Nonlinear gauge invariance and WZW-like action for NS-NS superstring field theory

We complete the construction of a gauge-invariant action for NS-NS superstring field theory in the large Hilbert space begun in arXiv:1305.3893 by giving a closed-form expression for the action and nonlinear gauge transformations. The action has the WZW-like form and vertices are given by a pure-gauge solution of heterotic string field theory in the small Hilbert space.

There exists an alternative formulation of superstring field theory: large space theory [16][17][18][19][20][21][22]. Large space theories are formulated by utilizing the extended Hilbert space of (ξ, η, φ) [23] and the WZW-like action including no explicit insertions of picture-changing 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 picture-changing 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 [24,25].
The cancellation of singularities can also occur in the small Hilbert space. Recently, by the brilliant works of [26,27], it is revealed how to obtain gauge-invariant insertions of picturechanging operators into (super-) string products in the small Hilbert space: the NS and NS-NS sectors of superstring field theories in the small Hilbert space is completely formulated. In this paper, we find that using the elegant technique of [27], one can construct the WZW-like action for NS-NS superstring field theory in the large Hilbert space.
A pure-gauge solution of small-space theory is the key concept of WZW-like 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 NS-NS sector. There is an attempt to construct non-vanishing interaction terms of NS-NS string fields utilizing a pure-gauge solution G B of bosonic closed string field theory [19]. However, the construction is not complete: the nonlinear gauge invariance is not clear and the defining equation of G B is ambiguous. To obtain nonlinear gauge invariances, we have to add appropriate terms to these interaction terms defined by G B at each order. Then, the ambiguities of vertices are removed and we obtain the defining equation of a suitable pure-gauge solution G L , which we explain in the following sections.
In this paper, we complete this construction begun in [19] by determining these additional terms which are necessitated for the nonlinear gauge invariance and by giving closed-form expressions for the action and nonlinear gauge transformations in the NS-NS sector of closed superstring field theory. We propose the action where Ψ t is an NS-NS string field Ψ plusη-exact terms and G L is a pure-gauge solution to the NS heterotic string equation of motion in the small Hilbert space of right movers. The action has the WZW-like form and the almost same algebraic properties as the large-space action for NS open and NS closed (heterotic) string field theory [18].
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 gauge-invariant insertions of picture-changing operators [26,27] and provide some useful properties of the (−, NS) closed superstring products. In section 4, first, we give the defining equation of G L and associated fields which are necessitated to construct the NS-NS action. Then, we derive the WZW-like expression for the action and nonlinear gauge transformations and show that η G L = 0 gives the NS-NS superstring equation of motion just as other large-space theories. We end with some conclusions.

Nonlinear gauge invariance
Let κ be the coupling constant of closed string fields. We expand an action S for NS-NS string field theory in powers of κ: S = 2 α ′ n κ n S n+2 . In the large Hilbert space, which includes the ξandξ-zero modes coming from bosonization of the βγ-andβγ-systems, the NS-NS string field Ψ is a Grassmann even, (total) ghost number 0, left-mover picture number 0, and right-mover picture number 0 state. The free action S 2 is given by where Q is the BRST operator, η is the zero mode of the left-moving current η(z), andη is the zero mode of the right-moving currentη(z) [17]. The bilinear is the c − 0 -inserted BPZ inner product: . For brevity, we use the symbol (G|p,p) which denotes that the total ghost number is G, the left-mover picture number is p, and the right-mover picture number isp. Then, ghost-and-picture numbers of Ψ, Q, η, andη are (0|0, 0), (1|0, 0), (1| − 1, 0), and (1|0, −1) respectively. Note that the inner product A, B gives a nonzero value if and only if the sum of A's and B's ghost-and-picture numbers is equal to (G|p,p) = (3| − 1, −1). Computing the variation of this action δS 2 = δΨ, ηQηΨ , we obtain the equation of motion QηηΨ = 0 and find that S 2 is invariant under the gauge transformation where Λ, Ω, and Ω are gauge parameters of Q-, η, andη-gauge transformations respectively.
We would like to construct nonzero and nonlinear gauge-invariant interacting terms S 3 , S 4 , S 5 , . . . using string fields Ψ belonging to the large Hilbert space. In the kinetic term S 2 , there exist three generators of gauge transformations: Q, η, andη. However, as we will see, only Qand η-gauge invariances are extended to be nonlinear andη-gauge invariance remains to be linear in our interacting theory. Consequently, with two nonlinear and one trivial gauge invariances, the theory is Wess-Zumino-Witten-likely formulated.
In section 2, starting with the action proposed in [19] and adding appropriate terms at each order of κ, we construct cubic and quartic terms S 3 , S 4 of the action, whose nonlinear gauge transformations of Q and η take WZW-like forms just as other large-space theories.

Cubic vertex
Let ξ andξ be the zero modes of ξ(z)-andξ(z)-ghosts respectively, and X and X be the zero modes of left-and right-moving picture-changing operators respectively. The (n + 2)-point interaction term S n+2 proposed in [19] includes ηΨ, (Q QΨ) n ,ηΨ to correspond to the result of first quantization, where [A 1 , . . . , A n ] is the bosonic string n-product and Q :=ηξ · Q ·ξη is a projected BRST operator. This term becomes ξV, (X XV) n ,ξV under partial gauge fixing Ψ = ξξ V (2|−1,−1) . However, naive use of this term makes nonlinear gauge invariance not clear. In this subsection, as the simplest example, we show that a gauge-invariant cubic term S 3 can be obtained by adding appropriate terms to ηΨ, [Q QΨ,ηΨ] = ηΨ, [ XQηΨ,ηΨ] . (2.3)

Cyclicity
It would be helpful to consider the cyclicity of vertices.
The upper index of (−1) A means the ghost number of A. When the (n + 1)-point action S n+1 is given by S n+1 = 1 (n+1)! Ψ, V n (Ψ n ) using a cyclic vertex V n , its variation becomes Then, if there exist a zero divisor of the state V n (Ψ n ), it generates gauge transformations. For example, bosonic string field theories and superstring field theories in the small Hilbert space are the case and their gauge transformations are determined by L ∞ -or A ∞ -algebras.
Next, we consider the following case: a vertex V ′ n is not cyclic but has the following property Nonzero W n implies that the cyclicity of V n is broken. For instance, this relation holds when V ′ n consists of a cyclic vertex with operator insertions: In this case, in general, a zero divisor of V ′ n + W n gives the generator of gauge transformations. However, when W n consists of lower V k<n , there exists a special case that the zero divisor of V n gives the generator of gauge transformations just as WZW-like 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 derivations of string products, such as ξ,ξ, X, X, and Q, makes nonlinear gauge invariances not clear. 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. Averaging these three terms, we obtain the cubic action satisfying (2.6) The variation of this cubic action is given by Gauge invariance δ 2 S 2 + δ 1 (κS 3 ) Let us determine second order gauge transformation δ 2 Ψ satisfying δ 2 S 2 + δ 1 (κS 3 ) = 0. For this purpose, it is rather suitable to use a pair of fundamental operators (Q, η,η,ξ) than to use a pair of composite operators such as (Q, η,η, Q). For example, V 1 (Ψ) = ηQηΨ appears in W 2 (Ψ 2 ) if and only if we use (Q, η,η,ξ) and furthermore, while Q[QΨ, Λ], ηQηΨ = 0, X[QηΨ, Λ], ηQηΨ = 0. The first order Q-gauge transformation of S 3 is given by Note that the zero mode X of the right-mover picture-changing operator is inserted cyclicly. We find that under the following second order gauge transformation the cubic term S 3 of the action is gauge invariant: δ 2,Λ S 2 + δ 1,Λ (κS 3 ) = 0. For brevity, we define the following new string product which includes the zero mode X of the right-mover picture-changing operator Note that when we use this new product, the cubic term S 3 of the action is given by under the following Q-gauge transformations Similarly, we find that S 2 + κS 3 is gauge invariant under η-andη-gauge transformations Although it naively looks like nonlinearη-gauge invariance of the action, it is essentially linear.

Quartic vertex
We can construct the quartic term S 4 and, in principle, higher interaction terms S n>4 by repeating the same procedure. More precisely, starting with ηΨ, [( XQηΨ) 2 ,ηΨ] , adding appropriate terms for (2.6), and imposing the gauge invariance δ 3 S 2 + δ 2 (κS 3 ) + δ 1 (κ 2 S 4 ) = 0, we obtain the quartic term S 4 . First, we consider the gauge invariance To be gauge invariant, the first order gauge transformation of κ 2 S 4 has to cancel δ 3 S 2 + δ 2 (κS 3 ). Note that δ 2 (κS 3 ) is given by Therefore, we have to consider the following 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 holds Hence, the three product L 2+2 (A, B, C) defined by This new three product L 2+2 (A, B, C) possesses the symmetric property and the derivation property of Q. Note that, however, the operator η does not act as a derivation of L 2+2 (A, B, C).
Note that the following types of products have the Q-derivation property namely, ∆ Q M = ∆ Q N = 0. Therefore, L 1+3 is given by a linear combination of these M -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: QηΨ, C =ηΨ) and this term becomes ξV, [X XV, X XV,ξV] under the partial gauge fixing Ψ = ξξV, which is necessitated for the correspondence to the result of first quantization.

Quartic vertex S 4
Let us now consider the quartic vertex having the property (2.6) and determine the third order gauge transformation δ 3 Ψ. To obtain the gauge invariance δ 3 S 2 + δ 2 (κS 3 ) + δ 1 (κ 2 S 4 ) = 0, the quartic term S 4 has to include the term ηΨ, [QηΨ, QηΨ,ηΨ] L because the L ∞ -relation (2.32) is the only way to eliminate the term [A, B] L , C L appearing in δ 3 S 2 + δ 2 (κS 3 ). We therefore start with the following computation To obtain the quartic vertex having the property (2.6), the term Ψ, η [QηΨ,ηΨ] L ,ηΨ L is necessitated. The variation of this term becomes Hence, the quartic term S 4 defined by has the property (2.6) and its variation is given by we obtain the third order Q-gauge transformation Similarly, the third order η-andη-gauge transformations are given by Note, however, that since δ 3, Ω Ψ as well as δ 2, Ω Ψ isη-exact, redefiningη-gauge parameters as In principle, we can construct higher vertices S 5 , S 6 , . . . by repeating these steps at each order of κ. However, it is not easy to read a closed form expression by hand calculation 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.

Gauge-invariant insertions of picture-changing operators
In this section, we briefly review the coalgebraic description of string vertices [28][29][30] and how to construct NS superstring vertices [27]. Since the NS string products satisfies L ∞ -relations, the shifted NS string products satisfies L ∞ -relations up to the equation of motion.

Coalgebraic description of vertices
Let T (H) be a tensor algebra of the graded vector space H. As the quotient algebra of T (H) by the ideal generated by all differences of products we can construct the symmetric algebra S(H). The product of states in S(H) is graded commutative and associative as follows Note that the n-product of the identity map I : H → H on symmetric algebras is different from the n-tensor product or the identity I n on H n : In other words, I n is the sum of all permutations, I n gives the sum of equivalent permutations, and I k · I l is equivalent to the sum of different (k, l)-partitions of k + l.
The n string product L n (A 1 A 2 . . . A n ) ≡ [A 1 , . . . , A n ] defines a n-fold linear map L n : H n → H. We can naturally define a coderivation  where s is a real parameter and L(s) is the generating function for the bosonic string products

Gauge-invariant insertions
Let L (n) N +1 be a (N +1)-product including n-insertions of picture-changing operators. We consider a series of these inserted products where t is a real parameter and L Starting with these relations, we can construct the NS superstring products L(0, t) satisfying the L ∞ -relations and derivation properties of η, which we explain in this subsection.

Gauge-invariant insertions
To construct L(0, t) satisfying the L ∞ -relations and η-derivation conditions The solution of this pair of differential equations generates all products including appropriate insertions of picture-changing operators.
We can obtain the superstring L ∞ -relations (3.13) as a solution of the differential equation Therefore, we can always derive explicit forms of these inserted products as follows: 2 ]], 2 ]] , . . .

(3.24)
For example, we find that the lowest inserted product L 2 is given by 2 ]](A, B) where L

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 L n+1 gives the (n + 1)-product of NS (heterotic) superstring field theory in the small Hilbert space of left movers [27].
L ∞ -properties of right-mover NS string products Let G be a ghost-and-picture number (2|0, −1) state and A, A 1 , . . . , A n be arbitrary states. We can define a shifted BRST operator Q G and shifted right-mover NS string products 32) in the same manner as shifted bosonic string products. Provided that the state G shifting these products satisfies the equation of motion F(G) = 0 of NS (heterotic) string field theory in the small Hilbert space of right movers these shifted products satisfy (strong) L ∞ -relations 1 : Then, Q G becomes a nilpotent operator.

WZW-like expression
In this section, first, we gives the defining equations of a formal pure-gauge G L and associated fields Ψ t , Ψ η , Ψ δ . These are functions of NS-NS string fields Ψ and become key ingredients of our construction. Then, we present a closed form expression of WZW-like action for NS-NS string field theory, the equation of motion, and the gauge invariance of the action.

Pure-gauge G L and 'large' associated field Ψ X
The NS-NS string field Ψ is a Grassmann-even and ghost-and-picture number (0|0, 0) state living in the large Hilbert space of left-and-right movers: ηΨ = 0 andηΨ = 0.
A pure-gauge G L of right-mover NS theory We can build a formal pure-gauge solution G L of NS heterotic string field theory in the small Hilbert space of right-movers with a finite gauge parameterηΨ living in the left-mover large and right-mover small Hilbert space by successive infinitesimal gauge transformations. The puregauge field G L = G L [ηΨ] is a function ofηΨ, defined by the τ = 1 solution of the differential equation with the initial condition G L [0] = 0, where τ ∈ [0, 1] is a real parameter connecting 0 and G L [ηΨ]. (See also [5,18], Appendix A, and Appendix B.) Solving the defining equation (4.1) and setting τ = 1, we obtain the explicit form of the pure-gauge G L ≡ G L [τ = 1] as follows Note that G L is a Grassmann even and ghost-and-picture number (2|0, −1) state satisfying ηG L = 0 andηG L = 0, the fieldηΨ is a Grassmann odd and ghost-and-picture number (1|0, −1) state satisfying η(ηΨ) = 0 andη(ηΨ) = 0, and the n-product [A 1 , . . . , A n ] L is a Grassmann odd and ghost-and-picture number (3 − 2n|0, n − 1) product satisfying ∆ η/η [A 1 , . . . , A n ] L = 0.
An associated field ψ X with derivation X In the rest, we simply write G L [τ ] for the intermediate pure-gauge field rather than G L [τηΨ]. There exists a special string field ψ X , so-called an associated field, satisfying where X is a derivation of our right-mover (−, NS) string products [A 1 , . . . , A n ] L : , the defining equation of ψ X is given by (4.5) with the initial condition ψ X [0] = 0. Note that ψ X [τ ], as well as G L [τ ], is a function of τηΨ and τ is a real parameter connecting 0 and ψ X := ψ X [1]. The associated filed ψ X carries ghost-andpicture number (G X + 1|p X ,p X − 1), where (G X |p X ,p X ) is that of X.
A 'large' associated field Ψ X These pure-gauge field G L and associated field ψ X belong to the left-mover large and rightmover small Hilbert space: η G L = 0, η ψ X = 0, andη G L =η ψ X = 0. Sinceη-cohomology is trivial in the large Hilbert space of left-and-right movers, there exist large fields G L and Ψ X such that (4.6) Note that the relationη XG L = −η Q G L Ψ X holds because of (−) X X(ηG L ) = Q G L (ηΨ X ) and ηG L = 0. Hence, up to Q G L -andη-exact terms, these large-space fields G L and Ψ X satisfy and the defining equations (up to Q G L -andη-exact terms) of G L and Ψ X are given by with the initial conditions G L [τ = 0] = 0 and Ψ X [τ = 0] = 0. As well as G L and ψ X , large fields G L and Ψ X are also functions of (Ψ,ηΨ). Here τ is a real parameter connecting 0 andηΨ. With the initial condition Ψ X [τ = 0] = 0, we find that the first few terms in Ψ X are given by Note that the large associated field Ψ X has the same ghost-and-picture number as X.
A t-parametrized large field Ψ t Let Ψ(t) be a t-parametrized path connecting Ψ(0) = 0 and Ψ(1) = Ψ. The above defining equations of G L , ψ X , G L , and Ψ X hold not only for the field Ψ but for the t-parametrized field Ψ(t). Hence, we can built t-parametrized ones G L (t), ψ X (t), G L (t), and Ψ X (t) by replacing Ψ with Ψ(t) in the defining equations of G L , ψ X , G L , and Ψ X . For example, for X = ∂ t , solving (4.9) with replacement of Ψ and setting τ = 1, we obtain the t-parametrized field Ψ t ≡ Ψ ∂t (t) which appears in the action for NS-NS string fields with general t-parametrization. Note that this Ψ t has the same ghost-and-picture number (0|0, 0) as NS-NS string field Ψ, and the equatioñ ηΨ t =ηΨ holds for the linear path Ψ(t) = tΨ.

Wess-Zumino-Witten-like action
L be the expansion of the pure-gauge G L in powers of κ. Here, we propose a large-space WZW-like action utilizing the pure-gauge G L (t) and the large associated field Ψ t .
The generating function for V n (Ψ n ) Recall that the kinetic term S 2 = 1 2 Ψ, V 1 (Ψ) is given by V 1 (Ψ) = ηQηΨ, which is equivalent to ηG is also given by 4!·η G Note that all coefficients of V n+1 and G (n) L match by the t-integral.
The generating function for W n (Ψ n ) δ be the expansion of the associated field Ψ δ in powers of κ, where 'δ' is the variation operator. Recall that the variation of S 2 is given by δS 2 = δΨ, V 1 (Ψ) + W 1 (Ψ) = δΨ, ηQηΨ , which means W 1 (Ψ) = 0. In section 2, we also determined W 2 and W 3 , as well as V 2 and V 3 , appearing in the calculation of the variation δS 3 and δS 4 . Recall that W 2 is given by 17) which is equivalent to Ψ (1) δ , V 1 (Ψ) , and W 3 is given by 18) which is equivalent to Ψ (1) Similarly, the following relation holds Hence, the associated field Ψ δ The WZW-like action Let Ψ(t) be a t-parametrized NS-NS string field satisfying Ψ(0) = 0 and Ψ(1) = Ψ. Replacing NS-NS string fields Ψ with t-parametrized NS-NS string fields Ψ(t) in (4.1) and (4.9), we obtain t-parametrized pure-gauge and large associated fields: G L (t), Ψ t , and Ψ η (t). WZW-like NS-NS action consists of these fields, which we explain in the rest. Since the relation (4.21) holds and Q G L , η, andη are nilpotent operators, the state η G L is a Q G L -, η-, andη-exact state.
Thus, we propose the following WZW-like action for NS-NS string field theory which reduces to (4.16) or the familiar WZW form (see Appendix B) if we set Ψ(t) = tΨ. Note that the (n + 1)-point vertex includes n insertions ofη and the action S is invariant under the linear 2 gauge transformation δ Ω ′ Ψ =η Ω ′ .
2 Note that, however, GL and Ψt include a lot ofηΨ and as seen in 4.3, it does not mean that there are no gauge transformations including nonlinear terms ofηΨ.
The equation of motion is given by (4.24) which is derived in subsection 4.3. Although the action includes the integral over a real parameter t, the action S, the variation δS, the equation of motion η G L = 0, and gauge transformations are independent of the t-parametrization or t-parametrized path Ψ(t).
For this purpose, we prove that the variation δS does not includes t and is given by we find that the following equation holds for any t Similarly, since η G L = 0, [[η, Q G L ]] = 0, and ψ X = η Ψ X , we obtain Hence, the variation δS of the WZW-like action S is given by which does not include t-parametrized fields. The equation of motion is, therefore, given by (4.24) and it is independent of t-parametrization of fields.
Since η G L is a Q G L -, η-, andη-exact state, we find that the action is invariant under the following nonlinear Q-and η-gauge transformations and linearη-gauge transformation Ψ δ = Q G L Λ + η Ω +η Ω, (4.29) where Λ, Ω, and Ω are gauge parameter fields whose ghost-and-picture numbers are (−1|0, 0), (−1|1, 0), and (−1|0, 1) respectively. Note that Ψ δ is an invertible function of δΨ, at least in the expansion in powers of κ as follows For instance, an explicit expression for Q-gauge transformation δ Λ Ψ and η-gauge transformation δ Ω Ψ are given by These gauge transformations are nonlinear. Note, however, that sinceη-gauge transformation obtained from Ψ δ Ω =η Ω consists ofη-exact terms, it is equivalent to the linearη-gauge transformation where Ω ′ is a redefinedη-gauge parameter 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 Wess-Zumino-Witten-likely formulated with two nonlinear gauge invariances.

Conclusion
In this paper, we proposed WZW-like expressions for the action and nonlinear gauge transformations in the NS-NS sector of superstring field theory in the large Hilbert space. Although the action uses t-parametrized large fields Ψ(t) satisfying Ψ(0) = 0 and Ψ(1) = Ψ, it does not depend on t-parametrization. Vertices are determined by a pure-gauge 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 right-moving picture-changing operators [27].

Gauge equivalent vertices
We used the (−, NS) string products, namely, the right edge points at the diamonds of products in Figure 5.1 of [27]. It would be possible to write the large-space NS-NS action utilizing another but gauge-equivalent products in [27] instead of the (−, NS) string products.

Ramond sectors
We have not analyzed how to incorporate the R sector(s). Our large-space NS-NS action has the almost same algebraic properties as the large-space action for NS closed string field theory. Thus, we can expect that the method proposed in [22] also goes in the NS-NS case.
It is very important to obtain clear understandings of the geometrical meaning of theory, gauge fixing [31,32], the relation between two formulations: large-and small-space formulations. However, our large-space 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 A Heterotic theory in the small Hilbert space The action for heterotic string field theory in the small Hilbert space of right movers is given by where the NS heterotic string field Φ is a ghost-and-picture number (2|0, −1) state in the small Hilbert space of right movers and right-moving picture-changing operators X inserted product [A 1 , . . . , A n ] L given by [27] carries ghost-and-picture number (3 − 2n|0, n − 1). This action is invariant under the following gauge transformation [5,6] where λ is a gauge parameter carrying ghost-and-picture number (1|0, −1). Just as bosonic theory [4,6], the equation of motion is given by QΦ + and a pure-gauge G L is constructed by infinitisimal gauge transformations [5,18]. Therefore, G L is defined by the τ = 1 value solution G L ≡ G L [τ = 1] of the following differential equation where c − 0 = 1 2 (c 0 −c 0 ) and X = Q, η,η.
The Maurer-Cartan element A pure-gauge solution G L satisfies the equation of motion F(G L ) = 0 of NS heterotic string field theory in the small Hilbert space of right movers. Using the defining equation of G L , we find that which leads to the differential equation ∂ τ F = F,ηΨ L G L with the initial condition F(0) = 0. Hence, G L satisfies F(G L ) = 0 and Q G L is a nilpotent operator. (See also [18].) The standard WZW form Recall that when there exist higher sting products [A 1 , . . . , A n ] L (n > 2), a field-strength-like object f XY ≡ Xψ Y −(−) XY Y ψ X +(−) X κ[ψ X , ψ Y ] L G L is not zero f XY = 0 but a Q G L -exact state: Q G L f XY = 0, where X and Y are derivation operators satisfying [[X, Y ]] = 0. Let be a large field-strength-like object satisfyingηF XY = (−) X f XY . Utilizing this F ηt and the relation Ψ t , Q G L ψ η = Ψ η , ∂ t G L , our WZW-like action can be rewritten as Recall also that the linear t-parametrization Ψ(t) = tΨ gives Ψ t = Ψ up toη-exact terms. When we identify τ and t, the defining equation of ψ X becomes ∂ t ψ X = XηΨ + κ[ηΨ, ψ X ] L G L , which impliesη ∂ t Ψ X −(−) X XΨ+κ[Ψ, ψ X ] L G L = 0. Hence, provided that Ψ(t) = tΨ, we obtaiñ ηF ηt = 0 and the action reduces to the familiar WZW-form: