JHEP04(2015)032 Springer April 8, 2015 ty in Gauge March 6, 2015 even the two : : p February 9, 2015 : = 2 . 1)-forms described and = 1 supersymmetric N p − Published N p Accepted = 2 10.1007/JHEP04(2015)032 Received − D D doi: .C.L.A., Los Angeles, CA, U.S.A. r multiplet or a massive linear e systems should play a role for ms in one theory translate into er-Higgs in ss theory is self-dual. This state r, for l massive Published for SISSA by -forms and ( p [email protected] , dual pairs of a,c massive . 3 1502.01650 The Authors. and A. Sagnotti and Duality, Supersymmetry Breaking, Duali c

1 We consider , a,b, [email protected] On leave of absence from Department of Physics and Astronomy, U 1 INFN - Laboratori Nazionali diVia Frascati, Enrico Fermi 40, Frascati,Scuola I-00044 Normale Italy Superiore andPiazza INFN, dei Cavalieri 7, Pisa, I-56126E-mail: Italy Department of Physics, CERNGeneva Theory 23, Division, CH-1211 Switzerland b c a Keywords: Born-Infeld action, which describes(tensor) either multiplet with a a massive Born-Infeldthe vecto mass-like massive term. gravitino multiplet Thes obtained from a partial sup non-linear structures coincide whenof the affairs non-linear finds massle a natural realization in the four-dimensiona Theories ArXiv ePrint: Open Access Article funded by SCOAP Massive Born-Infeld and other dual pairs Abstract: S. Ferrara by non-linear Lagrangians, wherenon-linear non-linear mass-like curvature terms ter in the dual theory. In particula JHEP04(2015)032 1 2 5 7 8 10 11 11 early -form )-form p p = 2 two- − = 2 vector ty of being D N N ] has the further l 3 1)-form gauge field , 2 − 1)-form gauge fields, ] of their ( p re interactions present − 17 − p D he partial breaking of Su- heories by some additional − the four-dimensional Born- mass generation for ng [ ty of its bosonic counterpart, D = 2 Supersymmetry partially n rigid theories with extended where half of the N . This fate can result from a scalar = 1 Supersymmetry and gives a large ]. It is the aim of this paper to explore N 16 – 13 – 1 – , 7 , 6 ]. 12 – 9 , 7 -form gauge fields naturally arise in Superstrings and are cl p ]. Alternatively, this action is seen to arise from an 8 – 4 -form gauge field. Or, equivalently, of their ( p ], which in the bosonic case possesses the remarkable proper 1 = 1 [ = 1 supersymmetric completion of the Born-Infeld action [ N N + 1)-form field strength. As a result, the non-linear curvatu = 1 superspace formulation p The The supersymmetric Born-Infeld action inherits the proper N to a ( multiplet, i.e. a chiral multiplet, haspotential been that integrated admits out amass vacuum to with the unbroken chiral multiplet [ derivative action, quadratic in the Maxwell field strength, field strength to a geometric couplings that are connected to the Stueckelberg in that it isthe self-dual mass in generation Superspace mechanism [ induced in these non-linear t electric-magnetic self-dual. remarkable property of being the Goldstone action for broken to gauge fields. They admit a dual formulation in terms of ( where the dual mass term results from a Green-Schwarz coupli Non-linear theories of 1 Introduction 6 Massive supersymmetric Born-Infeld and7 its superspace dua Discussion A The dual massive Born-Infeld-like action 3 Self-dualities in even dimensions 4 Self-dual massive dualities in5 odd dimensions Contents 1 Introduction 2 Massless and massive dualities relevant in the descriptionpersymmetry. of The brane latter dynamicsSupersymmetry. incarnation as also models A presents prototype for itselfInfeld of t i action these [ non-linear theories is JHEP04(2015)032 5 4 ], = 23 ecall m (2.1) (2.2) ], whereby even, where 19 , p we move to the = 1 superspace m. In section 18 6 for N The other involves p , p th a “mostly positive” = 2 ⋆ B elevant reen-Schwarz coupling. In imensions, which are only orm formulation realizes a assive absorbing a massless D h are a symmetry for ∧ oundary terms. In section nally, in terms of the vector ], which is only available for , p enon is provided by the mas- dual, instead, describes a mas- 1 f the mass term present in the re gauge and is absorbed by the B 21 it derivation of the dual massive m given by the same Born-Infeld − ggs mechanism [ 2 , tion. In section p -interacting massive vector multi- 2 near mass term is determined by we describe massless and massive m Λ 20 d 2 = ) + p take the universal form p B p ( B +1 p ], so that the non-linear interactions maintain – 2 – contains our conclusions and some prospects for , δ B ⋆ H 13 7 p -forms ∧ p ) dB p ] and by Pilch, Townsend and van Nieuwenhuizen [ B = 22 ( +1 +1 p odd. These are simple extensions of the systems considered p H p H ]. There are again two dual formulations. One provides the 2 24 2 k , = = 1 supersymmetric theories in four dimensions, and briefly r 19 L + 1 and N p even, a particular case being the massive Born-Infeld syste = 2 p D ] with a non-linear, rather than quadratic, curvature part. we describe the notion of self-dual actions [ 26 and 3 , p 25 The plan of this paper is as follows. In section A particularly interesting situation presents itself when = 2 Green-Schwarz term [ where the field strength and the gauge transformations, whic 2 Massless and massive dualities Let us beginsignature, by the recalling free that, actions for in massive the language of forms and wi the same supersymmetric Born-Infeldmultiplet field action strength. built Finally, section up,future developments, origi while the appendix containsBorn-Infeld an action. explic massive case in superspace, and to this endsuperspace we generalization of introduce a the Stueckelbergplet r action, [ a self a self-interacting massive linear multiplet, whose non-li which are also providing mutually dual descriptions, up to b dualities in a generalsection setting where the massD is induced by a G Stueckelberg-Higgs phenomenon for the massivesive vector. antisymmetric tensor Its theory with aaction, non-linear mass albeit ter with the vector fieldantisymmetric strength tensor that field. becomes pu Thea latter realizes massless the antisymmetric anti-Hi tensorvector. (dual to a scalar) becomes m their functional form. Asive notable Born-Infeld realization system of in this phenom four dimensions, where the one-f by Deser, Jackiw and Teitelboim [ possible for we concentrate on the superspace self-duality of the massless Born-Infeld ac we turn to the peculiar self-dual gauge-field systems in odd d other. in one formulation are dual to non-linear generalizations o the non-linear system can be self-dual [ 0, read JHEP04(2015)032 . ). 2 − p 2.2 (2.5) (2.9) (2.4) (2.7) (2.6) (2.3) (2.8) − (2.10) D , and is 2 A , . This is − 2 , with the ling. This p )] 1 − − 1 p − , . Let us also − D p − ] p 1 − p D − − , dA D 2)-form p into a first-order C D ...µ p − A 2 ich can be reached . 1 − = A ) translate into the µ D ( − ...µ )] p 1 p p Λ 1 A 1 − , and let us begin by 1 2.2 − − µ − − p and 1 µ [ in Supergravity. It will p D D B . − − p − F p D A 2 p D [ ions on the nature of the and p ∂ D − B − 2 F p p ...µ D A or = L 1 − ( , ) and ( B µ p p p D 2 + dA − B and -dimensional duality between a B − 2.1 . 1 ! ...µ . D p 2 dC p 1 D p − = p 2 − F p µ [ m ∧ 2 B − D p − dB 2 p 1) − D − L − +1 − D = p A ) leads finally to the dual representation D D ⋆ dC F +1 ( H ∧ p ) + +1 p ∧ 2.5 dC p ) p + , δ B p 2 1) − ...µ B ] H 1 and − ( B − µ p +1 ( = +1 ( p p – 3 – p − +1 H − p +1 D +1 p ...µ dB p ⋆ H +1 2 = p , so that in flat space its solution is given in eq. ( dC µ ⋆ H p ∧ = 2 mH B ...µ ∧ starts out as an unconstrained field, and the massless +1 k ⋆ H 1 1 1 p ) µ 2 2 +1 µ [ +1 p p H ⋆ ⋆B k ) + p +1 , which is unconstrained to begin with, as we have stressed, ∂ dimensions and with the given “mostly positive” signature, H B p p = H ( H and a corresponding “magnetic” ( +1 2 B H D 2 p L ( of mass dimension, and one is thus led to p k +1 + 1) p 2 H B + 1)! +1 m k p = p H p 2 L 2 k ) and making use of eq. ( = ( 2 ( ⋆ H -form − = ∧ 2.6 +1 p p ) ) is recovered once one enforces the field equation of = L p p ...µ B L 1 2.1 B ( , but let us add to this action principle a Green-Schwarz coup µ 2 +1 H L p H -dependent expressions “electric” -form 2 p 2 p k is a dimensionless coupling. In components eqs. ( = k Let us also recall that, in Our starting point is quite familiar and plays a central role We can now show that this Lagrangian admits two dual forms, wh Let us now move on to consider a pair of gauge forms, In this first-order action L encompassed by the “master action” for any This Lagrangian describes a pair of massless gauge fields, precisely the Bianchi identity for carries along a coefficient considering a Lagrangian of the type corresponding field strengths yields where the antisymmetrizations have strength one. refrain, for the timeLagrangian being, from making any definite assumpt which is gauge invariant up to a total derivative. Substituting in eq. ( and Conversely, integrating out familiar It involves the corresponding field strengths and with an inverted overall coefficient. massless serve to fix our notation, and concerns the standard turning the description of either of the two gauge fields version of eq. ( JHEP04(2015)032 al m. ) the 0 one (2.12) (2.15) (2.11) (2.14) (2.13) 2.9 → , k ] , p −  2 D 2 − − F [ p p − − ) with L 6= 0 the Lagrangian D D + 0 this Lagrangian de- m  2.10 dC dC 1 ing in an eventual more 1 ∧ − m → p . orrespondence of this type hich carry three degrees of + h link vectors to vectors or +1 e degrees of freedom carried  m on, and thus their masses, in p dE 1 1 erturbative excitations. These binomial identity − H oncerns, for instance, a massive = 0 and 1)-form, while if also would pertain to massless ones, − pread belief that the underlying p − p , , considering p k − − + p ) is a Stueckelberg realization, a D p D  1 − p  − A − D p m B 2  2.10 − F − = D , replacing eq. ( ⋆ . 2 D pD − 1  +1 A 1 ∧ 1) p − − D p ∧ − 2 H  − − D ( 2  D p −  +1 − p p A ∧ – 4 – − = D − p , D − is then constrained to be pure gauge. D  mH )] cases that we shall shortly reach from eq. ( D 2 1 1)-form. Finally, if  1 dC F + − − − p − p + 1 , the end result, p m − − p − +1 )]  D ) +1 p D 1 + 2 D p p −  − A 1 A B p H − p -form and a massless ( ( massive D − ⋆ H ( − p p p describe, of course, identical numbers of degrees of freedo fields, a correspondence that reflects the well-known binomi D ∧ D − − +1 p 1  D D A ( − +1 F A p [ p p ) reflects the fact that eq. ( ⋆ H − 2 −  H D D L ∧ 2 2 2 2 F k ) k A + 2.11 [ p m massless is unconstrained. Notice that in the limit 2 2 B = + L ( +1 and L p +1 6= 0, a massless = p p H k L B H 2 2 k = The identity ( Let us now move to a first-order form for Let us stress that in the Conversely, one can go to a first-order form for If one now integrates out L which underlies the more familiar massless dualities. identity Moreover these identical numbers coincideby with corresponding the sum of th This is the counterpart, in the massive construction, of the phenomenon ought of takecomplete the formulation. form of a huge spontaneous break property that we shallplays shortly a role make in manifest. ,modes in Notice the are that behavior indeed of a described, the in c massivebut the p they light-cone occur formalism, in as combinationsa that Stueckelberg build realization. up an This extra fact dimensi has long led to a wides two fields electromagnetic potential and afreedom. massive It two-form, is bothtwo-forms sharply of to different scalars. w from the massless ones, whic form. In four dimensions, the duality that we are exploring c involves a Stueckelberg-like mass term for where now scribes, for is left with a single massless ( becomes empty, since the field JHEP04(2015)032 -like (3.1) (3.3) (3.2) (2.16) (2.18) (2.17) es the = 2, the two , . Integrating . . p  C m 1

 − 2  p ,  p 2  − µν dE L 2 D e L F  − is the Born-Infeld scale ∂ ∂F p µν µν , µ p ending on However, as we have seen, e B F F is thus the master action − ple would obtain eliminating  that for brevity we can just , dimensions the massive dual D ssible. Taking = 0 pD µν 4 ) F 4 ve Born-Infeld vector, F µ p D inear corrections. 1) 2 µν  − − − L ] − 4 D ( p 4 , the counterpart of the parameter L µ − g ∂ m µν ρσ ∂F  -dimensions. Note, however, that this D − F F g, µ, F F ( − D [ , and we would like to consider the case µν µν µν 1 dual 1 BI L , F B − F 2 A L p 2  4 L µ µν + dE µνρσ F = µ ǫ 2 – 5 – ) +  4 − A p µ 1 + 4 1 m p : µ B − 2 r ( p B A − L 1 + 2 − 2 +1 k dE pD p 1 r m µνρ 2 1) and one-form −  H − − ⋆ H p − 2 2 ( 2 1 g B B ∧ = µ  µνρ m 8 ) 2 p L pD is a left derivative, while the resulting Lagrangian involv H 2 g 2 B 2 µ 1) ( k 12 8 C ≡ ) = 3)-form. L − ( + − +1 − p = 2 this could describe a massive two-form with a Born-Infeld m were the Born-Infeld action in = H  D 2 p g, µ, F 2 L ( k dual dual , , BI = 2 2 L L is a Born-Infeld action. L L 2 now leads to the condition L p − D Reverting to the conventional notation, our starting point As above, a massive variant of the Born-Infeld action princi F after moving to a first-order form where it is unconstrained. that accompanies the two-form kinetic term. The parameter corresponding Legendre transform of k out where we have introduced a dimensionless parameter action is not self-dual away from four dimensions, since in where In particular, for mass term if where the derivative of where where we have performed a partial integration in the term dep in which 3 Self-dualities in even dimensions In four dimensions an even stronger instance of duality is po fields at stake are a two-form of a two-form is a ( factor, with mass-squared dimension, which sizes the non-l H the additional field isdisplay just the a gauge-fixed standard Proca-like Stueckelberg Lagrangian mode, for the so massi JHEP04(2015)032 ). + ess ), as 3.6 (3.6) (3.8) (3.7) (3.5) (3.4) m B 3.1 hat can and forms p . # ] 2 e Green-Schwarz even , Infeld system for dual formulation,  21 . ,  µν .  . In these cases, the ) e p p B 20 C # B ( , , with 2 µν p  µν 15 B . e F , , m B  variance of the four-dimen- ′ = 2 ,F tion would translate into the ture as the dual gauge field, 8 e of four-dimensional duality F reproduction as above. The invariant combination 14 s term and reads 2 = 0 -forms g  he transformation in eq. ( ndre transform of eq. ( µ D eof as the Born-Infeld system, , µ g 4 p 4 ′  4 s µ g on [ ) under duality rotations implies µ . m = ρσ  2 ′ 4 C F 16 and !) 3.8 L p − p , µ µν ( = F 1 g p µν 4 − ) is precisely that the Born-Infeld action = 2 1 g B 2 = m B ′ 3.5 F D + g µν = 2  B µ p L 8 ! – 6 – ρσ 4 L L ∂F BI ∂ p g ∂

2 L 2 ∂F p ) µ L m = 1 + B µν . The self-duality of the massless Born-Infeld theory ∂ L 4 ( ∂F  ∂ s p µν ) ∂F F µνρ B − A 4 1 + (  − 1 H ] s ) µν " p B 2 − , and therefore for ( 2 µνρσ g p ǫ 1 µ 0. In this fashion the limiting Lagrangian describes a massl 8 " µνρ − , with ). This is actually a special manifestation of a phenomenon t 1 2 = g , µ , F → H 2 g , µ , F − D g 2 [ p  µ 3.4 L k 12 8 m L = BI ...µ + − 1 L p µ = A L is the dual gauge field. On the other hand, in the presence of th 6= 0), one can eliminate the vector altogether and work, in the . In these cases one can indeed formulate analogs of the Born- C m p 1)-forms This property of self-reproduction reflects, in fact, the in Let us stress that the invariance of actions like ( In the massless case, the self-duality of the Born-Infeld ac The reason for the peculiar result in eq. ( before letting ...µ − 1 µ p sional massless Maxwell system,under or duality rotations, of which generalizations translates ther into the conditi that they are self-reproducing, in the massive case, under t possesses the property ofsummarized reproducing in itself eq. ( under the Lege condition that The very form ofrelations, this which condition finds reflects a the direct counterpart symplectic in natur all dimension solely in terms of the two-form two-form, dual to a scalar, and a dual massless vector The massless limit can bedC recovered reintroducing the gauge where term ( then implies that the dual action involves a Born-Infeld mas and therefore one cancorresponding condition contemplate would the read possibility of a self- field strength one starts from possesses the same index struc ( occur whenever F JHEP04(2015)032 . − odd F self- (4.1) (4.3) (4.2) + F vanishes real ]. In this ∼ e F 2 28 F F . . -dimensional duality p p ]. p B B lt of Deser, Jackiw and 23 +1. The purpose of this ∧ ∧ tring Theory, both in six p oposed long ago by Pilch, ) , the product p xpect that a relation to the + 1. It describes the same +1 ction principles [ ing world-sheet, although it , and moreover p that have been associated to B p = 2 r the 2 , ( he massive case. ± even, one can impose reduces to a constant. H odd ed in [ p on-linear completion of a mass F D p +1 2 p m = 2 p ts -form. On the other hand, for ) the Hubbard-Stratonovich trans- H + p D +1. Furthermore, the corresponding p 2 m ⋆ B p m 4.2 , the second term would be a total with C p = p ) implies that, in the massive case, self- ∧ = 2 p ) + ) + 2, with p p = 2 p D . In these cases a non-linear completion of a . One possible form can indeed be deduced B 2.13 dB B ( p p 2 self-duality condition halving the correspond- ( D k = 2 +1 – 7 – +1 p p ≡ odd D real even ) H is duality invariant, simply because dualities act as p ], for arbitrary ⋆ H  + B 2 ∧ 23 ( p dimensions ) F p +1 p ⋆ C -forms in odd dimensions B -forms, which halve their propagating degrees of freedom. L ( p odd p H ∧ 2 +1 p k , the proper p C p H 2 1)-forms with B 2 k 1 2 k − 2 p = = ] can be extended in general to describe dual pairs. -form L L p 23 ]. ), and reads 22 ] and [ 2.13 -forms and ( ) is a generalization of the classic three-dimensional resu 22 p 4.2 To begin with, the very structure of eq. ( Another formulation for massive self-dual fields was also pr One can exhibit this relation performing in eq. ( For a massive On the other hand, in even dimensions eq. ( duality conditions on massless from eq. ( As a result, if the field strength is self-dual the4 Lagrangian Self-dual massive dualitiesIt in is odd well dimensions known that in even dimensions These systems play anand important in role ten in dimensions,was Supergravity as recognized and well long in as ago in S that they the do description not of admit the conventional a str Townsend and van Nieuwenhuizen [ which is clearly possible only in ing degrees of freedom is formation Templeton [ dual systems can only exist for section we would like to elaborate on their counterparts in t identically, and any Lagrangian opposite rescalings on the self-dual and antiself-dual par number of propagating degrees ofpreceding freedom, one so exist, that although one apparently would this e was not notic Consequently, similar results hold, in the massive case, fo section is to describethem how in the [ two types of action principles Lagrangians can only be formulated for between From this expression one can see that, for even derivative, so that one would be describing a massless kinetic term turns, afterterm a for duality the transformation, dual into field a with n an identical functional form. p JHEP04(2015)032 , → V = 2 (5.1) (4.5) (4.6) (5.2) (4.4) A ) then N leads to 4.2 p = 1 vector B N belongs to a linear = 0). The two-form , with one degree of α µν dB L because of the residual ts are inverted, so that, ˙ counterpart in an “anti- a massive antisymmetric B ress massive Born-Infeld α , this system is expected = e fermion, must therefore rsymmetric context, since auge transformation ten by the gravitino of the D ( e role of Goldstone fermion , H . = 1 massive multiplet, which p α eak coupling duality, while the dom). he Goldstone action for on . Substituting in eq. ( belongs to a real superfield L N dC L A ∧ = 1 super-Higgs effect of partially , p ), while integrating out Λ = 0) ˙ N C α Λ V, 4.2 d . α D m 1 → 2 2 D ]. The Born-Infeld system, which contains + m k 2 p − 27 C D = p = 2 1 m Λ( – 8 – ⋆ C N ) = − phys ∧ V ( = M p fermion [ α p C + Λ + 2 1 B W 2 k V 1 2 → = V = 1 superspace the vector = 1. It thus describes the self interactions of an L N N belongs to a spinor chiral multiplet B one would simply recover eq. ( p belongs to the chiral multiplet C dA -forms. p = 1 superspace formulation . On the other hand, the three-form field strength dλ In four-dimensional This relation becomes particularly interesting in the supe N + the condition while the two-form which is invariant under the superspace gauge transformati freedom, eats a massless vector carrying two degrees of free in the rigid casebecome a massive. massless This vectorsystems provides as for in partner general of a the Goldston motivation to add leads to the dual Lagrangian up to Λ-dependent boundarylike all terms. preceding examples, Notice thisphysical that has mass the both is flavor coefficien in of both a strong-w cases Higgs” mechanism for the latter (in the sense that a massless 5 As we have seen,tensor, in four and dimensions the a Stueckelberg massive mechanism vector for is the dual to former finds a Integrating out This is theA superspace counterpart of the familiar Maxwell g field strength where the last term gives rise only to total derivatives in spontaneously broken to the supersymmetric extension of the Born-Infeld action is t multiplet whose fermionic component,for the the gaugino, broken Supersymmetry. playsto th be When a coupled key to ingredient Supergravity in models for the also contains two vectors and a spin- Supersymmetry the massive gravitino must complete an broken Supersymmetry. Consequently, thebroken gaugino must Supersymmetry, be which ea then becomes massive. In fact, JHEP04(2015)032 ) F 5.5 (5.9) (5.7) (5.8) (5.5) (5.6) (5.4) (5.3) xwell yields (5.11) (5.10) α . Only the , which is ) when , W α . Eq. ( L ˙ α F 5.9

) D roportional to , . 2 c . F D

˙ 1 α  . f eq. ( . + h c D ) and translates into . With a slight abuse of . dz α µν = ) in the first-order form 3.7 + W B α α . ) + h B 5.6 D ion takes the form q. ( α 2 V y of the dual field strength M , ( ( → 2 i D iM α B − W = 0 , D + = ), while integrating out F 2 . 

, ˙ F α g α 1 

) . 5.6 . α 2  L W = 0 . Z D for both. ˙ c 2 W ( α ˙ . α V 1 ins the tensor field + α g = 0 α α D D M W L

D + + h W ˙ − 2  α 2 4 2 ) + α α 1 D D g V D F α L W – 9 – ( -component. Notice that this is trivially satisfied ∂ − = L α 2 + ∂W = F 2 α α L D 2 α → W g 2 1  is the super field strength of the chiral multiplet M M , D 2 i α M D α ) L according to α L ≡ = + D V α ( M and letting α L L 2 D  BI

D  L ℑ , it is now convenient to recast eq. ( 2 V ∂ ∂W 2 2 W , D ], the superspace Born-Infeld action enjoys a self-duality , due to the superspace identity 2 ]. This expression clearly reduces to the supersymmetric Ma g , µ , W 6 Z . 15  , , D 3 ] F

one recovers the original action ( ) 14 15 = , V D , ( 6 V g , µ , W 2 BI  14 = 0, and the leading non-linear order correction is clearly p L F is invariant under the gauge transformation W ) , which is related to F L L V is given in [ ( picks the imaginary part of the = 2 ), F ℑ BI = 1 Maxwell Electrodynamics, on account of the special form o D W L V Proceeding as in section In superfield language, the supersymmetric Born-Infeld act 2 Strictly speaking, the linear multiplet ( g N 1 1 α 2 µ The equation of motionW then follows from the Bianchi identit introducing a dual potential latter ought to be called tensor multiplet, because it conta vanishes. Integrating out the condition [ where action for is the superspace counterpart of the gauge transformation the condition multiplet Note that where language, however, we use loosely the term linear multiplet As discussed in [ and satisfies the two constraints in the direct counterpart of what we have seen in components in e for any real superfield JHEP04(2015)032 ) 6.1 (6.5) (6.2) (6.1) (6.4) (6.6) (6.3) , and , thus . The U 2 µνρ H , al F Maxwell term }| . c . term in eq. ( ,  . alar mass terms, but )  metric completion of ) V )] + h ity of section ( V , ˙ D ] and, more recently, for α ( are real superfields. The V ˙  α calar and a vector in the ( ) 29 W ultiplet contains a physical . α V , ). W V  ) tain four bosonic degrees of the linear multiplet. L ( , which is dual to the massive M the form ˙ ) α − V tiplet becomes dual to a chiral 5.6 ( + V and = 0 rary function Φ. In particular, in )= a massive vector multiplet. Mak- α ), the supersymmetric Proca-like ( W α , U α T ( ) U ′ ˙ 3.2 α Φ V

mL ( D [  ). The notation is somewhat concise, theories in [ α ) α 2 U g , µ , W (  R 2.15 ′ g , µ , W ) and ( i W  Φ + BI { , which requires the introduction of the dual BI L U R T, 2.14 α ) contains in general non-linear interactions of + g , µ , W L ) reflects the freedom to dress the tensor kinetic  – 10 – gives L − + W D ( ) + U

) BI L ψ T ) + U L  , one can turn it into the Proca-like form ˙ α = mV ) is Φ( BI  m V ) + − W L , + α U 2.13 m V is the superfield extension of an unconstrained ( ) = T m V L − L U -term, as we have seen in eq. ( ( + is generally dressed by a scalar function. U F ψ ]. µ 2 T ) + = Φ( A U 24 g , µ , W L  = Φ( BI = Φ( ). To begin with, however, notice that one can integrate out L L D L V ), with the last term replaced by a standard quadratic super- ( ) + α L ) contains supersymmetric generalizations of vector and sc 6.1 ( M ψ also contains an  is a linear multiplet Lagrange multiplier and mV ). It can be obtained integrating by parts the Green-Schwarz BI = , was considered in connection with becomes L α L L 3.5 L Varying the action with respect to The dual supersymmetric Born-Infeld action is the supersym Notice that, in this richer setting, W α Combining all these terms, the first-order Lagrangian takes gauge field replacing the first two terms with the Legendre transform the presence of the arbitrary function Φ( eq. ( term with an arbitrary function of the scalar field present in since and then going to a first-order form for mass term that extends and W models of inflation, in [ counterpart of the Lagrangian ( the massive scalar presentscalar in of the the massive massivefreedom linear vector (a multiplet, multiplet. scalar anddual Both a vector multiplets multiplet). tensor con In inmultiplet, the the while massless linear the limit vector multiplet, the multiplet and linear becomes mul a self-dual. s We now turn to the supersymmetric version of the massive dual 6 Massive supersymmetric Born-Infeld and its superspace du where where Φ( which is the supersymmetric extension of eq. ( Lagrangian ( This is the supersymmetric Stueckelberg representationing of use of the gauge invariance of scalar, which is the very reasonthis for the supersymmetric presence generalization of of the arbit eqs. ( also a scalar kinetic term. As we have stressed, the massive m JHEP04(2015)032 ) D re- and also V α (6.7) ( (A.1) p α W BI M L = 2 he massive ), D ) n the partial 6= 0 5.6 , B ( m 2  ) 1)-forms and their H D 2 V − k 12 ( p ˙ α S. Stefi). The authors − − M ty.  ve multiplet with respect D non-linear curvature terms + ould be identified with the + t the massive linear multi- 2 duality property, this leaves A.S. is on sabbatical leave, ˙ α rangian  es, in general, non-derivative r correction to the linear mul- with the field strength ) he same form in the other. In L hat can emerge for one finally gets A ( . Notice that for = 1 Supersymmetry the gravitino e , m BI F D )

) L N ˙ α D A V ( -forms and ( L ( p ˙ F α α  L M 4 α 4 acquires a Born-Infeld-like mass term, whose µ + L ). α α α − L – 11 – L 3.2 ) 6 A g 2 -term and, as we have seen in eq. ( ( 4 , m L 2 µ D m 2 F γδ , but also non-linear curvature interactions encoded in µ 2 g α 4 B µ , ) 1 g A mL (  1 + αβ BI r = 2 Goldstone multiplet, whose fermionic part is eaten by the F L and using the self-duality of − N α 1 ) + αβγδ  W ǫ L 2 ( 4 2 m g ψ µ 8 − = ). Systems of this type are expected to emerge as subsectors i = 1 super- of extended Supergravity, where t = -term. L L ( F L N ψ → This implies, in particular, that the first-order non-linea Integrating over = 2 even. In these cases, if the massless dynamics enjoys a self- Let us begin by considering the four-dimensional master Lag supported in part bywould like Scuola to Normale thank Superiore the CERN and Ph-Th by Unit INFN for (I. A the kind hospitali The dual massive Born-Infeld-like action gravitino that becomes massive. Acknowledgments We are grateful to P. Aschieri for stimulating discussions. leftover portion of the multiplet contains in fact two massive vectors. One of them w gravitino must be accompanied byto the the other unbroken Supersymmetry. members In of four-dimensional a massi N a function in one formulation andthe non-linear supersymmetric non-derivative terms case of theplet t above dual phenomenon to indicates ainteractions tha that Born-Infeld build massive up vector the multiplet very same possess Born-Infeld action an imprint, in the massive theories, as a duality map between placed by the tensor multiplet p tiplet mass term is proportional to where both contributions contain a dual descriptions, with emphasis on the peculiar features t contains an 7 Discussion In this article we have considered pairs of massive can be shifted away, so that the superfield bosonic counterpart can be found in eq. ( JHEP04(2015)032 µν ut B (A.2) (A.3) (A.4) (A.7) (A.6) (A.5) ommons . e B  e F 2 B 4 σ F 2 µ σ 4 2 µ 2 , and a gauge field g and the introduction + 2 ch emerges explicitly µν 1 + λσ 2 µ F F  sive Lagrangian for 2 B . to a quadratic expression. . , δ , 2 + σ C e 2 e λ F B m redited. 2 . One can then integrate out γ B 4 2 F ∂ F B 2 σ H . g 2 and 2 σ µ 4 µ αβ 4 g λ 2 µν 2 λ − F 4 −    B + 2 2 2 2 2 e 4 4 4 B 2 σ ]. The first multiplier eliminates the − σ σ σ 2 µ µ g µ µ 4 4 4 αβγδ 2 2 B B , 21 2 ǫ  µ ,   λ σ 4 1 e B − λ 2 2 4 ). 1 +  1 + 1 + 4 4 1 13 σ σ . The Green-Schwarz coupling proportional 2 µ µ −  √  B  4 4 4 µν , σ − 2 µ σ 3.2 e 7 4 B γδ µ − 2 2 µνρ − − 2 B  λ – 12 – H 1 1 + λ λ   1 + √ 1 2 αβ m g 2 2  √  F B 2 2 − m m 2 , as in [ − λ λ 2 4 4 g σ m = λ g g λ µ αβγδ 2 ǫ 16 √ g ), which permits any use, distribution and reproduction in − − µν 4 and  m F − = = = 2 ) ) into λ with the corresponding field strength − 2 g 2 e e B F B ) follows after integrating out µ 2 ( µ F A.1 2 16 2 g F F A 3.5 2 H − µ λ σ , the last term can be gauged away, and one can then integrate o CC-BY 4.0 ) k 4 12 B m ( This article is distributed under the terms of the Creative C − − 2 = H L k 12 − ). The first step involves the transition to a first-order for = 3.5 results in the introduction of a mass term for the vector, whi L with the corresponding field strength Here we would like to outline the steps leading to the dual mas , obtaining m , arriving finally at the Lagrangian of eq. ( µν µν integrating it by parts and turning to a first-order form for All in all, this turns eq. ( This result determines the three bilinears In the presence of H to B any medium, provided the original author(s) and source are c of two Lagrange multipliers square root, while the second reduces the term depending on of eq. ( F The final result of eq. ( involving a gauge field and the Lagrangian then takes the form Open Access. Attribution License ( JHEP04(2015)032 , , , l , , ry ] , in rigid ] Gauge (2014) 065 ]. = 1 global 12 = 10 N (2000) 034 Supersymmetry D (1986) 370 SPIRE = 2 ]. PoS(tmr2000)022 to 03 ]. , IN N JHEP = 2 ][ , = 2 hep-th/9811232 [ N , SPIRE B 180 N hep-th/9512180 ]. SPIRE JHEP [ IN , IN Generalized Born-Infeld [ ][ supersymmetry, constrained SPIRE ]. , = 4 ]. nyan, IN [ R. Roiban, D Phys. Lett. (1996) 275 (1999) 106001 , SPIRE (1984) 117 arXiv:1202.0014 SPIRE [ Nonlinear selfduality in even dimensions IN ]. ]. 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