Journal of Modern Physics, 2012, 3, 110-115
http://dx.doi.org/10.4236/jmp.2012.31015 Published Online January 2012 (http://www.SciRP.org/journal/jmp)
String Gauge Symmetries in the Conformally Gauge-Fixed
Polyakov D1 Brane Action in the Presence of
Background Gauge Fields*
Usha Kulshreshtha1, Daya Shankar Kulshreshtha2
1Department of Physics, Kirori Mal College, University of Delhi, Delhi, India
2Department of Physics and Astrophysics, University of Delhi, Delhi, India
Received September 5, 2011; revised November 6, 2011; accepted December 1, 2011
ABSTRACT
Recently we have studied the instant-form quantization (IFQ) and the light-front quantization (LFQ) of the conformally
gauge-fixed Polyakov D1 brane action using the Hamiltonian and path integral formulations. The IFQ is studied in the
equal world-sheet time framework on the hyperplanes defined by the world- sheet time σ0 = τ = constant and the LFQ in
the equal light-cone world-sheet time framework, on the hyperplanes of the light-front defined by the light-cone
world-sheet time . The light-front theory is seen to be a constrained system in the sense of
Dirac in contrast to the instant-form theory. However, owing to the gauge anomalous nature of these theories, both of
these theories are seen to lack the usual string gauge symmetries defined by the world-sheet reparametrization invari-
ance (WSRI) and the Weyl invariance (WI). In the present work we show that these theories when considered in the
presence of background gauge fields such as the NSNS 2-form gauge field

constant   



,B




or in the presence of

1U

,A
gauge field



,C


and the constant scalar axion field


= constant 
, then they are seen to possess the usual string gauge
symmetries (WSRI and WI). In fact, these background gauge fields are seen to behave as the Wess-Zumino/Stueckel-
berg fields and the terms containing these fields are seen to behave as Wess-Zumino or Stueckelberg terms for these
theories.
Keywords: Light-Front Quantization; Hamiltonian Quantization; Path Integral Quantization; Constrained Dynamics;
Constraint Qua nt i zat i on; Gau ge Sym metry; String Gau ge Symmetry; String T he ory ; D1-Brane Actions;
Polyakov Action; Light-Cone Quantization
Study of D-brane actions [1-19] is a domain of wider
interest in string theories. Polyakov action does not in-
volve any square root and is in particular, simpler to
study. Recently, we have studied IFQ of this action [12]
for the D1 brane in the conformal gauge (CG), using the
Hamiltonian [20] and path integral [21-25] formulations
in the instant-form (IF) of dynamics (on the hyperplanes
defined by the world-sheet (WS ) ti me )
[26,27]. We have also studied its LFQ [13-19] using the
light-front (LF) dynamics (on the hyperplanes of the LF
defined by the light-cone (LC) WS time
0=


== 

constant ) [13-19,26-32].
The LF theory [13-19] is seen to be a constrained sys-
tem in the sense of Dirac [20], which is in contrast to the
corresponding IF theory [12], where the theory remains
unconstrained in the sense of Dirac. The LF theory is
seen to possess a set of twenty six second-class contraints
[13-19]. Further, the conformally gauge-fixed Polyakov
D1 brane action (CGFPD1BA) describing a gauge-non-
invariant (GNI) theory (being a gauge-fixed theory) is
seen to describe a gauge-invariant (GI) theory in the pre-
sence of an antisymmetric NSNS 2-form gauge field


,B

[13].


Recently we have shown [13] that this NSNS 2-form
gauge field behaves like a Wess-Zumino (WZ) field and
the term involving this field behaves like a WZ term for
the CGFPD1BA [13]. We have also studied the Hamilto-
nian [20] and path integral [21-25] formulations of the
CGFPD1BA with and without a scalar dilaton field in the
IF [12] as well as in the LF [13-19] dynamics. In both the
above cases the theory is seen (as expected) to be gauge-
noninvariant (GNI), possessing a set of second-class
constraints in each case, owing to the conformal gauge-
fixing [1-8, 12 - 19] of the theory.
*Part of this work was presented as an Invited Contributed Talk by
DSK at the International Light-Cone Conference LC2010: Relativistic
Hadronic and Particle Ph
y
sics
,
Valencia
,
S
p
ain
,
June 14-18
,
2010.The CGFPD1BA being GNI does not respect the usual
C
opyright © 2012 SciRes. JMP

U. KULSHRESHTHA ET AL. 111


(string) gauge symmetries defined by the WS reparametri-
zation invariance (WSRI) and the Weyl invariance (WI).
However, in the presence of a constant 2-form gauge
field B


it is seen [13] to describe a gauge-inavriant
(GI) theory respecting the usual string gauge symmetries
defined by the WSRI and the WI [13].
The IF and LF Hamiltonian and path in tegral formula-
tions of this theory in the presence of the constant 2-form
gauge field B


have been studied by us in Ref. [13]. In
the present work, we consider the question of the string
gauge symmetries associated with the Polyakov D1 brane
action in the presence of some other background fields
such as the gauge field

1U

,A





and the con-
stant scalar axion field

,C


h
[19]. The Polyakov D1
brane action in a d-dimensional curved background


2
=dSL
is defined by [1-8,12-19]:


 (1a)
=2
T
LhhG







XX

 (1b)

=det , =hhG

 



1, 1,, 1
 
,1 , ,=01d IFQ   

=, IFQ
(1c)

 =diag

,=0,1,. , =2,3,ii 
(1d)
(1e)

,=,,. , =2,3,,1 , ,ii d    (1f)
Here

,


  are the two parameters describing the
worldsheet (WS). The overdots and pr imes would denote
the derivatives with respect to

and

. T is the string
tension. G


is the induced metric on the WS and

,X


 are the maps of the WS into the d-di-
mensional Minkowski space and describe the strings
evolution in space-time [1-16]. h


are the auxiliary
fields (which turn out to be proportional to the metric
tensor



of the two-dimensional surface swept out by
the string). One can think of as the action describing
d massless scalar fields S

X

in two dimensions moving
on a curved background h


. Also because the metric
components h


are varied in above equation, the 2-
dimensional gravitational field h


is treated not as a
given background field, but rather as an adjustable quan-
tity coupled to the scalar fields [1-8,12-19]. The action
has the well-known three local gauge symmetries
given by the 2-dimensional WSRI and WI [1-8,12-19] as
follows:
S

=XXXX









XX
 


(2a)
=





hh
 

(2b)
=hh
 





hh
 (2c)
=hh

 


  
 
 

 

(2d)
hh



where the WSRI is defined f
(2e)
or the two parameters


,


 ; and the WI and is specified by a func-
tion


,

  . In the following we would, however,
-called orthonormal gauge where one sets
=1
work in the so

. Also for the CGFPD1BA one makes use of the
fact that the 2-dimensional metric h


is also specified
ee independent functions as it is a symmetric 22 by thr

metric. one can therefore use these ge symmetries of
the theory to choose h
gau


to be of a particular form
the IFQ (on the hyperplanes defined by 0== in
x
tcon-
stant) as follows:
:= , :=hh


 

(3)
For the IF dynamics we take [1-12]

(4a)
1 0
==
01
h
 






(4b)
with
1 0
==
01
h
 






=det= 1hh


 (5)
In LF formulation we use the Light-
ables defin ed by: Cone (LC) vari-



01
:=; :=2XXX 


In this case for the LFQ (on the hyperplane
by == constant)x

we take:
(6)
s defined
012
=12






:= 0 h



(7a)
02
:==2 0 h
 






(7b)
with

=det =12hh

  (8)
Now the action S
 in the CG (in the IF an
reads[1-3,9-16]: d LF) finally
2
=d
NN
SL

 (9a)
=2
NT
LX

X











=0,1; =0,1, ; =2,3,,ii  

=,; =,, ; =2,3,,25 LFQii   
The action
(9b)

25 IFQ (9c)

(9d)
N
S is the CGFPD1BA. The above
in IFQ [12] is seen to be an unconstrained system an
LF is
action
d in
Q [13,14] itseen to possess a set of 26 second-class
constraints [13] implying that the corresponding theories

U. KULSHRESHTHA ET AL.
112


=XX
 


are seen to be gauge anomalous and GNI and therefore
they do not possess the local gauge symmetries defined
by the WSRI and the WI. The Hamiltonian and path in-
tegral formulations of this CGFPD1BA defined by the
action
N
S have been studied by us Ref. [12]. When this
above action is considered in the presence of a Scalar
Dilatonld in the IFQ as well as in the LFQ then also it is
seen to possess sets of second-class constraints implying
that it remains GNI theory which does not respect the string
gauge symmetries: the WSRI and t he WI [1- 8,12-19] .
This action is thus seen to lack the local gauge sym-
metries. This is in contrast to the fact that the orig



fie
inal
action S
 had the local gauge symmetries and was
therefore GI. The theory defined by the action
N
S, on
the othehand describe GNI. This is not surprising at all
because the theory defined by
r
N
S is afterall (for-
mally) gauge-fixed theory and consequently not expected
to be GI anyway. Infact, the IF thy defined by
con
eor
N
S is
seen to be unconstrained [12] wherea s the LF theory is seen
to possess a set of 26 second-class constraints [13-19]. In
both the cases theory does not respect the usual local
string gauge symmetries defined by WSRI and WI.
We now consider this CGFPD1BA in the presence of a
constant background antisymmetric 2-form gauge field
B


studied earlier by Schmidhuber, de Alwis and Sato,
Tseytlin and Abou Zeid and Hull and others defined by
,13]:
2
=d , =
[1-8
I
IICB
SL LLL 


 (10a)

==
2
CNT
LL X

X





(10b) 


 


 

=2
BT
LB








 (10c)

2 01
, =



(10d)
=1, =constant10   


0
=0
B
B
B




(10e) := , BXXB





IFQ
01 10
==BB B (10f)


(10g) ==BB
 LFQB 

, =2,3,ii 

3,,25 LFQ
sess (only) one
27 first-class con-
st

,25 IFQ (10h) ,=0,1, =0,1,  

(10i) ,=, , =,, , =2,ii    
The above action is seen to pos
class constraint in IFQ and a set of first-
raints in LFQ [13-19]. Accordingly the theory in both
the cases is seen to three local gauge symmetries given
by the two-dimensional WSRI and the WI:
=XXXX





 (11a)


=hhh
 

(11b)
h





 (11c)

=hhhh


 
 
 


 

 (11d)

=BB BB







=BB

 

(11e)


 (11f)



hh



It is important to recolle
field B
(11g)
ct here that the 2-form gauge


a constant anti-symmetric tensor field in the world-
heet s. In Ref. [13], we have studied the Hamiltonian
and path integral formulations of this theory under the
gauge 0B
is a scalar field in th
is pace
e target-space whereas it

[13].
In this work we investigate the string gauge symme-
tries ofPD1BA CGF describing a GNI theory in the
presence of a


1U gauge field
 
,AA


  and a
constant scalar axion field


,CC

  [19] and show
that the CGFPA describing a GNeing a
gauge-fixed theory) is seenI theory when
considered in the presence of above background fields.
We also show that the
D1BI theory (b
to describe a G


1U gauge field


,A


 and
the constant scalar axion field


,C


are both seen to
bemino[19] and
een to b
have like the Wess-Zu (WZ) fields the
term involving these fields is sehave like a WZ
term for the CGFPD1BA [19]. Here the field
A

is a
scalar field in the target space and a vector field in the
WS space and the axion field C is a constant scalar field
in both the target space as well as in the WS space [19].
We find that the resulting th eory obtained in the abo ve
manner describes a GI system respecting the usual string
gauge symmetr ies defined by the WSRI and the WI. It is
seen that the axion field C and the

1U gauge field
A

, in the resulting theory behave like th e WZ fields and
the term involving these fields behavee a WZ term
the CGFPD1BA [19].
The situation in the present case is seen to be exactly
analogous to a theory whe
s lik
for
re one consid ers the CGF PD1-
BA in the presence of a 2-form gauge field
B
B

as studied
by us in Ref. [13], where the field B


behaves like a
WZ field and the term involving this field bees like a
WZ term for the CGFPD1BA [13].
The CGFPD1BA in the presence of a constant back-
ground scalar axion field C and an U
hav

1 gauge field
A

is defined by [1-8,19]:

2
=d , =
I
IIC
SL LL 
 (1
A
L2a)
 
==
2
CNT
LL XX




(12b) 






U. KULSHRESHTHA ET AL. 113

C
F

=2
AT
L


 



 (12c)

2
=1 , =c onsta   

=
nt (12d)
01
= ,
10
F
AA






(12e) 






01
== IFQF
01
fF  (12f)


LFQF

 (12g)
1,, =2,3,ii 
3,,25 LFQ (12i)
of
are the canonical
01
,,
==fF

,=0,1, =0,  

,25 IFQ (12h)

,=, , =,,, =2,ii    
In IFQ the theory is seen to possess a set
class constraints: 3 first-

0
12 3
=0 , =0, =0
c
ETC 
(13)
where

01
,,PE

 and C 
momenta conjugate respectively to
X
AA

a
atrix b
2(14b)
=0
nd C.
Now thPoissiorackets of the con-
straints i
 is seen to be singular implying that the con-
straints i
 form a set of first-class constraints and that
the theory described by the above action is a GI theory
[13-19]. The LFQ of this theory alsorevels that the LF
theory possesses a set of 29 first-class constraints:
1=0


 (14a)

=0TC 


e mof the n
3c

 (14c)

2X

 


(14d)
4=0
T
P











=0
X






52
T
P





 
 (14e)

0,
2
i
ii
T
PX






 


= =2,3,,25.
i (14f)
where P, P, i
P, c ,


and
menta canonically conjugate respectively

 are th
to e mo-
,
X
X

,
,,
i
X
CA
 and
A
.
Accordingly the theory in both the cases seen to osss
three local gauge symmetries given by the two -
p e
dimen

=XX






hh
 




(15c)
=
hhh
sional WSRI and the WI defined by [ 1-19]
=XXX X





 (15a)
(15b)
=hh


h



 
 






 (15d)
=
A
AAA



 




=
(15e)
A
A






 (15f)



=CC CC




= CC



(15g)





hh
(15h)







The above theory is thus see
three local gauge symmetries d
sional WSRI and the WI in bot
In conclusion, the Polyakov
di
(15i)
n to be GI possessing the
efined by the two-dimen-
h the IF and LF dynamics.
D1 brane action in a d-
mensional courved background h


defined by S
 is
GI and it possesses the well-known three local string
gauge symmetries.
However, under conformal gauge-fixing, the CGFP-
D1BA is no longer GI as expectednd it also doesnot
possess the local string gauge symmetries being a
gauge-fixed theory.
a
Hovever, this GNI theory when con-
sidered in the presence of a contant background scalar
axiom field C and an


1U gauge field
A

it is seen
to become a GI theory possessing the three local string
gauge symmetries [19].
The scalar axion field C and the


1U gauge field
A

are seen to behave the WZ fields and the term like


A
L involving these fields is seen to behave like a WZ
term for the CGFPD1BA [19], which in the absence of
th seco
stra
does
is term is seen to possess a set of nd-class con-
ints and consequently describes a GNI theory which
not respect the local string gauge symmetries.
The situation in th e present case is analogous to a th e-
ory where one considerers the CGFPD1BA in the pres-
ence of a constant 2-form gauge field B


which be-
haves like a WZ field and the term involving this field
be
heory,”
[2] L. Brink and les of String The-
ory,” Plenum
y Particle
haves like a WZ term for the CGFPD1BA [13].
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