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SU-ITP-97-26
hep-th/9705118
May 15, 1997
Volkov-Akulov theory and D-branes1
Renata Kallosh 2
Physics Department, Stanford University, Stanford, CA 94305-4060, USA
kallosh@physics.stanford.edu
ABSTRACT
The action of supersymmetric Born-Infeld theory (D-9-brane in a Lorentz covariant
static gauge) has a geometric form of the Volkov-Akulov-type. The first non-linearly
realized supersymmetry can be made manifest, the second world-volume supersymmetry
is not manifest. We also study the analogous 2 supersymmetries of the quadratic action
of the covariantly quantized D-0-brane. We show that the Hamiltonian and the BRST
operator are build from these two supersymmetry generators.
This article is dedicated to the memory of D. V. Volkov whose insights into the nature of super-
symmetry and geometry proved to be enlightening for few generations of high-energy physicists.
His ideas inspired the most active developments in theoretical physics over the last quarter of the
century.
Extended objects with global supersymmetry have local κ-symmetry. This symmetry is difficult
to quantize in Lorentz covariant gauges keeping finite number of fields in the theory. A revival of
interest to κ-symmetric objects is due to the recent discovery of D-p-branes [1], κ-symmetric non-
linear effective actions and/or equations of motion for D-branes [2, 3, 4, 5, 6, 7] and the M-5-brane
1Contribution to Supersymmetry and Quantum Field Theory, International Seminar dedicated to the memory of
D.V. Volkov, Kharkov State University (Kharkov, Ukraine) January 5-7, 1997.
2This work is supported by the NSF grant PHY-9219345.
1

action [8, 9, 10], complementing the κ-symmetric superstring and M-2-brane [11, 12]. A nice review
on worldvolume actions in a doubly supersymmetric geometric approach initiated by D. V. Volkov
is presented in these Proceedings [13].
The new issues in quantization of D-branes have been analyzed recently in [3, 7, 14].
The κ-symmetric D-brane action in the flat background geometry3 consists of the Born-Infeld-
Nambu-Goto term S1 and Wess-Zumino term S2:
SDBI + SWZ = T
(
−
∫
dp+1σ
√
−det(Gµν + Fµν) +
∫
Ωp+1
)
. (1)
Here T is the tension of the D-brane, Gµν is the manifestly supersymmetric induced world-volume
metric
Gµν = ηmnΠmµ Πnν , Πmµ = ∂µXm − θ¯Γm∂µθ , (2)
and Fµν is a manifestly supersymmetric Born-Infeld field strength (for p even) 4
Fµν ≡ Fµν − bµν =
[
∂µAν − θ¯Γ11Γm∂µθ
(
∂νXm −
1
2
θ¯Γm∂νθ
)]
− (µ↔ ν) . (3)
When p is odd, Γ11 is replaced by τ3 ⊗ I. The action has global supersymmetry
δǫθ = ǫ, δǫXm = ǫ¯Γmθ . (4)
and local κ-supersymmetry:
δXm = θ¯Γmδθ = −δθ¯Γmθ, δθ¯ = κ¯(1 + Γ), (5)
and
Γ = ea2Γ′(0)e−
a
2 , a =
{
+12YjkγjkΓ11 IIA ,
−12Yjkγjkσ3 ⊗ 1 IIB .
(6)
Here Γ′(0) is the product structure, independent on the BI field, (Γ
′
(0))
2 = 1, tr Γ′(0) = 0). All
dependence on BI field F = “ tan ”Y is in the exponent [7]. The matrix Γ11 in IIA and σ3 ⊗ 1 in
IIB theory anticommute with Γ′(0) and with Γ. Therefore in the basis where Γ11 and σ3 ⊗ 1 are
diagonal, Γ′(0) and Γ are off-diagonal. We introduce a 16× 16-dimensional matrix γˆ which does not
depend on BI field.
Γ′(0) =



0 γˆ
γˆ−1 0


 , Γ =



0 γˆeaˆ
(γˆeaˆ)−1 0


 , (7)
where
aˆ =
{
+12Yjkγjk IIA ,
−12Yjkγjk IIB .
(8)
3We use notation of [3].
4We define spinors for even p as θ = θ1 + θ2 where θ1 ≡ 12 (1 + Γ11)θ and θ2 ≡ 12 (1− Γ11)θ.
2

The fact that Γ is off-diagonal and that the matrix γeaˆ is invertible is quite important and the
significance of this for covariant quantization of D-branes was already discussed in [3, 7, 14]. In
particular this allows us to consider only irreducible κ-symmetry transformations by imposing a
Lorentz covariant condition on κ¯ of the form
κ¯1 = 0 κ¯2 6= 0 IIA (9)
κ¯2 = 0 κ¯1 6= 0 IIB . (10)
In this way we have an irreducible 16-dimensional κ-symmetry since the matrix γˆeaˆ is invertible,
acting as
δθ¯1 = κ¯2γˆeaˆ δθ¯2 = κ¯2 δXm = −κ¯2Γmθ2 IIA (11)
δθ¯1 = κ¯1 δθ¯2 = κ¯1(γˆeaˆ)−1 δXm = −κ¯1Γmθ1 IIB . (12)
Recently a covariant gauge fixing κ-symmetry of D-branes has been discovered [3]. The fermionic
gauge is of the form
θ2 = 0 IIA θ1 ≡ λ (13)
θ1 = 0 IIB θ2 ≡ λ . (14)
Note that our choice of irreducible κ-symmetry (which is not unique) was made here with the
purpose to explicitly eliminate θ2 (θ1) in IIA (IIB) case using δθ¯2 = κ¯2 ( δθ¯1 = κ¯1). The gauge-fixed
action has one particularly useful property: the Wess-Zumino term vanishes in this gauge [3]. We
are left with the reparametrization invariant action:
Sκ−fixed = −
∫
dp+1σ
√
−det(Gµν + Fµν) , (15)
Gµν = ηmnΠmµ Πnν , Πmµ = ∂µXm − λ¯Γm∂µλ , (16)
Fµν = [∂µAν − λ¯Γm∂µλ
(
∂νXm −
1
2
λ¯Γm∂νλ
)
]− (µ↔ ν) . (17)
This action (18) has a local reparametrization symmetry and a 32-component global supersym-
metry. The form of the action is such that it can be brought to the form close to the one discovered
by Volkov-Akulov [15]. Consider for example a D-9-brane in a static gauge [3] Xµ = σµ :
Sg.f. = −
∫
d10σ
√
−det(Gµν + Fµν) , (18)
where
Gµν = eµmeνnηmn (19)
and
eµm = δµm − λ¯Γm∂µλ . (20)
3

If we introduce the 1-forms depending on fermion fields λ(σ)
em = dσµeµm[λ(σ)] = dσm + λ¯Γmdλ (21)
we can rewrite the supersymmetric Born-Infeld 9-brane action as
S =
∫
em0 ∧ em1 ∧ · · · ∧ em9
√
−det(ηmn + Fmn) (22)
where Fmn = eµmeνnFµν . In absence of the two-form Fmn the supersymmetric Born-Infeld action
is reduced to geometric action of Volkov-Akulov [15], generalized to d=10:
S =
∫
em0 ∧ em1 ∧ · · · ∧ em9 =
∫
d10σ dete[λ(σ)] (23)
This action depends only on fermions λ(σ) and has a non-linearly realized supersymmetry manifest,
since the 1-forms em are supersymmetric. The second supersymmetry of the supersymmetric Born-
Infeld 9-brane action (22) is not manifest. It explicit form can be obtained from the preservation
of the gauge-fixing condition for the kappa-symmetry.
The gauge-fixed actions of extended supersymmetric objects in static gauge are known to lead
to complicated non-linear actions. For example, the action of d=2 massive superparticle [16]
S = −M
∫
dt
(
[
−(X˙m − θ¯Γmθ˙)(X˙n − θ¯Γnθ˙)ηmn
]1/2 − 1 + θΓ3θ˙
)
m = 0, 1. (24)
quantized in the gauge Γ3θ = θ for κ-symmetry and the static gauge for reparametrization symme-
try, X0 = t, gives the kink effective action [16]
Sg.f = −M
∫
dt
(
[
1− φ˙2
]1/2 − 1 + i
4Mρρ˙
)
. (25)
Here the bosonic field is the remaining coordinate of the d=2 superparticle, φ = X1. The Hamil-
tonian associated with this action is also non-linear:
H = (p2 +M2)1/2 −M , (26)
p is the momentum canonical conjugate to X1.
Here we will perform a covariant quantization of the D-0-brane which is a d=10 generalization
of the d=2 massive superparticle [16]. Instead of the static gauge, which belongs to a class of
canonical gauges with the non-propagating ghosts, we will use a covariant gauge for reparametriza-
tion symmetry. Consider the κ-symmetric action of a D-0-brane [2, 3, 4, 5, 6]. D-0-brane action
does not have Born-Infeld field since there is no place for an antisymmetric tensor of rank 2 in
one-dimensional theory. The D-p-brane action for p = 0 case reduces to
S = −T
(∫
dτ
√
−Gττ +
∫
θ¯Γ11θ˙
)
. (27)
4

This action can be derived from the action of the massless 11-dimensional superparticle [4].
S =
∫
dτ√−gττgττ
(
X˙mˆ − θ¯Γmˆθ˙
)2 , mˆ = 0, 1, · · · , 8, 9, 10. (28)
We may solve equation of motion for X 1ˆ0 as P1ˆ0 = Z, where Z is a constant, and use Γ11 = Γ1ˆ0.
From this one can deduce a first order action
S =
∫
dτ
(
Pm(X˙m − θ¯Γmθ˙) +
1
2
V (P2 + Z2)− Zθ¯Γ11θ˙ + χ¯1d2
)
. (29)
We will show now that the D-0-brane action can be obtained from this one upon solving equations
of motion for Pm, V, χ1, and d2. Here V (τ) is a Lagrange multiplier, Z = T is some constant
parameter in front of the WZ term and P2 ≡ PmηmnPn. The chiral spinors χ1 and d2 are auxiliary
fields. They are introduced to close the gauge symmetry algebra off shell. To verify that this first
order action is one of the D-p-brane family actions given in (1) we can use equations of motion for
Pm
Pm = −
1
V (X˙
m − θ¯Γmθ˙) , (30)
and for the auxiliary fields χ1 = 0 and d2 = 0. The action (29) becomes
S =
∫
dτ
(
− 1
2V (X˙
m − θ¯Γmθ˙)2 + 1
2
V Z2 − Zθ¯Γ11θ˙
)
. (31)
Equation of motion for V is
V 2 = − 1Z2 (X˙
m − θ¯Γmθ˙)2 , (32)
and we can insert V = − 1Z
√
−(X˙m − θ¯Γmθ˙)2 back into the action (31) and get
S = −Z
(∫
dτ
√
−(X˙m − θ¯Γmθ˙)2 + Zθ¯Γ11θ˙
)
. (33)
This is the action (1) for D-0-brane at T = Z as given in (27).
The action (29) is invariant under the 16-dimensional irreducible κ-symmetry and under the
reparametrization symmetry. The gauge symmetries are (we denote ΓmPm = /P):
δθ¯ = κ¯2(Γ11Z + /P) , (34)
δXm = −ηPm − δθ¯Γmθ − κ¯2Γmd , (35)
δV = η˙ + 4κ¯2θ˙ + 2χ¯1κ2 , (36)
δχ¯ = κ¯2/˙P , (37)
δd = [P2 + Z2]κ2 . (38)
Here η(τ) is the time reparametrization gauge parameter and κ2(τ) = 12(1 − Γ11)κ(τ) is the 16-
dimensional parameter of κ-symmetry. The gauge symmetries form a closed algebra
[δ(κ2), δ(κ′2)] = δ(η = 2κ¯2/Pκ′2) . (39)
5

To bring the theory to the canonical form we introduce canonical momenta to θ and to V and
find, excluding auxiliary fields
L = PmX˙m + PV V˙ + P¯θθ˙ +
1
2
V (P2 + Z2) + PV ϕ+
(
P¯θ + θ¯(/P + ZΓ11)
)
ψ . (40)
We have primary constraints Φ¯ ≡ P¯θ + θ¯(/P + ZΓ11) ≈ 0 and PV = 0. The Poisson brackets for
32 fermionic constraints are
{Φ,Φ} = 2C(/P + Γ11Z) . (41)
We also have to require that the constraints are consistent with the time evolution {PV , H} = 0.
This generates a secondary constraint
t = P2 + Z2 . (42)
Thus the Hamiltonian is weakly zero and any physical state of the system satisfying the reparamet-
rization constraint is a BPS state M = |Z| since
P2 + Z2|Ψ〉 = 0 =⇒ Z2|Ψ〉 = −P2|Ψ〉 = M2|Ψ〉 . (43)
The 32 × 32 -dimensional matrix C(/P + Γ11Z) is not invertible since it squares to zero when the
reparametrization constraint is imposed. This is a reminder of the fact that D-0-brane is a d=11
massless superparticle. The 32 dimensional fermionic constraint has a 16-dimensional part which
forms a first class constraint and another 16-dimensional part which forms a second class constraint.
We notice that the Poisson brackets reproduce the d = 10, N = 2 algebra with the central charge
which can also be understood as d = 11, N = 1 supersymmetry algebra with the constant value of
P11 = Z.
We proceed with the quantization and gauge-fix κ-symmetry covariantly by taking θ2 = 0, θ1 ≡ λ
and find
Lκg.f. = Pm(X˙m − λ¯Γmλ˙) +
1
2
V (P2 + Z2) . (44)
The 16-dimensional fermionic constraint
Φ¯λ ≡ (P¯λ + λ¯/P) ≈ 0 (45)
forms the Poisson bracket
{Φαλ ,Φβλ} = 2(/PC)αβ . (46)
The matrix /PC is perfectly invertible as long as the central charge Z is not vanishing. The inverse
to (46) is
{Φα,Φβ}−1 |t=0 = [2(/PC)αβ]−1 =
(C/P)αβ
2P2
. (47)
This proves that the fermionic constraints are second class and that the fermionic part of the
Lagrangian
− λ¯/Pλ˙ ≡ −iλαΦαβλ˙β , Φαβ = −i(C/P)αβ , (48)
6

is not degenerate in a Lorentz covariant gauge. None of this would be true for a vanishing central
charge. Note that in the rest frame P0 = M, ~P = 0, hence
Φαβ = Mδαβ . (49)
For D-0-brane one can covariantly gauge-fix the reparametrization symmetry by choosing the V =
1 gauge and including the anticommuting reparametrization ghosts b, c. This brings us to the
following form of the action:
Lκ,ηg.f. = PmX˙m − λ¯/Pλ˙+
1
2
(P2 + Z2) + bc˙ . (50)
Now we can define Dirac brackets
{λ, λ¯}∗ = {λ, Φ¯}{Φ¯,Φ}−1{Φ , λ¯} =
/P
2P2
= − /P
2Z2 . (51)
The generator of the 32-dimensional supersymmetry is
ǫ¯Q = ǫ¯(/P + Γ11Z)λ . (52)
It forms the following Dirac bracket
[ǫ¯Q , Q¯ǫ′]∗ = ǫ¯(/P + Γ11Z) /P
2P2
(/P + Γ11Z)ǫ′ = ǫ¯ΓmˆPmˆǫ′ = ǫ¯(/P + Γ11Z)ǫ′ . (53)
We can also rewrite it in d=11 Lorentz covariant form
[ǫ¯Q , Q¯ǫ′]∗ = ǫ¯ΓmˆPmˆǫ′ = ǫ¯ /ˆPǫ′ , mˆ = 0, 1, · · · , 8, 9, 10, Z = P1ˆ0 , Γ11 = Γ1ˆ0 . (54)
This Dirac bracket realizes the d=11, N=1 supersymmetry algebra or, equivalently, d=10, N=2
supersymmetry algebra with the central charge Z.
One can also to take into account that the path integral in presence of second class constraints
has an additional term with
√
Ber{Φλ,Φλ} ∼
√
BerΦαβ [17]. It can be used to make a change of
variables
Sα = Φ1/2αβ λβ . (55)
The action becomes
L = PmX˙m − iSαS˙α + bc˙−H (56)
H = −1
2
(P2 + Z2) . (57)
7

The generators of global supersymmetry commuting with the Hamiltonian take the form
ǫ¯Q = ǫ¯(/P + Γ11Z)Φ−1/2S . (58)
Taking into account that {Sα, Sβ}∗ = − i2δαβ we have again realized d = 10, N = 2 supersym-
metry algebra in the form (53) or (54). The nilpotent BRST operator in this gauge where only
reparametrization ghosts are propagating is given by
QBRST = cH H = {b, QBRST} (QBRST )2 = 0 (59)
and here we used the fact that {b, c} = 1. In turn, the Hamiltonian (and therefore the BRST
operator) can be constructed from supersymmetry generators as follows:
H = 1
2
{Qα, Qβ}∗{Q¯α, Q¯β}∗ =
1
2
(C/ˆP)αβ(/ˆPC)αβ = −
1
2
/ˆP2 = −1
2
(/P2 + Z2) (60)
The terms with anticommuting fields Sα can be rewritten in a form where it is clear that they
can be interpreted as world-line spinors,
L = Pm∂0Xm + S¯αρ0∂0Sα + bc˙−H . (61)
Here S¯α = iSαρ0 and (ρ0)2 = −1, ρ0 = i being a 1-dimensional matrix.
Thus, we have the original 10 coordinates Xm and their conjugate momenta Pm, and a pair
of reparametrization ghosts. There are also 16 anticommuting world-line spinors S, describing 8
fermionic degrees of freedom. The Hamiltonian is quadratic. The ground state with M2 = Z2 is
the state with the minimal value of the Hamiltonian. Thus for the D-superparticle one can see that
the condition for the covariant quantization is satisfied in the presence of a central charge which
makes the mass of a physical state non-vanishing. The global supersymmetry algebra is realized in
a covariant way, as different from the light-cone gauge.
Thus we have found that covariantly quantized D-0-brane has a quadratic action with the
physical state being the BPS state M = |Z|. The resulting supersymmetry generator is d = 10
Lorentz covariant and the Dirac bracket of the quantized theory form d = 10, N = 2 supersymmetry
algebra with a central charge.
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