Notes on the WessZuminoWittenlike structure: L ∞ triplet and NSNS superstring field theory
Received: March
Notes on the WessZuminoWittenlike structure: triplet and NSNS superstring eld theory
Open Access 0 1 2
c The Authors. 0 1 2
0 Yukawa Institute for Theoretical Physics, Kyoto University
1 Na Slovance 2 , Prague 8 , Czech Republic
2 Institute of Physics, the Czech Academy of Sciences
In the NSNS sector of superstring eld theory, there potentially exist three nilpotent generators of gauge transformations and two constraint equations: it makes the gauge algebra of type II theory somewhat complicated. In this paper, we show that every NSNS actions have their WZWlike forms, and that a triplet of mutually commutative L products completely determines the gauge structure of NSNS superstring eld theory via its WZWlike structure. We give detailed analysis about it and present its characteristic
triplet; and; eld; theory; String Field Theory; Superstrings and Heterotic Strings

properties by focusing on two NSNS actions proposed by [1] and [2].
Contents
1 Introduction 2 3 4
Two triplets of L1
WZWlike action
Two constructions
A General WZWlike action based on (Lc; Lc~ ; Lp)
Introduction
alent to A1=L
1 actions given by [2]. In this paper, we focus on the NSNS sector and
1 triplet,
theories of [1{12].1
rst two
extend gauge symmetry. However, to be gauge invariant, a state
appearing in the action
must satisfy the constraint equation: H (z)
= 0 .
S[ ] =
where Q is the BRST operator and hA; Bi
hAjc0 jBi is the BPZ inner product with
c~0)insertion. An NSNS string eld
is total ghost number 0, leftmoving
denote gauge parameter elds. We thus have three nilpotent gauge
form. To see constraints, it is helpful to consider the kinetic term3 of the L
1 action [2],
S[ ] =
1, and
= 0 and
An NSNS string eld
is total ghost number 2, leftmoving picture number
rightmoving picture number
1 state satisfying two constraint equations:
= 0. One can
nd that if and only if
satis es constraints, the action has gauge
invariance under
= Q
where the gauge parameter
also satis es constraints:
= 0 and ~
= 0 . In [2], starting
1 relations,
and gave a full action whose interacting terms satisfy L
1 relations. When we include all
interactions, to be gauge invariant (or to be cyclic), a state
appearing in the L
must satisfy two constraint equations:
= 0 and ~
= 0 . From these analysis, we
1 action
state space consists of states belonging to the kernels of both
always impose (b0
= (L0
= 0 for all closed superstring eld
and ~ as the small Hilbert space HS. We
3Note that these two free actions are equivalent each other with linear partial gauge
xing or trivial
uplift. For example, recall that
of (1.1a) is obtained by an embedding of
of (1.2a) such as ~
= .
invariant action.
1 action as follows. Let ' be a dynamical string
eld. We rst consider a state
~['], which will be a functional of ', satisfying two constraint equations,
Then, using this
~['], a gauge invariant action whose onshell condition is given by
~['] = 0 ;
~['] = 0 :
~['{] = 0
taking ' =
of (1.2a), it reduces to the original L
is completely described by a triplet of L
1 product ( ; ~ ; LNS;NS). Likewise, every known
1 action of [2]. Namely, L
1 formulation
by their L
section 6 and appendix A.
2, we nd that the NSNS superstring product LNS;NS has two dual L
1 products:
We will see that as well as , ~, or LNS;NS, these L
Then, one can consider the constraint equations provided by these L
and L~ :
1 products have nice algebraic properties.
= 0 ;
~['{] ~ = 0 :
Using a state
~['] satisfying these constraint equations, which will be a functional of some
~['] = 0 :
which we explain in section 3. The L
1 triplet (L ; L~ ; Q) determines this WZWlike
struc~['], and we give two explicit forms of this key functional
~['] in section 4. As
which is our main result.
single functional form which consists of single functionals
~['] and elementally operators.
given in section 4. Thirdly, we clarify the relation to L
1 theory: we nd that our WZWlike
action and the L
1 action are o shell equivalent. Then we give a short discussion about
o shell duality of equivalent L
1 triplets. Finally, we discuss the relation to the earlier
on a general (nonlinear) L
1 triplet (Lc; Lc~ ; Lp) . We show that as well as other known
WZWlike actions, it also satis es the expected properties.
Two triplets of L1
In this section, we present two triplets of mutually commutative L
1 products. The L
We write the graded commutator of two coderivations D1 and D2 as
D1; D2
D1 D2
Note that it satis es Jacobi identity exactly (without L
1 homotopy terms):
D1; [[ D2 ; D3 ]] + ( )D1(D2+D3) D1; [[ D2; D3 ]]
Original L1 triplet: ( ; ~ ; LNS;NS).
the ofshell condition of the L
L1products ( ; ~ ; LNS;NS), which is the
As we explained, the constraint equations and
1 action is described by a triplet of mutually commutative
rst one of two L
1 triplets. The other L
let us recall how this L
1 product LNS;NS was constructed. In [2], they introduced a
generating function L(s; s~ ; t) for a series of L
1 products, and required that L(0; 0; 0)
determine the gauge invariance. They called this
(s; s~ ; t) as a gauge product. The
NSsolving the recursive equations,
; (s; s~ ; t) ;
~ ; (s; s~ ; t) ;
(s; s~ ; t), and
vice versa. This L(s; s~ ; t) is a series of L
1 products with operator insertions satisfying
(s; s~ ; t) can
be determined by solving the recursive equation,
L(s; s~ ; t) ; (s; s~ ; t) ;
which ensures L
superstring L
(s; s~ ; t) , the NSNS
L(s = 0; s~ = 0; t = 1) :
We write Ln for the nth product of LNS;NS as follows,
Ln A1 ; : : : ; ; An
1 LNS;NS A1 ^ : : : ^ An :
(s; s~ ; t), how
1 product
LNS;NS(t) =
LNS;NS(t) ; (t)
(s = 0; s~ = 0; t). Hence, we
LNS;NS = P! exp
1 algebras. In this form, L
form, we nd two dual L
1 products for LNS;NS and a dual of the L
1 triplet ( ; ~ ; LNS;NS) .
with two L
two dual L
these dual L
1 products
1 products as follows,
Gb ~ Gb 1
One can quickly nd that these products satisfy L
because of ( )2 = 0 for
= Gb ( Gb 1 Q Gb)
G 1 =
( Gb 1 Q Gb) Gb 1 =
because of [[ LNS;NS; ]] = 0 for
commutativity:
= 0
= 0 ;
= ; ~) :
It is owing to an invertible cohomomorphism Gb, and thus the L
1 triplet (L ; L~ ; Q) has
equivalence of WZWlike actions governed by equivalent L
this paper, we write the nth product of L
as follows,
1 triplets (See section 5.). In
[A1; : : : ; An] := 1 Gb
( = ; ):
Nilpotent relations and derivation properties
some details of related properties. The dual L
1 product L
= ; satis es L
permutation. Likewise, L
Q + Q L
Q A1; : : : ; An
+ X( )A1+ +Ak 1 A1; : : : ; QAk; : : : ; An
= 0 ;
where the upper index of ( )
A means the grading of A, namely, the total ghost number of
A. The commutativity L L~ + L~ L
= 0 provides
2 = 0 :
1; 2= ;e
= 0 ;
The lowest relation of (2.3c) is just
nd that the second lowest relation of (2.3c) is given by
A A ; B ~
+ ~ A ; B
+ ~ A ; B
A A ; ~ B
= 0;
algebra of L for
= ; ~,
MCL (A)
zA ; :}:: ; A{
and in the WZWlike action. Likewise, we often refer MCQ(A)
element for Q . There is an natural operation, a shift of the products, in L
QA as the MaurerCartan
1 algebras. For
any state A, the Ashifted products are de ned by
One can check that with MCL (A), the Ashifted products satisfy weak L
1 relations:
X( )j jh B (1); : : : ; B (k) A; B (k+1); : : : ; B (n) A
i
MCL (A); B1; : : : ; Bn A:
then the Ashifted products exactly satisfy the L
1 relations. Similarly, one can consider
1; 2= ;e
1
B (1) ; : : : ; B (k) A ; B (k+1) ; : : : ; B (n) A
MCL (A) ; B1 ; : : : ; Bn A
MCL~(A) ; B1 ; : : : ; Bn A
WZWlike action
we need are two functional elds and their algebraic relations.
Algebraic ingredients
A functional eld
['] satisfying these constraint equations plays the most important
role, which we call a puregaugelike (functional)
eld. With this functional
~['], the
' of the theory.
WZWlike functional eld. Let
~['] be a Grassmann even, ghost number 2,
leftmoving picture number
1, and rightmoving picture number
1 state in the
leftandright large Hilbert space:
~ satis es the constraint equations:
~ 6= 0. We call this
~ a puregaugelike (functional) eld
= 0;
{ ~ = 0:
In other words,
~['] gives a solution of the MaurerCartan equations for the both dual
products (2.1a) and (2.1b). Therefore, two
~[']shifted products again have L
1 relations
and commute each other. One can de ne two linear operators D
and D~ acting on any
state A by
( = ; );
and two bilinear products of any states A and B by
( = ; ):
+ D A ; B
A A ; D B
= 0:
(D )2A = 0 ;
( = ; ):
D D~ + D~ D
A = 0;
D A ; B ~
A A ; D B ~
+D~ A ; B
D~ A ; B
A A ; D~ B
= 0 :
L~ : namely,
D D A1; : : : ; An
X( )D(A1+ +Ak 1) A1; : : : ; D Ak; : : : ; An
( = ; )
dynamical string eld satisfy the Leibniz rule for these L1products L
and L~, one can
nd D (D
~) = 0 and D~(D
imply that with some (functional) state
D['] belonging to the leftandright large Hilbert
space H, we have
D D
['] = D D~ D['] ;
the WZWlike relation. Note that the existence of the (functional) state
D['] is ensured
Hilbert space H . We call this
D['] satisfying (3.3) as an associated (functional) eld.
and rightmoving picture p, the associated eld
D['] has the same quantum numbers: its
We started with the L
1 triplet (L ; L~ ; Q) and obtained the above algebraic
ingre? As we
~['] = 0 :
relations in NSNS superstring eld theory.
Action, equations of motion, and gauge invariances
Let ' be a dynamical NSNS string
functionals of given dynamical string
eld ', we can construct a WZWlike action for
['] and
D['] as
NSNS string eld theory:
S ~['] =
t['(t)]; Q
~['(t)] ;
4In section 5, we will see this fact again.
t['(t)] denotes
eld. As we will see, using the variational associated (functional) eld
D['] with D = ,
the variation of this action is given by tindependent form:
S ~['] =
['] = D
The equation of motion is given by tindependent form
['] = D D~ Q['] =
Q['] = 0:
which we explain in the rest.5
Variation of the action
D0 A ; B
= ( )D0A A ; D0 B ;
A ; B ; C
= ( )AB B ; A ; C
= ( )A(B+C) B ; C ; A
= 0 ;
( = ; ~):
t and B = D
, the relation (3.2d) provides
+ [D~A; B]
+ [D A; B]~
+ [A; D~B]
= 0:
We prove that when we have WZWlike functional elds
~['] and
D['] which
a direct computation of the variation of the action:
S ~['] =
t['(t)]; Q
~['(t)] +
5These computations are similar to those of the earlier WZWlike action [9].
B's total ghost, leftmoving picture, and rightmoving picture numbers are 3,
1, respectively.
~)i plus extra terms:
~) = h t; Q D~D
= h t; D~D Q
= h
~ i + h Q; D~D [ t; D
Q; [D~D
+ [D~ t; D
Likewise, we nd the rst term of the variation becomes h@t
~i plus extra terms:
~ =
t; D~D
+ D~[ t; D~D
+ D~[ t; D~D
t; D~D
= h@t
+ D~[D~D
t; D~D
= @t
Q; D~ [D
+ [ t; D~D
+ D~[ t; D~D
(3:11a) + (3:11b) = @t
S ~['] =
['(t)]; Q
~['(t)] =
In summary, for xed L
1 triplet (L ; L~ ; Q), we rst consider a functional
~
satisfying constraint equations (3.1a) and (3.1b) de ned by two of it, L
and L~. Next, using
which gives a half input of the action. Lastly, using
~, we consider the MaurerCartan
element of the remaining L
1 product Q, which provides the onshell condition (3.4) and
WZWlike action (3.5).
As we showed in section 3, when two states
D['] satisfying (3.3) are obtained,
In this section, we present two di erent expressions of these
D using two di erent
dynamical string elds
. It gives two di erent realisations of our WZWlike action,
Through these constructions, we also see that once we have
~['] explicitly as a
functional of ', the other functional
D['] can be derived from
~[']. It would suggest
next section.
for a NSNS dynamical string eld belonging to the small Hilbert space:
= 0
= 0. This
is a Grassmann even, total ghost number 2, leftmoving picture number
1, and rightmoving picture number
1 state.
Puregaugelike (functional) eld
As a functional of , the puregaugelike
~[ ] can be constructed by
1 Gb e^
moving picture numbers and this
~[ ] has correct quantum numbers as a
puregaugeequations (3.1a) and (3.1b).
~[ ] is given by using the grouplike element, the following relation holds:
e^ ~['] = 0 ;
( = ; ~):
Because of (2.1a) and (2.1b), one can quickly nd that (4.1) satis es
e^ ~[ ] = ( Gb
= Gb
= 0 ;
= ; ~) ;
equality, we used the properties of the dynamical string
= 0 and ~
= 0.
, it is the origin of all algebraic relations
of WZWlike theory.
Associated (functional)
D[ ]. Similarly, as functionals of , the associated
(functional) eld
D =
D[ ] with D = @t or D =
can be constructed by
and the associated (associated) eld
Q[ ] can be given by
1 Gb
1 Gb Q ~ e^
where Q ~ is a coderivation operation which we will de ne below.
Recall that
~ satis es the constraint equations (3.1a) and (3.1b), and thus D
D satisfying (3.3). One can
derive an explicit form of the functional
D[ ] from
~[ ] in this manner.
Using the graded commutator of two coderivations D1 and D2,
D1 ; D2
D1 D2
= ; ~ . Note that I
= Gb
= 0
; Gb 1 D Gb
= 0 :
Namely, the coderivation Gb 1 D Gb commutes with both
exactness and ~exactness, there exist a coderivation D ~ such that
and ~ . Hence, because of
Using D ~ and the properties of the dynamical string eld,
= 0 and ~
= 0, we nd
; [[ ~ ; D ~ ]]
1 Gb D ~ (e^ ) ^ e^ 1 Gb(e^ ) ^ e^ 1 Gb(e^ ) :
Note that with (4.1), the linear operator D for
= ; can be written as
= 1L
= ; e) :
We thus nd that if we de ne the associated eld
D[ ] by the following functional of ,
deed holds:
1 Gb D ~ (e^ ) ;
[ ] =
D D~ D[ ]:
We write
Hilbert space:
6= 0, ~ 6= 0, and ~ 6= 0. This
has total ghost number 0, leftmoving
picture number 0, and rightmoving picture number 0.
Puregaugelike (functional) eld
the following di erential equation,
~[ ]. Let us consider the solution
with the initial condition
D ( )A
; ( = ; ~) :
A puregaugelike (functional) eld
~[ ] is obtained as the
= 1 value solution
~[ ; ] = D ( )D~( )
[ = 1; ]:
We check that this
~[ ] satis es (3.1a) and (3.1b). For this purpose, we set
( = ; ~):
Because of the initial condition
and (2.4a), we obtain the following linear di erential equation
MCL ( ) = D ( ) @
= ( )j j MCL ( ) ; D~( )
where ( )j j denotes
and +1 for
and (3.1b) and gives a proof that (4.4) is a puregaugelike (functional)
eld. By the
~[ ] = ~
In this parametrisation, the properties of the dynamical string eld
L~ and L~, and to have a puregaugelike eld
[ ] as a functional of
. (Note that
~ 6= 0 makes possible
Associated (functional) eld
We consider the following di erential equation
D[ ; ] = ( )DD
with the initial condition
(functional) eld
D[ ] is obtained by the
= 1 value solution of (4.6),
D[ = 1; ] :
As D exacts and D~exacts does not a ect in the rst slot of (3.5), this
D is determined
up to these. To prove (4.7) satisfy (3.3), we set
D D~ D[ ; ] + ( )
D[ ]. Using (3.2) and (4.3), we nd
= n
+ D D~ @
= D D~ ; D~ D
+ D D~ @
= D~ ; I( )
+ D D~ @
= D D~
+ D D~ @
+ (D )
= D D~
From the third equal to the forth equal, we used the following identity:
D[ ; ] satis es (4.6) up to D exacts and D~exacts, we have
@ I( ) =
D~( )A ; B
t[ (t)] =
@t (t) +
(t); ~ @t (t)
On the D exacts and D~exacts.
We found a de ning equation (4.6) of
D[ ]. Since
the following identity
which provides another expression of (4.8):
@ I( ) =
It ensures that as a de ning equation of
D[ ; ], we can also use
+ D D~ @
D =
D~( )D ( ), one may
compute as
@ I( ) = @
+ D D~ @
However, we have the following identity
Comparing (4.8) and (4.10) with (4.9), we also nd
0 =
On the small associated
We constructed two functionals
~['] and
D[']. It is su cient to give a WZWlike
D~ D[']:
~['] = 0;
~['] = 0:
( )D1D2D2D1, one can nd8
~ = D
D~ = D~
D1 D2 ~ ( )D1D2D2 D1 ~ ( )D1
= D exact;
= D~exact:
Smallspace parametrisation. It is easy to obtain these in terms of
because the
D are given by
1 Gb D e^ ; D[ ] = 1 Gb D ~ e^ ;
where we used coderivations D and D ~ such that
D[ ] as the
= 1 value solutions,
D~[ = 1; ];
D[ = 1; ];
of the following di erential equations
D~[ ; ] = D D~( )
D[ ; ] = D D ( )
8They follow from direct computations
D1D2 ~ = ( )D2D1 D
with the initial conditions
D~[ = 0; ] = 0 and
D[ = 0; ] = 0. The minus sign
the equation
= [D D~ ;
= D~ @
We can therefore obtain
D~ satisfying (4.12) without using
D and (4.11).
When we start with
D and (4.6), does D
D of (4.11) satisfy the above di
eren
D ( ) D[ ; ] =
+ D~ D
D[ ] and start with these di erential equations, can we
On the D exactness and D~exactness.
We can only specify the large associated
(functional) eld
D up to D  and D~exact terms, and these ambiguities do not
conhave operators F
and Fe ~ de ned by
These F
as follows,
which satisfy D F + F D
and Fe ~ consist of the puregaugelike (functional) eld
and ~. Using these pieces, one can construct
~['] and operators L ,
D['] via
D['] and
D~['] =
we see in the next section.
The author thanks to T.Erler for comments.
Single functional form
As we found, two or more types of functional elds
D['] appear in the WZWlike
consists of the single functional
~['] and elementally operators. It may be helpful in the
gauge xing problem.
! : : : (exact) ;
Furthermore, since
= 0,
= 0, ~ +
~ = 0, and
= 0 hold, we
have the direct sum decomposition of the large state space H as follows:
H =
Likewise, the existence of (4.13) satisfying D F + F
= 1 and D~ Fe ~+ Fe ~D~ = 1
! : : : (exact) ;
decomposition using these exact sequences? To achieve this, we consider
Fe ( Fe 1
One can quickly nd that as well as (4.13), this F and its inverse F
1 also provide
= F
D~ = F ~ F
D F + F D = 1 ; D~ F~ + F~ D~ = 1 ; ( F
Furthermore, now, these operators all are constructed from single F , we have
D D~ + D~ D
= 0; D F~ + F~ D
= 0; D~ F + F D~ = 0;
F F~ + F~ F
= 0;
H = D D~ H
F F~ H :
Since Q
~ = D F D~F~(Q
~) and D~D
t = @t
~, using this F , we nd
S ~['] =
t['(t)]; Q
~['(t)]; F F~ Q
~['(t)] :
It consists of the single functional
~['] and elementary operators L , L~, , , ~, ~, and
F [[D; D ]]F
+ [[D ; F
D F ]] .
Equivalence of two constructions
equivalence of S ~[ ] and S ~[ ] follows if we consider the identi cation
~[ ] =
which consists of
~ and elementally operators.
Since both actions have the same
See also [1, 19{22].
their Fock spaces
Field relation.
Note that the identi cation of states (5.2) provides the identi cation of
e^ ~[ ] = e^ ~[ ] ;
Under the identi cation (5.2), by acting @t, we have
t[ ] =
t[ ] + D exacts + D~exacts:
^ t[ (t)] = Gb e^ (t)
= e^ ~[']
~['] = e^ ~[']
Since cohomomorphism Gb is invertible, we obtain the following eld relation
= 1
= 1
which can be directly derived from (5.2).
Relation to L1 theory
We write
0 for the dynamical string
eld of the L
1 action proposed in [2]. As well as
belongs to the small Hilbert space:
dynamical string eld , we constructed an action
We will show that this S [ ] is exactly o shell equivalent to the L
S ~[ ] =
1 Gb
@t (t) ^ e^ (t) ; Q 1 Gb e^ (t)
SL1 [ 0] =
1 z
n=1 (n + 1)! h ~ 0; Ln+1( 0; :}:: ; {0; 0) :
1 action,
0(t) be a path connecting
0(0) = 0 and
0(1) =
0, where t 2 [0; 1] is a real
SL1 [ 0] =
SL1 [ 0(t)] =
@t 0(t); 1LNS;NS e^ 0(t) :
we obtained a proof that the L
1 action SL1 [ 0] proposed in [2] is equivalent to our S ~[ ].
S ~[ ] and S ~[ ] both are equivalent to that of L
1 formulation. See also [1, 22]
by ( ; ~ ; LNS;NS). We thus consider a functional
WZWlike reconstruction of L1 action. In the L
~['] which satis es two constraint
equations de ned by
1 (e^ ~[']) =
1 ~ (e^ ~[']) = ~
~['] = 0;
~['] = 0:
~['] =
~) = 0 and ~ (D
D['], we have the
WZWlike relation,
andright large Hilbert space. Using
~['], we can consider the MaurerCartan element
for the remaining L
1 products LNS;NS :
1 LNS;NS(e ~[']) = Q
Note that there also exists an associated eld
L['] such that
1 LNS;NS(e^ ~[']) =
SL1 ['] =
t['(t)]; 1 LNS;NS(e^ ~['(t)])
t['(t)]; ~ L['(t)] :
WZWlike manner. In particular, since
and ~ are linear L
1 products, their shifted
prodWe notice that if we set ' =
= ~
of the functional,
, because of the triviality of  and ~cohomology. Similarly,
if we use ' =
, it also implies
~ . While its smallspace parametrisation is
just the L
smallspace one.
O shell duality of L1 triplets.
As we mentioned, when Gb is cyclic in the BPZ
inner product, (2.2) ensures not only the equivalence of L
1 triplets but also the o shell
appearing the action of [10, 12] because of their smallspace constraints.
Cartanlike element in the correlation function :
MC (A)
thus obtain
MCQ(A) =
MCQ(A) =
MCL(A0) ;
t~ t[']. Using a measure factor d
dt @ t~ , we can express the WZWlike action (3.5) as
S ~ =
MCQ(A) ;
SL1 =
MCL(A0) :
~['] + t~ t['], the WZWlikely extended L
1 action (5.7) can be written as
(L ; L~ ; Q) and the (WZWlikely extended) L
1 triplet
1 action (5.7) based on the L
WZWlike structure.
Relation to the earlier WZWlike theory
eld. Let GL be a state which has ghost number
2, leftmoving picture number 0, and rightmoving picture number
1 state in the large
Hilbert space. When this GL satis es
;NS = 0;
~ GL = 0;
and ~ : namely D L ;NS
( )DL ;NS D = 0 and D ~
( )D ~ D = 0. For example, one
we consider
( )DD GL =
0D['0] is a functional of the
dynamical string
eld, which has the same ghost, leftmovingpicture, and
rightmovingpicture numbers as d. We call this
0D['0] satisfying (5.12) as an associated (functional)
of (5.11a) and (5.11b).
In [9], using these GL['0] and
D['], a WZWlike action was given by
0t['0(t)];
GL['0(t)] :
We write
0t['0(t)] for the associated eld
While the
0 in the leftandright large Hilbert space in [9], if one
on its WZWlike structure.
0 with the smallspace string eld ,
0 = ~
the action (5.13) reduces to the L
1 action based on their asymmetric construction of [2].
lower order.
Conclusion
We presented that a triplet of mutually commutative L
1 products (Lc; Lc~ ; Lp) completely
constraint equations,
1 Lc e^ cc~['] =
1 Lc~ e^ cc~['] =
cc~[']; : : : ; cc~['] = 0;
cc~[']; : : : ; cc~['] = 0;
and introducing a functional
cc~['] of some dynamical string eld ' satisfying these
conScc~['] =
MCLp (A) =
t['(t)]; 1Lp e^ cc~['(t)] ;
1 Lp e^ cc~['] =
cc~[']; : : : ; cc~['] = 0 :
One can prove its gauge invariance using the functional
cc~['] and algebraic relations
derived from the mutual commutativity of the L
1 triplet (Lc; Lc~ ; Lp),11 without using
and ( ; ~ ; LNS;NS) which provide the L
form, one can say that to study its L
1 triplet is equivalent to know the gauge structure of
NSNS superstring eld theory. In this paper, we focused on two L
1 triplets (L ; L~ ; Q)
1 action of [2]. Particularly, we presented detailed
WZWlike theory.
Acknowledgments
Republic, under the grant P201/12/G028.
General WZWlike action based on (Lc; Lc~ ; Lp)
(Lc; Lc~ ; Lp) . In this appendix, we prove that the general WZWlike action,
Scc~['] =
t['(t)]; 1Lpe^ cc~['(t)] ;
is expected from the result of [19]. Actually, with deep insights, one can
nd a pair of (nonlinear) A1
sectors [24].
has topological parameter dependence: its variation is given by
Scc~['] =
is invariant under the gauge transformations generated by Lc , Lc~, and Lp ,
Then, because of the nilpotency of L
1 triplet (Lc; Lc~ ; Lp) , the general WZWlike action
['] = 1 Lce^ cc~[']
any coderivation D commuting with Lc and Lc~ , we nd
( )DD Lc0e^ cc~['] = Lc0e^ cc~[']
^ 1De^ cc~['] = 0 ;
Hence, since  and ~cohomology are trivial, there exist a state
DcDc~ D[']
1LcLc~e^ cc~[']
where we de ned Dc0A
like relation for a general L
1 triplet (Lc; Lc~ ; Lp) , which provides
cc~ =
@ cc~ =
which are mutually commute with Lc and Lc~ , we nd
1 D1 D2 e^ cc~['] = 1 D1 e^ cc~[']
^ 1( )D2Lc Lc~ e^ cc~[']
(c0 = c ; c~) :
D['] such that
= ( )D2 1Lc Lc~ D1e^ cc~[']
= ( )D2 1Lc Lc~ e^ cc~[']
^ 1D1 e^ cc~['] ^ D2[']
+ ( )D2 1Lc Lc~ e^ cc~[']
cc~) , (3.2), and (3.9) . Using these, we nd a half of the variation is
t; 1 Lp e^ cc~ ^ DcDc~
; Dc~Dc 1 Lp e^ cc~ ^ t
; 1LcLc~ e^ cc~ ^ DcDc~ Lp ^ t :
We notice that these computation can be carried out by replacing Q
~ = D D~ Q
t; 1 Lpe^ cc~ =
= @t Dc~Dc
1LcLc~ e^ cc~ ^ DcDc~
; 1LcLc~ e^ cc~ ^ DcDc~ Lp ^ t :
Note that this term can be also obtained by replacing
Q of (3.11b) with
Lp . Hence, we
obtain the desired result
['(t)]; 1Lpe^ cc~['(t)] :
1 triplet (Lc; Lc~ ; Lp) in the completely same way. In general,
eld rede nitions Ub
terms of L
1 algebras, it is just described by an L
1 morphism between two L
1 triplets,
under any string
eld rede nitions. Thus, as a gauge theory, it may capture general eld
theoretical properties of superstrings.
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