CC BY
24
UDC 530.1; 539.1
A note about fermionic equations of AdS4 x CP3 superstring
D. V. Uvarov
NSC Kharkov Institute of Physics and Technology, Kharkov, 61108, Ukraine.
E-mail: d_ [email protected]
It is proved that when 8 fermions associated with the supersymmetries broken by the AdS4 x CP3 superbackground are gauged away by using the k—symmetry corresponding equations obtained by variation of the AdS4 x CP3 superstring action are contained in the set of fermionic equations of the OSp(4|6)/(SO(1, 3) x U(3)) sigma-model.
Keywords: AdS4 x CP3 superstring, OSp(4|6)/(SO(1, 3) x
motion.
1 Introduction
The motivation for studying IIA superstring theory on the AdS4 x CP3 superbackground comes from the AdS4/CFT3 correspondence in the formulation by Aharony, Bergman, Jafferis and Maldacena fl] conjecturing that in the limit k5 ^ ^ 1 it describes grav-
ity dual of the D = 3 N =6 superconformal Chern-Simons-matter theory with U(N)k x U(N)_k gauge symmetry. In contrast to the maximally supersymmet-ric AdS5 x S5 superbackground on which propagates the IIB superstring involved into the AdS5/CFT4 correspondence [2] the AdS4 x CP3 superbackground preserves only 24 of 32 space-time supersymmetries that complicates the structure of the superstring action [3] and verification of the duality conjecture (for recent review see, e.g. [4]).
However, within the AdS4 x CP3 superspace there exists a (10|24) —dimensional subsuperspace isomorphic to the OSp(4|6)/(SO(1,3) x U(3)) supercoset manifold on which the 2d sigma-model can be constructed [5], [6] using the supercoset approach elaborated to obtain the AdS5 x S5 superstring action
[7]- [10] 1. One of important properties of such a OSp(4|6)/(SO(1,3) x U(3)) sigma-model is that its equations of motion are manifestly classically integrate [5], [6] similarly to the AdS5 x S5 superstring case [12]. The OSp(4|6)/(SO(1,3) x U(3)) sigmamodel action is known to arise from the complete AdS4 x CP3 superstring action [3] when 8 fermionic coordinates associated with the broken supersymmetries are gauged away. As usual when imposing the gauge conditions on the level of the action non-trivial equations of motions for the gauged out variables should not be lost. It was proved in [13] that to leading order in the fermionic coordinates parametrizing the OSp(4|6)/(SO(1,3) x U(3)) supermanifold
U(3)) sigma-model, osp(4|6) superalgebra, equations of
corresponding equations of the AdS4 x CP3 superstring constitute a subset of fermionic equations of the OSp(4|6)/(SO(1,3) x U(3)) sigma-model. Here using that the OSp(4|6)/(SO(1,3) x U(3)) sigma-model equations are formulated in terms of the osp(4|6) Car-tan forms we give a parametrization independent proof i.e. to all orders in the fermionic coordinates.
Organization of this note is the following. After recalling the Z4—grading of the osp(4|6) superalgebra that is relevant to constructing geometric constituents of the OSp(4|6)/(SO(1,3) x U(3)) superspace we prove that equations of motion corresponding to variation of the action of massless AdS4 x CP3 superparticle on 8 fermionic superspace coordinates associated with the supersymmetries broken by the superbackground in the limit when those coordinates are put to zero are contained in the set of fermionic equations for the massless superparticle on the OSp(4|6)/(SO(1,3) x U(3)) supermanifold. The proof is then generalized to the case of AdS4 x CP3 superstring.
2 osp(4|6) superalgebra and OSp(4|6)/(SO(1, 3)x U(3))
At the heart of group-theoretic approach to description of geometry of symmetric coset spaces G/H lies identification of the g/h coset Cartan forms with the vielbein components and of the stability algebra h Cartan forms with the connection 1-form of the manifold G/H. For the supercoset manifolds like OSp(4|6)/(SO(1, 3) x U(3)) one on which the classically integrable sigma-models can be defined [12],
[14]- [17] important role is played by Z4 automorphism of the isometry superalgebra g that generalizes Z2 automorphisms of the isometry algebras of corresponding bosonic coset manifolds. So that the (anti)commutation relations of g can be cast into the
1Another approach to the description of OSp(4|6)/(SO(1, 3) x U(3)) sigma-model based on introduction of the pure spinor variables was considered in [11].
Z4 —graded form
\g(j), g(k)} g(j+k)mod4
(1)
with all the generators divided into 4 eigenspaces according to their Z4 eigenvalues
T(fl(k)} = ik0(k), k = 0,1, 2, 3.
[Mo'm' ,Mo>ri>} = Mm/ni,
\Mmlnl ,M0lkl} = Vnl kl M0lml Vml kl M0lnl
[Mklll, Mmlnl} Vklnl Mllml Vklml Mllnl
and
\Ta,Tb} = \Ta,Vbc} = [Vab, Vcd} =
Vllnl Mkl ml + nllml Mkl.
i(Vab + sba Vcc},
—i5caTb, \Ta,Vbc}= i5^Tc
i(ôbVad — SdVcb}.
The so(2,3} generators M0lm^d Mmlnl (m',n' = 0,..., 3) are related to the D = 3 conformai group generators as
M0
0lm
2 (Pm + ^-m^ M0l3 = —D,
M3m = 2 (Km — Pm).
By analogy with the osp(4l6) generators they can be arranged in the form
C(d} = C(0) + C(2) + C(d + C(з),
(8)
(2)
where each summand takes value in respective eigenspace under the Z4 grading
For the osp(4l6} superalgebra relevant Z2—graded form of so(2,3} ~ sp(4} and su(4} ~ so(6} isome-try algebras of AdS4 = SO(2,3}/SO(l, 3} and CP3 = SU(4}/U(3} manifolds is given by the relations
C,
(0)
2G3mM3m + GmnMmn + &abVba,
C
Cm =
(2) = 2G0mM0lm
(1) = W(l)aQ(l)M + ^(l)MaQ (1)aa,
+ AD + QaTa +naTa,
(9)
c,
(3) = W(3)^Q(3)^ + W(3)
Iaa
Q (3)aa.
(3)
2 (um (d}+cm (d}},
So that the Cartan forms G0 m(d) — 1'
A(d^d (d), Qa(d) associated with the g(2) genera-
tors are identified with the OSp(4l6)/(SO(1, 3) x U(3)) supervielbein bosonic components and fermionic Car-tan forms
(4)
^(1)^(d} = 2 (^a(d} + lXl¿(d}}, U(3)%(d} = 2 (L^aa(d} — lXa(d}}
(10)
(5)
and c.c. are identified with the OSp(4l6)/(SO(1, 3) x U(3)) supervielbein fermionic components. Accordingly Cartan forms for the g(0) generators G3m(d) = -2(um(d) - cm(d)), Gmn (d) md Qab(d) are identified with the connection 1-form on the OSp(4\6)/(SO(1, 3) x U(3)) supermanifold.
Thus the 0(o) eigenspace is spanned by the generators Mmint 'And Vab, while 0(2) by the generators M0/m> and Ta, Ta. Using the isomorphism bet ween the osp(4l6) superalgebra and D = 3 N = 6 superconformal algebra that is the symmetry algebra of the ABJM gauge theory [18] one can span g(1) mid g(3) eigenspaces by
Q(i)a = Q°a + lSa, Q (i)aa = Q aa lSaa;
(l)a a a
Q(3)aa = Qaa — S
) aa aa v^aat
Q (3) aa Q aa + ISaa,
where Qa^, QMa and Sa, Spa are the generators of D N = 6 super-Poincare and superconformal symmetries carrying SL(2, R) spinor index ^ = 1, ^d SU(3) (anti)fundamental representation index a = 1, 2, 3 in
SO(6)
the SU(3) representations 6 = 3 © 3.
Left-invariant osp(4l6) Cartan forms in conformal basis are defined by the relation [19]
C(d} = G-1 dG = Lvm(d}Pm + cm(d}Km + A(d}D + Qa(d}Ta +na(d}Ta + "a(d}Qaa + &aa(d)Q aa + Xaa(d}Saa
+ Xaa(d}Sa + Gm
l(d}Mmn + Qab(d}Vba
3 OSp(4l6}/(SO(l, 3} x U(3}} superparticle
Massless superparticle action on the OSp(4l6}/ (SO(1,3} x U(3}} supercoset manifold [6]
sp
dT (Gt
0lm
+ At AT +QTa^ra}, (11)
(6)
3
constructed out of the world-line pullbacks of Cartan forms from the C(2) eigenspace, is by definition invariant under the OSp(4l6) global symmetry acting as the left group multiplication on G G OSp(4l6)/(SO(1, 3) x U(3)). Action variation w.r.t. Lagrange multiplier e(r) produces the mass-shell constraint
Gt 0 mGT 0 m + At At + QTa^ = 0.
(12)
(7)
Applicability of the supercoset approach implies that V-ra = 0 [5], [3] so that GT°'mGT0'm + ATAT = GT ■ GT+A2 = 0. Variation of the action (11) is contributed by variations of the constituent so(2,3)/so(1,3) © su(4)/u(3) Cartan forms that can be obtained from the general expression
SF( d} = dig F + lg dF
(13)
0
upon substitution in the second summand Maurer-Cartan equations for C(2) Cartan forms [19]
dG0'm- 2G3m A A-2Gmnh G0' n
+ 2lS(i)a A lS(i)1
dA+ 2G3m A G0’ m
+ (1 ^ 3} = 0,
+ 2s(i)a a s(i)a— (1 ^3} =o,
dQa + lQb A (Qba + SbaQcc}
— 2leabcs(i)a A S(i)ac + (1 ^ 3} = 0
dQa + l(Qab + saQcc} A Qb
S(i) Tb
S(i)TVb
where the 12 x 12 matrix of rank 8 equals
Sha mTav
-lSaeacb Q
lSa £acbQT
S
b "°t v
22
a
TV
—iGt 0lmamav + At sa, IGt 0’m<Tmav + At sa
obey the relation
mTavmtVx = Sa(Gt ■ Gt + A2}.
S(3) Tb
S(3)TVb
with
M(3)X =
— sa m Ta v lSaeacbQt
îsa,£acbQT
SamTav
or the mass-to-zero limit of the DO-brane [21] the equations of motion for gauged away fermions require special attention. It appears that they are non-trivial and can be brought to the form [20]
Q
b v
T S(i,3)Tb
= QT
bS(i,3)T
vb
0.
(21)
(14)
Below we shall show that 8 equations (21) are contained in the set of equations (15), (19).
Consider in detail the system of equations (15). Applying rank 8 projection matrix
n
( Sb sa
I ^ av
(i) l l m't^v___£acbQ
V Gt Gt + A2 £ Qtc
Gt Gt +AT
sasaa
£acbQT
(22)
S(i)ac + (1 ^ 3} = 0.
Then taking the contractions of the Cartan forms (8) with the variation symbol lg as independent parameters yields the set of superparticle equations of motion [20].
Here we concentrate on the fermionic equations that can be brought to the form
that satisfies n31) — 3n21) + 2n^) = 0 and recalling that GT ■ GT + A“2 = 0 allows to bring (15) to the block-diagonal form
a Q Q b
It v*LTa*LT
0
0
—mS TavQT a QTb
(i)T
vb
0.
Further using (18) yields that the system (23) is equivalent to 4 equations
(15)
Qt S(i)T6 — QTbS(i)T
vb
0
(24)
(16)
(17)
(18)
coinciding with (21). Analogously it can be shown that the system (19) includes equations
nTbu(3)TVb = ^Tb^(3)TVb = 0. (25)
4 OSp(4l6)/(SO(1, 3) x U(3)) sigma-model
The OSp(4l6)/(SO(1, 3) x U(3)) sigma-model [5], [6] action in conformal basis for the osp(4l6)/(so(1, 3) x u(3)) Cartan forms can be written as [19]
Ss-m = J d2e(Lkin + LWz}
(26)
Similarly equations for Cartan forms corresponding to the generators from the g(3) eigenspace read
with the kinetic and Wess-Zumino Lagrangians given by
Lkin = — iY%j (G<0 mGj0 m + AiAj + QiaQj^j
(19)
(20)
— 2 Ytj (I (sim + cim '}(s'jl + j }
(27)
+ AiAj + Qia Qja}
and
Lwz = —£lj (S(i)ia£avS(3)jva + (1 ^ 3})
\£ij (Siaa£avSjva + Xiaa£avSjl)
(28)
At the same time since the OSp(4\6)/(SO(1,3) x U(3)) superparticle action comes about upon gauging away 8 coordinates from the broken supersymmetries sector in the action for massless superparticle on the AdS4 x CP3 superbackground [20] that can be viewed as the zero-mode limit of the AdS4 x CP3 superstring [3]
Fermionic equations of the OSp(4l6)/(SO(1,3) x U(3)) sigma-model [5], [6] can be cast into the following form [19] generalizing superparticle equations (15), (19)
V+ Mm/i1 S(ij
S(i)j
vb
(29)
m^v
c
v
S(i)Tb
0
c
a
m
Tv
0
c
AdS4 x CP3
and
V-M(3)iSÎ
S(3)jb
S(3)jvb
(30)
and
M(3)
ab id v
sa miav -lsa£ acbQ
—sa m iav lsa£acbQic
—lsa acbQi —saam iav
lsva acbQic
samiav
V+j Qibs(i) jvb
Vij Q
+ QibS (i) j
vb
0,
V-j QibS(3) jvb = V-j QibS(3) jvb = 0.
V± f-
VIFT
V
± = 2 \ Yt<tT!
^tt
FT = yttFt + (j™ ± 1}Fa
one can write Eqs. (29), (30) in the form
M(i) + and
aa a v
S(i)-b
vb
S(i)-
S(3)+b S(3) + vb
s\m+av
-lsa £acbQ
+ c
— lsa£ acbQ+c
— sa m + av
that is similar to the superparticle equations (15), (19). Note that application of the projectors V± to the Vi-rasoro constraints gives [19]
where V± = 2 (ylj ±ej) are (anti)self-dual world-sheet 12 x 12
world-sheet vector index i = (t, a) are defined as
G± 0 mG± 0 m + A2 + Q±aQ± a = 0
(39)
generalizing the superparticle mass-shell condition (12)
m±av
relations
(31)
(32)
m±avm±vx = sa(G± ■ G± + A±}
(40)
Apart from these equations similarly to the superparticle case there remain non-trivial equations for 8 fermions from the sector of broken supersymmetries that can be brought to the form [22]
(33)
In the remaining part of this section we shall show that they are contained in the set of fermionic equations (29), (30) of the OSp(4l6)/(SO(1,3) x U(3)) sigmamodel. Recalling that the action of the projectors V± on a vector can be factorized as
(34)
(35)
(36)
(37)
generalizing (18). Thus repeating the analysis performed for the superparticle case we conclude that Eqs. (33) are contained in the system of OSp(4\6}/(SO(1, 3} x U(3}} sigma-model fermionic equations (29), (30).
5 Conclusion
We have proved that in the partial «—symmetry gauge in which 8 fermionic coordinates correspond-
AdS4 x CP3
superbackground are set to zero equations of motion
AdS4 x CP3
action w.r.t to such coordinates are contained in the fermionic equations of the OSp(4l6}/(SO(1, 3} x U(3}} sigma-model generalizing the argument of Ref. [13]. This provides yet another necessary consistency check of the OSp(4l6}/(SO(1,3} x U(3}} sigma-model approach to the description of the certain sector of AdS4 x CP3
such a partial «—symmetry gauge all the non-trivial AdS4 x CP3
to those derivable from the OSp(4l6} / (SO(1,3} xU (3}} sigma-model action.
The situation changes when the gauge condition is relaxed to allow non-trivial dynamics in the sector of broken supersymmetries. This lifts restrictions on admissible string motions imposed in the framework of the OSp(4l6}/(SO(1,3} x U(3}} sigmamodel approach and also equations of motion for the fermions related to the broken supersymmetries become independent of those for the fermions associated with the unbroken supersymmetries of the background. Both sets are involved in proving integrabil-AdS4 x CP3
tions beyond the OSp(4l6}/(SO(1,3} x U(3}} supercoset [131, [2312, [221, [201.
and
?(3)-av =
—sa m-av lsa£acbQ-c
lsa acbQ
sam-av
(38)
Acknowledgement
The author is grateful to A.A. Zheltukhin for interesting discussions.
2 Generalizations to the superstring models on other superbackgrounds part of which admits supercoset description were examined in [24], [23], [25].
0
c
0
0
c
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Received 01.10.2012
Д. В. Уваров
ЗАМЕТКА О ФЕРМИОННЫХ УРАВНЕНИЯХ AdS4 x CP3 СУПЕРСТРУНЫ
Доказывается, что в случае, когда 8 фермионов, отвечающих нарушенным AdS^xCP3 бекграундом суперсимметриям, обращаются в нуль выбором калибровки для к—симметрии, уравнения, следующие из действия AdS4 xCP3 суперструны варьированием по ним, содержатся в наборе фермионных уравнений движения OSp(4|6)/(SO(1, 3) x U(3)) сигма-модели.
Ключевые слова: суперструна типа ПА, AdS4 x CP3 супербекграунд, OSp(4|6)/(SO(1, 3) x U(3)) сигма-модель, уравнения движения.
Уваров Д. В., кандидат физико-математических наук, старший научный сотрудник.
Харьковский физико-технический институт НАН Украины.
Ул. Академическая, 1, Харьков, Украина, 61108.
E-mail: [email protected]; [email protected]
