ ⋆ ( G + aQ ) − F ] = 0 , (3.13) E A ≡ d [ ⋆ ( F + aP )] + ( − ( p + d + pd ) da ∧ Q = 0 , (3.14) E B ≡ d [ ⋆ ( G + a Q )] + da ∧ P = 0 , (3.15) E R ≡ d [ a ⋆ ( F + aP )] − ( − ( p + d + pd ) da ∧ G = 0 , (3.16) E S ≡ d [ a ⋆ ( G + a Q )] − da ∧ F = 0 . (3.17)We then proceed, much as under (2.17), by forming the combination E R − aE A ≡ da ∧ [ ⋆ ( F + aP ) − ( − ( p + d + pd ) ( G + aQ )] = 0 . (3.18)Taking Hodge dual of the above and then using identity (B.8), we get ι da [( F + aP ) − ⋆ ( G + aQ )] = 0 . (3.19)We also have E S − aE B ≡ da ∧ [ ⋆ ( G + aQ ) − ( F + aP )] = 0 . (3.20)Acting with da ∧ on (3.19), acting with ι da on (3.20), adding the results and using theprojection-rejection identity (2.1), we get( F + aP ) = ⋆ ( G + aQ ) (3.21)As in the previous section, this relation, derived from the equations of motion (3.14-3.17), ensures that the equation of motion for a given by (3.13) is identically satisfied.We now proceed to reveal the dynamical content of (3.2). Plugging the expressionfor ⋆ ( F + aP ) derived from the above relation into (3.14), and plugging the aboveexpression for ⋆ ( G + aQ ) into (3.15) we get da ∧ dS = 0 and da ∧ dR = 0 (3.22)– 11 –espectively. The general solution of these equations is obtained as in appendix C andreads R = dU + da ∧ V, S = dW + da ∧ X, (3.23)where U and V are arbitrary p -forms and W and X are arbitrary ( d − p − R , dU is gauged away by the Maxwellian shift symmetry (3.6)and da ∧ V is gauged away using symmetry (3.8). And in the above solution for S , dW is gauged away by transformation (3.7) and da ∧ X is gauged away using symmetry(3.9). Then we get R = S = 0 . (3.24)Plugging this solution for R and S into relation (3.21) we get F = ⋆G (3.25)Thus, despite starting with a theory containing four form fields, the gauge symmetriesreduce the physical degrees of freedom down to a single Maxwellian p -form, with electricand magnetic field strengths F and G respectively. As for the selfdual case of section 2, by applying the field redefinitions A old = A new − aR and B old = B new − aS , one can recast (3.2) in a form where the derivatives of theauxiliary forms R and S do not appear in the Lagrangian: L = ( F − da ∧ R ) + ( G − da ∧ S ) + 2 da ∧ S ∧ F − G ∧ da ∧ R. (3.26)With respect to these variables, the gauge transformations take the following form: δa = 0 , δA = dU , δB = 0 , δR = 0 , δS = 0 ; (3.27) δa = 0 , δA = 0 , δB = dV , δR = 0 , δS = 0 ; (3.28) δa = 0 , δA = 0 , δB = 0 , δR = da ∧ U , δS = 0; (3.29) δa = 0 , δA = 0 , δB = 0 , δR = 0 , δS = da ∧ V ; (3.30) δa = 0 , δA = da ∧ U , δB = 0 , δR = − dU , δS = 0 ; (3.31) δa = 0 , δA = 0 , δB = da ∧ V , δR = 0 , δS = − dV ; (3.32) δa = ϕ , δA = ϕR , δB = ϕS , δR = ϕ ( ∂a ) ι da [ dR + ( − ( p + d + pd ) ⋆ dS ] ,δS = ϕ ( ∂a ) ι da ( ⋆dR + dS ) . (3.33)– 12 –hile this form of the Lagrangian looks less conventional than (3.2) relative totextbook field theories, it has the advantage that the equations of motion for R and S are purely algebraic. One can then immediately solve these equations, substitute theresult back into the Lagrangian, and thus obtain a nonpolynomial PST-like formulationwith only two forms A and B that connects to the considerations of [16, 17, 30, 34].We give the relevant derivations in appendix D. D = 3For clarity, we present the simplest example of a polynomial democratic Lagrangianconsisting of a scalar ( p = 0) ϕ and a vector field ( d − p − A µ in three dimensions.This may be useful for readers more accustomed to the index notation. (Conversionformulas for switching between form and index notation are given in appendix A). Thefield strengths are F µ = ∂ µ ϕ , G µν = 2 ∂ [ µ A ν ] . (3.34)The polynomial Lagrangian is L = F + G + ( da ∧ R ) + ( da ∧ S ) − F ∧ ⋆ ( da ∧ R ) + 2 ⋆ F ∧ ⋆ ( da ∧ S ) ≡ F µ F µ + 12 G µν G µν + ∂ µ aR ∂ µ aR + ∂ [ µ a S ν ] ∂ µ a S ν − F µ ∂ µ aR + ǫ ρµν F ρ ∂ µ a S ν (3.35)where R is a rank-0 (scalar) auxiliary field, S is a rank-1 (vector) auxiliary field and F µ = F µ + ǫ µνρ G νρ . The equation of motion for the field R is da ∧ ⋆ ( da ∧ R − F ) = 0 , or ∂ µ a ∂ µ aR − ∂ µ a F µ = 0 . (3.36)This is solved, up to gauge transformations (3.29), by R = ∂ µ a ( ∂a ) F µ . (3.37)Analogously for S , S = − ι v ⋆ F p ( ∂a ) , or S ρ = − p ( ∂a ) ǫ µνρ F µ v ν . (3.38)Substituting solutions (3.36) and (3.38) into Lagrangian (3.35) we get, L = F µ F µ + 12 G µν G µν − v µ F µ F ν v ν − v [ µ F ν ] F ν v µ = F µ F µ + 12 G µν G µν − v µ F µ F ν v ν − v µ G µν G ρν v ρ (3.39)where G µν = ǫ µνρ F ρ . The Lagrangian above is manifestly in a PST-like form.– 13 – Discussion
We have revisited the polynomial Lagrangian formulation for chiral p -forms proposed in[43] and demonstrated that it leads to a compact, efficient treatment of selfduality. Wehave then extended this formulation to the case of polynomial democratic Lagrangianswhere the dual electric and magnetic potentials appear on the same footing. Due to ourreliance on the differential form notation, the formulation is universally valid for formsof all ranks in any number of dimensions. We conclude with a summary of relatedsituations where analogous approaches could be of value. This summary focuses onclassical field theories, though having polynomial formulations at hand could also bebeneficial for quantization.One motivation for developing simple and tractable theories of free fields is that itmay aid a subsequent inclusion of interactions. Interactions of chiral fields are stronglyconstrained [21], but a PST-like theory for DBI interactions of chiral forms is known[23]. It could be interesting to look for a polynomial version of this theory along thelines similar to our current treatment. Another interesting question is the numberof auxiliary PST scalars in the theory. Since the auxiliary PST scalar is required tohave properties of a global coordinate, both in the original PST theory and in itspolynomial version of [43], a natural inclination is to add extra scalars of this kind on amany-dimensional manifold. One particular construction of this sort has been proposedin [36]. It could be interesting to polynomialize it with the techniques of [43] and lookfor formulations with a still larger number of PST scalars compared to [36].Turning to problems involving democratic formulations with explicit electric andmagnetic potentials, a foundational question that goes back to [1] is developing fieldtheories including both electric and magnetic matter. There are no known ways to givea consistent Lagrangian descriptions when both electrically and magnetically chargedclassical fields are present, though the theories become consistent upon quantization,subject to the Dirac condition on electric and magnetic charges [8, 30]. This paradoxicalsituation would be worth revisiting.Finally, there has been a recent surge of interest [41, 42, 45, 48, 50, 52] in in-teracting theories of p-form fields with electric-magnetic duality symmetries, see alsoearlier related works [54–56]. It could be worthwhile to approach such theories from aperspective similar to our current treatment. Acknowledgements
We have benefitted from discussions with Hiroshi Isono, Akash Jain, Dmitri Sorokinand Arkady Tseytlin. Research of S.B. and O.E. is supported by the CUniverse research– 14 –romotion initiative (CUAASC) at Chulalongkorn University. K.M. is supported by theEuropean Union’s Horizon 2020 research and innovation programme under the MarieSk lodowska-Curie grant number 844265.
A Conversion between differential calculus and index notation
In the following, the Levi-Civita tensor is understood as ǫ µ µ ...µ d = p | g | ε µ µ ...µ d , ǫ µ µ ...µ d = 1 p | g | ε µ µ ...µ d , (A.1)given in terms of the metric deteminant g = det( g µν ) and the Levi-Civita symbol ε µ µ ...µ d = +1 if ( µ , µ , ..., µ d ) is an even permutation of (1 , , ..., d ) − µ , µ , ..., µ d ) is an odd permutation of (1 , , ..., d )0 otherwise . (A.2) Component representation: A ( p ) = 1 p ! (cid:0) A [ µ µ ...µ p ] (cid:1) dx µ ∧ dx µ ∧ ...dx µ p . (A.3) Exterior product: A ( p ) ∧ B ( q ) = 1( p + q )! (cid:18) ( p + q )! p ! q ! A [ µ µ ...µ p B ν ν ...ν q ] (cid:19) dx µ ∧ ...dx µ p ∧ dx ν ∧ ...dx ν q . (A.4) A ( p ) ∧ B ( q ) ∧ ... ∧ C ( r ) = 1( p + q + ...r )! (cid:18) ( p + q + ...r )! p ! q ! ...r ! A [ µ ...µ p B ν ...ν q ...C ρ ...ρ r ] (cid:19) dx µ ∧ ...dx µ p ∧ dx ν ∧ ...dx ν q ... ∧ dx ρ ∧ ...dx ρ r . (A.5) Interior product: ι v ( A ( p ) ) = 1( p − (cid:0) v µ A [ µ µ ...µ p ] (cid:1) dx µ ∧ dx µ ... ∧ dx µ p . (A.6) Hodge dual: ⋆ ( A ( p ) ) = ( ⋆A ) ( d − p ) = 1( d − p )! (cid:18) p ! ǫ µ µ ...µ d A µ µ ...µ p (cid:19) dx µ ( p +1) ∧ ...dx µ d . (A.7)– 15 – xterior derivative: d ( A ( p ) ) = 1( p + 1)! (cid:16) ( p + 1) ∂ [ µ A µ ...µ ( p +1) ] (cid:17) dx µ ∧ dx µ ∧ ...dx µ ( p +1) . (A.8) Integral
On a d -dimensional manifold, for a form of the top rank d , the Hodge dual is a scalar.One can thus use the following definition for integrals of such forms via the ordinaryinvariant integral of a scalar: Z C ( d ) ≡ Z d d x p | g | d ! ǫ µ µ ...µ d C µ µ ...µ d . (A.9)Note that the explicit p | g | is cancelled by the one contained in the Levi-Civita tensoras per (A.1), making this integration formula metric independent. It can be understoodas simply R d d x C ...d . B Exterior calculus identities
The following identities are used extensively in the calculations of this work. Here, u and v are 1-forms (the same letters are used for the dual vectors), A and C are p -forms,and B is a q -form. The number of dimensions is d . Exterior derivative identities: dd = 0 (nilpotency) , (B.1) d ( A ∧ B ) = dA ∧ B + ( − p A ∧ dB (Leibniz rule) . (B.2) Commutation in the exterior product: A ∧ B = ( − pq B ∧ A. (B.3) Hodge star identities: ⋆ ⋆ A = sgn( g )( − p ( d − p ) A, (B.4) A ∧ ⋆C = C ∧ ⋆A (where both A and C have the same rank) , (B.5)where sgn( g ) = − Interior product identities: ι u ι v = − ι v ι u , ι v ι v = 0 , (B.6) ι v ( A ∧ B ) = ι v A ∧ B + ( − p A ∧ ι v B (Leibniz rule) , (B.7)– 16 – v ⋆ A = ⋆ ( A ∧ v ) , (B.8) ⋆ ( ι v A ) = ( − ( p − v ∧ ⋆A = ( − ( d − ⋆ A ∧ v, (B.9) ι v ( v ∧ A ) + v ∧ ι v A = A (projection-rejection decomposition) , (B.10)with v = 1 assumed in the last line. C General solution of the equation da ∧ dB = 0 We consider the general equation da ∧ dB = 0 (C.1)where B is a p -form in d dimensions and a is a scalar field that has the properties ofa good global coordinate. In particular, the level surfaces of a defined by equations a = const provide a globally nondegenerate foliation of R d , and each slice a = consthas the topology R d − .The key observation is that (C.1) has a natural and simple restriction to the levelsurfaces a = const. Then the easiest way to analyze (C.1) is to go to a coordinatesystem where one of the coordinates, which we shall call x , is simply chosen as a .We shall call the remaining coordinates, parametrizing the constant a slices, x i with i = 1 , . . . , d −
1. In this coordinate system, (C.1) becomes ∂ [ i B ⊥ i ··· i p ] = 0 (C.2)where B ⊥ denotes the restriction of B on the surface a = const (a p -form in R d − )given by the components of B along the surface a = const, B ⊥ i ··· i p = B i ··· i p . Theabove equation simply says that B ⊥ is closed on the a = const slice. Since the slice isisomorphic to R d − , this immediately implies that B ⊥ is exact, and the most generalsolution of (C.1) is B ⊥ i ··· i p = ∂ [ i C i ··· i p ] , (C.3)where C is an arbitrary ( p − x , since the equations on different slices are completelyindependent). The solution above is equivalent to the following coordinate-invariantstatement in the original Minkowski space: da ∧ ( B − dC ) = 0 . (C.4)For any form A , da ∧ A = 0 implies that A = da ∧ E for some form E . This is easilyseen by first expressing A using the projection-rejection identity (2.1) and then taking– 17 –nto account that da ∧ A = 0 . Hence, the most general solution of the above equation,and of (C.1), is B = dC + da ∧ E, (C.5)where C and E are arbitrary. D PST-like form of the democratic Lagrangian
D.1 Derivation
Lagrangian (3.26) can be rewritten as follows: L = F + G + ( da ∧ R ) + ( da ∧ S ) − F ∧ ⋆ ( da ∧ R ) + 2 da ∧ S ∧ F (D.1)where F = ( F − ⋆G ). It has the following variation upto total derivatives: δ L = − δa d [ R ∧ ⋆ ( da ∧ R − F ) + S ∧ {F + ⋆ ( da ∧ S ) } ] − − p δA ∧ [ d ⋆ ( F − da ∧ R ) + ( − ( p + d + pd ) da ∧ dS ] − − ( d − p ) δB ∧ [ d ⋆ ( G + ( − ( p + d + pd ) da ∧ S ) − da ∧ dR ]+ 2 ( − p δR ∧ da ∧ ⋆ ( da ∧ R − F ) + 2 ( − ( d − p ) δS ∧ da ∧ [ F + ⋆ ( da ∧ S )] . (D.2)The equations of motion of the auxiliary fields R and S are δ L δR ≡ da ∧ ⋆ ( da ∧ R − F ) = 0 and δ L δS ≡ da ∧ [ F + ⋆ ( da ∧ S )] = 0 . (D.3)The above equations can be rewritten as following: R = 1( ∂a ) ( ι da F + da ∧ ι da R ) and S = − ( − pd ∂a ) ( ι da ⋆ F + da ∧ ι da S ) . (D.4)Due to the gauge freedom coming from symmetries (3.29) and (3.30), the above equa-tions imply that the following equations also hold: R = 1( ∂a ) ( ι da F + da ∧ C ) and S = − ( − pd ∂a ) ( ι da ⋆ F + da ∧ D ) . (D.5)where C is an arbitrary ( p − D is arbitrary ( d − p − R the second term can be gauged away using gauge symmetry (3.29) andthe second term in the above solution for S can be gauged away by transformation(3.30). The we get the following solutions for R and S : R = ι da F ( ∂a ) and S = − ( − pd ι da ⋆ F ( ∂a ) . (D.6)– 18 –n plugging the above solutions into polynomial Lagrangian (D.1) we get the demo-cratic PST-like Lagrangian (D.7): L = F + G − ( ι v F ) − ( ι v ⋆ F ) (D.7)where v = da/ p ( ∂a ) , a is a scalar, F = dA , A is a p -form gauge potential, G = dB , B is a ( d − p − F ≡ F − ⋆G .The variation of Lagrangian (D.7) can be derived in the following form with meth-ods typical of the PST theory (see, for instance, [37]), though this derivation is consid-erably more demanding than for the polynomial case: δ L = ( − p δa d (cid:18) v p ( ∂a ) ∧ ι v F ∧ ι v ⋆ F (cid:19) − ( − p δA ∧ d ( v ∧ ι v ⋆ F ) − ( − ( d − p ) δB ∧ d ( v ∧ ι v F ) . (D.8)One has, in particular, to keep in mind the following variation of v : δv = ι v ( v ∧ dδa ) p ( ∂a ) . (D.9)Lagrangian (D.7) is invariant under the following gauge transformations: δa = 0 , δA = dU , δB = 0 ; (D.10) δa = 0 , δA = 0 , δB = dV ; (D.11) δa = 0 , δA = da ∧ U , δB = 0 ; (D.12) δa = 0 , δA = 0 , δB = da ∧ V ; (D.13) δa = ϕ , δA = ϕ p ( ∂a ) ι v F , δB = ( − ( p + d + pd ) ϕ p ( ∂a ) ι v ⋆ F ; (D.14)where ϕ is a 0-form, U is a ( p − V is a ( d − p − D.2 Dynamics
The equations of motion are as follows: δ L δa ≡ d (cid:18) v p ( ∂a ) ∧ ι v F ∧ ι v ⋆ F (cid:19) = 0 , (D.15) δ L δA ≡ d ( v ∧ ι v ⋆ F ) = 0 , (D.16) δ L δB ≡ d ( v ∧ ι v F ) = 0 . (D.17)– 19 –hen equations (D.16) and (D.17) hold, (D.15) holds identically. This can be seenusing the relations d (cid:18) p ( ∂a ) (cid:19) ∧ v = dv p ( ∂a ) , d (cid:18) v p ( ∂a ) (cid:19) = 2 dv p ( ∂a ) . (D.18)One can rewrite (D.17) as da ∧ d (cid:18) ι v F p ( ∂a ) (cid:19) = 0 . (D.19)As per analysis of appendix C, the general solution to (D.17) is v ∧ ι v F = − da ∧ dX, (D.20)where X is an arbitrary ( p − v ∧ ι v F transforms as v ∧ ι v F → v ∧ ι v F − da ∧ dU = − da ∧ dX − da ∧ dU. (D.21)Fixing the value of the gauge field U as U = − X we get v ∧ ι v F = 0 . (D.22)The general solution to (D.16) is v ∧ ι v ⋆ F = d ( da ∧ Y ) = − da ∧ dY. (D.23)where Y is an arbitrary ( d − p − v ∧ ι v ⋆ F transforms as v ∧ ι v ⋆ F → v ∧ ι v ⋆ F + ( − ( p + pd ) da ∧ dV = − da ∧ dY + ( − ( p + pd ) da ∧ dV. (D.24)Fixing the value of the gauge field V as V = ( − ( p + pd ) Y we get v ∧ ι v ⋆ F = 0 . (D.25)Using the projection-rejection identity (2.1), as well as (2.2), (D.22) and (D.25) imply F = 0, and hence F = ⋆G. (D.26)This shows that, for all solutions of the equations of motion, the forms F and G aredual to each other, up to gauge redundancy.When we take d = 2 p + 2, p to be even and G = F , the Lagrangian in (D.7)becomes L = 2 [ F − ( ι v F ) ] . (D.27)This is twice the usual PST Lagrangian [22].– 20 – eferences [1] D. Zwanziger, Local Lagrangian quantum field theory of electric and magnetic charges,
Phys. Rev. D (1971) 880.[2] S. Deser and C. Teitelboim, Duality transformations of Abelian and non-Abelian gaugefields,
Phys. Rev. D (1976) 1592.[3] N. Marcus and J. H. Schwarz, Field theories that have no manifestly Lorentz-invariantformulation,
Phys. Lett. B (1982) 111.[4] W. Siegel,
Manifest Lorentz invariance sometimes requires nonlinearity,
Nucl. Phys. B (1984) 307–316.[5] A. R. Kavalov and R. L. Mkrtchian,
Lagrangian of the selfduality equation and d=10,N=2b supergravity,
Sov. J. Nucl. Phys. (1987) 728.[6] R. Floreanini and R. Jackiw, Selfdual fields as charge density solitons,
Phys. Rev. Lett. (1987) 1873.[7] M. Henneaux and C. Teitelboim, Dynamics of chiral (selfdual) p-forms,
Phys. Lett. B (1988) 650.[8] M. Blagojevi´c and P. Senjanovi´c,
The quantum field theory of electric and magneticcharge,
Phys. Rept. (1988) 233.[9] K. Harada,
The chiral Schwinger model in terms of chiral bosonization,
Phys. Rev.Lett. (1990) 139.[10] A. A. Tseytlin, Duality symmetric formulation of string world sheet dynamics,
Phys.Lett. B (1990) 163.[11] B. McClain, F. Yu and Y. S. Wu,
Covariant quantization of chiral bosons and
OSp (1 , | symmetry, Nucl. Phys. B (1990) 689.[12] C. Wotzasek,
The Wess-Zumino term for chiral bosons,
Phys. Rev. Lett. (1991) 129.[13] A. A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars,
Nucl. Phys. B (1991) 395.[14] J. H. Schwarz and A. Sen,
Duality symmetric actions,
Nucl. Phys. B (1994) 35arXiv:hep-th/9304154.[15] A. Khoudeir and N. Pantoja,
Covariant duality symmetric actions,
Phys. Rev. D (1996) 5974 arXiv:hep-th/9411235.[16] P. Pasti, D. P. Sorokin and M. Tonin, Note on manifest Lorentz and general coordinateinvariance in duality symmetric models,
Phys. Lett. B (1995) 59arXiv:hep-th/9503182 [hep-th].[17] P. Pasti, D. P. Sorokin and M. Tonin,
Duality symmetric actions with manifest – 21 – pace-time symmetries,
Phys. Rev. D (1995) 4277 arXiv:hep-th/9506109.[18] P. Pasti, D. P. Sorokin and M. Tonin, Space-time symmetries in duality symmetricmodels, in Gauge theories, applied supersymmetry, quantum gravity (Leuven, 1995) pp.167–176 arXiv:hep-th/9509052.[19] A. A. Tseytlin,
Selfduality of Born-Infeld action and Dirichlet three-brane of type IIBsuperstring theory,
Nucl. Phys. B (1996) 51 arXiv:hep-th/9602064.[20] F. P. Devecchi and M. Henneaux,
Covariant path integral for chiral p-forms,
Phys.Rev. D (1996) 1606 arXiv:hep-th/9603031 [hep-th].[21] M. Perry and J. H. Schwarz, Interacting chiral gauge fields in six-dimensions andBorn-Infeld theory,
Nucl. Phys. B (1997) 47 arXiv:hep-th/9611065.[22] P. Pasti, D. P. Sorokin and M. Tonin,
On Lorentz invariant actions for chiral p-forms,
Phys. Rev. D (1997) 6292 arXiv:hep-th/9611100.[23] I. A. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti, D. P. Sorokin and M. Tonin, Covariant action for the superfive-brane of M-theory,
Phys. Rev. Lett. (1997) 4332arXiv:hep-th/9701149.[24] M. Cederwall and A. Westerberg, Worldvolume fields, SL(2,Z) and duality: the typeIIB three-brane,
JHEP (1998) 004 arXiv:hep-th/9710007.[25] I. A. Bandos, N. Berkovits and D. P. Sorokin, Duality symmetric eleven-dimensionalsupergravity and its coupling to M-branes,
Nucl. Phys. B (1998) 214arXiv:hep-th/9711055.[26] A. Maznytsia, C. R. Preitschopf and D. P. Sorokin,
Duality of selfdual actions,
Nucl.Phys. B (1999) 438 arXiv:hep-th/9805110.[27] P. Pasti, D. P. Sorokin and M. Tonin,
Harmonics, notophs and chiral bosons,
Lect.Notes Phys. (1999) 97 arXiv:hep-th/9807133.[28] M. Roˇcek and A. A. Tseytlin,
Partial breaking of global D = 4 supersymmetry,constrained superfields, and three-brane actions,
Phys. Rev. D (1999) 106001arXiv:hep-th/9811232 [hep-th].[29] R. Manvelyan, R. Mkrtchian and H. J. W. Muller-Kirsten, On different formulations ofchiral bosons,
Phys. Lett. B (1999) 258 arXiv:hep-th/9901084.[30] K. Lechner and P. A. Marchetti,
Duality invariant quantum field theories of chargesand monopoles,
Nucl. Phys. B (2000) 529 arXiv:hep-th/9906079 [hep-th].[31] S. M. Kuzenko and S. Theisen,
Supersymmetric duality rotations,
JHEP (2000) 034arXiv:hep-th/0001068 [hep-th].[32] Y. G. Miao, R. Manvelyan and H. J. W. Mueller-Kirsten, Selfduality beyond chiralp-form actions,
Phys. Lett. B (2000) 264 arXiv:hep-th/0002060 [hep-th]. – 22 –
33] D. Sorokin,
Lagrangian description of duality-symmetric fields,
NATO Sci. Ser. II (2002) 365.[34] P. Pasti, D. Sorokin and M. Tonin, Covariant actions for models with non-lineartwisted self-duality,
Phys. Rev. D (2012) 045013 arXiv:1205.4243 [hep-th].[35] C. Bunster and M. Henneaux, Duality invariance implies Poincar´e invariance,
Phys.Rev. Lett. (2013) 011603 arXiv:1208.6302 [hep-th].[36] S.-L. Ko, D. Sorokin and P. Vanichchapongjaroen,
The M5-brane action revisited,
JHEP (2013) 072 arXiv:1308.2231 [hep-th].[37] H. Isono, Note on the self-duality of gauge fields in topologically nontrivial spacetime,
PTEP (2014) 093B05 arXiv:1406.6023 [hep-th].[38] A. Sen,
Covariant action for type IIB supergravity,
JHEP (2016) 017arXiv:1511.08220 [hep-th].[39] H. Afshar, E. Esmaeili and M. M. Sheikh-Jabbari, Asymptotic symmetries in p -formtheories, JHEP (2018) 042 arXiv:1801.07752 [hep-th].[40] A. Sen, Self-dual forms: action, Hamiltonian and compactification,
J. Phys. A (2020) 084002 arXiv:1903.12196 [hep-th].[41] G. Buratti, K. Lechner and L. Melotti, Duality invariant self-interactions of abelianp-forms in arbitrary dimensions,
JHEP (2019) 022 arXiv:1906.07094 [hep-th].[42] G. Buratti, K. Lechner and L. Melotti, Self-interacting chiral p-forms in higherdimensions,
Phys. Lett. B (2019) 135018 arXiv:1909.10404 [hep-th].[43] K. Mkrtchyan,
On covariant actions for chiral p -forms, JHEP (2019) 076arXiv:1908.01789 [hep-th].[44] N. Lambert, (2,0) Lagrangian structures, Phys. Lett. B (2019) 134948arXiv:1908.10752 [hep-th].[45] P. K. Townsend,
An interacting conformal chiral 2-form electrodynamics in sixdimensions,
Proc. Roy. Soc. Lond. A (2020) 20190863 arXiv:1911.01161 [hep-th].[46] P. K. Townsend,
Manifestly Lorentz invariant chiral boson action,
Phys. Rev. Lett. (2020) 101604 arXiv:1912.04773 [hep-th].[47] E. Andriolo, N. Lambert and C. Papageorgakis,
Geometrical aspects of an Abelian(2,0) action,
JHEP (2020) 200 arXiv:2003.10567 [hep-th].[48] I. Bandos, K. Lechner, D. Sorokin and P. K. Townsend, A non-linear duality-invariantconformal extension of Maxwell’s equations,
Phys. Rev. D (2020) 121703arXiv:2007.09092 [hep-th].[49] Y. Bertrand, S. Hohenegger, O. Hohm and H. Samtleben,
Toward exotic 6Dsupergravities, arXiv:2007.11644 [hep-th]. – 23 –
50] B. P. Kosyakov,
Nonlinear electrodynamics with the maximum allowable symmetries,
Phys. Lett. B (2020) 135840 arXiv:2007.13878 [hep-th].[51] P. Vanichchapongjaroen,
Covariant M5-brane action with self-dual 3-form, arXiv:2011.14384 [hep-th].[52] I. Bandos, K. Lechner, D. Sorokin and P. K. Townsend,
On p-form gauge theories andtheir conformal limits, arXiv:2012.09286 [hep-th].[53] C. A. Cremonini and P. A. Grassi,
Self-dual forms in supergeometry I: the chiral boson, arXiv:2012.10243 [hep-th].[54] I. Bia lynicki-Birula,
Nonlinear electrodynamics: variations on a theme by Born andInfeld, in Quantum theory of particles and fields: birthday volume dedicated to Jan
Lopusza´nski (World Scientific, 1984), pp. 31-48.[55] I. Bia lynicki-Birula,
Field theory of photon dust,
Acta Phys. Polon. B (1992) 553.[56] D. Chru´sci´nski, Strong field limit of the Born-Infeld p-form electrodynamics,
Phys.Rev. D (2000) 105007 arXiv:hep-th/0005215.(2000) 105007 arXiv:hep-th/0005215.