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E. A. Ivanov, B. M. Zupnik. Auxiliary superfields in N = 2 Born-Infeld theory
UDC 530.1; 539.1
AUXILIARY SUPERFIELDS IN N =2 BORN-INFELD THEORY
E. A. Ivanov, B. M. Zupnik
Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Reg., Russia. E-mail: eivanov,zupnik@theor.jinr.ru
Some time ago the authors proposed a new approach to self-dual electrodynamics and its superextensions, based on introducing auxiliary (super)fields which ensure linearization of the duality transformations. In the talk this universal method is applied for analysis of the self-duality properties of the N =2 Born-Infeld (BI) action with the nonlinearly realized "hidden" N = 4 supersymmetry. Its self-duality is proved up to the 10th order in the superfield strengths. The conjecture about the general structure of the N = 2 BI action modified by the auxiliary chiral superfields is put forward.
Keywords: supersymmetry, duality, superGeld.
1 Motivations: why Born-Infeld?
The BI theory is the renowned nonlinear extension of Maxwell theory:
LBI(v, f) = 1 - V1 + V + V+(1/4)(v - V)2,
V + V = o(Fmn) , V - V
_ 771 pmn
2 F mn F •
It possesses many remarkable properties.
It is the first known example of nontrivial self-dual model of nonlinear electrodynamics.
The BI actions are necessary ingredients of the static-gauge D brane world volume actions.
The super D branes actions enjoy linearly realized worldvolume supersymmetries, the relevant worldvolume supermultiplets being the vector ones, with the gauge field among the component fields. This poses the problem of supersymmetrizing the BI action.
D
supersymmetry should manifest itself as the second nonlinear supersymmetry of such super BI actions.
To date, two superextended BI actions with the second nonlinearly realized supersymmetry are known in the superfield approach.
N = 1
It describes the "space-filling" D3 brane and possesses N = 1 N = 2
as "N = 2/N = 1" BI action.
N = 2
[3-6]. It describes D3 brane in D = 6 and exhibits a nonlinearly realized N = 4/N = 2 supersymmetry.
While the N = 2/N = 1 BI action is known in a closed superfield form, no such a formulation was given for the N = 4/N = 2 BI action so far. There exists N = 2
but it does not possess any extra supersymmetry.
It seems very interesting to further elaborate on the N = 4/N = 2 BI action and try to find a closed form for it. One more intriguing problem IS cUS follows.
N = 2/N = 1 be self-dual like its bosonic counterpart. The same N = 4/N = 2
N = 2
strengths. The question is whether this self-duality extends to all orders and how it is related to the hidden supersymmetry.
Recently, a new framework was developed for
N=
1, N = 2 extensions [8-10]. It proceeds from the formulation proposed in [11] and includes, as a new ingredient, auxiliary tensorial fields and their superfield
N = 4/N = 2
BI action in this general approach, with the hope that it will allow to give a general proof of self-duality of this action and find an ansatz for the latter beyond
N = 2
strengths. Some steps toward this goal were recently-made in [12]. Major part of the talk will be based on this work.
We will start with recalling the salient features of self-duality in electrodynamics and explain basics of the new approach with auxiliary tensorial fields. Then we will discuss how to put the N = 4/N = 2 BI theory into this framework and to which new understanding this gives rise.
2 Self-duality: general setting
U(N) duality invariance is an on-shell symmetry of a wide class of the nonlinear electrodynamics models including the renowned Born-Infeld theory. It generalizes the free-case O(2) symmetry between the equations of motion and Bianchi identities (Fmn =
dm An dn Am 7 Fmn i ^
E.O.M. : dmFmn = 0
SF„
uF fiF
2^-mnpq
—uFm
Fpq):
In the nonlinear case: Pmn = 2 Hmr,
SPm
uFm
SFm
— uPm
There should be valid the self-consistency condition (the so called GZ condition) [13,14]:
PP — FF = 0.
Recently, there came about a rebirth of interest in the duality-invariant theories [15-17]. The basic reason
together with Bianchi identity, are covariant under O(2) transformations
SVafi = -iuVafi, 6Fap = iw(Fap —2Vap), Sv = —2iwv. The algebraic equation of motion for Vap, 1 dE
Fa/3 = Vaj3 + 2 QVOP = Val3 (1 + Ev) '
is O(2) covariant if and, only if
vEv — vEp = 0 ^ E(v, v) = E(a), a := vv .
The meaning of this constraint is that E(v, V) O(2)
SuE = 2iu(vEp — vEv) = 0 .
This is none other than the GZ constraint:
is the hypothetical important role of the generalized f2 + p2 _ p2 _ p2 = o
duality symmetry in analyzing the UV properties of extended 4D supergravities.
We will use the bispinor formalism:
vEv- vET/ = 0 .
Fmn ^ (Faß, Faß), f = FaßFa
'aß 7 f :
F a ß f
a ß ■
L(f, f) = - 2(f + f) + Lint(f7 f) 7
The auxiliary equation can be now written as
Faß — Vaß = VaßV E' .
It serves to express Vaß, Vaß in terms of Faß, Faß.
Vaß (F) = Faß G(f,f), G(f,f) = 2 —Lv = (1+VEa)
-1
€ Pa ß (F) — dß Paß (F) = 0 7 et Faß ß — dß Faß = 0,
Pa
dL
aß lQp aß &u Faß = uPaß,
Su Paß
-uFr
aß7
FaßFaß+PaßPaß —c.c = 0 7 ^ f—4f(Lv)2 —C.C = 0 7
aß F sd
Lsd = 2 (PF — PF) + I(f f) 7 SuI(f,f)=0 .
After substituting these expressions back into L we obtain the corresponding selfdual Lsd((, (p)
Lsd(p, () = — 1 (( + ((1 — aE2«) + 8a2E33 + E(a), ((,() 2 1 + aE2 + ( ) *
where a is related to p, (V by the algebraic equation
(1 + aE2a)2 ((9 = a[(( + (p)Ea + (1 — aE2a )2]2 .
How to determine the invariant I((, ()? Our approach We find 1 () = E(a) — 2aEa, a = v((* ()v((*
O(2)
of nonlinear electrodynamics without derivatives on the
O(2)
E(a)
Let us introduce the auxiliary unconstrained fields real quartic combination of the auxiliary fields. This Va(S and Va0 and write the extended Lagrangian in the universality is the basic advantage of the approach with
tensorial auxiliary fields.
As an instructive example, we consider the Born-Infeld action in the new setting. The BI model has a more simple description in terms of the new variables:
with the auxiliary tensorial fields provides an answer.
3 Formulation with bispinor auxiliary fields
! int
Va ß
(F7 V^representation as C(V,F ) = C2(V,F) + E (v,v)7
L2(V7 F) = \(f + f)+ v + V — 2 (V ■ F + V ■ F)7
with v = V27 V = V2. Here, E(v7 V) is the nonlinear interaction involving only auxiliary fields. Dynamical equations of motion
€ Pa ß (V7F) — 3ßa Paß (V7 F )=0 7
Paß (F7 V) = i^LPOP = i(Faß — 2Vaß\
EBI(a) = IBI(b) — 2b IB1 (b) 7
I
BI
2b j Bi
7 h
2
4b
(b — 1)2 ' (1 — b)4 ' b
ff b2 + [2ff — (f + f + 2)2] b + ff = 0 4ff
=>
b
[2(1 + Q) + f + f]2
Bianchi : dmFmn = 0
E.O.M. : dmPmn = 0
Bianchi : dmFmn = 0
Q(v) = \/1 + V + f + (1/4)(v — 9c)2 • The density A0, together with other superfields An,
.. ... ... , can be expressed in terms of W, W by imposing an
After substituting this into the general formula for . c ., , r .r . . , ...
TcH, , , . _T T . . lnhmte set of N = 4 covanant constraints Lcd(v, <f) the standard B! Lagrangian is recovered
LBI ( f) I /1+ +-+(1/4)(-^ A0 -W 2 - 2A0D4A0 - D4 ^ ^ An°nAn =0 ,
L (V,^) = 1 -V1 + V + f+(1/4)(V - f)2 • 2 n=i 2
□Ai + ••• = 0,......□nAn + • • • = 0,
How to supersymmetrize the bispinor formulation?
The basic idea [9,10] is to embed tensorial auxiliary where dots stand for some nonlinear terms. Solving it fields into chiral auxiliary N =1, 2 superfields:
Vap (x) ^ Ua(x, 0, 0) = Va(x) + 03 Va/3 (x) + • • • ,
V«3(x) ^ U(x, 0\ () = v(x) + (030ak)Va3(x) + • • • ,
Dy Ua(x, 0, 0) = 0 , DY iU (x,0j ,fk )=0 •
The (W, URepresentation for the N =1 self-dual actions is defined as
S(W,U) = J d6Z ^UW - 1U2 - 1W^ +c.c.
+ 1Jd8zU2U2 E(u,u,g,9),
u = 1D2U2, u = 1D2U2, g = DaUa • 8 8 a
N = 1 U(1)
invariant interaction
Einv = F(B,A, C)+ F(B,A,C), A := uu, C := gg , B := ug2 , B := ug2 •
A0
A04) = 1 D4(W2W2),
A0
2
=4 D4
2
W2W2(D4W2 + ۥۥ) - -W^W3 9
In this way, the interaction IBI was found in [4] up to the 8th order. Recently [12], the next, 10th order term, was explicitly computed.
Inspecting the explicit structure of the perturbative terms in A0 and the respective terms in IBI, we found A0
Ao = X + R + Y • (1)
Here, X is defined by the equation [5-7]
X = W2 + 1X D4X (2)
and accounts for all terms without □.The part R accounts for all terms involving only box operator
R = 2D4 ¿(-1)r
1
(n!)
rWn^n-2>V n.
The remaining piece Y collects, in its perturbative
D4, D9 4, □
N=2
A0
these pieces? Is it possible to prove, at least up to
N = 4/N = 2 is simultaneously self-dual? The latter property was proved in [5,6] up to the 8th order. The reformulation N = 2
turns out to be helpful for getting the answers.
4 N = 2 BI theory: the (W, W) formulation
Sbi(W) = S2(W) + Ibi(W) = 1 j d8Z Ao + ac : Ibi(W) = y di2ZLbi(W),
where
Lbi = £ L(2n), Ao(W) = £ A02n) = W2 + 2D4Lbi•
n=2
n=i
W A0
an infinite-dimensional multiplet of the spontaneously N = 4
1 1 - i
Sf W = f (1 - 1D 4a40) + ^ /□A0 + ^D %/Dkad«„ A0,
Sf A0 = 2fW + -/□Ai + -Da/DkadaaAi:
Sf An = 2f An-i + 77 f □An+i + 7 Da/DkadaaAn+i:
N = 2
The (W, U) representation of the general action of N = 2
S(W,U) = S6(W, U)+1(U), I(U) = J d8ZL(U) +C.C.,
/1 1
d8Z (UW - ^U2 - 1W2) + c.c. •
The interaction L(U) is a local functional of U, U. The U
S
S
f := c + 2*0^ , n > 1 • U = W + — , — := J(U, U) = D4J(U, U), (3)
S
allows one to eliminate auxiliary superfields and recover the nonlinear W action. The N =2 self-duality condition [5] becomes
/ ^U % = / A. Up g.
This is just the condition of the invariance of the functional X(U) under the U (1) transformations SuU = —iuiU , SuU = iwU.
Thus any self-dual system of N = 2 electrodynamics can be reformulated as a system with the off-shell action S(W,U) = Sb(W,U) + X(U), so that the interaction X(U) is U(1) duality invariant. N = 2
reformulation, it is self-dual.
So, one way to prove that the N = 4/N = 2 BI
(U, W)
to show that XBI(U) is U(1) invariant.
The goal is to write the N = 4/N = 2 BI action as
SBI (W, U) = Sb(W, U) + Xbi (U),
where, in accordance with the triple decomposition (1) of the (W, W) BI action,
Xbi(U) = Xx(U) + Xr(U) + Xy (Y).
The first term generates the action associated with the chiral superfield X. It is self-dual [6]. The term
Ir(U) = J d8ZCn(U) + c.c. ,
= - D R 2
4 E
n=3
(-1)
(n!)
-unon-2u
is the only structure capable to reproduce the highestderivative ^contributions in the W representation. At last, Xy (U) stands for possible further corrections. The term (U) should reproduce the action
Sx(W) = 4 J d8ZX(W) + \ j d8ZX(W).
After some work with introducing auxiliary superfields, this action can be equivalently represented as
Sx (W, U,N)= Sb(W, U) + Xx(U2,N),
Xx(U2,N) = 1 i dl2Z[u2N + U2N - NN V 4 J I 1 — 4 nn J
= D 4N.
U(1)
SuN = —2iw N. Thus indeed SX(W) corresponds to N = 2
UN
SX(W,U,N). On the other hand, eliminating N,
N — (1 — 1 nn)U2 + 1(1 — 1 nn)D4
NN n
(1 — 4 nn)2
0,
U
Xx(U2)= Xx(U2,N(U2)) = ^J d12ZC^n)(U, U).
A few lowest recursive terms are
44) = -U2U2, = —1U2U2 A
4
x
16
LX2) = U2U2 (BE + B2 + B2), A := (D4U2)(D4U2), B := D4D4(UV2U2).
They are capable to restore the original action SX(W) up to the 18th order upon eliminating U, U1. U
same superfields A, B, B makes it probable that the whole XX (U) can be written as a sum of the well defined terms related by some general recurrence formula.
( U , W ) N = 2
10th order
Our aim is to present the auxiliary interaction XBI(U) which reproduces the (W, W) form of the BI action up to the 10th order, i.e. the sum of four terms
T = I (4) + I(6) + I(8) + I (10) ibi = 1 BI + 1 bi + 1 bi +1 bi .
(U, W)
Sbi = Sb + Xx + Xr + Xy ,
1 — -(D4U 2)(D 4U2)
Xx(U) = 1J d12ZU2U2
Xr(u) = 1 i d12Z ( — 2U3uU3 + -U4u2U4 R ' 8 V 9 72
1
-U5U3U5
1800
Xy(U) = ^J d12Z \ (U3U2D4U2UD4U3 +U 2U3D 4U2 aD4U3) + \u3D4U2u(;UA3D4U2)
Xy
U
manifestly U(1) invari^t, so the N = 4/N = 2 BI action is self-dual up to the 10th order. The number of terms in this interaction is much smaller compared W
U, W
all possible terms and give a general proof of the self-duality of the N = 4/N = 2 BI action.
'in [18] it was restored up to the 14th order by directly solving eq. (2).
1
7 Summary and outlook
All the duality invariant systems of nonlinear electrodynamics and its N = 1,2 superextensions admit an off-shell formulation with the auxiliary bispinor fields or their superfield counterparts. This formulation reduces the self-duality constraints to U(1)
interaction.
This universal method was applied for analysis of
N = 2 N = 4
Its self-duality was proved up to the 10th order in the
(U, W)
representation of this action to the same order. The
(U, W)
form of the BI action was put forward.
The new closed auxiliary-superfield representation was found for the action SX (W) as a necessary-constituent of the full N = 4/N = 2 BI action.
Some further lines of study:
(a) It would be interesting to inquire whether the
(U, W)
promoted to all orders in the auxiliary superfields U, U (U, W)
N = 4/N = 2 BI action.
(b) The closely related problem is to understand
N = 4
(U, W)
(c) It is tempting to find some general framework for the "irregular" Y terms of the N = 2 BI action. Perhaps, they could be understood as a recursive solution of some nonlinear superfield equation.
Acknowledgement
E.I. thanks the Organizers of the Conference QFTG 2014 for the kind hospitality in Tomsk. This research has been partially supported by the RFBR grants No. 12-02-00517 and No. 13-02-91330.
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Received 08.11.2014
E. А. Иванов, В. M. Зупник ВСПОМОГАТЕЛЬНЫЕ СУПЕРПОЛЯ В N =2 ТЕОРИИ БОРНА-ИНФЕЛЬДА
Некоторое время назад в работах авторов был развит новый подход к описанию само-дуальной электродинамики и ее суперрасширений на основе введения вспомогательных (супер)полей, обеспечивающих линейность преобразований дуальности. В данном докладе этот универсальный метод применяется для анализа свойств само-дуальности суперполевого действия N = 2 теории Борна-Инфельда (Б.-И.) с нелинейно реализованной "скрытой" N = 4 суперсимметрией. Самодуальность этого действия доказана до 10-го порядка по суперполевым напряженностям. Вы-
N = 2
суперполями.
Ключевые слова: суперсимметрия, дуальность, суперполе. Иванов Е. А., доктор физико-математических наук, профессор.
Объединенный институт ядерных исследований, Лаборатория теоретической физики им. Н. Н. Боголюбова.
Ул. Жолио-Кюри, 08, 141980 Дубна, Московская область, Россия. E-mail: eivanov@theor.jinr.ru
Зупник Б. М., доктор физико-математических наук, профессор.
Объединенный институт ядерных исследований, Лаборатория теоретической физики им. Н. Н. Боголюбова.
Ул. Жолио-Кюри, 08, 141980 Дубна, Московская область, Россия. E-mail: zupnik@theor.jinr.ru
