We build up the anticommutator algebra for the fermionic coordinates of open
superstrings attached to branes with antisymmetric tensor fields. We use both
Dirac quantization and the symplectic Faddeev Jackiw approach. In the
symplectic case we find a way of generating the boundary conditions as zero
modes of the symplectic matrix by taking a discretized form of the action and
adding terms that vanish in the continuous limit. This way boundary conditions
can be handled as constraints.Comment: Revision: passage from discrete to continuous clarified, comment on
  previous results using Dirac quantization included, typos corrected. Version
  to appear in Phys. Lett. 

Ardalan

Banerjee

Barcelos-Neto

Braga

Cresus F.L. Godinho

Faddeev

Haggi-Mani

Hofman

Lindström

Montani

Nelson R.F. Braga

Polchinski

Romero

Seiberg

Sheikh-Jabbari

Wotzasek

Physics Letters B

English

arXiv

Crossref

AbstractWe build up the anticommutator algebra for the fermionic coordinates of open superstrings attached to branes with antisymmetric tensor fields. We use both Dirac quantization and the symplectic Faddeev–Jackiw approach. In the symplectic case we find a way of generating the boundary conditions as zero modes of the symplectic matrix by taking a discretized form of the action and adding terms that vanish in the continuous limit. This way boundary conditions can be handled as constraints

Braga, Nelson R.F.

Godinho, Cresus F.L.

Elsevier - Publisher Connector 

Fermionic anticommutators for open superstrings in the presence of antisymmetric tensor field 

Physics Letters B 570 (2003) 111–117
www.elsevier.com/locate/npe
Fermionic anticommutators for open superstrings in the presence
of antisymmetric tensor field
Nelson R.F. Braga a, Cresus F.L. Godinho b
a Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21945-970 Rio de Janeiro, Brazil
b Centro Brasileiro de Pesquisas Físicas, Rua Dr Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
Received 18 June 2003; received in revised form 11 July 2003; accepted 14 July 2003
Editor: T. Yanagida
Abstract
We build up the anticommutator algebra for the fermionic coordinates of open superstrings attached to branes with
antisymmetric tensor fields. We use both Dirac quantization and the symplectic Faddeev–Jackiw approach. In the symplectic
case we find a way of generating the boundary conditions as zero modes of the symplectic matrix by taking a discretized form
of the action and adding terms that vanish in the continuous limit. This way boundary conditions can be handled as constraints.
 2003 Elsevier B.V.
PACS: 11.25.-w
1. Introduction
Non-commutativity of spacetime and its consequences for quantum field theory have been one of the main
objects of interest for theoretical particle physicists in the last years. A general discussion and an important
list of references can be found in [1]. An important source of non-commutativity of space in string theory
[2,3] is the presence of an antisymmetric constant tensor field along the D-brane [4] world volumes (where
the string endpoints are located). The quantization of strings attached to branes involves mixed (combination
of Dirichlet and Neumann) boundary conditions. This makes the quantization procedure more subtle since the
quantum commutators/anticommutators must be consistent with these boundary conditions. For the bosonic string
coordinates, the non-commutativity at end points has already received much attention. Many important aspects
have been discussed and the commutators have been explicitly calculated (see, for example, [5–11]).
In contrast, the complete canonical structure (anticommutators) for the fermionic coordinates when antisym-
metric tensor fields are present has not yet been presented explicitly, although some important aspects have already
been discussed [5,12,13]. As we will see here, the requirement of consistency with boundary conditions affect the
structure of the anticommutation relations at the string endpoints even in the absence of any external field. We start
calculating the complete Dirac antibrackets for the fermionic coordinates consistent with the boundary conditions
E-mail addresses: braga@if.ufrj.br (N.R.F. Braga), godinho@cbpf.br (C.F.L. Godinho).
Open access under CC BY license.0370-2693  2003 Elsevier B.V.
doi:10.1016/j.physletb.2003.07.011
  Open access under CC BY license.
112 N.R.F. Braga, C.F.L. Godinho / Physics Letters B 570 (2003) 111–117of strings attached to branes with antisymmetric tensor field. We will do this by introducing a discretization along
the string spacelike coordinate. Such a construction make it transparent the behavior of the anticommutators at
the end points. The string boundary conditions will lead to a discontinuity in the anticommutator at the endpoints.
Also it will emerge that, in contrast to the standard canonical form, the anticommutator between the fermionic
components ψµ(+),ψ
µ
− is not vanishing at the endpoints even in the absence of the antisymmetric field.
Then we consider the symplectic quantization scheme and develop a procedure of generating the fermionic
string boundary conditions as constraints directly from the symplectic matrix. We will consider again a
discretization of the string world sheet spatial coordinate. In a previous article [11] we found the boundary
conditions for the bosonic string from the corresponding symplectic matrix by means of some field redefinitions.
Here we will improve such a procedure making use of the fact that a finite number of terms that vanish in the
continuous limit may be added to the discretized form of the action. By choosing appropriate terms we will get
the boundary conditions as zero modes of the fermionic symplectic matrix. Then the anticommutators will be
calculated in the standard way.
2. The model
Let us start with a superstring coupled to an antisymmetric tensor field living on a brane. Considering just the
coordinates along the brane, the action can be represented in superspace as
(1)S = −i
8πα′
∫
Σ
d2σ d2θ
(DYµDYµ +Fµν DYµρ5DYµ),
where the superfield
Yµ
(
σa, θ
)=Xµ(σa)+ θ¯ψµ(σa)+ 1/2θ¯θBµ(σa)
contains the bosonic and fermionic spacetime string coordinates. In components the action reads1
(3)S = 1
4πα′
∫
Σ
d2σ
(
ηµν∂aX
µ∂aXν + abFij ∂aXi∂bXj −BµBµ − iψ¯µρa∂aψµ + iFµνψ¯µρbab∂aψν
)
.
The bosonic and fermionic sectors decouple. We will consider just the fermionic sector once the bosonic sector
was already discussed [5–11]. The fermions are Majorana and can be represented as
(4)ψµ =
(
ψ
µ
(−)
ψ
µ
(+)
)
.
So that the fermionic sector reads
(5)S0 = −i4πα′
∫
Σ
dτ dσ
(
ψ
µ
(−)∂+ψ(−)µ +ψµ(+)∂−ψ(+)µ −Fµνψµ(−)∂+ψν(−) +Fµνψµ(+)∂−ψν(+)
)
.
The minimum action principle δS = 0 leads to a volume term that vanishes when the equations of motion hold
and also to a surface term:
(6)(ψµ(−)(ηµν −Fµν)δψν(−) −ψµ(+)(ηµν +Fµν)δψν(+))∣∣π0 = 0.
1 Our conventions are
(2)ρ1 =
(
0 i
i 0
)
, ρ0 =
(
0 −i
i 0
)
, ρ5 ≡ ρ0ρ1, ∂± = ∂0 ± ∂1.
N.R.F. Braga, C.F.L. Godinho / Physics Letters B 570 (2003) 111–117 113It is not possible to find non-trivial boundary conditions involving ψµ
(−) and ψ
µ
(+) that makes this surface term
vanish. However, the solution to this problem shows up when we take into account a result from Ref. [12] (see also
[14]). There it was shown that in order to keep supersymmetry unbroken at the string endpoints it is necessary to
include a boundary term to the action. Actually, considering the boundary term (6), we realize that it is impossible
even to solve the boundary condition unless some extra term is added to the action. The interesting thing is that the
same kind of term proposed in [12] in order to restore SUSY at the end points
(7)SBound = i2πα′
∫
Σ
dτ dσ
(Fµνψµ(+)∂−ψν(+))
will make it possible to find a solution to the boundary condition. Adding this term to S0 the total action reads
(8)S = −i
4πα′
∫
Σ
dτ dσ
(
ψ
µ
(−)E
νµ∂+ψ(−)ν +ψµ(+)Eνµ∂−ψ(+)ν
)
,
where Eµν = ηµν +Fµν . The corresponding boundary term coming from δS = 0 is now
(9)(ψµ(−)Eνµδψν(−) −ψµ(+)Eνµδψν(+))∣∣π0 = 0.
This condition is satisfied imposing the constraint that preserve supersymmetry [5]
(10)Eνµψν(+)(0, τ )= Eµνψν(−)(0, τ ),
(11)Eνµψν(+)(π, τ )= λEµνψν(−)(π, τ ),
at the endpoints σ = 0 and σ = π , where λ=±1 with the plus sign corresponding to Ramond boundary condition
and the minus corresponding to the Neveu–Schwarz case. We will only consider the endpoint σ = 0 in our
calculations. The results for σ = π have the same form.
Now considering the total fermionic action S we want to incorporate the boundary conditions (10) in a quantum
formulation of the theory. That means: we want to calculate anticommutators that are consistent with these
boundary conditions. Following the approach successfully applied to the bosonic sector (see, for example, [6–11])
we will consider a discrete version of the string in which we replace the continuous coordinate σ with range (0,π)
by a discrete set corresponding to intervals of length . Representing the fermionic coordinates at the endpoints of
the N intervals as: ψν0(−),ψ
ν
1(−), . . . ,ψ
ν
N(−);ψν0(+),ψν1(+), . . . ,ψνN(+) , the discretized form of the Lagrangian reads
(12)
L= −i
4πα′
(
ψ
µ
0(−)Eνµ∂0ψ
ν
0(−) + ψµ1(−)Eνµ∂0ψν1(−) + · · · + ψµ0(−)Eνµ
ψν1(−) −ψν0(−)

+ ψµ1(−)Eνµ
ψν2(−) −ψν1(−)

+ · · · + ψµ0+Eνµ∂0ψν0+ + ψµ1+Eνµ∂0ψν1+ + · · ·
− ψµ0+Eνµ
ψν1+ −ψν0+

− ψµ1+Eνµ
ψν2+ −ψν1+

+ · · ·
)
.
The original theory is recovered by taking the limit → 0.
3. Dirac quantization
The equal time canonical antibrackets for the original continuous fermionic fields are:{
ψ
µ
(+)(σ ),ψ
ν
(+)(σ
′)
}= {ψµ(−)(σ ),ψν(−)(σ ′)}=−2πiα′ηµνδ(σ − σ ′),
(13){ψµ(+)(σ ),ψν(−)(σ ′)}= 0.
114 N.R.F. Braga, C.F.L. Godinho / Physics Letters B 570 (2003) 111–117So that the canonical antibrackets for the corresponding discrete fermionic variables become
{
ψ
µ
i(+),ψ
ν
j (+)
}= {ψµ
i(−),ψ
ν
j (−)
}=−2πiα′δij ηµν

,
(14){ψµi(+),ψνj (−)}= 0.
The discrete version of the boundary condition (10), that we will impose as a constraint in the Dirac formalism, is
Ωµ ≡ Eνµψν0(+) −Eµνψν0(−).
So, the matrix of constraints is
(15)Mµν ≡ {Ωµ,Ων} = −4πiα
′

(
ηµρ −FνµFνρ
)= −4πiα′

(
1−F2)
µρ
,
and the Dirac (anti-) brackets are calculated in the standard way:
(16){A,B}D = {A,B} − {A,Ωµ}M−1µν {Ων,B}.
For the coordinates ψµi± with i = 0, corresponding to points inside the string they will be equal to the Poisson
brackets but for the boundary coordinates we get:
(17){ψµ0(+),ψν0(+)}= {ψµ0(−),ψν0(−)}=−πiα
′ηµν

,
(18){ψµ0(+),ψν0(−)}=−πiα
′

(
ηµγ +Fµγ )([1−F2]−1)
γρ
(
ηρν +Fρν).
The anticommutators (17) agree with the results found previously in Ref. [5]. Now we can obtain the continuous
version of our results. Once the anticommutators of ψµi(±) for i = 0 are not changed by the Dirac quantization,
inside the string (0  σ  π ) the anticommutators keep their canonical form. Then, for the boundary points, we
use the fact that the mapping between continuous and discrete expressions involve the mapping of Kronecker and
Dirac deltas in the following way: δij /⇔ δ(σi − σj ) (note that expressions (17), (18) involve a factor δ00 = 1).
The anticommutators of the points inside the string and on the boundary may be accommodated in one single
expression if we introduce a parameter β such that β = 1/2 for σ = σ ′ = 0 or β = 1 elsewhere. The continuous
limit of the Dirac antibrackets is then
(19){ψµ(+)(σ ),ψν(+)(σ ′)}= {ψµ(−)(σ ),ψν(−)(σ ′)}=−2βπiα′ηµνδ(σ − σ ′),
(20){ψµ(+)(σ ),ψν(−)(σ ′)}=−πiα′(ηµγ +Fµγ )([1−F2]−1)γρ(ηρν +Fρν)δ(σ − σ ′)
for σ = σ ′ = 0 and zero elsewhere except for the other endpoint σ = σ ′ = π where the same kind of relation holds
but with a sign depending on choosing Ramond or Neveu–Schwarz boundary conditions. It is important to note
that the anticommutator (20) does not vanish even in the absence of the antisymmetric tensor field. This result is
consistent with the boundary condition (10) that relates ψµ(+) and ψν(−) at the string endpoints.
4. Symplectic quantization
Let us now see how the fermionic anticommutators can be calculated using the symplectic Faddeev–Jackiw
quantization [15]. We need particularly the analysis of constraints and gauge symmetries in the symplectic
quantization developed in [16–18].
We consider a Lagrangian that is first order in time derivatives (if the original Lagrangian is not in this form one
can introduce auxiliary fields and change it to first order).
(21)L0 = a0k (q)∂τqk − V (q),
N.R.F. Braga, C.F.L. Godinho / Physics Letters B 570 (2003) 111–117 115where qk are the generalized coordinates of the system. For bosonic variables the symplectic matrix is defined as
(22)f 0kl =
∂a0l
∂qk
− ∂a
0
k
∂ql
.
If it is non-singular we define the commutators of the quantum theory (if there is no ordering problem for the
corresponding quantum operators) as
(23)[A(q),B(q)]= ∂A
∂qk
(
f 0
)−1
kl
∂B
∂ql
.
If the matrix (22) is singular we find the zero modes that satisfy f 0klvαl = 0 and the corresponding constraints:
(24)Ωα = vαl
∂V
∂ql
≈ 0.
Then we introduce new variables λα and add a new term to the kinetic part of Lagrangian
(25)L1 = a0k (q)q˙k + λ˙αΩα − V (q)≡ a1r (q˜) ˙˜qr − V (q),
where we introduced the new notation for the extended variables: q˜r = (qk, λα). We find now the new matrix f 1rs
(26)f 1rs =
∂a1s
∂q˜r
− ∂a
1
r
∂q˜s
.
If f 1 is not singular we define the quantum commutators as
(27)[A(q˜),B(q˜)]= ∂A
∂q˜r
(
f 1
)−1
rs
∂B
∂q˜s
.
This process of incorporating the constraints in the Lagrangian is repeated until a non-singular matrix is found.
In the present case we are dealing with fermionic string coordinates. For fermionic variables Ψi we define
(28)aΨi =
∂L
∂(∂τ )Ψi
,
as in the bosonic case, but the symplectic matrix takes the form
(29)fΨiΨj =
∂aΨj
∂Ψi
+ ∂aΨi
∂Ψj
.
The procedure of incorporating constraints then is the same as in the bosonic case.
In the previous section, the boundary conditions did not show up directly from the Dirac procedure. That means,
the method of quantization itself did not generate the boundary conditions. We had to impose them as additional
constraints. In the symplectic approach, in contrast, we will find the boundary conditions from the zero modes of
the symplectic matrix. We do not get this result if we use directly action (12) as our starting point. However, we
note that the individual terms in this discrete form of the action tend to zero in the limit → 0. So if we remove or
add a finite number of them we do not change the continuous limit → 0 corresponding to the original action of
Eq. (8). However the symplectic matrix in the discrete variables changes and this will make it possible to find the
boundary conditions as zero modes of the symplectic matrix. A possible way to do this is to include in the action
the extra term
(30)i
4πα′
ψ
µ
0(+)Eµν
(
ψν1(−) −ψν0(−)
)
,
that vanishes in the limit → 0 and then redefine the variables as
ψ
µ
(±)i ≡
ψ˜
µ
(±)i√

(i = 0), ψµ(0)+ ≡
ψ˜
µ
(0)+√

, ψ
µ
(0)(−) ≡ ψ˜µ(0)(−)
116 N.R.F. Braga, C.F.L. Godinho / Physics Letters B 570 (2003) 111–117in the Lagrangian and we will see that this will make it possible to generate the appropriate boundary condition
that mixes the (+) and (−) components.
The symplectic matrix takes the form
(31)


ψ˜ν0(+) ψ˜
ν
0(−) ψ˜
ν
1(+) ψ˜
ν
1(−) ψ˜
ν
2(+) ψ˜
ν
2(−) · · ·
ψ˜
µ
0(+) −2gµν 0 0 0 0 0 · · ·
ψ˜
µ
0(−) 0 −2gµν 0 0 0 0 · · ·
ψ˜
µ
1(+) 0 0 −2gµν 0 0 0 · · ·
ψ˜
µ
1(−) 0 0 0 −2gµν 0 0 · · ·
ψ˜
µ
2(+) 0 0 0 0 −2gµν 0 · · ·
ψ˜
µ
2(−) 0 0 0 0 0 −2gµν · · ·
. . . . . . . . . . . . . . . . . . . . . . . .


(times a factor 1/i4πα′). In the limit → 0 this symplectic matrix becomes singular. The zero mode corresponds
to the vector
(32)


0
1
0
...

 ,
and the corresponding constraints come from
(33)∂V
∂ψ˜
µ
0(−)
= 0.
Considering the inclusion of the extra crossed term of Eq. (30) in the potential V , we find that in the → 0 limit
the constraints, returning to the original variables, are
(34)Ωµ = Eµνψν0(−) −ETrµνψν0(+) = 0.
Then we introduce a Lagrange multiplier λµ and include the term λ˙µΩµ in the Lagrangian. Returning to the
original fermionic variables, the symplectic matrix becomes
(35)


ψν0(+) ψ
ν
0(−) ψ
ν
1(+) ψ
ν
1(−) ψ
ν
2(+) ψ
ν
2(−) · · · λν
ψ
µ
0(+) −2gµν 0 0 0 0 0 · · · −Eµν
ψ
µ
0(−) 0 −2gµν 0 0 0 0 · · · Eνµ
ψ
µ
1(+) 0 0 −2gµν 0 0 0 · · · 0
ψ
µ
1(−) 0 0 0 −2gµν 0 0 · · · 0
ψ
µ
2(+) 0 0 0 0 −2gµν 0 · · · 0
ψν2(+) 0 0 0 0 0 −2gµν · · · 0
. . . . . . . . . . . . . . . . . . . . . . . . 0
λµ −Eνµ +Eµν 0 0 0 0 · · · 0


(again times a factor 1/i4πα′). Inverting this matrix we find the anticommutators
(36){ψµ0(+),ψν0(+)}= {ψµ0(−),ψν0(−)}= −πiα
′

gµν,
(37){ψµ0(+),ψν0(−)}=−πiα
′ (
ηµγ +Fµγ )([1−F2]−1)
γρ
(
ηρν +Fρν).
N.R.F. Braga, C.F.L. Godinho / Physics Letters B 570 (2003) 111–117 117The anticommutators (36) agree with the previous result from [5] and reproduce the result obtained in the Dirac
quantization.
5. Conclusion
We have calculated the fermionic anticommutators at a string endpoint by Dirac and symplectic quantization.
In both cases, the discretization of the string spatial coordinate made it more easy to handle the discontinuity in the
antibrackets associated to the effect of the boundary conditions. In the symplectic case we found a way of getting
the boundary conditions from the symplectic matrix by adding terms that vanish in the continuous limit.
Acknowledgements
The authors are partially supported by CNPq, FAPERJ and FUJB (Brazilian Research Agencies).
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Fermionic anticommutators for open superstrings in the presence of
  antisymmetric tensor field

Fermionic anticommutators for open superstrings in the presence of antisymmetric tensor field

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