N=1 Supersymmetric Born-Infeld theory in the component...

N=1 Supersymmetric Born-Infeld theory in thecomponent approach

Sergey Krivonos

Joint Institute for Nuclear Research, Dubna, Russia

Selected Topics in Theoretical High Energy Physics, September 21 - 24, Tbilisi

in collaboration with S.Bellucci, N.Kozyrev and A.Sutulin

S. Krivonos (JINR, Dubna) Born-Infeld Tbilisi, Sept. 21 - 24, 2015 1 / 25

Plan

Introduction

Coset approach: Bosonic membrane in D = 4

Adding supersymmetry

Comments on D-branes

D3-brane: Sketch of Bagger & Galperin results

Curing the constraints

D3-brane: component action

Conclusion


Introduction

Coset approach

It is a well known fact that a domain wall spontaneously breaks the Poincaré

invariance of the target space down to the symmetry group of the world volume

subspace. This breaking results in the appearing of the Goldstone bosons associated

with spontaneously broken symmetries. When we are dealing with the purely bosonic

p-branes this information is enough to construct the corresponding action. From the

mathematical point of view, the most appropriate approach to describe a partial

breaking of Poincaré symmetry is the nonlinear realization (or coset) method.

Schematically the coset approach works as follows.



Bosonic membrane in D = 4

As a simplest example let us consider the nonlinear realization of D = 4 Poincarégroup in its coset over d = 3 Lorentz group SO(1,2), which should correspond tomembrane in D = 4 space-time.In d = 3 notation the D = 4 Poincaré algebra contains the following set of generators:

Pab,Z ,Mab,Kab ,

a, b = 1, 2 being the d = 3 SL(2,R) spinor indices. Here, Pab and Z are D = 4translation generators, the generators Mab form d = 3 Lorentz algebra so(1,2) andthe generators Kab belong to the coset SO(1, 3)/SO(1,2).Therefore, we will define the coset element as follows

g = exabPab eq(x)ZeΛab(x)Kab .

Here,

xab are space-time coordinates, while the remaining coset parameters are

Goldstone fields.



The transformation properties of the coordinates and fields with respect to allsymmetries can be found by acting from the left on the coset element g by thedifferent elements of D = 4 Poincaré group. In what follows, we will need the explicitform only for Kab automorphism transformations which read:

Automorphism Kab transformations g0 = er abKab

δxab = −2qrab, δq = −4rabxab, δλab = rab − 4λacrcdλdb.

The (important) Cartan forms

g−1dg = ΩP + ΩZ + ΩK + ΩM

have the following explicit form(λab =

tanh(√

2Λ2)√

2Λ2Λab

)

ΩZ =1 + 2λ2

1 − 2λ2

[dq +

41 + 2λ2

λabdxab]

Z ,

ΩP ≡ ΩabP Pab =

[dxab +

21 − 2λ2

λab(

dq + 2λcddxcd)]

Pab



In accordance with the general theorem (Inverse Higgs phenomenon) formulated byE.Ivanov and V.I.Ogievetsky, in order to reduce the number of independent superfieldsone has to impose the constraints (δΩZ = 0)

ΩZ = 0 ⇒ ∂abq = −4λab

1 + 2λ2.

Thus, we expressed the fields λab through space-time derivatives of q(x). Therefore,

the bosonic field q(x) is the only essential Goldstone field needed for this case of the

partial breaking of the global symmetry. The above constraints are covariant under all

symmetries, they do not imply any dynamics.



It is very important that the form ΩP defines the vielbein Babcd , connecting the

covariant world volume coordinate differentials ΩP and the world volume coordinatedifferential dx as

(ΩP) = dxabBabcdPcd

with

Bcdab = δ

(ca δ

d)b − 4

λabλcd

1 + 2λ2.

Therefore, the simplest invariant bosonic action reads

Sbos =

∫d3x det B =

∫d3x

1 − 2λ2

1 + 2λ2,

or in terms of ∂abq

Sbos =

∫d3x

√1 −

12∂abq ∂abq.

This is the static gauge Nambu-Goto action for the membrane in D=4.




The supersymmetric generalization of the coset approach involves into the game newspinor generators Q and S which extend the D-dimensional Poicaré group to thesupersymmetric one

Q,Q ∼ P, S,S ∼ P, Q,S ∼ Z .

The most interesting cases are those when the Q supersymmetry is kept unbroken,while the S supersymmetry is supposed to be spontaneously broken. When#Q = #S we are facing the so called 1/2 Partial Breaking of Global Supersymmetrycases, which most of all interesting supersymmetric domain walls belong to. Onlysuch cases of supersymmetry breaking will be considered in this talk.Now, all our symmetries can be realized by group elements acting on the cosetelement

g = exPeθQeq(x,θ)Zeψ(x,θ)SeΛ(x,θ)K .

The main novel feature of the supersymmetric coset is the appearance of the

Goldstone superfields q(x , θ), ψ(x , θ),Λ(x , θ).



The Cartan forms can be constructed similarly to the bosonic case

g−1dg = ΩP + ΩQ + ΩZ + ΩS + ΩK + ΩM .

The most important form ΩZ reads

ΩZ =1 + 2λ2

1 − 2λ2

[dq + ψadθa +

41 + 2λ2

λabdxab]

Z ,

where the covariant differential dxab has the form

dxab = dxab +12θ(adθb) +

12ψ(adψb).

Defining the semi-covariant derivatives ∇ab and ∇a

dxab ∂

∂xab+ dθa ∂

∂θa= dxab∇ab + dθa∇a

and imposing the same constraint ΩZ = 0 we will get

∇abq +4

1 + 2λ2λab= 0, ∇aq − ψa = 0.

Thus, the bosonic superfield q is the only essential Goldstone superfield we need for

this case.S. Krivonos (JINR, Dubna) Born-Infeld Tbilisi, Sept. 21 - 24, 2015 9 / 25


The last step we can make within the coset approach is to write the covariantsuperfield equations of motion. This can be achieved by imposing the followingconstraint on the Cartan form:

ΩS | = 0 ⇒ (a) ∇aψa = 0 , (b) ∇(aψb) = −2λab .

where | denotes the ordinary dθ- projection of the form ΩS .These equations imply the proper dynamical equation of motion

∇a∇aq = 0.

This equation is also covariant with respect to all symmetries, and its bosonic limit (forq(x) ≡ q(x , θ)|θ=0) reads

∂ab

∂abq√

1 − 12∂q · ∂q

= 0 ,

which corresponds to the “static gauge” form of the D = 4 membrane Nambu-Gotoaction

S =

∫d3x

(1 −

√1 −

12∂abq∂abq

).

Thus, our equations indeed describe the supermembrane in D = 4.



Unfortunately, this similarity between purely bosonic and supersymmetric cases is notcomplete due to the fact that the standard methods of nonlinear realizations fail toconstruct the superfield action! The main reason for this is simple: all that we have athands are the covariant Cartan forms, which we can construct the superfieldinvariants from, while the superspace Lagrangian is not invariant. Instead it is shiftedby the full spinor derivatives under unbroken and/or broken supersymmetries.Funny enough, if we instead will be interested in the component action, then it can beconstructed almost immediately within the nonlinear realization approach due to thefollowing important features:

all physical components, i.e. q|θ=0 and ψa|θ=0, are among the “coordinates” ofour coset as the θ = 0 parts of the corresponding superfields,

under spontaneously broken supersymmetry the superspace coordinates θa donot transform at all. Therefore, the corresponding transformation properties ofthe fermionic components ψa|θ=0 are the same as in the Volkov-Akulov model,where all supersymmetries are supposed to be spontaneously broken,

Finally, the θ = 0 component of our essential Goldstone superfield q(x , θ) doesnot transform under spontaneously broken supersymmetry and, therefore, itbehaves like a “matter” field within the Volkov-Akulov scheme.



As the immediate consequences of these features we conclude that

The fermionic components ψa|θ=0 may enter the component action eitherthrough det(E) (to compensate the transformation of volume d3x undersupersymmetry) or through the covariant derivatives ∇ab only,

The “matter” field – q|θ=0 may enter the action only through covariant derivatives∇abq.

Thus, the unique candidate to be the component on-shell action, invariant with respectto spontaneously broken supersymmetry (Sa) reads

S = α

∫d3x + β

∫d3x (det E)F [∇abq∇abq],

with an arbitrary, for the time being, function F and

Ecdab =

12(δc

aδdb + δd

a δcb) +

14(ψc∂abψ

d + ψd∂abψc).

All other interactions between the bosonic component q and the fermions ofspontaneously broken supersymmetry ψa are forbidden! Moreover, knowledge of thebosonic action further restrict our action to be

S = (α+ 1)∫

d3x − α

∫d3x (det E)−

∫d3x (det E)

√1 −

12∇abq∇abq.

The role of the unbroken supersymmetry is just to fix the parameter α = 1.




The above construction works for any p-brane embedded in any target-space. Whenwe are dealing with the gauge fields, i.e. with the D-branes, the situation is less clear,despite the knowledge of the explicit actions. The main question, which is stillunsolved, is that we do not know which symmetry is spontaneously broken in thesecases. Moreover, the field-strengths can not be treated as the Goldstone fields. So,we have the following situation

P-branes D-branes

All fields are the Goldstone oneThe field strengths havenon-Goldstone nature

The bosonic actionmay be easily constructed

The explicit bosonic actioncan not be constructed within nonlinearrealization approach

The irreducibility constraintsmay be easily covariantized

Bianchi identities and the equations ofmotion may be covariantized only together

The equations of motionmay be easily covariantized



Space-filling D3-brane

The space-filling D3 brane, i.e. N = 1 supersymmetric Born-Infeld theory describesthe system with the N = 2 → N = 1 partial spontaneous breaking of d = 4supersymmetry with the vector supermultiplet as the Goldstone one.Starting from N = 2 supersymmetry algebra

Qα,Qα

= 2Pαα,

Sα,Sα

= 2Pαα,

where Qα and Sα are the supersymmetry generators and Pαα is the generator of fourdimensional translations, one may realize this algebra in the coset parameterized as

g = eixααPααei(θαQα+θ

αQα)ei(ψαSα+ψ

αSα).

Thus, the coset element does not contain the bosonic fields.



Using the Cartan forms one may define the covariant derivatives

∇αα =(

E−1)ββαα

∂ββ, Eββαα = δβαδ

βα − iψβ∂ααψ

β − iψβ∂ααψβ ,

∇α = Dα − i(ψβ∇αψ

β +ψβ∇αψβ)∂ββ,

∇α = Dα − i(ψβ∇αψ

β +ψβ∇αψβ)∂ββ,

where the flat covariant derivatives are defined as follows

Dα =∂

∂θα− iθα∂αα, Dα = −

∂

∂θα+ iθα∂αα,

Dα,Dα

= 2i∂αα.

These covariant derivatives obey the following relations

∇α,∇β = −2i(∇αψ

γ∇βψγ +∇βψ

γ∇αψγ)∇γγ ,

∇α,∇α

= 2i∇αα − 2i

(∇αψ

γ∇αψγ +∇αψ


[∇α,∇ββ

]= 2i

(∇αψ

γ∇ββψγ +∇ββψ


[∇αα,∇ββ

]= 2i

(∇ααψ

γ∇ββψγ +∇ααψ

γ∇ββψγ)∇γγ .



Now one has to find the covariant version of the flat constraints selecting theN = 1, d = 4 vector multiplet Wα,W α

DαWα = 0, DαW α = 0 (a), DαWα + DαW α = 0 (b).

It was shown by Bagger & Galperin that the chirality constraints can be directlycovariantized as

∇αψα = 0, ∇αψα = 0.

The reason why these constraints are correct is that they are just the d θα and dθα

parts of the Cartan forms ωSα and ωSα, respectively:

ωSα = dθβ∇βψ

α + d θβ∇βψα + dxbc∇bcψ

α

ωSα = dθβ∇βψα + d θβ∇βψα + dxbc∇bcψα

Remembering, that the Cartan forms are invariant with respect to all transformationsof N = 2, d = 4 super Poincaré group, we conclude that these constraints are alsocovariant. Note, that with our constraints taken into account one may obtain

∇α,∇β =∇α,∇β

= 0,

and thus the subsequent action of the corresponding covariant derivatives on the

constraints will not produce the new ones.



The situation with the Bianchi identities is more complicated. Indeed, after imposingthe covariant chirality constraints the dθα and d θα parts of the Cartan forms ωS

α andωSα contain only the quantities

Vαβ = ∇αψβ and Vαβ = −∇αψβ.

Of course, one may impose the following covariant constraints

∇αψα = 0 and ∇α

ψα = 0.

But the problem is that these constraints put our theory on-shell. Unfortunately, onecan not impose the constraint

∇αψα +∇α

ψα = 0,

because the first and second summands belong to the different Cartan forms and,

therefore, being covariant can differ on some again covariant multipliers. As the result,

acting on this equation by ∇2 or ∇2 we will get the equations of motion instead of

identities. One may suppose that for the construction of the on-shell component action

of D3 brane this is not the problem and one may use the above equations, which

contain the Bianchi identities and the equations of motion. However, in such situation

we will be unable to find a proper Bianchi identities on the bosonic components of

V(αβ),V (αβ) and therefore it will be impossible to find the invariant action.



The idea how to find the covariant Bianchi identities is rather simple. Let us suppose that weintroduced into game some additional bosonic superfield φ which enter the new Cartan forms as

ΩαS ∼ dψα − iφdθα + . . . , ΩS α ∼ dψα + iφd θα + . . . ,

where . . . denotes the all possible terms of the higher orders in φ. Note, that such modificationsis possible if the new generator U commutes with the supersymmetry generators Q and S asfollows

[U,Qα] = Sα, [U,Sα] = Qα,[

U,Qα]

= −Sα,[

U,Sα]

= −Qα.

Therefore, the forms ωQ and ωQ will also changed that results in the new covariant derivatives∇ → D. Now, the covariant chirality constraints will be replaced by new ones

Dαψα = 0, Dαψα = 0,

while the second constraints (equations of motion and Bianchi identities) will be modified as

Dαψα = iφ, D

αψα = −iφ.

Therefore, the covariant Bianchi identities will acquire the form

Dαψα + D

αψα = 0,

while the equationDαψ

α− D

αψα = 2iφ

will relate the first component of the superfield φ with the auxiliary field of the N = 1, d = 4

off-shell vector supermultiplet. In other wordsnow we are in the off-shell situation and the newly

defined Bianchi identities are fully covariant.



To realize the idea sketched above, we extend the coset element g as

g → g = eixααPααei(θαQα+θ

αQα)ei(ψαSα+ψ

αSα)eiφU ,

where U is the generator defined above, while φ is a new corresponding Goldstonesuperfield. Such modification of the coset element gives rize to the modification of theCartan forms. Firstly, the new Cartan forms corresponding to the generators ofunbroken supersymmetry ΩQ ,ΩQ read

ΩαQ = cosφ dθα − i sinφ dψα, ΩQ α = cosφ d θα + i sinφ dψα.

The modifications of these Cartan forms forced us to introduce the new covariant (withrespect to N = 2, d = 1 Poincaré group and U transformations) derivatives Dα,Dαimplicitly defined as

∇α = cosφ Dα − i sinφ ∇αψβDβ + i sinφ ∇αψβD

β ,

∇α = cosφ Dα − i sinφ ∇α

ψβDβ + i sinφ ∇α

ψβDβ .

Secondly, the forms ΩS ,ΩS corresponding to the generators of broken supersymmetryare also modified as

ΩαS =1

cosφ

[ΩβQ(Dβψ

α − iδαβ sinφ)+ ΩQ βD

βψα + ωββP Dββψ

α],

ΩS α =1

cosφ

[ΩQ β

(Dβψα + iδβα sinφ

)+ ΩQ

βDβψα + ωββP Dββψα

].



Now, one may impose the constraints on these forms. Nullifying the ΩQ projection ofthe form ΩS and the ΩQ projection of the form ΩS we will get the modified chiralityconstraints

Dαψα = 0, Dαψα = 0,

while nullifying the traces of the ΩQ projection of the form ΩS and the ΩQ projection ofthe form ΩS will results in the constraints

Dαψα = 2i sinφ

Dαψα = −2i sinφ

⇒ Dαψα + D

αψα = 0

Thus, our the constraints have the form we supposed above.Using the explicit form of the covariant derivatives, these constraints can be rewrittenas

∇αψα = 0, ∇αψα = 0 and∇αψ

α

1 + 12∇

βψγ ∇βψγ+

∇αψα

1 + 12∇

βψγ∇βψγ

= 0

The second expression provides the suitable covariant version of the Bianchi

identities.



Bosonic Bianchi identityIt is follows from the previous analysis, that going on-shell means nullifying thesuperfield φ:

φ = 0 ⇒ ∇αψα = ∇α

ψα = 0.

Thus, the on-shell vector supermultiplet contains the following physical components

ψα = ψα|, ψα = ψα|, Vαβ = ∇(αψβ)|, V αβ = −∇(αψβ)|.

These definitions have to be supplied by the Bianchi identity needed to treat the fieldsVαβ,V αβ as the components of the field strength. Before the construction of theBianchi identities in a full generality, it is instructive to derive their bosonic limit whichreads

∂βαVβα + V γ

β V γα ∂γγVβ

α

1 + 12 V 2

−∂αβV β

α + V γα V γ

β∂γγV β

α

1 + 12 V 2

= 0.

It has been already clarified in Phys.Lett. B482 (2000) 233, that these conditions canbe transformed in the conventional form of the Bianchi identities:

∂βα

[Vβα

1 + 12 V 2

1 − 14 V 2V 2

]− ∂αβ

[V βα

1 + 12 V 2

1 − 14 V 2V 2

]= 0.

Thus, the bosonic fields Vαβ and V αβ are indeed the components of the field strength,

as it should be.S. Krivonos (JINR, Dubna) Born-Infeld Tbilisi, Sept. 21 - 24, 2015 21 / 25


Complete Bianchi identityThe derivation of the complete Bianchi identities is more involved. After quite lengthlycalculations one may find that passing to the vector notation

F AB =12ǫABCDFCD, W AB =

12ǫABCDWCD,

where

F AB =12

[(σAB)αβ

Fαβ +(σAB)αβ

F αβ

]

and

W AB =12

[(σAB)αβ 1 + 1

2 V 2

1 − 14 V 2V 2

Vαβ +(σAB)αβ 1 + 1

2 V 2

1 − 14 V 2V 2

V αβ

],

one may check that the proper field strength acquires rge form

F AB = detE(E−1

)A

C

(E−1

)B

DW CD,

such that∂AF AB = 0.

Here,Eββαα = Eββ

αα|θ=0 = δβαδβα − iψβ∂ααψ

β − iψβ∂ααψβ.



Within our approach, the construction of the invariant component on-shell action canbe performed in two steps

Firstly, one has to construct the most general action invariant with respect tobroken supersymmetry

Secondly, the action has to be fixed by imposing its invariance with respect tounbroken supersymmetry.

Knowing the transformation properties of the coordinates and superfields underbroken supersymmetry

δSxαα = i(εαψ

α+ εαψα

), δSθα = δS θα = 0, δSψα = εα, δSψα = εα,

one may immediately conclude that the most general component action which isinvariant under these transformations reads

S =

∫d4x detE G(V 2,V 2).

Here, the function G is an arbitrary function which explicit form has to be fixed by the

invariance of this action with respect to unbroken supersymmetry.



To fix the function G in the action one has to know the transformation properties of thecomponents under unbroken supersymmetry, which defined in a standard way as

δQ f = −(ǫαDα + ǫαDα

)f|θ→0 = −

(ǫα∇α + ǫα∇

α)

f|θ→0 + Hλλ∂λλf ,

Hλλ = −i(ǫγ ψλVλ

γ − ǫγψλV γλ

).

Calculating the variation of the action one may find it is equal to zero (modulo Bianchiidentities) if

S =

∫d4x det E

1 +

(1 + 1

2 V 2) (

1 + 12 V 2

)

1 − 14 V 2V 2

.

This action is invariant with respect to broken and unbroken supersymmetries.Note, that being written in the terms of W AB defined previously, this action acquiresthe familiar form of the Born-Infeld action

S =

∫d4x det E

[1 +

√1 +

12

WABW AB −1

16

(WABW AB

)2],

and thus it gives the component action of the famous N = 1 supersymmetric

Born-Infeld theory


Conclusion

Conclusion

Thus, following our approach

We derive the component on-shell action of the space-filling D3-brane, akaN = 1 supersymmetric Born-Infeld action

We introduce an additional Goldstone superfield associated with the generator ofthe automorphism transformation of the fermionic part of N = 2,d = 4 superPoincaré algebra. The first component of this superfield is the auxiliary field ofthe vector supermultiplet and thus, being in off-shell situation, one may find theoff-shell, covariant Bianchi identities.

The D3-brane on-shell component action has very simple form being written interms of derivatives covariant with respect to spontaneously brokensupersymmetry. Similarly to the cases of p-branes, it mimics the bosonicBorn-Infeld action.

The present results and the idea to introduce some additional Goldstone superfields

to go off-shell are the good start point to re-consider N = 2, d = 4 Born-Infeld theory,

with hidden spontaneously broken N = 2 supersymmetry within the nonlinear

realization approach.


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