A ∞ structure from the Berkovits formulation of open superstring field theory
aa r X i v : . [ h e p - t h ] M a y LMU-ASC 30/15UT-Komaba/15-1 A ∞ structure from the Berkovits formulationof open superstring field theory Theodore Erler, Yuji Okawa and Tomoyuki Takezaki Arnold Sommerfeld Center, Ludwig-Maximilians UniversityTheresienstrasse 37, 80333 Munich, Germany [email protected] Institute of Physics, The University of TokyoKomaba, Meguro-ku, Tokyo 153-8902, Japan [email protected], [email protected]
Abstract
By formulating open superstring field theory based on the small Hilbert space of the supercon-formal ghost sector, an action for the Neveu-Schwarz sector with an A ∞ structure has recentlybeen constructed. We transform this action to the Wess-Zumino-Witten-like form and showthat this theory is related to the Berkovits formulation of open superstring field theory basedon the large Hilbert space by partial gauge fixing and field redefinition. ontents A ∞ structure 6 A ∞ structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Construction of the multi-string products . . . . . . . . . . . . . . . . . . . . . . 12 t -dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Field redefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Embedding to the large Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . 22 B.1 The expansion of the multi-string products . . . . . . . . . . . . . . . . . . . . . 28B.2 Translation into string field theory conventions . . . . . . . . . . . . . . . . . . . 31B.3 The action up to quartic interactions . . . . . . . . . . . . . . . . . . . . . . . . 34B.4 Field redefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Introduction
Can we consistently quantize string field theory? This is one of the fundamental questions whenwe work on string field theory. While the gauge structure of string field theory is complicated,open bosonic string field theory [1] and closed bosonic string field theory [2, 3, 4, 5, 6, 7] havebeen quantized based on the Batalin-Vilkovisky formalism [8, 9]. However, the quantization ofthe bosonic string is formal because of the presence of tachyons. How about the quantizationof superstring field theory?Among various formulations of superstring field theory, the Berkovits formulation for theNeveu-Schwarz (NS) sector of open superstring field theory [10] has been quite successful. It isbased on the large Hilbert space of the superconformal ghost sector [11, 12], and various analyticsolutions have been constructed [13, 14, 15, 16, 17, 18, 19]. The construction of a classicalmaster action in the Batalin-Vilkovisky formalism, however, has turned out to be formidablycomplicated for the Berkovits formulation [20, 21, 22, 23, 24]. Why is it so complicated?In bosonic string field theory, the equation of motion and the gauge transformation can bothbe written in terms of the same set of string products. The string products satisfy the set ofrelations called A ∞ [25, 26, 27, 28, 29, 30] for the open string and L ∞ [7, 31, 32] for the closedstring. These structures play a crucial role in the Batalin-Vilkovisky quantization, and they areclosely related to the decomposition of the moduli space of Riemann surfaces. The source ofthe difficulty for the Batalin-Vilkovisky quantization in the Berkovits formulation can be seenin the free theory by comparing the equation of motion and the gauge transformations. Theequation of motion takes the form Qη Φ = 0 , (1.1)where Φ is the open superstring field in the large Hilbert space, Q is the BRST operator, and η isthe zero mode of the superconformal ghost η ( z ). On the other hand, the gauge transformationsare given by δ Φ = Q Λ + η Ω , (1.2)where Λ and Ω are the gauge parameters. In the equation of motion, the product of Q and η appears, while the sum of the term generated by Q and the term generated by η appears inthe gauge transformations. This difference can be thought of as the source of the difficulty.Working in the large Hilbert space obscures the relation to the supermoduli space of super-Riemann surfaces, and it might be one possible reason underlying the difficulty. The approachto incorporating the Ramond sector into the Berkovits formulation proposed in [33] is alsocomplicated, and it might also be related to our insufficient understanding of the connectionbetween the large Hilbert space and the supermoduli space of super-Riemann surfaces.If we formulate open superstring field theory based on the small Hilbert space, the equation2f motion of the free theory is Q Ψ = 0 , (1.3)where Ψ is the open superstring field in the small Hilbert space, and the gauge transformationis given by δ Ψ = Q Λ , (1.4)where Λ is the gauge parameter. The equation of motion and the gauge transformation areboth written in terms of the BRST operator Q , and this seems to be promising for constructingstring products satisfying the A ∞ relations in the interacting theory. In fact, the covering ofthe supermoduli space of super-Riemann surfaces is more closely related to formulations basedon the small Hilbert space. It had long been thought, however, that a regular formulationbased on the small Hilbert space would be difficult because of singularities coming from localpicture-changing operators.Recently, it was demonstrated in [34] that a regular formulation of open superstring fieldtheory based on the small Hilbert space can be obtained from the Berkovits formulation bypartial gauge fixing. In the process of the partial gauge fixing, an operator ξ satisfying therelation { η, ξ } = 1 (1.5)is used, and we can realize such an operator by a line integral of the supercoformal ghost ξ ( z ).For instance, we can choose the zero mode ξ of ξ ( z ) to be ξ . While a line integral of ξ ( z ) doesnot have simple transformation properties under conformal transformations, the partial gaugefixing guarantees that the resulting theory is gauge invariant. The BRST transformation ofthe line integral of ξ ( z ) yields a line integral of the picture-changing operator, and singularitiesassociated with local insertions of the picture-changing operator are avoided in this approach.The equation of motion and the gauge transformation can be systematically derived from thoseof the Berkovits formulation, but it turned out that the resulting theory does not exhibit the A ∞ structure.Once we recognize that a line integral of ξ ( z ) can be used in constructing a gauge-invariantaction of open superstring field theory, we do not necessarily start from the Berkovits formula-tion. In [35], an action with an A ∞ structure based on the small Hilbert space was constructedusing a line integral of ξ ( z ) as a new ingredient. Because of the A ∞ structure of the the-ory, the quantization based on the Batalin-Vilkovisky formalism is straightforward just as inopen bosonic string field theory. The construction was further generalized to the NS sector ofheterotic string field theory and the NS-NS sector of type II superstring field theory in [36].We now have two successful formulations for the NS sector of open superstring field theory.The Berkovits formulation is beautifully constructed based on the large Hilbert space, and thetheory constructed in [35] is based on the small Hilbert space and exhibits the A ∞ structure. In3his paper we show that the two theories are related by partial gauge fixing and field redefinition.In the rest of the introduction, we summarize the relation we found.Let us start with the description of the Berkovits formulation. We denote the open super-string field in the large Hilbert space by e Φ. The action takes the Wess-Zumino-Witten-like(WZW-like) form given by [10] S = − h e − e Φ ( ηe e Φ ) , e − e Φ ( Qe e Φ ) i− Z dt h e − e Φ( t ) ∂ t e e Φ( t ) , { e − e Φ( t ) ( Qe e Φ( t ) ) , e − e Φ( t ) ( ηe e Φ( t ) ) } i , (1.6)where e Φ(1) = e Φ, e Φ(0) = 0, h A, B i is the BPZ inner product of A and B , and all string productsare defined by the star product [1]. It was shown in [37] that this action can be written as S = − Z dt h η ( e − e Φ( t ) ∂ t e e Φ( t ) ) , e − e Φ( t ) ( Qe e Φ( t ) ) i , (1.7)and it can be further transformed to S = − Z dt h A t ( t ) , QA η ( t ) i (1.8)with A η ( t ) = ( η e e Φ( t ) ) e − e Φ( t ) , A t ( t ) = ( ∂ t e e Φ( t ) ) e − e Φ( t ) . (1.9)We will use this form of the action in this paper. The dependence of e Φ( t ) on t is topological,and the action is a functional of e Φ, which is the value of e Φ( t ) at t = 1. The action has nonlineargauge invariances given by A δ = Q Λ + D η Ω , (1.10)where Λ and Ω are gauge parameters and A δ = ( δe e Φ ) e − e Φ , D η Ω = η Ω − ( ηe e Φ ) e − e Φ Ω − Ω ( ηe e Φ ) e − e Φ . (1.11)All these properties of the action follow from the relations ηA η ( t ) = A η ( t ) A η ( t ) , ∂ t A η ( t ) = ηA t ( t ) − A η ( t ) A t ( t ) + A t ( t ) A η ( t ) (1.12)together with the fact that the cohomology of η is trivial.In [34], the gauge invariance associated with Ω was used to impose the following conditionon e Φ for partial gauge fixing: ξ e Φ = 0 . (1.13)The string field e Φ satisfying this condition can be written as e Φ = ξ e Ψ (1.14)4 ✖✕✗✔ field redefinition ✲✛ e Φ ✖✕✗✔ partial gauge fixing ❄ partial gauge fixing ❄ Ψ ✖✕✗✔ e Ψ ✖✕✗✔ field redefinition ✲✛ Figure 1: The relation between the theory with the A ∞ structure in terms of Ψ in the smallHilbert space and the Berkovits formulation in terms of e Φ in the large Hilbert space.with e Ψ in the small Hilbert space. We replace e Φ( t ) in the action by ξ e Ψ( t ) satisfying e Ψ(1) = e Ψand e Ψ(0) = 0 to obtain S = − Z dt h ( ∂ t e ξ e Ψ( t ) ) e − ξ e Ψ( t ) , Q (( η e ξ e Ψ( t ) ) e − ξ e Ψ( t ) ) i . (1.15)The dependence of e Ψ( t ) on t is topological, and we can regard this as an action of e Ψ in thesmall Hilbert space. The action is invariant under the residual gauge transformation after thepartial gauge fixing.Now the string fields A η ( t ) and A t ( t ) satisfying (1.12) are parameterized by e Ψ( t ) in the smallHilbert space. We show in this paper that the action with the A ∞ structure constructed in [35]can be brought to the form (1.8), where A η ( t ) and A t ( t ) satisfying (1.12) are parameterizeddifferently in terms of Ψ( t ) in the small Hilbert space. Since the relations (1.12) are satisfied,the dependence on t is topological, and the action is a functional of Ψ, which is the value ofΨ( t ) at t = 1. We show that the action in terms of Ψ and the action in terms of e Ψ are relatedby field redefinition and establish the equivalence of the theory obtained by the partial gaugefixing of the Berkovits formulation and the theory with the A ∞ structure.We further show that the action in terms of Ψ( t ) can be obtained from an action in termsof Φ( t ) in the large Hilbert space by the partial gauge fixing with the two string fields beingrelated as Φ( t ) = ξ Ψ( t ). Therefore, the theory constructed in [35] with the A ∞ structure interms of Ψ in the small Hilbert space can also be understood as being related to the Berkovitsformulation in terms of e Φ by field redefinition from e Φ to Φ and by the partial gauge fixingΦ = ξ Ψ. See figure 1.While transforming the action with the A ∞ structure constructed in [35] to the WZW-likeform is crucial in showing its relation to the Berkovits formulation in this paper, the equivalenceof the theory with the A ∞ structure and the theory obtained from the Berkovits formulation5y the partial gauge fixing can also be shown from a different perspective. In an accompanyingpaper [38], the equivalence is shown by establishing that the theory obtained from the Berkovitsformulation by the partial gauge fixing can be derived by a finite gauge transformation throughthe space of A ∞ structures just as in the case of the theory with the A ∞ structure constructedin [35].The rest of the paper is organized as follows. In section 2, we review the construction of the A ∞ structure in [35]. We begin in subsection 2.1 by summarizing all the algebraic ingredients tobe used in this paper. We then briefly review the basics of the A ∞ structure in subsection 2.2.The construction of the A ∞ structure in [35] is reviewed in subsection 2.3, and in this subsectionwe also present several new relations which are essential in this paper. In section 3, we showthat open superstring field theory with the A ∞ structure in the preceding section is related tothe Berkovits formulation by partial gauge fixing and field redefinition. We first transform theaction with the A ∞ structure to the WZW-like form in subsection 3.1, and we further write theaction in a form with a topological dependence on the parameter t of the WZW-like action insubsection 3.2. We then establish its equivalence by field redefinition to the theory we obtainfrom the Berkovits formulation by the partial gauge fixing in subsection 3.3, and we embedthe theory to the large Hilbert space in subsection 3.4. Section 4 is devoted to conclusionsand discussion. In appendix A, we summarize a few identities regarding cohomomorphisms. Inappendix B, we translate the A ∞ conventions into standard string field theory conventions, andwe demonstrate the equivalence of the action with the A ∞ structure and the action we obtainfrom the Berkovits formulation by the partial gauge fixing up to quartic interactions explicitly. A ∞ structure In this subsection we present all the algebraic ingredients to be used in this paper. We considerthe NS sector of open superstring field theory. An open superstring field is a state in the confor-mal field theory which consists of the matter sector, the bc ghost sector, and the superconformalghost sector. We describe the superconformal ghost sector using ξ ( z ), η ( z ), and φ ( z ) [11, 12].We denote the BPZ inner product of string fields A and B by h A, B i . It obeys h B, A i = ( − AB h A, B i . (2.1)Here and in what follows a state in the exponent of − Q and the zero mode6f η ( z ) which we denote by η . They are Grassmann odd and obey Q = 0 , η = 0 , { Q, η } = 0 , h QA, B i = − ( − A h A, QB i , h ηA, B i = − ( − A h A, ηB i . (2.2)The cohomology of the operator η is trivial because there exists a Grassmann-odd operator ξ which satisfies { η, ξ } = 1 . (2.3)The existence of such an operator is crucial in this paper, and we need to make a choice whenwe explicitly construct various string products to be discussed. However, the results of thepaper are independent of the choice of ξ , and we only use the relation (2.3). When we make achoice of ξ , we use a line integral of ξ ( z ) to realize ξ , and we further assume that ξ = 0 , h ξA, B i = ( − A h A, ξB i . (2.4)For example, we can choose ξ to be the zero mode ξ of ξ ( z ).We say that a string field A is in the small Hilbert space when A is annihilated by η : ηA = 0 . (2.5)For a pair of string fields A and B in the small Hilbert space, we define the BPZ inner product hh A, B ii in the small Hilbert space by hh A, B ii = h ξ A, B i . (2.6)It obeys hh B, A ii = ( − AB hh A, B ii . (2.7)We can use the operator ξ to relate the two BPZ inner products as hh A, B ii = h ξA, B i . (2.8)Products of string fields are defined by the star product [1]. The star product is associative:( A A ) A = A ( A A ) . (2.9)The operators Q and η act as derivations with respect to the star product: Q ( AB ) = ( QA ) B + ( − A A ( QB ) , η ( AB ) = ( ηA ) B + ( − A A ( ηB ) . (2.10)These are all the ingredients we will use in this paper.7 .2 A ∞ structure We briefly review the basics of the A ∞ structure in this subsection. See [38] for more compre-hensive explanations.Consider a two-string product V ( A , A ) and a three-string product V ( A , A , A ) with ǫ ( V ( A , A )) = ǫ ( A ) + ǫ ( A ) , ǫ ( V ( A , A , A )) = ǫ ( A ) + ǫ ( A ) + ǫ ( A ) + 1 , (2.11)where ǫ ( A ) denotes the Grassmann parity of A . We say that V ( A , A ) is a Grassmann-even product and V ( A , A , A ) is a Grassmann-odd product. They are assumed to have thefollowing cyclic properties: hh A , V ( A , A ) ii = hh V ( A , A ) , A ii , hh A , V ( A , A , A ) ii = − ( − A hh V ( A , A , A ) , A ii , (2.12)where A , A , A , and A are string fields in the small Hilbert space. Consider an action givenby S = − hh Ψ , Q Ψ ii − g hh Ψ , V (Ψ , Ψ) ii − g hh Ψ , V (Ψ , Ψ , Ψ) ii + O ( g ) , (2.13)where Ψ is a Grassmann-odd string field in the small Hilbert space and g is the couplingconstant. Because of the cyclic properties in (2.12), the variation of the action is given by δS = − hh δ Ψ , Q Ψ ii − g hh δ Ψ , V (Ψ , Ψ) ii − g hh δ Ψ , V (Ψ , Ψ , Ψ) ii + O ( g ) , (2.14)and the action is invariant up to O ( g ) under the gauge transformation given by δ Λ Ψ = Q Λ + g (cid:0) V (Ψ , Λ) − V (Λ , Ψ) (cid:1) + g (cid:0) V (Ψ , Ψ , Λ) − V (Ψ , Λ , Ψ) + V (Λ , Ψ , Ψ) (cid:1) + O ( g ) (2.15)if V ( A , A ) and V ( A , A , A ) satisfy QV ( A , A ) − V ( QA , A ) − ( − A V ( A , QA ) = 0 ,QV ( A , A , A ) − V ( V ( A , A ) , A ) + V ( A , V ( A , A ))+ V ( QA , A , A ) + ( − A V ( A , QA , A ) + ( − A + A V ( A , A , QA ) = 0 (2.16)for states A , A , and A in the small Hilbert space.These relations are part of the A ∞ relations. To describe the A ∞ structure, it is convenientto introduce degree for a string field A denoted by deg( A ). It is defined bydeg( A ) = ǫ ( A ) + 1 mod 2 , (2.17)where ǫ ( A ) is the Grassmann parity of A . We define ω ( A , A ), M ( A , A ), M ( A , A , A ) by ω ( A , A ) = ( − deg( A ) hh A , A ii ,M ( A , A ) = ( − deg( A ) V ( A , A ) ,M ( A , A , A ) = ( − deg( A ) V ( A , A , A ) . (2.18)8ote that the sign factor for M ( A , A , A ) is different. The inner product ω ( A , A ) is gradedantisymmetric: ω ( A , A ) = − ( − deg( A ) deg( A ) ω ( A , A ) . (2.19)We have deg( QA ) = deg( A ) + 1 , deg( M ( A , A )) = deg( A ) + deg( A ) + 1 , deg( M ( A , A , A )) = deg( A ) + deg( A ) + deg( A ) + 1 , (2.20)and we say that Q , M , and M are degree odd. The BPZ property of Q and the cyclicproperties (2.12) are translated into ω ( A , QA ) = − ( − deg( A ) ω ( QA , A ) ,ω ( A , M ( A , A )) = − ( − deg( A ) ω ( M ( A , A ) , A ) ,ω ( A , M ( A , A , A )) = − ( − deg( A ) ω ( M ( A , A , A ) , A ) , (2.21)and the A ∞ relations (2.16) are written as QM ( A , A ) + M ( QA , A ) + ( − deg( A ) M ( A , QA ) = 0 ,QM ( A , A , A ) + M ( M ( A , A ) , A ) + ( − deg( A ) M ( A , M ( A , A ))+ M ( QA , A , A ) + ( − deg( A ) M ( A , QA , A ) + ( − deg( A )+deg( A ) M ( A , A , QA ) = 0 . (2.22)We construct multi-string products based on the star product. Associated with the starproduct, we define m ( A , A ) by m ( A , A ) = ( − deg( A ) A A . (2.23)The two-string product m is degree odd,deg( m ( A , A )) = deg( A ) + deg( A ) + 1 , (2.24)and it has the following cyclic property: ω ( A , m ( A , A )) = − ( − deg( A ) ω ( m ( A , A ) , A ) . (2.25)The associativity of the star product (2.9) and the derivation property of Q and η with respectto the star product (2.10) can be stated in terms of m as m ( m ( A , A ) , A ) + ( − deg( A ) m ( A , m ( A , A )) = 0 ,Q m ( A , A ) + m ( QA , A ) + ( − deg( A ) m ( A , QA ) = 0 ,η m ( A , A ) + m ( ηA , A ) + ( − deg( A ) m ( A , ηA ) = 0 . (2.26)9or the description of the A ∞ structure, it is further convenient to consider linear operatorsacting on the vector space T H defined by T H = C ⊕ H ⊕ H ⊗ ⊕ H ⊗ ⊕ . . . , (2.27)where C is the space of complex numbers, H is the state space of the boundary conformal fieldtheory, and H ⊗ n is the space obtained by tensoring n copies of H . For example, we define Q acting on T H associated with the BRST operator Q , which is degree odd, as follows: Q , Q A = QA , Q ( A ⊗ A ) = QA ⊗ A + ( − deg( A ) A ⊗ QA , Q ( A ⊗ A ⊗ A ) = QA ⊗ A ⊗ A + ( − deg( A ) A ⊗ QA ⊗ A + ( − deg( A )+deg( A ) A ⊗ A ⊗ QA , ... (2.28)With this definition, we find Q = 0 . (2.29)We similarly define η for η , which is also degree odd, and find η = 0 , Q η + η Q = 0 . (2.30)We denote the commutator of C and C graded with respect to degree by [ C , C ]. Then therelations in (2.29) and (2.30) can be written as[ Q , Q ] = 0 , [ η , η ] = 0 , [ Q , η ] = 0 . (2.31)For the two-string product M ( A , A ) and the three-string product M ( A , A , A ), which areboth degree odd, we define M and M , respectively, by M , M A = 0 , M ( A ⊗ A ) = M ( A , A ) , M ( A ⊗ A ⊗ A ) = M ( A , A ) ⊗ A + ( − deg( A ) A ⊗ M ( A , A ) , M ( A ⊗ A ⊗ A ⊗ A ) = M ( A , A ) ⊗ A ⊗ A + ( − deg( A ) A ⊗ M ( A , A ) ⊗ A , + ( − deg( A )+deg( A ) A ⊗ A ⊗ M ( A , A ) , ... (2.32)10nd by M , M A = 0 , M ( A ⊗ A ) = 0 , M ( A ⊗ A ⊗ A ) = M ( A , A , A ) , M ( A ⊗ A ⊗ A ⊗ A ) = M ( A , A , A ) ⊗ A + ( − deg( A ) A ⊗ M ( A , A , A ) , ... (2.33)The A ∞ relations can then be compactly written as[ Q , M ] ( A ⊗ A ) = 0 , [ Q , M ] ( A ⊗ A ⊗ A ) + 12 [ M , M ] ( A ⊗ A ⊗ A ) = 0 . (2.34)Consider an action written in terms of a set of degree-odd n -string products M n ( A , A , . . . , A n )satisfying ω ( A , M n ( A , A , . . . , A n +1 )) = − ( − deg( A ) ω ( M n ( A , A , . . . , A n ) , A n +1 ) (2.35)as S = − ω (Ψ , Q Ψ) − ω (Ψ , M (Ψ , Ψ)) − ω (Ψ , M (Ψ , Ψ , Ψ)) + . . . = − ∞ X n =1 n + 1 ω (Ψ , M n ( Ψ , Ψ , . . . , Ψ | {z } n )) , (2.36)where M = Q . (2.37)The A ∞ relations can be expressed by introducing M n acting on T H for the n -string product M n ( A , A , . . . , A n ) as [ M , M ] = 0 , (2.38)where M = ∞ X n =1 M n (2.39)with M = Q . (2.40)The operators such as M n are called coderivations .We construct M satisfying the A ∞ relations from the star product. We introduce thecoderivation m associated with m . The associativity of the star product and the derivationproperty of Q and η with respect to the star product in (2.26) can be stated in terms of m as[ m , m ] = 0 , [ Q , m ] = 0 , [ η , m ] = 0 . (2.41)11 .3 Construction of the multi-string products In [35] the NS sector of open superstring field theory with the A ∞ structure was constructed.We introduce M ( s ) defined by M ( s ) = ∞ X n =0 s n M n +1 . (2.42)The A ∞ relations are [ M ( s ) , M ( s ) ] = 0 , (2.43)and the condition that the string field M n (Ψ , Ψ , . . . , Ψ) is in the small Hilbert space can bestated as [ η , M ( s ) ] = 0 . (2.44)In the construction of [35], the coderivation M ( s ) is characterized by the differential equation dds M ( s ) = [ M ( s ) , µ ( s ) ] (2.45)with the initial condition M (0) = Q . (2.46)The degree-even coderivation µ ( s ) is arbitrary at the moment except that the correspondingmulti-string products µ n have the following cyclic property: ω ( A , µ n ( A , A , . . . , A n +1 )) = − ω ( µ n ( A , A , . . . , A n ) , A n +1 ) . (2.47)We will discuss the conditions we impose on µ ( s ) later. The solution to the differential equationcan be written as M ( s ) = G − ( s ) Q G ( s ) , (2.48)where G ( s ) is the path-ordered exponential given by G ( s ) = P exp (cid:20) Z s ds ′ µ ( s ′ ) (cid:21) (2.49)with the following ordering prescription: P [ µ ( s ) µ ( s ) ] = (cid:26) µ ( s ) µ ( s ) for s < s , µ ( s ) µ ( s ) for s > s . (2.50)It obeys the differential equation dds G ( s ) = G ( s ) µ ( s ) (2.51)12ith the initial condition G (0) = 1. It then follows from the structure of M ( s ) that the A ∞ relations are satisfied:[ M ( s ) , M ( s ) ] = [ G − ( s ) Q G ( s ) , G − ( s ) Q G ( s ) ] = G − ( s ) [ Q , Q ] G ( s ) = 0 , (2.52)where we used [ Q , Q ] = 0.If [ η , µ ( s ) ] = 0, the resulting theory is related to the free theory by field redefinition. Toobtain a nontrivial interacting theory, we need [ η , µ ( s ) ] = 0, while the condition [ η , M ( s ) ] = 0is satisfied. In the construction of [35], the coderivation µ ( s ) is characterized by[ η , µ ( s ) ] = m ( s ) , (2.53)where m ( s ) is a degree-odd coderivation obeying the differential equation dds m ( s ) = [ m ( s ) , µ ( s ) ] (2.54)with the initial condition m (0) = m . (2.55)The solution to the differential equation is given by m ( s ) = G − ( s ) m G ( s ) . (2.56)Let us calculate [ η , G ( s ) ] to show that the condition [ η , M ( s ) ] = 0 is satisfied when weconstruct M ( s ) from µ ( s ) characterized this way. We have[ η , G ( s ) ] = Z s ds ′ G (0 , s ′ ) [ η , µ ( s ′ ) ] G ( s ′ , s ) = Z s ds ′ G (0 , s ′ ) m ( s ′ ) G ( s ′ , s ) , (2.57)where G ( s , s ) = P exp (cid:20) Z s s ds ′ µ ( s ′ ) (cid:21) . (2.58)It follows from (2.56) that G (0 , s ′ ) m ( s ′ ) = G ( s ′ ) m ( s ′ ) = m G ( s ′ ) = m G (0 , s ′ ) , (2.59)and we find Z s ds ′ G (0 , s ′ ) m ( s ′ ) G ( s ′ , s ) = Z s ds ′ m G (0 , s ′ ) G ( s ′ , s ) = s m G ( s ) . (2.60)We thus obtain the following important relation:[ η , G ( s ) ] = s m G ( s ) . (2.61)13sing this relation, we can show in the following way that the condition [ η , M ( s ) ] = 0 issatisfied:[ η , M ( s ) ] = − G − ( s ) [ η , G ( s ) ] G − ( s ) Q G ( s ) − G − ( s ) Q [ η , G ( s ) ]= − s G − ( s ) m Q G ( s ) − s G − ( s ) Q m G ( s ) = − s G − ( s ) [ m , Q ] G ( s ) = 0 , (2.62)where we used [ m , Q ] = 0.The path-ordered exponential G ( s , s ) is called a cohomomorphism . A cohomomorphism H generates a field redefinition H (Ψ) when it acts on a group-like element 1 / (1 − Ψ), which isdefined for a degree-even state Ψ by11 − Ψ = ∞ X n =0 Ψ ⊗ Ψ ⊗ . . . ⊗ Ψ | {z } n = 1 + Ψ + Ψ ⊗ Ψ + Ψ ⊗ Ψ ⊗ Ψ . . . , (2.63)as follows: H (Ψ) = π H − Ψ , (2.64)where π is the projector to the one-string sector. We only use a few identities regardingcohomomorphisms in this paper, and they are summarized in appendix A. See [38] for moredetailed discussions. In this section we show that open superstring field theory with the A ∞ structure in the precedingsection is related to the Berkovits formulation by partial gauge fixing and field redefinition. The action with the A ∞ structure in the preceding section can be written as S = − ω (Ψ , Q Ψ) − ω (Ψ , M (Ψ , Ψ)) − ω (Ψ , M (Ψ , Ψ , Ψ)) + . . . = − Z dt (cid:20) ω (Ψ , Q t Ψ) + ω (Ψ , M ( t Ψ , t Ψ)) + ω (Ψ , M ( t Ψ , t Ψ , t Ψ)) + . . . (cid:21) = − Z dt ∞ X n =1 ω (Ψ , M n ( t Ψ , t Ψ , . . . , t Ψ | {z } n )) = − Z dt ω (cid:16) Ψ , π M − t Ψ (cid:17) , (3.1)where M = M (1). The two important relations we derived in subsection 2.3 are M = G − Q G (3.2)14ith G = G (1) = P exp (cid:20) Z ds µ ( s ) (cid:21) , (3.3)and [ η , G ] = m G , (3.4)which is (2.61) with s = 1.We define the inner product ω L in the large Hilbert space by ω L ( ξ A , A ) = − ω ( A , A ) . (3.5)It is related to the BPZ inner product in the large Hilbert space as ω L ( A , A ) = ( − deg( A ) h A , A i , (3.6)and the inner product ω L ( A , A ) is graded antisymmetric: ω L ( A , A ) = − ( − deg( A ) deg( A ) ω L ( A , A ) . (3.7)In terms of the inner product ω L , we have S = Z dt ω L (cid:16) ξ Ψ , π M − t Ψ (cid:17) . (3.8)Since M consists of Q and µ n , we can use the cyclic property of µ n to bring this action to theform S = − Z dt X i h A ( i ) t ( t ) , QA ( i ) η ( t ) i (3.9)with some t -dependent string fields A ( i ) η ( t ) and A ( i ) t ( t ). It will turn out that the summationover i is not necessary, and the action will be brought to the form S = − Z dt h A t ( t ) , QA η ( t ) i . (3.10)Let us first introduce the operator ξ∂ t as a one-string product and denote the correspondingcoderivation by ξ t . We then write the action as S = Z dt ω L (cid:16) π ξ t − t Ψ , π M − t Ψ (cid:17) . (3.11)Using the identity (A.16) in appendix A, we have ω L (cid:16) π ξ t − t Ψ , π M − t Ψ (cid:17) = ω L (cid:16) π G ξ t − t Ψ , π G M − t Ψ (cid:17) = ω L (cid:16) π G ξ t − t Ψ , π Q G − t Ψ (cid:17) = ω L (cid:16) π G ξ t − t Ψ , Q π G − t Ψ (cid:17) , (3.12)15here we also used (3.2). The action can therefore be written as S = Z dt ω L (cid:16) π G ξ t − t Ψ , Q π G − t Ψ (cid:17) , (3.13)and we identify A η ( t ) and A t ( t ) as A η ( t ) = π G − t Ψ , A t ( t ) = π G ξ t − t Ψ = π G (cid:16) − t Ψ ⊗ ξ Ψ ⊗ − t Ψ (cid:17) . (3.14)In the context of the relation between cohomomorphisms and field redefinitions mentioned atthe end of section 2, A η ( t ) is interpreted as the string field we obtain from t Ψ by the fieldredefinition associated with the cohomomorphism G . This field redefinition, however, is not amap to the small Hilbert space because [ η , G ] = 0.An important property of the string field A η ( t ) is that it obeys the following equation forany t : ηA η ( t ) = A η ( t ) A η ( t ) . (3.15)This can be shown as follows: ηA η ( t ) = π η G − t Ψ = π [ η , G ] 11 − t Ψ = π m G − t Ψ= π m − A η ( t ) = A η ( t ) A η ( t ) , (3.16)where we used (3.4) and the identity (A.5) in appendix A. The relation (3.15) is analogous tothe equation of motion QA + A = 0 in open bosonic string field theory with the bosonic stringfield A . In this case, however, the cohomology of η is trivial, and we can use ξ as a homotopyoperator satisfying { η, ξ } = 1. Therefore, the string field A η ( t ) satisfying (3.15) can be writtenin the pure-gauge form A η ( t ) = ( ηe ξ e Ψ( t ) ) e − ξ e Ψ( t ) (3.17)with e Ψ( t ) in the small Hilbert space.Furthermore, since the string field A η ( t ) satisfies the relation (3.15) for any t , an infinitesimalchange in t has to be implemented by a gauge transformation generated by η . It is convenientto define the corresponding covariant derivative D η ( t ) by D η ( t ) Φ = η Φ − A η ( t ) Φ + ( − Φ Φ A η ( t ) . (3.18)It is nilpotent because of (3.15): D η ( t ) = 0 . (3.19)It acts as a derivation with respect to the star product, D η ( t ) (Φ Φ ) = ( D η ( t ) Φ ) Φ + ( − Φ Φ ( D η ( t ) Φ ) , (3.20)16nd it is BPZ odd: h Φ , D η ( t ) Φ i = − ( − Φ h D η ( t ) Φ , Φ i . (3.21)Using the covariant derivative D η ( t ), ∂ t A η ( t ) can be written as ∂ t A η ( t ) = D η ( t ) A t ( t ) = ηA t ( t ) − A η ( t ) A t ( t ) + A t ( t ) A η ( t ) . (3.22)As alluded by the notation, A t ( t ) in (3.14) satisfies this relation. To prove this, let us calculate ηA t ( t ). We find ηA t ( t ) = π η G ξ t − t Ψ = π [ η , G ξ t ] 11 − t Ψ = π [ η , G ] ξ t − t Ψ + π G [ η , ξ t ] 11 − t Ψ= π m G ξ t − t Ψ + π G ∂ t − t Ψ , (3.23)where ∂ t is the coderivation corresponding to ∂ t as a one-string product. We can use theidentity (A.6) to transform the first term in the second line as π m G ξ t − t Ψ = π m (cid:16) − A η ( t ) ⊗ A t ( t ) ⊗ − A η ( t ) (cid:17) = A η ( t ) A t ( t ) − A t ( t ) A η ( t ) , (3.24)while the second term is π G ∂ t − t Ψ = ∂ t π G − t Ψ = ∂ t A η ( t ) . (3.25)We have thus shown (3.22) with A t ( t ) in (3.14).Note that A t ( t ) satisfying (3.22) for given A η ( t ) is not unique. Suppose that A (1) t ( t ) and A (2) t ( t ) both satisfy (3.22). Then the difference ∆ A t ( t ) = A (1) t ( t ) − A (2) t ( t ) is annihilated by D η ( t ): D η ( t ) ∆ A t ( t ) = 0 . (3.26) t -dependence When we write the action in the form (3.13), we can replace t Ψ by Ψ( t ) with a general t -dependence satisfying Ψ(1) = Ψ , Ψ(0) = 0 . (3.27)While the action is written in terms of Ψ( t ), the t -dependence is topological and the action isa functional of Ψ. This can be shown in the following way.We first define A η ( t ) by A η ( t ) = π G − Ψ( t ) . (3.28)17his satisfies ηA η ( t ) = A η ( t ) A η ( t ) (3.29)for any Ψ( t ) because we can replace t Ψ by Ψ( t ) in the proof (3.16). As in the case of the linear t -dependence, an infinitesimal change in t has to be implemented by a gauge transformation asfollows: ∂ t A η ( t ) = D η ( t ) A t ( t ) = ηA t ( t ) − A η ( t ) A t ( t ) + A t ( t ) A η ( t ) . (3.30)We can show that A t ( t ) given by A t ( t ) = π G ξ t − Ψ( t ) (3.31)satisfies (3.30) because again we can replace t Ψ by Ψ( t ) in the proof of the relation (3.22).Similarly, under the variation δ Ψ( t ), the change δA η ( t ) has to be implemented by a gaugetransformation with some gauge parameter A δ ( t ) as follows: δA η ( t ) = D η ( t ) A δ ( t ) = ηA δ ( t ) − A η ( t ) A δ ( t ) + A δ ( t ) A η ( t ) . (3.32)We can think of a map from Ψ( t ) to ξδ Ψ( t ) as an action of a one-string product, and we denotethe corresponding coderivation by ξ δ . We can then show that A δ ( t ) given by A δ ( t ) = π G ξ δ − Ψ( t ) (3.33)satisfies (3.32) as in the proof of the relation (3.22).We will also need a relation between δA t ( t ) and ∂ t A δ ( t ) which follows from the equation δ ∂ t A η ( t ) = ∂ t δA η ( t ). Starting with (3.30), we find δ ∂ t A η ( t ) = η δA t ( t ) − δA η ( t ) A t ( t ) − A η ( t ) δA t ( t ) + δA t ( t ) A η ( t ) + A t ( t ) δA η ( t )= D η ( t ) δA t ( t ) − δA η ( t ) A t ( t ) + A t ( t ) δA η ( t )= D η ( t ) δA t ( t ) − ( D η ( t ) A δ ( t )) A t ( t ) + A t ( t ) ( D η ( t ) A δ ( t )) . (3.34)Starting with (3.32), we find ∂ t δA η ( t ) = η ∂ t A δ ( t ) − ∂ t A η ( t ) A δ ( t ) − A η ( t ) ∂ t A δ ( t ) + ∂ t A δ ( t ) A η ( t ) + A δ ( t ) ∂ t A η ( t )= D η ( t ) ∂ t A δ ( t ) − ∂ t A η ( t ) A δ ( t ) + A δ ( t ) ∂ t A η ( t )= D η ( t ) ∂ t A δ ( t ) − ( D η ( t ) A t ( t )) A δ ( t ) + A δ ( t ) ( D η A t ( t )) . (3.35)We therefore have D η ( t ) F δt ( t ) = 0 , (3.36)where F δt ( t ) = δA t ( t ) − ∂ t A δ ( t ) − A δ ( t ) A t ( t ) + A t ( t ) A δ ( t ) . (3.37)18et us now consider the variation of the action. The action is given by S = − Z dt h A t ( t ) , QA η ( t ) i . (3.38)Under the variation δ Ψ( t ), we have δ h A t ( t ) , QA η ( t ) i = h δA t ( t ) , QA η ( t ) i + h A t ( t ) , Q δA η ( t ) i . (3.39)Let us rewrite the first term on the right-hand side using F δt ( t ) in (3.37): h δA t ( t ) , QA η ( t ) i = h ∂ t A δ ( t ) , QA η ( t ) i + h A δ ( t ) A t ( t ) − A t ( t ) A δ ( t ) , QA η ( t ) i + h F δt ( t ) , QA η ( t ) i . (3.40)The last term on the right-hand side actually vanishes because QA η ( t ) can be written in theform QA η ( t ) = − D η ( t ) A Q ( t ) . (3.41)This can be shown as follows. First, the coderivation M satisfies the condition [ η , M ] = 0 sothat it can be written as M = − [ η , ξ Q ] (3.42)with some coderivation ξ Q . We then have QA η ( t ) = π Q G − Ψ( t ) = π G M − Ψ( t ) = − π G [ η , ξ Q ] 11 − Ψ( t )= − π G η ξ Q − Ψ( t ) = − π η G ξ Q − Ψ( t ) + π m G ξ Q − Ψ( t )= − ηA Q ( t ) + A η ( t ) A Q ( t ) + A Q ( t ) A η ( t ) = − D η ( t ) A Q ( t ) (3.43)with A Q ( t ) = π G ξ Q − Ψ( t ) . (3.44)In appendix B, we explicitly write QA η ( t ) in this form to a few orders in Ψ for demonstration.Since D η ( t ) F δt ( t ) = 0, we find h F δt ( t ) , QA η ( t ) i = − h F δt ( t ) , D η ( t ) A Q ( t ) i = h D η ( t ) F δt ( t ) , A Q ( t ) i = 0 . (3.45)For the second term on the right-hand side of (3.39), we use (3.32) and (3.30) to find h A t ( t ) , Q δA η ( t ) i = h A t ( t ) , Q ηA δ ( t ) i − h A t ( t ) , Q ( A η ( t ) A δ ( t ) − A δ ( t ) A η ( t ) ) i = h ηA t ( t ) , QA δ ( t ) i + h QA t ( t ) , A η ( t ) A δ ( t ) − A δ ( t ) A η ( t ) i = h ∂ t A η ( t ) , QA δ ( t ) i + h A η ( t ) A t ( t ) − A t ( t ) A η ( t ) , QA δ ( t ) i + h QA t ( t ) , A η ( t ) A δ ( t ) − A δ ( t ) A η ( t ) i = h A δ ( t ) , Q ∂ t A η ( t ) i − h QA δ ( t ) , A η ( t ) A t ( t ) − A t ( t ) A η ( t ) i + h QA t ( t ) , A η ( t ) A δ ( t ) − A δ ( t ) A η ( t ) i . (3.46)19ince h A δ ( t ) A t ( t ) − A t ( t ) A δ ( t ) , QA η ( t ) i = − h Q ( A δ ( t ) A t ( t ) − A t ( t ) A δ ( t ) ) , A η ( t ) i = − h QA δ ( t ) , A t ( t ) A η ( t ) − A η ( t ) A t ( t ) i + h QA t ( t ) , A δ ( t ) A η ( t ) − A η ( t ) A δ ( t ) i , (3.47)we find δ h A t ( t ) , QA η ( t ) i = h ∂ t A δ ( t ) , QA η ( t ) i + h A δ ( t ) , Q ∂ t A η ( t ) i = ∂ t h A δ ( t ) , QA η ( t ) i . (3.48)This shows that the t -dependence of the action is topological: δS = − Z dt ∂ t h A δ ( t ) , QA η ( t ) i = − h A δ , QA η i , (3.49)where A η = A η (1) = π G − Ψ , A δ = A δ (1) = π G ξ δ − Ψ = π G (cid:16) − Ψ ⊗ ξδ Ψ ⊗ − Ψ (cid:17) . (3.50)Let us verify that this form of the variation of the action coincides with the original formwe started with. The variation of the action can be written as δS = − h A δ , QA η i = ω L (cid:16) π G ξ δ − Ψ , Q π G − Ψ (cid:17) = ω L (cid:16) π G ξ δ − Ψ , π Q G − Ψ (cid:17) = ω L (cid:16) π G ξ δ − Ψ , π G M − Ψ (cid:17) . (3.51)Using the identity (A.16), we have δS = ω L (cid:16) π ξ δ − Ψ , π M − Ψ (cid:17) = ω L (cid:16) ξ δ Ψ , π M − Ψ (cid:17) = − ω (cid:16) δ Ψ , π M − Ψ (cid:17) . (3.52)The equation of motion is thus given by π M − Ψ = 0 , (3.53)and the action is invariant under the gauge transformation given by δ Λ Ψ = π M (cid:16) − Ψ ⊗ Λ ⊗ − Ψ (cid:17) . (3.54)20 .3 Field redefinition To summarize, we have shown that the action constructed in [35] with the A ∞ structure canbe written in the form S = − Z dt h A t ( t ) , QA η ( t ) i , (3.55)where A η ( t ) = π G − Ψ( t ) , A t ( t ) = π G ξ t − Ψ( t ) = π G (cid:16) − Ψ( t ) ⊗ ξ∂ t Ψ( t ) ⊗ − Ψ( t ) (cid:17) , (3.56)and A η ( t ) and A t ( t ) satisfy ηA η ( t ) = A η ( t ) A η ( t ) , (3.57) ∂ t A η ( t ) = D η ( t ) A t ( t ) = ηA t ( t ) − A η ( t ) A t ( t ) + A t ( t ) A η ( t ) . (3.58)We show in this subsection that this action in terms of Ψ is related to the action in terms of e Ψwhich is obtained from the Berkovits formulation by the partial gauge fixing [34], S = − Z dt h e A t ( t ) , Q e A η ( t ) i (3.59)with e A η ( t ) = ( η e ξ e Ψ( t ) ) e − ξ e Ψ( t ) , e A t ( t ) = ( ∂ t e ξ e Ψ( t ) ) e − ξ e Ψ( t ) , (3.60)by the field redefinition determined from A η ( t ) = e A η ( t ) . (3.61)In terms of the string fields Ψ( t ) and e Ψ( t ), the relation (3.61) is written as π G − Ψ( t ) = ( η e ξ e Ψ( t ) ) e − ξ e Ψ( t ) . (3.62)Using this relation between Ψ( t ) and e Ψ( t ), we can write A t ( t ) in terms of e Ψ( t ). The stringfields A η ( t ) and A t ( t ) written in terms of e Ψ( t ) satisfy (3.58). On the other hand, e A t ( t ) alsosatisfies ∂ t A η ( t ) = D η ( t ) e A t ( t ) = η e A t ( t ) − A η ( t ) e A t ( t ) + e A t ( t ) A η ( t ) (3.63)because of the relation (3.61). As we explained in (3.26), the difference ∆ A t ( t ) = A t ( t ) − e A t ( t )is therefore annihilated by D η ( t ): D η ( t ) ∆ A t ( t ) = 0 . (3.64)It then follows that the difference does not contribute to the action because h ∆ A t ( t ) , QA η ( t ) i = − h ∆ A t ( t ) , D η ( t ) A Q ( t ) i = h D η ( t ) ∆ A t ( t ) , A Q ( t ) i = 0 , (3.65)21here we used (3.41). We have thus shown that the action (3.55) written in terms of Ψ( t ) ismapped to S = − Z dt h ( ∂ t e ξ e Ψ( t ) ) e − ξ e Ψ( t ) , Q ( ( η e ξ e Ψ( t ) ) e − ξ e Ψ( t ) ) i (3.66)by the field redefinition determined from (3.62). Since the t -dependence is topological in bothactions, the action (3.55) is a functional of Ψ and the action (3.66) is a functional of e Ψ. Thestring fields Ψ and e Ψ of the two theories are related by π G − Ψ = ( η e ξ e Ψ ) e − ξ e Ψ . (3.67) The theory in terms of e Ψ( t ) is obtained from the Berkovits formulation in terms of e Φ( t ) by thepartial gauge fixing. The string fields of the two theories are related by e Φ( t ) = ξ e Ψ( t ) . (3.68)In fact, the theory in terms of Ψ( t ) also allows a description in terms of a string field Φ( t ) inthe large Hilbert space, where the string fields of the two theories are related byΦ( t ) = ξ Ψ( t ) . (3.69)By replacing Ψ( t ) by η Φ( t ) and ξ∂ t Ψ( t ) by ∂ t Φ( t ) in (3.56), we have A η ( t ) = π G − η Φ( t ) , A t ( t ) = π G (cid:16) − η Φ( t ) ⊗ ∂ t Φ( t ) ⊗ − η Φ( t ) (cid:17) . (3.70)Apparently, the action in terms of Φ( t ) does not contain ξ , but we use ξ to realize G . While theoperator ξ does not have simple transformation properties under conformal transformations, therelations (3.57) and (3.58) are satisfied and they guarantee that the action has gauge invariancesin the large Hilbert space.The variation of the action is given by δS = ω L (cid:16) δ Φ , π M − η Φ (cid:17) . (3.71)The equation of motion is π M − η Φ = 0 , (3.72)and the action is invariant under the gauge transformation given by δ Φ = π M (cid:16) − η Φ ⊗ Λ ⊗ − η Φ (cid:17) + η Ω , (3.73)22here Λ and Ω are gauge parameters. Note that the gauge transformation generated by η islinear even in the interacting theory. If we choose Φ( t ) = t Φ, the action can be written as S = 12 ω L (Φ , Qη Φ) + ∞ X n =2 n + 1 ω L (Φ , M n ( η Φ , η Φ , . . . , η Φ | {z } n ))= 12 ω L (Φ , Qη Φ) + 13 ω L (Φ , M ( η Φ , η Φ)) + 14 ω L (Φ , M ( η Φ , η Φ , η Φ)) + . . . . (3.74)For any gauge-invariant theory in terms of Ψ in the small Hilbert space, we can trivially embedit to the theory with Φ in the large Hilbert space by replacing Ψ with η Φ. While the form (3.74)of the action can be directly obtained by this procedure, an important point in our discussionis that the actions written in terms of Ψ( t ) and Φ( t ) both take the WZW-like form. We have considered four theories in this paper. (See figure 1 in the introduction.) In theBerkovits formulation the string field e Φ is in the large Hilbert space, and the theory in termsof e Ψ in the small Hilbert space is obtained from the Berkovits formulation by the partial gaugefixing [34]. We have also seen that the theory with the A ∞ structure constructed in [35] writtenin terms of Ψ in the small Hilbert space can be obtained from a theory written in terms of Φin the large Hilbert space by the partial gauge fixing. The actions of these four theories can bewritten in the following common form: S = − Z dt h A t ( t ) , QA η ( t ) i , (4.1)where and A η ( t ) and A t ( t ) satisfy ηA η ( t ) = A η ( t ) A η ( t ) , (4.2) ∂ t A η ( t ) = ηA t ( t ) − A η ( t ) A t ( t ) + A t ( t ) A η ( t ) . (4.3)The string fields A η ( t ) and A t ( t ) are parameterized by Φ( t ), Ψ( t ), e Φ( t ), and e Ψ( t ) as A η ( t ) = π G − η Φ( t ) , A t ( t ) = π G (cid:16) − η Φ( t ) ⊗ ∂ t Φ( t ) ⊗ − η Φ( t ) (cid:17) , (4.4) A η ( t ) = π G − Ψ( t ) , A t ( t ) = π G (cid:16) − Ψ( t ) ⊗ ξ∂ t Ψ( t ) ⊗ − Ψ( t ) (cid:17) , (4.5) A η ( t ) = ( η e e Φ( t ) ) e − e Φ( t ) , A t ( t ) = ( ∂ t e e Φ( t ) ) e − e Φ( t ) , (4.6) A η ( t ) = ( η e ξ e Ψ( t ) ) e − ξ e Ψ( t ) , A t ( t ) = ( ∂ t e ξ e Ψ( t ) ) e − ξ e Ψ( t ) . (4.7)23hese string fields are related by the partial gauge fixing as follows:Φ( t ) = ξ Ψ( t ) , e Φ( t ) = ξ e Ψ( t ) . (4.8)The two string fields Ψ( t ) and e Ψ( t ) in the small Hilbert space are related by π G − Ψ( t ) = ( η e ξ e Ψ( t ) ) e − ξ e Ψ( t ) . (4.9)The t dependence in these actions is topological, and the actions are functionals of Ψ, e Ψ, Φ, and e Φ, which are values of Ψ( t ), e Ψ( t ), Φ( t ), and e Φ( t ), respectively, at t = 1. The field redefinitionbetween Ψ and e Ψ is determined by π G − Ψ = ( η e ξ e Ψ ) e − ξ e Ψ . (4.10)The relation between Φ and e Φ in the large Hilbert space is essentially determined by therelation between Ψ and e Ψ in the small Hilbert space, but there are many interesting aspects tobe explored for the embedding of the theory with the A ∞ structure to the large Hilbert spaceand its relation to the Berkovits formulation. We hope to continue research in this direction.Another important direction to extend the results in this paper will be to consider general A ∞ structures with nonassociative two-string products, as discussed in [36]. Once we succeed inthis generalization, we expect that the generalization to heterotic string field theory would bestraightforward, and we hope to further extend the results in this paper to type II superstringfield theory. Use of the large Hilbert space of the superconformal ghost sector has been an importantingredient in formulating superstring field theory, and we believe that we have gained insightinto its relation to the A ∞ structure. Another crucial aspect in formulating superstring fieldtheory would be its relation to the supermoduli space of super-Riemann surfaces [42], and whenwe combine all the insight from the large Hilbert space, the A ∞ structure, and the supermodulispace of super-Riemann surfaces, we expect to have more fundamental understanding towardscompletion of formulating superstring field theory. Acknowledgments
The work of T.E. was supported in part by the DFG Transregional Collaborative ResearchCentre TRR 33 and the DFG cluster of excellence Origin and Structure of the Universe. Thework of Y.O. was supported in part by Grant-in-Aid for Scientific Research (B) No. 25287049and Grant-in-Aid for Scientific Research (C) No. 24540254 from the Japan Society for thePromotion of Science (JSPS). See [39, 40, 41] for recent discussions on closed superstring field theory. Coderivations and cohomomorphisms
In this appendix we summarize identities regarding cohomomorphisms used in this paper.See [38] for more detailed discussions.For a degree-even state Ψ, we define a group-like element 1 / (1 − Ψ) by11 − Ψ = ∞ X n =0 Ψ ⊗ Ψ ⊗ . . . ⊗ Ψ | {z } n = 1 + Ψ + Ψ ⊗ Ψ + Ψ ⊗ Ψ ⊗ Ψ . . . . (A.1)The action of a coderivation C on 1 / (1 − Ψ) is given by C − Ψ = 11 − Ψ ⊗ (cid:16) π C − Ψ (cid:17) ⊗ − Ψ . (A.2)Consider the cohomomorphism H given by H = G ( s , s ) = P exp (cid:20) Z s s ds ′ µ ( s ′ ) (cid:21) . (A.3)It generates a field redefinition H (Ψ) when it acts on 1 / (1 − Ψ) as follows: H (Ψ) = π H − Ψ . (A.4)In the rest of this appendix, we prove the following identities: H − Ψ = 11 − H (Ψ) , (A.5) H C − Ψ = 11 − H (Ψ) ⊗ (cid:16) π H C − Ψ (cid:17) ⊗ − H (Ψ) , (A.6) ω L (cid:16) π C − Ψ , π C − Ψ (cid:17) = ω L (cid:16) π H C − Ψ , π H C − Ψ (cid:17) , (A.7)where C , C , and C are arbitrary coderivations.Let us begin with (A.5). This is trivially satisfied when s = s . We prove (A.5) by showing G ( s − ∆ s, s ) 11 − Ψ = 11 − G ( s − ∆ s ; Ψ) + O (∆ s ) (A.8)with G ( s ; Ψ) = π G ( s, s ) 11 − Ψ (A.9)when G ( s, s ) 11 − Ψ = 11 − G ( s ; Ψ) (A.10)is satisfied. From the definition of the path-ordered exponential, G ( s − ∆ s, s ) is given by G ( s − ∆ s, s ) = G ( s, s ) + ∆ s µ ( s ) G ( s, s ) + O (∆ s ) . (A.11)25ssuming (A.10), the left-hand side of (A.8) is then G ( s − ∆ s, s ) 11 − Ψ = G ( s, s ) 11 − Ψ + ∆ s µ ( s ) G ( s, s ) 11 − Ψ + O (∆ s )= 11 − G ( s ; Ψ) + ∆ s µ ( s ) 11 − G ( s ; Ψ) + O (∆ s )= 11 − G ( s ; Ψ) + ∆ s − G ( s ; Ψ) ⊗ (cid:16) π µ ( s ) 11 − G ( s ; Ψ) (cid:17) ⊗ − G ( s ; Ψ) + O (∆ s ) . (A.12)On the other hand, G ( s − ∆ s ; Ψ) is given by G ( s − ∆ s ; Ψ) = π G ( s, s ) 11 − Ψ + ∆ s π µ ( s ) G ( s, s ) 11 − Ψ + O (∆ s )= G ( s ; Ψ) + ∆ s π µ ( s ) 11 − G ( s ; Ψ) + O (∆ s ) , (A.13)and 1 / (1 − G ( s − ∆ s ; Ψ)) is11 − G ( s − ∆ s ; Ψ)= 11 − G ( s ; Ψ) + ∆ s − G ( s ; Ψ) ⊗ (cid:16) π µ ( s ) 11 − G ( s ; Ψ) (cid:17) ⊗ − G ( s ; Ψ) + O (∆ s ) . (A.14)We have thus shown (A.8), assuming (A.10). This completes the proof of (A.5).Since we have shown (A.5) for any s and s , let us now take a derivative with respect to s to obtain H µ ( s ) 11 − Ψ = 11 − H (Ψ) ⊗ (cid:16) π H µ ( s ) 11 − Ψ (cid:17) ⊗ − H (Ψ) . (A.15)This holds for arbitrary µ ( s ), so we can replace µ ( s ) by C and obtain (A.6). Since everythingother than C is degree even, the relation holds when C is degree odd as well. This completesthe proof of (A.6).Finally, let us consider (A.7). When s = 0 and s = 1, we have ω L (cid:16) π C − Ψ , π C − Ψ (cid:17) = ω L (cid:16) π G C − Ψ , π G C − Ψ (cid:17) (A.16)with G = P exp (cid:20) Z ds ′ µ ( s ′ ) (cid:21) . (A.17)We use this relation in this paper. The relation (A.7) can be shown by proving ∂∂s ω L (cid:16) π H C − Ψ , π H C − Ψ (cid:17) = 0 . (A.18)26ince ∂∂s H = − µ ( s ) H = − ∞ X n =0 s n µ n +2 H , (A.19)it is sufficient to show ω L (cid:16) π µ n H C − Ψ , π H C − Ψ (cid:17) = − ω L (cid:16) π H C − Ψ , π µ n H C − Ψ (cid:17) (A.20)for any n . The left-hand side of (A.20) is ω L (cid:16) π µ n H C − Ψ , π H C − Ψ (cid:17) = ω L (cid:16) π µ n − H (Ψ) H C (Ψ) 11 − H (Ψ) , H C (Ψ) (cid:17) = n − X k =0 ω L (cid:16) µ n ( H (Ψ) , . . . , H (Ψ) | {z } k , H C (Ψ) , H (Ψ) , . . . , H (Ψ) | {z } n − − k ) , H C (Ψ) (cid:17) , (A.21)where H C (Ψ) = π H C − Ψ . (A.22)Note that H (Ψ) is degree even. When A is degree even, we have ω L ( µ n ( A, A , . . . , A n − ) , A n ) = − ω L ( A, µ n ( A , . . . , A n ) ) = ω L ( µ n ( A , . . . , A n ) , A ) . (A.23)We use this formula to find n − X k =0 ω L (cid:16) µ n ( H (Ψ) , . . . , H (Ψ) | {z } k , H C (Ψ) , H (Ψ) , . . . , H (Ψ) | {z } n − − k ) , H C (Ψ) (cid:17) = ω L (cid:16) µ n ( H C (Ψ) , H (Ψ) , . . . , H (Ψ) | {z } n − ) , H C (Ψ) (cid:17) + n − X k =1 ω L (cid:16) µ n ( H C (Ψ) , H (Ψ) , . . . , H (Ψ) | {z } n − − k , H C (Ψ) , H (Ψ) , . . . , H (Ψ) | {z } k − ) , H (Ψ) ) (cid:17) = − n − X k =0 ω L (cid:16) H C (Ψ) , µ n ( H (Ψ) , . . . , H (Ψ) | {z } n − − k , H C (Ψ) , H (Ψ) , . . . , H (Ψ) | {z } k ) (cid:17) = − ω L (cid:16) π H C − Ψ , π µ n H C − Ψ (cid:17) . (A.24)We have thus shown the relation (A.20). This completes the proof of (A.7). B Equivalence of the actions up to quartic interactions
In this appendix we demonstrate the equivalence of the theory with the A ∞ structure con-structed in [35] and the theory obtained from the Berkovits formulation by partial gauge fix-ing [34] up to quartic interactions by constructing the field redefinition explicitly. We also27ranslate the descriptions of the two-string products and the three-string products into thestandard language in string field theory. B.1 The expansion of the multi-string products
We have seen in subsection 2.3 that the coderivation M ( s ) can be written as M ( s ) = G − ( s ) Q G ( s ) (B.1)with G ( s ) satisfying [ η , G ( s ) ] = s m G ( s ) . (B.2)In this subsection, we present an explicit expression of M ( s ) expanded in s to O ( s ) anddemonstrate that these relations are satisfied.Let us first solve the differential equation dds M ( s ) = [ M ( s ) , µ ( s ) ] (B.3)with the initial condition M (0) = Q (B.4)by expanding M ( s ) and µ ( s ) in s as follows: M ( s ) = ∞ X n =0 s n M n +1 = Q + s M + s M + s M + O ( s ) , (B.5) µ ( s ) = ∞ X n =0 s n µ n +2 = µ + s µ + s µ + O ( s ) . (B.6)Since dds M ( s ) = M + 2 s M + 3 s M + O ( s ) , (B.7)the differential equation (B.3) expanded in s to O ( s ) gives M = [ Q , µ ] , M = 12 [ Q , µ ] + 12 [ M , µ ] , M = 13 [ Q , µ ] + 13 [ M , µ ] + 13 [ M , µ ] . (B.8)As can be seen from these equations, the differential equation (B.3) determines the form of M n written in terms of Q , µ , µ , . . . , µ n . For M and M , we find M = 12 [ Q , µ ] + 12 [ [ Q , µ ] , µ ] , M = 13 [ Q , µ ] + 13 [ [ Q , µ ] , µ ] + 16 [ [ Q , µ ] , µ ] + 16 [ [ [ Q , µ ] , µ ] , µ ] . (B.9)28et us compare these expressions for M , M , and M with the form (B.1). The path-ordered exponential G ( s ) can be expanded in s as follows: G ( s ) = P exp (cid:20) Z s ds ′ µ ( s ′ ) (cid:21) = 1 + Z s ds µ ( s ) + Z s ds Z s ds µ ( s ) µ ( s )+ Z s ds Z s ds Z s ds µ ( s ) µ ( s ) µ ( s ) + O ( s )= 1 + s µ + s (cid:18) µ + 12 µ µ (cid:19) + s (cid:18) µ + 13 µ µ + 16 µ µ + 16 µ µ µ (cid:19) + O ( s ) . (B.10)Its inverse G − ( s ) is G − ( s ) = P exp (cid:20) Z s ds ′ µ ( s ′ ) (cid:21) = 1 + Z s ds µ ( s ) + Z s ds Z s ds µ ( s ) µ ( s )+ Z s ds Z s ds Z s ds µ ( s ) µ ( s ) µ ( s ) + O ( s )= 1 − s µ + s (cid:18) − µ + 12 µ µ (cid:19) + s (cid:18) − µ + 16 µ µ + 13 µ µ − µ µ µ (cid:19) + O ( s ) . (B.11)Then the expansion of G − ( s ) Q G ( s ) is given by G − ( s ) Q G ( s ) = Q + s N + s N + s N + O ( s ) , (B.12)where N = Q µ − µ Q = [ Q , µ ] , N = Q (cid:18) µ + 12 µ µ (cid:19) − µ Q µ + (cid:18) − µ + 12 µ µ (cid:19) Q = 12 [ Q , µ ] + 12 [ [ Q , µ ] , µ ] , N = Q (cid:18) µ + 13 µ µ + 16 µ µ + 16 µ µ µ (cid:19) − µ Q (cid:18) µ + 12 µ µ (cid:19) + (cid:18) − µ + 12 µ µ (cid:19) Q µ + (cid:18) − µ + 16 µ µ + 13 µ µ − µ µ µ (cid:19) Q = 13 [ Q , µ ] + 13 [ [ Q , µ ] , µ ] + 16 [ [ Q , µ ] , µ ] + 16 [ [ [ Q , µ ] , µ ] , µ ] . (B.13)29e have thus verified that M , M , and M in (B.8) and (B.9) are reproduced by the expansionof G − ( s ) Q G ( s ).Let us next consider the coderivation µ ( s ). The conditions on µ ( s ) are characterized by[ η , µ ( s ) ] = m ( s ) (B.14)and the differential equation for m ( s ) dds m ( s ) = [ m ( s ) , µ ( s ) ] (B.15)with the initial condition m (0) = m . (B.16)We expand µ ( s ) as in (B.6) and m ( s ) as m ( s ) = ∞ X n =0 s n m n +2 = m + s m + s m + O ( s ) (B.17)to obtain [ η , µ ] = m , (B.18) m = [ m , µ ] , (B.19)[ η , µ ] = m , (B.20) m = 12 [ m , µ ] + 12 [ m , µ ] , (B.21)[ η , µ ] = m . (B.22)The coderivation µ is constrained to satisfy (B.18). The cohomology of η is trivial, and wecan use ξ to construct µ satisfying (B.18). We present an explicit realization of µ in the nextsubsection. Once we decide on µ , the coderivation m is determined by (B.19). Then thecoderivation µ is constrained to satisfy (B.20). The cohomology of η is again trivial, and wecan use ξ to construct µ satisfying (B.20). We present an explicit realization of µ in the nextsubsection. This way we can construct µ n and m n from (B.14) and (B.15).Finally, let us verify that the relation (B.2) is satisfied for the expansion of G ( s ) givenin (B.10). Using (B.18), (B.20), and (B.22), the commutator [ η , G ( s ) ] is given by[ η , G ( s ) ] = s m + s (cid:18) m + 12 ( m µ + µ m ) (cid:19) + s (cid:18) m + 13 ( m µ + µ m ) + 16 ( m µ + µ m )+ 16 ( m µ µ + µ m µ + µ µ m ) (cid:19) + O ( s ) . (B.23)30sing the relations (B.19) and (B.21), we find12 m + 12 ( m µ + µ m ) = 12 ( m µ − µ m ) + 12 ( m µ + µ m ) = m µ , (B.24)13 m + 13 ( m µ + µ m ) + 16 ( m µ + µ m ) + 16 ( m µ µ + µ m µ + µ µ m )= m (cid:18) µ + 12 µ µ (cid:19) . (B.25)We have thus confirmed (B.2) up to O ( s ). B.2 Translation into string field theory conventions
While the grading with respect to degree is convenient and efficient for describing the A ∞ structure, it is relatively unfamiliar and it would be useful to translate the results of this paperinto standard conventions used in string field theory. We consider two-string products andthree-string products in the rest of this appendix.In subsection 2.2, we have related M , M , and m to V , V , and the star product, re-spectively. For m , µ , and µ , we introduce V ξ , V ξ , and V ξξ , respectively. The relations aresummarized as follows: M ( A , A ) = ( − deg( A ) V ( A , A ) ,M ( A , A , A ) = ( − deg( A ) V ( A , A , A ) ,m ( A , A ) = ( − deg( A ) A A ,m ( A , A , A ) = ( − deg( A ) V ξ ( A , A , A ) ,µ ( A , A ) = ( − deg( A ) V ξ ( A , A ) ,µ ( A , A , A ) = ( − deg( A ) V ξξ ( A , A , A ) . (B.26)The cyclic properties are translated into h A , V ( A , A ) i = h V ( A , A ) , A i , h A , V ( A , A , A ) i = − ( − A h V ( A , A , A ) , A i , h A , A A i = h A A , A i , h A , V ξ ( A , A , A ) i = − ( − A h V ξ ( A , A , A ) , A i , h A , V ξ ( A , A ) i = ( − A h V ξ ( A , A ) , A i , h A , V ξξ ( A , A , A ) i = − h V ξξ ( A , A , A ) , A i . (B.27)The condition (B.18) ηµ ( A , A ) − µ ( ηA , A ) − ( − deg( A ) µ ( A , ηA ) = m ( A , A ) (B.28)31n µ is translated into the following condition on V ξ : ηV ξ ( A , A ) + V ξ ( ηA , A ) + ( − A V ξ ( A , ηA ) = A A . (B.29)One realization of V ξ ( A , A ) is V ξ ( A , A ) = 13 h ξ ( A A ) + ( ξA ) A + ( − A A ( ξA ) i . (B.30)The expression of M in (B.8) is translated into the following expression of the two-stringproduct V ( A , A ): V ( A , A ) = QV ξ ( A , A ) + V ξ ( QA , A ) + ( − A V ξ ( A , QA ) . (B.31)With the choice of V ξ ( A , A ) in (B.30), V ( A , A ) is V ( A , A ) = 13 h X ( A A ) + ( XA ) A + A ( XA ) i , (B.32)where X = { Q, ξ } . (B.33)When we use a line integral of ξ ( z ) to realize ξ , the operator X takes the form of a line integralof the picture-changing operator. The cubic interaction written in terms of V with this choicedoes not suffer from singularities coming from local insertions of the picture-changing operator.The three-string product m is determined by (B.19): m ( A , A , A ) = m ( µ ( A , A ) , A ) + m ( A , µ ( A , A )) − µ ( m ( A , A ) , A ) − ( − deg( A ) µ ( A , m ( A , A )) . (B.34)This is translated into the following expression for V ξ in terms of V ξ : V ξ ( A , A , A ) = V ξ ( A , A ) A − ( − A A V ξ ( A , A )+ V ξ ( A A , A ) − V ξ ( A , A A ) . (B.35)With the choice of V ξ in (B.30), V ξ ( A , A , A ) is V ξ ( A , A , A ) = 23 h ( ξ ( A A )) A − ( − A A ( ξ ( A A )) i . (B.36)The condition (B.20) ηµ ( A , A , A ) − µ ( ηA , A , A ) − ( − deg( A ) µ ( A , ηA , A ) − ( − deg( A )+deg( A ) µ ( A , A , ηA ) = m ( A , A , A ) (B.37)on µ is translated into the following condition on V ξξ : ηV ξξ ( A , A , A ) − V ξξ ( ηA , A , A ) − ( − A V ξξ ( A , ηA , A ) − ( − A + A V ξξ ( A , A , ηA )= V ξ ( A , A , A ) . (B.38)32ince V ξ is written in terms of V ξ , this condition can be understood as the following conditionbetween V ξξ and V ξ : ηV ξξ ( A , A , A ) − V ξξ ( ηA , A , A ) − ( − A V ξξ ( A , ηA , A ) − ( − A + A V ξξ ( A , A , ηA )= V ξ ( A , A ) A − ( − A A V ξ ( A , A ) + V ξ ( A A , A ) − V ξ ( A , A A ) . (B.39)When V ξ ( A , A , A ) is given, one realization of V ξξ ( A , A , A ) is V ξξ ( A , A , A ) = 14 h ξV ξ ( A , A , A ) − V ξ ( ξA , A , A ) − ( − A V ξ ( A , ξA , A ) − ( − A + A V ξ ( A , A , ξA ) i . (B.40)With the choices of V ξ in (B.30) and V ξξ in (B.40), the explicit expression for V ξξ ( A , A , A )is given by V ξξ ( A , A , A ) = 16 h ξ (( ξ ( A A )) A ) − ( − A ξ ( A ( ξ ( A A ))) − ( ξ (( ξA ) A ))) A − ( − A ( ξA ) ( ξ ( A A )) − ( − A ( ξ ( A ( ξA ))) A + A ( ξ (( ξA ) A )) − ( − A + A ( ξ ( A A )) ( ξA ) + ( − A A ( ξ ( A ( ξA ))) i . (B.41)The expression of M in (B.8) is translated into the following expression of the three-stringproduct V ( A , A , A ): V ( A , A , A ) = 12 h QV ξξ ( A , A , A ) − V ξξ ( QA , A , A ) − ( − A V ξξ ( A , QA , A ) − ( − A + A V ξξ ( A , A , QA )+ V ( V ξ ( A , A ) , A ) − ( − A V ( A , V ξ ( A , A ))+ V ξ ( V ( A , A ) , A ) − V ξ ( A , V ( A , A )) i . (B.42)Since V is written in terms of V ξ , the three-string product V is given in terms of V ξ and V ξξ by V ( A , A , A ) = 12 h QV ξξ ( A , A , A ) − V ξξ ( QA , A , A ) − ( − A V ξξ ( A , QA , A ) − ( − A + A V ξξ ( A , A , QA )+ QV ξ ( V ξ ( A , A ) , A ) − ( − A QV ξ ( A , V ξ ( A , A ))+ 2 V ξ ( QV ξ ( A , A ) , A ) − V ξ ( A , QV ξ ( A , A ))+ V ξ ( V ξ ( QA , A ) , A ) + ( − A V ξ ( V ξ ( A , QA ) , A ) − ( − A + A V ξ ( V ξ ( A , A ) , QA ) − ( − A V ξ ( QA , V ξ ( A , A )) − V ξ ( A , V ξ ( QA , A )) − ( − A V ξ ( A , V ξ ( A , QA )) i . (B.43)33his corresponds to the expression of M in (B.9). For a Grassmann-odd string field Ψ, thethree-string product V (Ψ , Ψ , Ψ) is V (Ψ , Ψ , Ψ) = 12 h QV ξξ (Ψ , Ψ , Ψ) − V ξξ ( Q Ψ , Ψ , Ψ) + V ξξ (Ψ , Q Ψ , Ψ) − V ξξ (Ψ , Ψ , Q Ψ)+ QV ξ ( V ξ (Ψ , Ψ) , Ψ) + QV ξ (Ψ , V ξ (Ψ , Ψ))+ 2 V ξ ( QV ξ (Ψ , Ψ) , Ψ) − V ξ (Ψ , QV ξ (Ψ , Ψ))+ V ξ ( Q Ψ , V ξ (Ψ , Ψ)) − V ξ ( V ξ (Ψ , Q Ψ) , Ψ) − V ξ ( V ξ (Ψ , Ψ) , Q Ψ)+ V ξ ( QV ξ (Ψ , Ψ) , Ψ) − V ξ (Ψ , V ξ ( Q Ψ , Ψ)) + V ξ (Ψ , V ξ (Ψ , Q Ψ)) i . (B.44) B.3 The action up to quartic interactions
In the rest of this appendix, we rescale Ψ to g Ψ, where g is the coupling constant, and multiplythe action by 1 /g . The action of open superstring field theory with the A ∞ structure can bewritten as S = S + g S + g S + O ( g ) , (B.45)where S = − h ξ Ψ , Q Ψ i = − Z dt h ξ Ψ , Qt Ψ i ,S = − h ξ Ψ , V (Ψ , Ψ) i = − Z dt h ξ Ψ , V ( t Ψ , t Ψ) i ,S = − h ξ Ψ , V (Ψ , Ψ , Ψ) i = − Z dt h ξ Ψ , V ( t Ψ , t Ψ , t Ψ) i . (B.46)Following subsection 3.1, let us transform this action to the WZW-like form. The cubic inter-action S is S = − Z dt h h ξ Ψ , QV ξ ( t Ψ , t Ψ) i + h ξ Ψ , V ξ ( Qt Ψ , t Ψ) i − h ξ Ψ , V ξ ( t Ψ , Qt Ψ) i i . (B.47)Using the cyclic property of V ξ , this can be written as S = − Z dt h ξ Ψ , QV ξ ( t Ψ , t Ψ) i + Z dt h V ξ ( ξ Ψ , t Ψ) − V ξ ( t Ψ , ξ Ψ) , Qt Ψ i . (B.48)34he quartic interaction S is written in terms of V in (B.44). Using the cyclic properties of V ξ and V ξξ , S can be written as S = − Z dt h ξ Ψ , QV ξξ ( t Ψ , t Ψ , t Ψ) + QV ξ ( V ξ ( t Ψ , t Ψ) , t Ψ) + QV ξ ( t Ψ , V ξ ( t Ψ , t Ψ)) i + Z dt h V ξ ( ξ Ψ , t Ψ) − V ξ ( t Ψ , ξ Ψ) , QV ξ ( t Ψ , t Ψ) i− Z dt h h V ξξ ( ξ Ψ , t Ψ , t Ψ) − V ξξ ( t Ψ , ξ Ψ , t Ψ) + V ξξ ( t Ψ , t Ψ , ξ Ψ) , Qt Ψ i + h V ξ ( V ξ ( ξ Ψ , t Ψ) , t Ψ) − V ξ ( V ξ ( t Ψ , ξ Ψ) , t Ψ) + V ξ ( V ξ ( t Ψ , t Ψ) , ξ Ψ) , Qt Ψ i− h V ξ ( ξ Ψ , V ξ ( t Ψ , t Ψ)) + V ξ ( t Ψ , V ξ ( ξ Ψ , t Ψ)) − V ξ ( t Ψ , V ξ ( t Ψ , ξ Ψ)) , Qt Ψ i i . (B.49)We find that the action is written in the form S = − g Z dt h A t ( t ) , QA η ( t ) i (B.50)with A η ( t ) = g t Ψ + g V ξ ( t Ψ , t Ψ)+ g h V ξξ ( t Ψ , t Ψ , t Ψ) + V ξ ( V ξ ( t Ψ , t Ψ) , t Ψ) + V ξ ( t Ψ , V ξ ( t Ψ , t Ψ)) i + O ( g ) ,A t ( t ) = g ξ Ψ − g h V ξ ( ξ Ψ , t Ψ) − V ξ ( t Ψ , ξ Ψ) i + g h V ξξ ( ξ Ψ , t Ψ , t Ψ) − V ξξ ( t Ψ , ξ Ψ , t Ψ) + V ξξ ( t Ψ , t Ψ , ξ Ψ)+ V ξ ( V ξ ( ξ Ψ , t Ψ) , t Ψ) − V ξ ( V ξ ( t Ψ , ξ Ψ) , t Ψ) + V ξ ( V ξ ( t Ψ , t Ψ) , ξ Ψ) − V ξ ( ξ Ψ , V ξ ( t Ψ , t Ψ)) − V ξ ( t Ψ , V ξ ( ξ Ψ , t Ψ)) + V ξ ( t Ψ , V ξ ( t Ψ , ξ Ψ)) i + O ( g ) . (B.51)Let us next write the action in terms of Ψ( t ) following subsection 3.2. For Ψ( t ) we define A η ( t ) and A t ( t ) by A η ( t ) = g Ψ( t ) + g V ξ (Ψ( t ) , Ψ( t ))+ g h V ξξ (Ψ( t ) , Ψ( t ) , Ψ( t )) + V ξ ( V ξ (Ψ( t ) , Ψ( t )) , Ψ( t )) + V ξ (Ψ( t ) , V ξ (Ψ( t ) , Ψ( t ))) i + O ( g ) (B.52)35nd A t ( t ) = g ξ∂ t Ψ( t ) − g h V ξ ( ξ∂ t Ψ( t ) , Ψ( t )) − V ξ (Ψ( t ) , ξ∂ t Ψ( t )) i + g h V ξξ ( ξ∂ t Ψ( t ) , Ψ( t ) , Ψ( t )) − V ξξ (Ψ( t ) , ξ∂ t Ψ( t ) , Ψ( t )) + V ξξ (Ψ( t ) , Ψ( t ) , ξ∂ t Ψ( t ))+ V ξ ( V ξ ( ξ∂ t Ψ( t ) , Ψ( t )) , Ψ( t )) − V ξ ( V ξ (Ψ( t ) , ξ∂ t Ψ( t )) , Ψ( t ))+ V ξ ( V ξ (Ψ( t ) , Ψ( t )) , ξ∂ t Ψ( t )) − V ξ ( ξ∂ t Ψ( t ) , V ξ (Ψ( t ) , Ψ( t ))) − V ξ (Ψ( t ) , V ξ ( ξ∂ t Ψ( t ) , Ψ( t ))) + V ξ (Ψ( t ) , V ξ (Ψ( t ) , ξ∂ t Ψ( t ))) i + O ( g ) . (B.53)These coincide with A η ( t ) = π G − g Ψ( t )= g Ψ( t ) + g µ ( Ψ( t ) ⊗ Ψ( t ) )+ g h µ ( Ψ( t ) ⊗ Ψ( t ) ⊗ Ψ( t ) ) + µ µ ( Ψ( t ) ⊗ Ψ( t ) ⊗ Ψ( t ) ) i + O ( g ) (B.54)and A t ( t ) = π G ξ t − g Ψ( t ) = π G (cid:16) − g Ψ( t ) ⊗ g ξ∂ t Ψ( t ) ⊗ − g Ψ( t ) (cid:17) = g ξ∂ t Ψ( t ) + g µ ( ξ∂ t Ψ( t ) ⊗ Ψ( t ) + Ψ( t ) ⊗ ξ∂ t Ψ( t ) )+ g h µ ( ξ∂ t Ψ( t ) ⊗ Ψ( t ) ⊗ Ψ( t ) + Ψ( t ) ⊗ ξ∂ t Ψ( t ) ⊗ Ψ( t ) + Ψ( t ) ⊗ Ψ( t ) ⊗ ξ∂ t Ψ( t ) )+ µ µ ( ξ∂ t Ψ( t ) ⊗ Ψ( t ) ⊗ Ψ( t ) + Ψ( t ) ⊗ ξ∂ t Ψ( t ) ⊗ Ψ( t ) + Ψ( t ) ⊗ Ψ( t ) ⊗ ξ∂ t Ψ( t ) ) i + O ( g ) (B.55)under the translation in the preceding subsection. From these expressions, we can explicitlyconfirm that ηA η ( t ) = A η ( t ) A η ( t ) ,∂ t A η ( t ) = ηA t ( t ) − A η ( t ) A t ( t ) + A t ( t ) A η ( t ) (B.56)are satisfied up to O ( g ). B.4 Field redefinition
We define e A η ( t ) by e A η ( t ) = ( η e g ξ e Ψ( t ) ) e − g ξ e Ψ( t ) . (B.57)36t can be expanded in g as e A η ( t ) = g e Ψ( t ) + g ξ e Ψ( t ) , e Ψ( t ) ] + g ξ e Ψ( t ) , [ ξ e Ψ( t ) , e Ψ( t ) ] ] + O ( g ) . (B.58)The string fields Ψ( t ) and e Ψ( t ) are related by A η ( t ) = e A η ( t ) . (B.59)We expand both sides in g to obtain g Ψ( t ) + g V ξ (Ψ( t ) , Ψ( t ))+ g h V ξξ (Ψ( t ) , Ψ( t ) , Ψ( t )) + V ξ ( V ξ (Ψ( t ) , Ψ( t )) , Ψ( t )) + V ξ (Ψ( t ) , V ξ (Ψ( t ) , Ψ( t ))) i + O ( g )= g e Ψ( t ) + g ξ e Ψ( t ) , e Ψ( t ) ] + g ξ e Ψ( t ) , [ ξ e Ψ( t ) , e Ψ( t ) ] ] + O ( g ) . (B.60)Let us expand the field redefinition from e Ψ( t ) to Ψ( t ) asΨ( t ) = f (1) + g f (2) + g f (3) + O ( g ) . (B.61)We then have f (1) = e Ψ( t ) ,f (2) + V ξ ( f (1) , f (1) ) = 12 [ ξ e Ψ( t ) , e Ψ( t ) ] ,f (3) + V ξ ( f (2) , f (1) ) + V ξ ( f (1) , f (2) )+ 12 h V ξξ ( f (1) , f (1) , f (1) ) + V ξ ( V ξ ( f (1) , f (1) ) , f (1) ) + V ξ ( f (1) , V ξ ( f (1) , f (1) )) i = 16 [ ξ e Ψ( t ) , [ ξ e Ψ( t ) , e Ψ( t ) ] ] . (B.62)The field redefinition is given by f (1) = e Ψ( t ) ,f (2) = 12 [ ξ e Ψ( t ) , e Ψ( t ) ] − V ξ ( e Ψ( t ) , e Ψ( t )) ,f (3) = 16 [ ξ e Ψ( t ) , [ ξ e Ψ( t ) , e Ψ( t ) ] ] − V ξ ( [ ξ e Ψ( t ) , e Ψ( t ) ] , e Ψ( t )) − V ξ ( e Ψ( t ) , [ ξ e Ψ( t ) , e Ψ( t ) ] ) − V ξξ ( e Ψ( t ) , e Ψ( t ) , e Ψ( t )) + 12 V ξ ( V ξ ( e Ψ( t ) , e Ψ( t )) , e Ψ( t )) + 12 V ξ ( e Ψ( t ) , V ξ ( e Ψ( t ) , e Ψ( t ))) . (B.63)The string field A t ( t ) written in terms of e Ψ( t ) is A t ( t ) = g A (1) t ( t ) + g A (2) t ( t ) + g A (3) t ( t ) + O ( g ) , (B.64)37here A (1) t ( t ) = ξ∂ t e Ψ( t ) ,A (2) t ( t ) = 12 ξ∂ t [ ξ e Ψ( t ) , e Ψ( t ) ] − ξ∂ t V ξ ( e Ψ( t ) , e Ψ( t )) − V ξ ( ξ∂ t Ψ( t ) , Ψ( t )) + V ξ (Ψ( t ) , ξ∂ t Ψ( t )) , (B.65)and A (3) t ( t ) = 16 ξ∂ t [ ξ e Ψ( t ) , [ ξ e Ψ( t ) , e Ψ( t ) ] ] − ξ∂ t V ξ ( [ ξ e Ψ( t ) , e Ψ( t ) ] , e Ψ( t )) − ξ∂ t V ξ ( e Ψ( t ) , [ ξ e Ψ( t ) , e Ψ( t ) ] ) − ξ∂ t V ξξ ( e Ψ( t ) , e Ψ( t ) , e Ψ( t ))+ 12 ξ∂ t V ξ ( V ξ ( e Ψ( t ) , e Ψ( t )) , e Ψ( t )) + 12 ξ∂ t V ξ ( e Ψ( t ) , V ξ ( e Ψ( t ) , e Ψ( t ))) − V ξ ( ξ∂ t [ ξ e Ψ( t ) , e Ψ( t ) ] , e Ψ( t )) + V ξ ( ξ∂ t V ξ ( e Ψ( t ) , e Ψ( t )) , e Ψ( t )) − V ξ ( ξ∂ t e Ψ( t ) , [ ξ e Ψ( t ) , e Ψ( t ) ]) + V ξ ( ξ∂ t e Ψ( t ) , V ξ ( e Ψ( t ) , e Ψ( t )))+ 12 V ξ ([ ξ e Ψ( t ) , e Ψ( t ) ] , ξ∂ t e Ψ( t )) − V ξ ( V ξ ( e Ψ( t ) , e Ψ( t )) , ξ∂ t e Ψ( t ))+ 12 V ξ ( e Ψ( t ) , ξ∂ t [ ξ e Ψ( t ) , e Ψ( t ) ]) − V ξ ( e Ψ( t ) , ξ∂ t V ξ ( e Ψ( t ) , e Ψ( t )))+ 12 V ξξ ( ξ∂ t e Ψ( t ) , e Ψ( t ) , e Ψ( t )) − V ξξ ( e Ψ( t ) , ξ∂ t e Ψ( t ) , e Ψ( t ))+ 12 V ξξ ( e Ψ( t ) , e Ψ( t ) , ξ∂ t e Ψ( t ))+ 12 V ξ ( V ξ ( ξ∂ t e Ψ( t ) , e Ψ( t )) , e Ψ( t )) − V ξ ( V ξ ( e Ψ( t ) , ξ∂ t e Ψ( t )) , e Ψ( t ))+ 12 V ξ ( V ξ ( e Ψ( t ) , e Ψ( t )) , ξ∂ t e Ψ( t )) − V ξ ( ξ∂ t e Ψ( t ) , V ξ ( e Ψ( t ) , e Ψ( t ))) − V ξ ( e Ψ( t ) , V ξ ( ξ∂ t e Ψ( t ) , e Ψ( t ))) + 12 V ξ ( e Ψ( t ) , V ξ ( e Ψ( t ) , ξ∂ t e Ψ( t ))) . (B.66)On the other hand, e A t ( t ) is expanded as e A t ( t ) = ( ∂ t e g ξ e Ψ( t ) ) e − g ξ e Ψ( t ) = g e A (1) t ( t ) + g e A (2) t ( t ) + g e A (3) t ( t ) + O ( g ) , (B.67)where e A (1) t ( t ) = ξ∂ t e Ψ( t ) , e A (2) t ( t ) = 12 [ ξ e Ψ( t ) , ξ∂ t e Ψ( t ) ] , e A (3) t ( t ) = 16 [ ξ e Ψ( t ) , [ ξ e Ψ( t ) , ξ∂ t e Ψ( t ) ] ] . (B.68)We find A t ( t ) = e A t ( t ) , (B.69)38nd the difference is∆ A t ( t ) = A t ( t ) − e A t ( t ) = g ( A (2) t ( t ) − e A (2) t ( t ) ) + g ( A (3) t ( t ) − e A (3) t ( t ) ) + O ( g ) . (B.70)In subsection 3.3, we have shown that the difference ∆ A t ( t ) is annihilated by the covariantderivative D η ( t ). Let us confirm this explicitly to O ( g ). Since D η ( t ) ∆ A t ( t ) = g η ( A (2) t ( t ) − e A (2) t ( t ))+ g (cid:16) η ( A (3) t ( t ) − e A (3) t ( t )) − [ e Ψ( t ) , A (2) t ( t ) − e A (2) t ( t ) ] (cid:17) + O ( g ) , (B.71)we need to show η ( A (2) t ( t ) − e A (2) t ( t )) = 0 , (B.72) η ( A (3) t ( t ) − e A (3) t ( t )) − [ e Ψ( t ) , A (2) t ( t ) − e A (2) t ( t ) ] = 0 . (B.73)For ηA (2) t ( t ) and η e A (2) t ( t ), we find ηA (2) t ( t ) = 12 ∂ t [ ξ e Ψ( t ) , e Ψ( t ) ] − ξ∂ t { e Ψ( t ) , e Ψ( t ) } − ∂ t V ξ ( e Ψ( t ) , e Ψ( t )) + ξ∂ t ( e Ψ( t ) e Ψ( t )) − ( ξ∂ t Ψ( t )) Ψ( t ) + V ξ ( ∂ t Ψ( t ) , Ψ( t )) + Ψ( t ) ( ξ∂ t Ψ( t )) + V ξ (Ψ( t ) , ∂ t Ψ( t ))= 12 ∂ t [ ξ e Ψ( t ) , e Ψ( t ) ] + [ Ψ( t ) , ξ∂ t Ψ( t ) ] = 12 [ ξ e Ψ( t ) , ∂ t e Ψ( t ) ] + 12 [ Ψ( t ) , ξ∂ t Ψ( t ) ] , (B.74)and η e A (2) t ( t ) = 12 [ e Ψ( t ) , ξ∂ t e Ψ( t ) ] + 12 [ ξ e Ψ( t ) , ∂ t e Ψ( t ) ] . (B.75)We have thus confirmed (B.72). The calculation of ηA (3) t ( t ) is tedious but straightforward. Theupshot is ηA (3) t ( t ) = 16 ∂ t [ ξ e Ψ( t ) , [ ξ e Ψ( t ) , e Ψ( t ) ] ] −
12 [ ξ∂ t e Ψ( t ) , [ ξ e Ψ( t ) , e Ψ( t ) ] ] + [ e Ψ( t ) , A (2) t ( t ) ] . (B.76)For η e A (3) t ( t ), we have η e A (3) t ( t ) = 16 [ e Ψ( t ) , [ ξ e Ψ( t ) , ξ∂ t e Ψ( t ) ] ] + 16 [ ξ e Ψ( t ) , [ e Ψ( t ) , ξ∂ t e Ψ( t ) ] ] + 16 [ ξ e Ψ( t ) , [ ξ e Ψ( t ) , ∂ t e Ψ( t ) ] ] . (B.77)It is not difficult to verify that the relation (B.73) is satisfied.Finally, let us write QA η ( t ) in the form QA η ( t ) = − D η ( t ) A Q ( t ) (B.78)39p to O ( g ). Since η (cid:16) g ξQ Ψ + g ξV (Ψ , Ψ) + g V ξ ( ξQ Ψ , Ψ) + g V ξ (Ψ , ξQ Ψ) (cid:17) = g Q Ψ + g V (Ψ , Ψ) + g ( ξQ Ψ) Ψ − g V ξ ( Q Ψ , Ψ) + g Ψ ( ξQ Ψ) + g V ξ (Ψ , Q Ψ)= g Q
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