Nonlinear gauge invariance and WZWlike action for NSNS superstring field theory
JHE
Open Access 0 1
c The Authors. 0 1
0 Komaba , Meguroku, Tokyo 1538902 , Japan
1 Institute of Physics, University of Tokyo
We complete the construction of a gaugeinvariant action for NSNS superstring field theory in the large Hilbert space begun in arXiv:1305.3893 by giving a closedform expression for the action and nonlinear gauge transformations. The action has the WZWlike form and vertices are given by a puregauge solution of NS heterotic string field theory in the small Hilbert space of right movers.
String Field Theory; Superstrings and Heterotic Strings; Gauge Symmetry

Nonlinear gauge invariance and
WZWlike action for
NSNS superstring field theory
1 Introduction
Nonlinear gauge invariance
Quartic vertex
WZWlike expression
Coalgebraic description of vertices
Gaugeinvariant insertions
NS string products
WessZuminoWittenlike action
Nonlinear gauge invariance
Gaugeinvariant insertions of picturechanging operators
B Some identities
A Heterotic theory in the small Hilbert space
Introduction
While bosonic string field theories have been wellunderstood [1–8], superstring field
theories remain mysterious. A formulation of supersymmetric theories in the early days [9],
which is a natural extension of bosonic theory, has some disadvantages caused by
picturechanging operators inserted into string products: singularities and broken gauge
invariances [10]. To remedy these, various approaches have been proposed within the same
There exists an alternative formulation of superstring field theory: large space
theory [18–26]. Large space theories are formulated by utilizing the extended Hilbert space of
operators. One can check the variation of the action, the equation of motion, and gauge
invariance without taking account of these operators. Of course, the action implicitly
includes picturechanging operators, which appear when we concretely compute
scattering amplitudes after gauge fixing. The singular behaivor of them is, however, completely
regulated and there is no divergence [28, 29].
The cancellation of singularities can also occur in the small Hilbert space. Recently,
by the brilliant works of [30, 31], it is revealed how to obtain gaugeinvariant insertions
of picturechanging operators into (super) string products in the small Hilbert space: the
NS and NSNS sectors of superstring field theories in the small Hilbert space is completely
formulated. In this paper, we find that using the elegant technique of [31], one can construct
the WZWlike action for NSNS superstring field theory in the large Hilbert space.
A puregauge solution of smallspace theory is the key concept of WZWlike
formulation of NS superstring field theory in the large Hilbert space, which determines the vertices
of theory, and we expect that it goes in the case of the NSNS sector. There is an attempt
to construct nonvanishing interaction terms of NSNS string fields utilizing a puregauge
solution GB of bosonic closed string field theory [22]. However, the construction is not
complete: the nonlinear gauge invariance is not clear and the defining equation of GB is
ambiguous. To obtain nonlinear gauge invariances, we have to add appropriate terms to
these interaction terms defined by GB at each order. Then, the ambiguities of vertices are
removed and we obtain the defining equation of a suitable puregauge solution GL, which
we explain in the following sections.
In this paper, we complete this construction begun in [22] by determining these
additional terms which are necessitated for the nonlinear gauge invariance and by giving
closedform expressions for the action and nonlinear gauge transformations in the NSNS
sector of closed superstring field theory. We propose the action
S =
the NS heterotic string equation of motion in the small Hilbert space of right movers. The
action has the WZWlike form and the almost same algebraic properties as the largespace
action for NS open and NS closed (heterotic) string field theory [20, 21].
This paper is organized as follows. In section 2 we show that cubic and quartic actions
can be determined by adding appropriate terms and imposing gauge invariance. In section
3, we briefly review the method of gaugeinvariant insertions of picturechanging
operaIn section 4, first, we give the defining equation of GL and associated fields which are
necessitated to construct the NSNS action. Then, we derive the WZWlike expression for the
string equation of motion just as other largespace theories. We end with some conclusions.
Nonlinear gauge invariance
string field theory in powers of κ: S = α2′ P
number 0, and rightmover picture number 0 state. The free action S2 is given by
S2 =
use the symbol (Gp, p˜) which denotes that the total ghost number is G, the leftmover
picture number is p, and the rightmover picture number is p˜. Then, ghostandpicture
that the inner product hA, Bi gives a nonzero value if and only if the sum of A’s and B’s
S2 is invariant under the gauge transformation
We would like to construct nonzero and nonlinear gaugeinvariant interacting terms
invariance remains to be linear in our interacting theory. Consequently, with two nonlinear
and one trivial gauge invariances, the theory is WessZuminoWittenlikely formulated.
In section 2, starting with the action proposed in [22] and adding appropriate terms at
Cubic vertex
zero modes of left and rightmoving picturechanging operators respectively. The (n +
2)to the result of first quantization, where [A1, . . . , An] is the bosonic string nproduct and
gauge invariance not clear. In this subsection, as the simplest example, we show that a
gaugeinvariant cubic term S3 can be obtained by adding appropriate terms to
Cyclicity. It would be helpful to consider the cyclicity of vertices. A (n + 1)point vertex
Vn is called a (BPZ) cyclic vertex if Vn satisfies
1
Sn+1 is given by Sn+1 = (n+1)! hΨ, Vn(Ψn)i using a cyclic vertex Vn, its variation becomes
For example, bosonic string field theories and superstring field theories in the small Hilbert
Next, we consider the following case: a vertex Vn′ is not cyclic but has the following
Nonzero Wn implies that the cyclicity of Vn is broken. For instance, this relation holds
this case, in general, a zero divisor of Vn′ + Wn gives the generator of gauge
transformations. However, when Wn consists of lower Vk<n, there exists a special case that the zero
divisor of Vn gives the generator of gauge transformations just as WZWlike formulations
of superstring field theories in the large Hilbert space, which is the subject of this paper.
Adding terms.
We know that naive insertions of operators which do not work as
derivaclear. We show that a cubic vertex satisfying the special case of (2.6) can be constructed
by adding appropriate terms and by imposing gauge invariances. Computing the variation
of (2.3), we obtain
this term is given by
= hδΨ, ηn[QQeΨ, η˜Ψ] + Qe[QΨ, η˜Ψ] + [Qη˜Ψ, QeΨ]
Averaging these three terms, we obtain the cubic action satisfying (2.6)
S3 =
The variation of this cubic action is given by
3 Qe[QΨ, η˜Ψ] + [QQeΨ, η˜Ψ] + [Qη˜Ψ, QeΨ] i
3 Qe[Ψ, ηQη˜Ψ] − [QeΨ, ηQη˜Ψ] − [η˜Ψ, ηQQeΨ] i
Qgauge transformation of S3 is given by
Note that the zero mode Xe of the rightmover picturechanging operator is inserted cyclicly.
We find that under the following second order gauge transformation
define the following new string product which includes the zero mode Xe of the rightmover
picturechanging operator
[A, B]L :=
Xe[A, B] + [XeA, B] + [A, XeB] .
This new product [A, B]L satisfies the same properties as original product [A, B], namely,
that when we use this new product, the cubic term S3 of the action is given by
S3 =
under the following Qgauge transformations
Quartic vertex
We can construct the quartic term S4 and, in principle, higher interaction terms Sn>4 by
for example, as follows
Therefore, we have to consider the following terms
[A, B]L, C L =
Xe [XeA, B] + [A, XeB], C + [XeA, B] + [A, XeB], XeC
Xe[XeA, B] + Xe[A, XeB], C + Xe[A, B], XeC + Xe Xe[A, B], C + Xe2[A, B], C ,
quartic term S4 has to include the following types of terms
Of course, we can repeat similar computations of above terms as we did in subsection 2.1.
However, there exists a reasonable shortcut. Note that, for example, the following relation
+ Xe ξ˜[QA, B], C + (−)AXe ξ˜[A, QB], C
+ (−)A+BXe ξ˜[A, B], QC + Xe Xe[A, B], C
= 0.
Hence, the three product L2+2(A, B, C) defined by
L2+2(A, B, C) := 9 · 2
1 n2 ξ˜Xe[A, B], C − ξ˜ Xe[A, B] + [XeA, B] + [A, XeB], C
− Xe [ξ˜A, B], C − Xe[ξ˜A, B], C − [ξ˜A, B], XeC
− (−)A
− (−)A+B
+ (B, C, A)terms + (C, A, B)terms
This new three product L2+2(A, B, C) possesses the symmetric property and the
deriva
1 n
3 · 2 Xe [A, B], C + ξ˜ [XeA, B] + [A, XeB] , C + [A, B], XeC o
L1+3) = 0.
Note that the following types of products have the Qderivation property
N (A, B, C) = Xe ξ˜[A, B], C + ξ˜ Xe[A, B], C ,
and N types of products, whose coefficients are fixed by the cancellation of the second line
of (2.29) and the sum of N type products:
L1+3(A, B, C) :=
+ (−)A [A, ξ˜XeB], C + (−)A+B [A, B], ξ˜XeC o
8 Xe XeA, B, C + Xe A, XeB, C + Xe A, B, XeC
+ XeA, XeB, C + XeA, B, XeC + A, XeB, XeC o
+ Xe [ξ˜A, B], C − [ξ˜A, XeB], C − [ξ˜A, B], XeC
+ (−)A Xe [A, ξ˜B], C − [XeA, ξ˜B], C − [A, ξ˜B], XeC
+ (B, C, A)terms + (C, A, B)terms .
define the following new three string product
[A, B, C]L := L2+2(A, B, C) + L1+3(A, B, C),
which satisfies the same properties as the original three product [A, B, C], namely, the
Quartic vertex S4. Let us now consider the quartic vertex having the property (2.6) and
is necessitated. The variation of this term becomes
Hence, the quartic term S4 defined by
S4 =
has the property (2.6) and its variation is given by
we obtain the third order Qgauge transformation
In principle, we can construct higher vertices S5, S6, . . . by repeating these steps at each
because higher order vertices consist of a lot of terms and each term has complicated
operator insertions. To obtain a closed form expression of all vertices in an elegant way,
we necessitate another point of view, which we explain in section 3.
Gaugeinvariant insertions of picturechanging operators
In this section, we briefly review the coalgebraic description of string vertices [32–34] and
Coalgebraic description of vertices
Let T (H) be a tensor algebra of the graded vector space H. As the quotient algebra of T (H)
S(H) is graded commutative and associative as follows
A1A2 = (−1)deg(A1)deg(A2)A2A1,
A1(A2A3) = (A1A2)A3,
Fock space of superstrings, and the Z2 grading, called degree, is just Grassmann parity.
The product of two multilinear maps L : Hn → Hl, M : Hm → Hk also becomes a map
L · M : Hn+m → Hk+l which acts as
L · M (A1A2 · · · An+m) =
X(−)σ(n,m)L(Aσ(1) · · · Aσ(n)) · M (Aσ(n+1) · · · Aσ(n+m)). (3.2)
Note that the nproduct of the identity map I : H → H on symmetric algebras is different
from the ntensor product or the identity In on Hn:
In ≡
1 Iz ·}·· {I = Iz ⊗ ·}·· ⊗ I .
In other words, In is the sum of all permutations, In gives the sum of equivalent
permutations, and Ik · Il is equivalent to the sum of different (k, l)partitions of k + l.
Hn → H. We can naturally define a coderivation Ln : S(H) → S(H) from the map
LnA = (Ln · IN−n)A,
LnB = 0,
or, more simply,
Gaugeinvariant insertions
consider a series of these inserted products
[[L(s), L(s)]] = 0,
L(s) =
L[m](t) :=
[[L1, Ln]] + [[L2, Ln−1]] + · · · + [[Ln, L1]] = 0,
where s is a real parameter and L(s) is the generating function for the bosonic string
N+1 be a (N + 1)product including ninsertions of picturechanging operators. We
where t is a real parameter and L(mn+)n+1 acts on S(H) as (3.4). Note that the upper index
[m] on L[m](t) indicates the deficit in picture number of the products relative to what is
needed for superstrings: L[0](t) is the sum of all superstring products and L[m](0) is the
(m + 1)product of bosonic strings. To associate the generating functions L[0](t) of the NS
superstring products with (3.8), we define the following generating function
L(s, t) :=
X smL[m](t) =
where s is a real parameter. Note that powers of t count the picture number, and powers of s
count the deficit in picture number. These two parameters t and s connect the generating
L(s, 0) for the bosonic string products.
Starting with these relations, we can construct the NS superstring products L(0, t)
satisderivation conditions
[[L(s, 0), L(s, 0)]] = 0,
[[L(0, t), L(0, t)]] = 0,
we have to solve the following pair of differential equations
The solution of this pair of differential equations generates all products including
appropriate insertions of picturechanging operators.
Further, the equation (3.15) leads to the equation
Therefore, when L(s, t) satisfies the equation (3.16), we obtain
[[L(s, t), L(s, t)]]
1 ∂
2 ∂s
string products. As a result, the pair of equations (3.15) and (3.16) generates inserted
Expanding (3.15) and (3.16) in powers of (s, t), we obtain the following formulae
L(mn++n1+)2 =
k=0 l=0
[[η, Ξm+n+2]] = (m + 1)L(mn+)n+2,
(n+1)
L(kk+)l+a, Ξm+n+2−k−l ,
(n−k+1)
m + n + 3
ξL(mn+)n+2 − L(mn+)n+2 ξ · Im+n+1
Therefore, we can always derive explicit forms of these inserted products as follows:
L(n0+)1 = given,
L(n2+)1 =
L(nn+)1 =
n − 1
L(n1+)1 = [[Q, Ξ(n1+)1]] + [[L(20), Ξ(n1)]] + · · · + [[L(n0), Ξ(21)]],
+ [[L(n0−)1, Ξ(32)]] + [[L(n1−)1, Ξ(31)]] + [[L(n1), Ξ(21)]] ,
[[Q, Ξ(nn+)1]] + [[L(21), Ξ(nn−1)]] + · · · + [[L(nn−1), Ξ(21)]] .
X2 A, B, C + [X2A, B, C, ] + [A, X2B, C, ] + [A, B, X2C]
For example, we find that the lowest inserted product L(1)
is given by
is given by
X[A, B] + [XA, B] + [A, XB] ,
L(32)(A, B, C) =
where L(30)(A, B, C) = [A, B, C] and
+ (−)A[[A, ξXB], C] + (−)A+B[[A, B], ξXC]o
X[XA, B, C] + X[A, XB, C] + X[A, B, XC]
+ [XA, XB, C] + [XA, B, XC] + [A, XB, XC]o
+ (−)A+B X[[A, B], ξC]−[[XA, B], ξC]−[[A, XB], ξC] o
+ (B, C, A)terms + (C, A, B)terms .
− (−)A
− (−)A+B
+ (B, C, A)terms + (C, A, B)terms
NS string products
The generating function L(0, t) of the superstring products, as well as that of bosonic ones
L(s, 0), has nice properties, which we explain in this subsection. Note that the above
this sense, we write L(nn+,01) for this L(nn+)1, an NS superstring product with insertions of
left moving picturechanging operators. L(nn+,01) gives the (n + 1)product of NS (heterotic)
superstring field theory in the small Hilbert space of left movers [31].
By construction, we can also obtain an NS superstring product L(n0+,n1) with insertions of
[A0, A1, . . . , An]L := L(n0+,n1)(A0, A1, . . . , An).
X(−1)σ(A) [Aσ(1), . . . , Aσ(k)]L, Aσ(k+1), . . . , Aσ(n)
L = 0.
Let G be a ghostandpicture
BRST operator QG and shifted rightmover NS string products
QG A ≡ [A ]GL := QA +
[A1, . . . , An]GL :=
F (G) ≡ QG +
n=1 (n + 1)!
Gn, G L = 0,
in the same manner as shifted bosonic string products. Provided that the state G shifting
theory in the small Hilbert space of right movers
X(−1)σ [Aσ(1), . . . , Aσ(k)]GL, Aσ(k+1), . . . , Aσ(n) GL = 0.
Then, QG becomes a nilpotent operator.
WZWlike expression
In this section, first, we gives the defining equations of a formal puregauge GL and
associents of our construction. Then, we present a closed form expression of WZWlike action for
NSNS string field theory, the equation of motion, and the gauge invariance of the action.
A puregauge GL of rightmover NS theory.
We can build a formal puregauge
solution GL of NS heterotic string field theory in the small Hilbert space of rightmovers
ψX[τ ] = X η˜Ψ + κ η˜Ψ, ψX[τ ] GLL[τ]
socalled an associated field, satisfying
ΨX[τ ] = (−1)XXΨ + κ η˜Ψ, ΨX[τ ] GLL[τ],
are given by
+ . . . . (4.10)
which appears in the action for NSNS string fields with general tparametrization. Note
WessZuminoWittenlike action
Let GL = P∞
n=0 κnG(n) be the expansion of the puregauge GL in powers of κ. Here,
L
we propose a largespace WZWlike action utilizing the puregauge GL(t) and the large
gaugeinvariant cubic vertex V2 of S3 = 31! hΨ, V2(Ψ2)i
Recall that the kinetic term S2 = 21 hΨ, V1(Ψ)i
V4(Ψ4) = η (Qη˜Ψ)3, η˜Ψ L + [(Qη˜Ψ)2, η˜Ψ]L, η˜Ψ L
(n + 1)point vertex Vn. Therefore, the puregauge solution GL
+ . . . , (4.15)
Note that all coefficients of Vn+1 and G(n) match by the tintegral.
L
n=1 (n + 1)!
explain in the rest. Since the relation
Thus, we propose the following WZWlike action for NSNS string field theory
2 Z 1
S =
which reduces to (4.16) or the familiar WZW form (see appendix B)
SΨ(t)=tΨ = − α′ hΨη, GLi + κ
The equation of motion is given by
which is derived in subsection 4.3. Although the action includes the integral over a real
Nonlinear gauge invariance
Here, we derive the equation of motion and the closed form expression of nonlinear gauge
L
= −h∂tΨδ(t) − κ[Ψδ(t), ψt]GL(t), QGL(t)ψη(t)i
L
= h∂tΨδ(t), η GL(t)i − κhη GL(t), [Ψδ(t), ψt]GL(t)i.
L
= hΨδ(t), η(QGL(t)ψt) + κ[η GL(t), ψt]GL(t)i
L
= hΨδ(t), ∂t η GL(t) i + κhη GL(t), [Ψδ(t), ψt]GL(t)i.
which does not include tparametrized fields. The equation of motion is, therefore, given
by (4.24) and it is independent of tparametrization of fields.
As a result, although the action has three generators of gauge transformations, since one
of these gauge invariances reduces to trivial, the resulting theory is
WessZuminoWittenlikely formulated with two nonlinear gauge invariances.
In this paper, we proposed WZWlike expressions for the action and nonlinear gauge
transformations in the NSNS sector of superstring field theory in the large Hilbert space.
Alit does not depend on tparametrization. Vertices are determined by a puregauge solution
of NS (heterotic) string field theory in the small Hilbert space of right movers, which is
constructed by NS closed superstring products (except for the BRST operator) including
insertions of rightmoving picturechanging operators [31].
Gauge equivalent vertices.
edge points at the diamonds of products in figure 5.1 of [31]. It would be possible to
write the largespace NSNS action utilizing another but gaugeequivalent products in [31]
instead of the (−, NS) string products.
Ramond sectors.
We have not analyzed how to incorporate the R sector(s). Our
largespace NSNS action has the almost same algebraic properties as the largespace action for
NS closed string field theory. Thus, we can expect that the method proposed in [25, 26]
also goes in the NSNS case.
It is very important to obtain clear understandings of the geometrical meaning of
theory, gauge fixing [35–37], the relation between two formulations: large and smallspace
formulations. However, our largespace formulation is purely algebraic and these aspects
remain mysterious.
Acknowledgments
The author would like to express his gratitude to the members of Komaba particle theory
group, in particular, Keiyu Goto and my supervisors, Mitsuhiro Kato and Yuji Okawa.
The author is also grateful to Shingo Torii. This work was supported in part by Research
Fellowships of the Japan Society for the Promotion of Science for Young Scientists
Heterotic theory in the small Hilbert space
The action for heterotic string field theory in the small Hilbert space of right movers is
S =
n=1 (n + 2)!
small Hilbert space of right movers and rightmoving picturechanging operators Xe inserted
action is invariant under the following gauge transformation [6, 7]
Just as bosonic theory [5, 7], the equation of motion is given by
n=1 (n + 1)!
and a puregauge GL is constructed by infinitisimal gauge transformations [6, 20, 21].
differential equation
Some identities
or heterotic string products, and a derivation operator X satisfy
hA, Bi = (−)(A+1)(B+1)hB, Ai,
hXA, Bi = (−)AXhA, XBi,
where c0− = 12 (c0 − c˜0) and X = Q, η, η˜.
The MaurerCartan element.
A puregauge solution GL satisfies the equation of
moUsing the defining equation of GL, we find that
F (GL) =
n=1 (n + 1)!
The standard
WZW form.
Recall that when there exist higher sting products
[A1, . . . , An]
> 2), a fieldstrengthlike object fXY
FXY ≡ XΨY + (−)(X+1)(Y +1)Y ΨX + κ[ΨX , ψY ]GL
L
S =
dt hFηt − ∂tΨη − κ[Ψt, ψη]GLL , GLi − hΨt, QGL ψηi
dth h∂tΨη, GLi + hΨη, ∂tGLi + κhΨt, [ψη, GL]GLL ii. (B.6)
= 0. Hence, provided that
SΨ(t)=tΨ = − α′ hΨη, GLi + κ
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