Super Yang-Mills action from WZW-like open superstring field theory including the Ramond sector
aa r X i v : . [ h e p - t h ] M a r Super Yang–Mills action from WZW-like open superstring fieldtheory including the Ramond sector
Mitsuru Asada and Isao Kishimoto Department of Physics, Niigata University, Niigata 950-2181, Japan ∗ Faculty of Education, Niigata University, Niigata 950-2181, Japan † Abstract
In the framework of WZW-like open superstring field theory (SSFT) including the Ramond (R)sector whose action was constructed by Kunitomo and Okawa, we truncate the string fields in boththe Neveu–Schwarz (NS) and R sectors up to the lowest massless level and obtain the ten-dimensionalsuper Yang–Mills (SYM) action with bosonic extra term by explicit calculation of the SSFT action.Furthermore, we compute a contribution from the massive part up to the lowest order and findthat the bosonic extra term is canceled and instead a fermionic extra term appears, which can beinterpreted as a string correction to the SYM action. This calculation is an extension to the R sectorof the earlier work by Berkovits and Schnabl in the NS sector. We also study gauge transformation,equation of motion, and spacetime supersymmetry transformation of the massless component fieldsinduced from those of string fields. ∗ Present address: [email protected] † E-mail: [email protected]
Introduction
It is expected that open superstring theory describes super Yang–Mills theory at low energy. Sometime ago, Berkovits and Schnabl [1] showed that the Yang–Mills action is derived from Berkovits’ Wess–Zumino–Witten (WZW)-like action for open superstring field theory [2] in the Neveu–Schwarz (NS)sector. Recently, Kunitomo and Okawa have constructed a complete action for open superstring fieldtheory including the Ramond (R) sector as an extension of the WZW-like action [3]. Therefore, itis natural to expect that the ten-dimensional super Yang–Mills (SYM) action can be derived from it.Furthermore, an explicit formula for spacetime supersymmetry transformation in terms of string fieldshas been proposed in Refs. [4, 5] and hence we can compare the induced transformation of the componentfields and the conventional supersymmetry transformation of SYM.As a first step toward the above issue, we adopt the level truncation method at the lowest level, whichcorresponds to the massless component fields due to the GSO projection. We perform explicit calculationsin the superstring field theory in both NS and R sectors and obtain the SYM action with extra bosonicterm, O ( A µ ) , in terms of the component fields [6]. We compute a contribution to the effective action ofmassless fields from the massive part of string fields in the zero-momentum sector, up to lowest orderwith respect to the coupling constant, and find that the extra O ( A µ ) is canceled and, instead, an extrafermionic term, O ( λ α ) , appears in the order of ( α ′ ) , which can be interpreted as a string correction toSYM.We also derive the induced gauge and supersymmetry transformations up to nonzero lowest order andfind that the resulting formulas are consistent with those of the conventional SYM.The organization of this paper is as follows. In the next section, we briefly review the action of opensuperstring field theory in terms of Kunitomo and Okawa’s formulation. In Sect. 3, we derive the explicitform of the level-truncated action at the lowest level. In Sect. 4, we evaluate the contribution from themassive part. In Sect. 5, we find the induced transformations from those of the superstring field theory.In Sect. 6, we give some concluding remarks. In the appendix we summarize our convention on spinfields, which is necessary for explicit computations. Let us first review Kunitomo and Okawa’s action [3] for open superstring field theory (SSFT).As an extension of Berkovits’ WZW-like action in the NS sector, Kunitomo and Okawa proposed acomplete action for open superstring field theory including the R sector as follows: S [Φ , Ψ] = S NS [Φ] + S R [Φ , Ψ] , (2.1) S NS [Φ] = − ˆ dt h A t ( t ) , QA η ( t ) i , (2.2) S R [Φ , Ψ] = − hh Ψ , Y Q Ψ ii − ˆ dt h A t ( t ) , ( F ( t )Ψ) i . (2.3)The action is a functional of string fields: Φ and Ψ . h , i and hh , ii are the BPZ inner product inthe large and small Hilbert space, respectively. Q is the BRST operator and η is the zeromode of η ( z ) : η = ¸ dz πi η ( z ) . We use the relation of the superconformal ghosts between ( β, γ ) and ( ξ, η, φ ) as β ( z ) = ∂ξe − φ ( z ) and γ ( z ) = e φ η ( z ) . Φ is a Grassmann even string field in the NS sector, whose ghostnumber ( n gh ) and picture number ( n pic ) are both zero, and expanded by the states in the large Hilbert This result has already been obtained in Ref. [1] in computation in the NS sector. A t ( t ) , A η ( t ) , and F ( t ) are defined by A t ( t ) = ( ∂ t e t Φ ) e − t Φ = Φ , A η ( t ) = ( ηe t Φ ) e − t Φ , (2.4) F ( t )Ψ = Ψ + Ξ { A η ( t ) , Ψ } + Ξ { A η ( t ) , Ξ { A η ( t ) , Ψ }} + · · · (2.5) = ∞ X k =0 Ξ { A η ( t ) , Ξ { A η ( t ) , . . . , Ξ { A η ( t ) , | {z } k Ψ } . . . }} , (2.6)where Ξ = Θ( β ) , the Heaviside step function of β . Ψ is a Grassmann odd string field in the Rsector with ( n gh , n pic ) = (1 , − / , and it is expanded by the states in the restricted small Hilbert space.Namely, the conditions for the R string field are η Ψ = 0 , XY
Ψ = Ψ . (2.7) X and Y are kinds of picture-changing operators with the picture number and − , respectively: X = − δ ( β ) G + δ ′ ( β ) b = { Q, Θ( β ) } , (2.8) Y = − c δ ′ ( γ ) , (2.9)where XY X = X and Y XY = Y hold and therefore XY gives a projector: ( XY ) = XY .In the following, we consider SSFT on the N BPS D9-branes in the flat ten-dimensional spacetime,and hence the string fields have the Chan–Paton factors implicitly and the GSO projection is imposedon the string fields.
Gauge transformation
The action in Eq. (2.1) is invariant under the following infinitesimal gaugetransformations: A δ g (Λ ,λ, Ω) = Q Λ + D η Ω + { F Ψ , F Ξ( { F Ψ , Λ } − λ ) } , A δ ≡ ( δe Φ ) e − Φ , (2.10) δ g (Λ ,λ ) Ψ = Qλ + XηF Ξ D η ( { F Ψ , Λ } − λ ) , (2.11)where F = F ( t = 1) , D η B = ηB − [ A η , B } , A η = A η ( t = 1) , (2.12)and the gauge parameter λ in the R sector is in the restricted small Hilbert space, namely, ηλ = 0 , XY λ = λ . (2.13) Λ and Ω are gauge parameters in the NS sector in the large Hilbert space. Equation of motion
Taking the variation of the action in Eq. (2.1) with respect to the string fields, Φ and Ψ , we have the equations of motion as follows: QA η + ( F Ψ) = 0 , Q Ψ +
XηF
Ψ = 0 . (2.14)The second equation is consistent with the condition for Ψ , which is in the restricted small Hilbert space. In Ref. [7], a better expression for Ξ has been proposed. pacetime supersymmetry In the ten-dimensional Minkowski spacetime, the SSFT action in Eq. (2.1)is invariant under the spacetime supersymmetry transformation, which is given by [4]: A δ S = e Φ S Ξ( e − Φ F Ψ e Φ ) e − Φ + { F Ψ , F Ξ A S } , A S = ( S e Φ ) e − Φ , (2.15) δ S Ψ =
XηF Ξ S A η , (2.16)where S is a derivation with respect to the star product of string fields and is given by a constant Weylspinor ǫ ˙ α and the spin field with n pic = − / : S = ǫ ˙ α ˛ dz πi S ˙ α ( − / ( z ) . (2.17)We use a bosonized formulation of spin fields as in the appendix for the following explicit computation. Here, we evaluate the SSFT action in Eq. (2.1) using the level truncation method explicitly. We definethe level of string fields as the eigenvalue of L = { Q, b } , except for a contribution from momentum. Weexpand the string fields with component fields up to the lowest level in both NS and R sectors. In the NS sector, we expand a string field Φ with ( n gh , n pic ) = (0 , in the large Hilbert space. Thelowest-level state is cξe − φ e ik · X (0) | i , (3.1)which corresponds to the tachyon, but it is excluded by the GSO projection of Eq. (A.5). The lowest-levelstates on the GSO projected space are e ik · X (0) | i , cξψ µ e − φ e ik · X (0) | i , c∂cξ∂ξe − φ e ik · X (0) | i , (3.2)which correspond to the massless level. The first one can be eliminated by the Ω -gauge transformationin Eq. (2.10), which can be rewritten as e − Φ δ g (Ω) e Φ = η ( e − Φ Ω e Φ ) and has an expression for finitetransformation: e Φ ′ = e Φ g with ηg = 0 , because it can be rewritten as e ik · X (0) | i = η (cid:0) ξe ik · X (0) | i (cid:1) .In the following, we take into account only the other two states and their component fields as a level-truncated string field Φ in the NS sector. Namely, we use a partial gauge-fixing condition, ξ Φ = 0 .After Ref. [1], we use the notation Φ A = ˆ d k (2 π ) A µ ( k ) V µA ( k )(0) | i , V µA ( k ) = cξe − φ ψ µ ( z ) e ik · X ( z, ¯ z ) , (3.3) Φ B = ˆ d k (2 π ) B ( k ) V B ( k )(0) | i , V B ( k ) = c∂cξ∂ξe − φ ( z ) e ik · X ( z, ¯ z ) , (3.4)where A µ ( k ) and B ( k ) are the Fourier modes of component bosonic fields in the ten-dimensional space-time. Another form of supersymmetry transformation has been proposed in Ref. [5]. The difference does not matter in ourpaper because we compute only the linearized one in Eq. (5.20). S NS [Φ] = ∞ X M,N =0 ( − N ( M + N )!( M + N + 2)! M ! N ! h Q Φ , Φ M ( η Φ)Φ N i = 12 h Q Φ , η Φ i + 13! ( h Q Φ , Φ η Φ i − h Q Φ , η Φ Φ i )+ 14! ( h Q Φ , Φ η Φ i − h Q Φ , Φ η Φ Φ i + h Q Φ , η Φ Φ i ) + O (Φ ) , (3.5)and we truncate the string field Φ in the NS sector to the sum of Eqs. (3.3) and (3.4): Φ = Φ A + Φ B .Using the BRST transformations [ Q, V µA ( k )] = − α ′ k c∂cξe − φ ψ µ e ik · X − √ α ′ c (: k · ψψ µ : + iα ′ ∂X µ + (: ηξ : + ∂φ ) k µ ) e ik · X + ηe φ ψ µ e ik · X , (3.6) [ Q, V B ( k )] = −√ α ′ c∂cξk · ψe − φ e ik · X + ( − ∂c + 2 c (: ηξ : + ∂φ )) e ik · X (3.7)on the real axis, where Q = ¸ dz πi ( c ( T m + T φ + T ξη ) + bc∂c + e φ ηG m − η∂ηe φ b ) , T m = − α ′ ∂X µ ∂X µ − ψ µ ∂ψ µ , and G m = i q α ′ ψ µ ∂X µ , we can evaluate the kinetic term h Q Φ , η Φ i . For the interactionterms, we use the explicit form of the conformal maps, which defines the n -string term. Namely, for A k = A k (0) | i ( k = 1 , , . . . , n ), we have h A , A · · · A n i = h g ( n )1 ◦ A (0) g ( n )2 ◦ A (0) · · · g ( n ) n ◦ A n (0) i UHP , (3.8) g ( n ) k ( z ) = h − ( e iπ k − − nn ( h ( z )) n ) = tan (cid:18) n arctan z + π n (2 k − − n ) (cid:19) , (3.9)with the map from the upper half-plane to the unit disk: h ( z ) = iz − iz . The normalization of the largeHilbert space is given by h ξ ( y ) 12 c∂c∂ c ( z ) e − φ ( w ) e ik · X ( x, ¯ x ) i UHP = (2 π ) δ ( k ) . (3.10)In particular, with respect to the φ -charge, the terms such as Φ n − η Φ Q Φ are linear combinations of e qφ with q ≤ − n because Φ A ∼ e − φ ; η Φ A ∼ e − φ ; Φ B ∼ e − φ ; η Φ B ∼ e − φ ; Q Φ A ∼ e − φ , , e φ and Q Φ B ∼ e − φ , , which imply that the higher-order interaction terms, O (Φ ) in Eq. (3.5), vanish for Φ = Φ A + Φ B . Then, we obtain S NS [Φ ] = ˆ d x Tr " α ′ A µ ∂ A µ + i √ α ′ B∂ µ A µ + B + i √ α ′ ∂ µ ˜ A ν [ ˜ A µ , ˜ A ν ] (3.11) + ˆ d x α ′ ( ( ∂ − ∂ ) +( ∂ − ∂ ) )Tr (cid:20) A µ ( x ) A ν ( x ) A µ ( x ) A ν ( x ) − A µ ( x ) A ν ( x ) A ν ( x ) A µ ( x ) (cid:21) x i = x , where the trace is taken over the indices of the Chan–Paton factors, T a ; A µ ( x ) and B ( x ) are given by A µ ( x ) = A aµ ( x ) T a = ˆ d k (2 π ) A ˙ µ ( k ) e ik · x , B ( x ) = B a ( x ) T a = ˆ d k (2 π ) B ( k ) e ik · x ; (3.12)and ˜ A µ is defined as ˜ A µ ( x ) = K − α ′ ∂ A µ ( x ) , K ≡ √ . (3.13)We note that the reality condition for the NS string field, bpz − ◦ Φ † = − Φ [8], implies the (anti-)Hermiticity of the component fields: A µ ( x ) † = A µ ( x ) and B ( x ) † = − B ( x ) .4ntegrating out the scalar component field B ( x ) in Eq. (3.11), or using the equation of motion for B , B = − i p α ′ / ∂ µ A µ , and taking the small-momentum limit: K − α ′ ∂ ∼ and α ′ ( ( ∂ − ∂ ) +( ∂ − ∂ ) ) ∼ ,we have S NS [Φ ] = − α ′ ˆ d x Tr (cid:18) ∂ µ A ν − ∂ ν A µ − i √ α ′ [ A µ, A ν ] (cid:19) − ˆ d x Tr[ 12 A µ A ν A µ A ν + A µ A ν A ν A µ ] . (3.14)The first line on the right-hand side corresponds to the ordinary Yang–Mills action and the second lineis the difference from it. Actually, it is known that the difference is canceled by the contribution frommassive component fields [1]. In the R sector, the string field Ψ , which has ( n gh , n pic ) = (1 , − / , is in the restricted small Hilbertspace. The lowest-level states are given by spin fields with ( − / -picture: cS α ( − / e ik · X (0) | i , cS ˙ α ( − / e ik · X (0) | i , (3.15)where α and ˙ α are spinor indices with components, and they correspond to the massless level. By theGSO projection in Eq. (A.5), only the state with the dotted spinor remains in our convention. Hence, weexpand the level-truncated string field Ψ as Ψ = ˆ d k (2 π ) λ ˙ α ( k ) V ˙ αλ ( k )(0) | i , V ˙ αλ ( k ) = cS ˙ α ( − / ( z ) e ik · X ( z, ¯ z ) , (3.16)where λ ˙ α ( k ) is the Fourier mode of a fermionic component field, which is the Weyl spinor in the ten-dimensional spacetime. This string field Ψ is Grassmann odd, and we find that it is indeed in therestricted small Hilbert space: η Ψ = 0 and XY Ψ = Ψ .Similarly to the NS sector, we derive an explicit expression of the action including the R string fieldin terms of the component fields. The action in Eq. (2.3) is expanded as S R [Φ , Ψ] = − hh Ψ , Y Q Ψ ii − h Φ , Ψ i + O (Φ Ψ ) , (3.17)and we truncate the string fields up to the lowest level: the NS string field Φ to Φ = Φ A + Φ B and theR string field Ψ to Ψ given in Eq. (3.16).The kinetic term in the R sector is evaluated using the normalization of the small Hilbert space: hh c∂c∂ c ( z ) e − φ ( w ) e ik · X ( x, ¯ x ) ii UHP = (2 π ) δ ( k ) . (3.18)It is convenient to use hh Ψ , Y Q Ψ ii = hh Ψ , Y mid Q Ψ ii [3] for calculation, where Y mid = Y ( i ) is the midpointinsertion of the conventional inverse picture-changing operator Y ( z ) = c∂ξe − φ ( z ) . With the BRSTtransformation [ Q, V ˙ αλ ( k )] = − α ′ k c∂cS ˙ α ( − / e ik · X − i √ α ′ k µ (Γ µ ) ˙ αβ ηcS β (1 / e ik · X (3.19) We note that Φ B does not contribute to the terms including the R string fields, up to the lowest level, as we see inSect. 3.2.
5n the real axis, we have − hh Ψ , Y mid Q Ψ ii = √ α ′ ˆ d x Tr h λ ˙ α (Γ µ C ) ˙ α ˙ β ∂ µ λ ˙ β i , (3.20)where λ ˙ α ( x ) = λ a ˙ α ( x ) T a = ˆ d k (2 π ) λ ˙ α ( k ) e ik · x . (3.21)For the cubic interaction term, we find h Φ B , Ψ i = 0 by counting the number of c ghosts. With a similarmanipulation to the NS sector, the remaining term is evaluated as − h Φ A , Ψ i = i √ ˆ d x Tr h ˜ A µ ˜ λ ˙ α (Γ µ C ) ˙ α ˙ β ˜ λ ˙ β i , (3.22)where ˜ λ ˙ α ( x ) = K − α ′ ∂ λ ˙ α ( x ) (3.23)as in Eq. (3.13). For higher interaction terms, we find that O (Φ Ψ ) in Eq. (3.17) with Φ = Φ and Ψ = Ψ vanishes thanks to the normalization in Eq. (3.10) because Φ A ∼ c, Φ B ∼ cc , and Ψ ∼ c ,with respect to the bc -ghost sector. From Eqs. (3.20), (3.22), and (3.17), for the small-momentum limit K − α ′ ∂ ∼ , we have obtained S R [Φ , Ψ ] = − √ α ′ ˆ d x Tr h i ˆ λ α ( C Γ µ ) αβ D µ ˆ λ β i , (3.24)where ˆ λ α = C α ˙ β λ ˙ β , (3.25) D µ ˆ λ = ∂ µ ˆ λ − i √ α ′ [ A µ , ˆ λ ] . (3.26)The above form of S R [Φ , Ψ ] just corresponds to the gaugino term of the ten-dimensional SYM action. Here, we consider a contribution to the effective action of massless fields, A µ and ˆ λ α , obtained in theprevious section, from the massive part of string fields, Φ and Ψ , in the same way as Ref. [1]. The SSFTaction with the coupling constant g , which is obtained by replacing g − S [ g Φ , g Ψ ] using Eq. (2.1) with S [Φ , Ψ] , can be expanded as S [Φ , Ψ] = − h η Φ , Q Φ i + g h η Φ , [Φ , Q Φ] i− hh Ψ , Y Q Ψ ii − g h Φ , Ψ i + O ( g ) . (4.1)In this section, we concentrate on the zero-momentum sector, and then the level-truncated string fields, Φ , Ψ , satisfy Qη Φ = 0 , Q Ψ = 0 because of Eqs. (5.17) and (5.18) with B = − i p α ′ / ∂ µ A µ . We alsonote that Φ satisfies ξ η Φ = Φ , namely, the partial gauge-fixing condition. Around the massless partof the string fields, Φ , Ψ , we expand the string fields Φ , Ψ as Φ + R, Ψ + S , where R and S are the6assive part in the NS and R sector respectively, and we have S [Φ + R, Ψ + S ] − S [Φ , Ψ ] = − h ηR, QR i + g h ηR, [Φ , Q Φ ] i − g h R, (Ψ ) i− hh S, Y QS ii − g hh S, { η Φ , Ψ }ii + · · · (4.2)from Eq. (4.1). Here, (+ · · · ) denotes the higher-order terms in g when R and S are assumed to be O ( g ) .Varying the above with respect to the massive part, we have equations of motion QηR = g (cid:18) − { η Φ , Q Φ } − (Ψ ) (cid:19) + · · · , QS = g ( − X { η Φ , Ψ } ) + · · · , (4.3)and these can be solved by using the propagators in Ref. [9] as R s = − g ξ b L (cid:18) η [Φ , Q Φ ] + (Ψ ) (cid:19) , S s = − g b XηL [Φ , Ψ ] , (4.4)up to lowest order in g , where R s satisfies the partial gauge-fixing condition: ξ R s = 0 .Expanding the massive part of string fields around Eq. (4.4) as R = R ′ + R s , S = S ′ + S s , we have S [Φ + R ′ + R s , Ψ + S ′ + S s ] − S [Φ , Ψ ]= 12 h QR ′ , ηR ′ i − hh S ′ , Y QS ′ ii− h QR s , ηR s i + 12 hh S s , Y QS s ii + · · · , (4.5)where linear terms with respect to R ′ and S ′ vanish and therefore the massive part decouples from themassless part. The second line on the right-hand side gives a contribution to the effective action of themassless fields, which can be computed as − h QR s , ηR s i + 12 hh S s , Y QS s ii = g h [Φ , Q Φ ] , b L { η Φ , Q Φ }i + g hh (Ψ ) , b L (Ψ ) ii− g h (Ψ ) , b L [Φ , Q Φ ] i − g h{ Q Φ , Ψ } , b L [Φ , Ψ ] i . (4.6)In the computation for the term including S s , we manipulated X as in Ref. [9]. Namely, we rewrote it as hh S s , Y QS s ii = g hh{ η Φ , Ψ } , b XL { η Φ , Ψ }ii = g h ξ η Φ , Ψ b XL { η Φ , Ψ }i + g h ξ η Φ , b XL { η Φ , Ψ } Ψ i = g h [Φ , Ψ ] , b XL { η Φ , Ψ }i = g h [Φ , Ψ ] , b L Q Ξ { η Φ , Ψ }i = g h [Φ , Ψ ] , (cid:18) − Q b L (cid:19) Ξ { η Φ , Ψ }i = g h [Φ , Ψ ] , Ξ { η Φ , Ψ }i − g h{ Q Φ , Ψ } , b L [Φ , Ψ ] i , (4.7)where the first term of the last expression vanishes due to the number of c -ghosts and the normalizationin Eq. (3.10). 7he first term of the right-hand side of Eq. (4.6) has been evaluated in Ref. [1]: h [Φ , Q Φ ] , b L { η Φ , Q Φ }i = − ˆ d x Tr[ A µ A ν A ρ A σ ] ˆ ∞ dt e − t ( e − t a − a ) (cid:18) η µρ η νσ ( a − + e − t a ) + η µσ η νρ ( a − − e − t a ) (cid:19) = 14 ˆ d x Tr[ 12 A µ A ν A µ A ν + A µ A ν A ν A µ ] , (4.8)where a = tan π √ − . (4.9)Equation (4.8) cancels the extra term of Eq. (3.14) in the NS sector. We note that the component field B in Φ does not contribute to the above thanks to the equation of motion, B = − i p α ′ / ∂ µ A µ in thelowest order in g , which is negligible in the zero-momentum sector. The remaining terms of the right-hand side of Eq. (4.6), which includes the R sector, do not shift the coefficient of the O ( A µ ˆ λ ˆ λ ) term ofEq. (3.24). Furthermore, we can neglect the third and fourth terms in Eq. (4.6), namely − h (Ψ ) , b L [Φ , Q Φ ] i = 0 , − h{ Q Φ , Ψ } , b L [Φ , Ψ ] i = 0 , (4.10)because of Eq. (3.10), noting that V µA ( k = 0) = cξe − φ ψ µ , [ Q, V µA ( k = 0)] = − i p /α ′ c∂X µ + ηe φ ψ µ , V ˙ αλ ( k =0) = cS ˙ α ( − / . The second term of Eq. (4.6) can be evaluated, in a similar way to Eq. (4.8), as hh (Ψ ) , b L (Ψ ) ii = 12 ˆ d x Tr[ λ ˙ α λ ˙ β λ ˙ γ λ ˙ δ ] hh cS ˙ α ( − / ∗ cS ˙ β ( − / , b L ( cS ˙ γ ( − / ∗ cS ˙ δ ( − / ) ii = 12 ˆ d x Tr[ λ ˙ α λ ˙ β λ ˙ γ λ ˙ δ ] h | cS ˙ α ( − / ( −√ cS ˙ β ( − / ( √ U b L U † cS ˙ γ ( − / ( 1 √ cS ˙ δ ( − / ( − √ | i = 12 ˆ d x Tr[ λ ˙ α λ ˙ β λ ˙ γ λ ˙ δ ] ˆ ∞ dt h S ˙ α ( − / ( − a ) S ˙ β ( − / ( 1 a ) S ˙ γ ( − / ( e − t a ) S ˙ δ ( − / ( − e − t a ) i× h | c ( − a ) c ( 1 a ) b c ( e − t a ) c ( − e − t a ) | i , (4.11)where a is given in Eq. (4.9) and U is given in Refs. [10, 11], which corresponds to the conformal map tan (cid:0) arctan z (cid:1) . For the bc -ghost sector, we have h | c ( − a ) c ( 1 a ) b c ( e − t a ) c ( − e − t a ) | i = − e − t ( e − t a − a ) (4.12)with the normalization h | c − c c | i = 1 . As for the φ -ghost and the spin field sector, we note that h S ˙ α ( − / ( z ) S ˙ α ( − / ( z ) S ˙ α ( − / ( z ) S ˙ α ( − / ( z ) i = δ ˙ A + ˙ A + ˙ A + ˙ A , Y p 14 ( ∂ µ A ν − ∂ ν A µ − ig [ A µ, A ν ]) − i ˆ λ T C Γ µ (cid:16) ∂ µ ˆ λ − ig [ A µ , ˆ λ ] (cid:17) + log (cid:16) √ − (cid:17) α ′ g (ˆ λ T C Γ µ ˆ λ )(ˆ λ T C Γ µ ˆ λ ) (cid:21) , (4.19)where the first line is the same as the action of ten-dimensional SYM and the second line can be regardedas an α ′ -correction due to a superstring.Concerning the α ′ -correction, we comment on the term of the form α ′ g Tr [ F µν F νσ F µσ ] for the fieldstrength, F µν = ∂ µ A ν − ∂ ν A µ − ig [ A µ, A ν ] , which was investigated in Refs. [13, 14]. In this section,we have included only the zero-momentum sector for contributions from the massive part, and then ifthe above term exists, it should appear in the form α ′ g Tr [ i [ A µ , A ν ][ A ν , A σ ][ A σ , A µ ]] , which is a higherorder in g than Eq. (4.19). In this sense, it is necessary to perform further computations including thenonzero-momentum sector or higher-order terms in g in order to compare the action from SSFT with thenon-Abelian Born–Infeld action [15]. 9 Induced transformations at the lowest order Here, we derive the gauge and spacetime supersymmetry transformations for massless component fieldsfrom those of string fields given in Refs. [3, 4, 5]. We perform explicit computations up to the lowestorder for simplicity. The linearized version of Eqs. (2.10) and (2.11) is given by δ (0) g Φ = Q Λ + η Ω , δ (0) g Ψ = Qλ . (5.1)At the massless level, we have Λ ε = ˆ d k (2 π ) ε ( k ) cξ∂ξe − φ e ik · X (0) | i , Ω ω = ˆ d k (2 π ) ω ( k ) ξe ik · X (0) | i (5.2)for the gauge transformation parameter string fields in the NS sector, Λ and Ω , and we have no states for λ -gauge transformation in the R sector at this level. In order to respect the partial gauge-fixing condition ξ Φ = 0 , we take the gauge parameter as ω ( k ) = ε ( k ) = ε a ( k ) T a and we have Q Λ ε + η Ω ε = ˆ d k (2 π ) (cid:16) − α ′ k ε ( k ) V B (0) − √ α ′ k µ ε ( k ) V µA (0) (cid:17) | i . (5.3)Then, from the linearized gauge transformation at the massless level, δ (0) ε (Φ A + Φ B ) ≡ Q Λ ε + η Ω ε , (5.4)we have obtained the induced transformation for component fields: δ (0) ε B ( x ) = α ′ ∂ ε ( x ) , δ (0) ε A µ ( x ) = i √ α ′ ∂ µ ε ( x ) , ε ( x ) = ˆ d k (2 π ) ε ( k ) e ik · x . (5.5)For the gaugino field ˆ λ α ( x ) , the transformation is trivial: δ (0) ε ˆ λ α = 0 at the linearized level. Inorder to get a nontrivial transformation, we should include the interaction term of SSFT in the gaugetransformation. Expanding Eq. (2.11) as δ g (Λ) Ψ = XηF Ξ D η { F Ψ , Λ } = Xη { Ψ , Λ } + O (ΦΨ) , (5.6)we define δ (1) ε Ψ = Xη { Ψ , Λ ε } . (5.7)Since the star product of Ψ and Λ ε has ( n gh , n pic ) = (0 , − / and its φ -charge is − / , we expand as { Ψ , Λ ε } = ˆ d k (2 π ) | ϕ ˙ α ( k ) i ( C − ) ˙ αα h ϕ c α ( − k ) , { Ψ , Λ ε }i + · · · (5.8)at the massless level, where ϕ ˙ α ( k ) = c∂c ξ∂ξ S ˙ α ( − / e ik · X (0) | i , ϕ c α ( k ) = cηS α (1 / e ik · X (0) | i , (5.9)10hich satisfy the normalization h ϕ c α ( k ) , ϕ ˙ α ( k ) i = C α ˙ α (2 π ) δ ( k + k ) . (5.10)Using the relations h ϕ c α ( k ) , { Ψ , Λ ε }i = ˆ d k (2 π ) ˆ d k (2 π ) ( − 1) [ λ ˙ α ( k ) , ε ( k )] × C α ˙ α K α ′ ( k + k + k ) (2 π ) d δ d ( k + k + k ) , (5.11) Xη | ϕ ˙ α ( k ) i = −V ˙ αλ ( k )(0) | i , (5.12)we have obtained δ (1) ε Ψ = ˆ d k (2 π ) δ (1) ε λ ˙ α ( k ) V ˙ αλ ( k )(0) | i + · · · , (5.13) δ (1) ε λ ˙ α ( k ) = ˆ d k (2 π ) ˆ d k (2 π ) [ λ ˙ α ( k ) , ε ( k )] K α ′ ( k + k + k ) (2 π ) d δ d ( k + k − k ) , (5.14)and hence the induced gauge transformation of the component field is δ (1) ε ˆ λ α ( x ) = − h ε ( x ) , ˆ λ α ( x ) i (5.15)for small momentum: K α ′ ( k + k + k ) ∼ . Equations (5.5) and (5.15) are consistent with the ordinarygauge transformation of the SYM. First, we derive explicit expressions for equations of motion for component fields from those for stringfields. The equations of motion in Eq. (2.14) are linearized as Qη Φ = 0 , Q Ψ = 0 . (5.16)At the massless level, we have Qη Φ = ˆ d k (2 π ) (cid:16) − (cid:0) B ( k ) − r α ′ k µ A µ ( k ) (cid:1) cηe ik · X (0) | i− √ α ′ (cid:0) k µ B ( k ) − r α ′ k A µ ( k ) (cid:1) c∂ce − φ ψ µ e ik · X (0) | i (cid:17) , (5.17) Q Ψ = ˆ d k (2 π ) λ ˙ α (cid:16) α ′ k c∂cS ˙ α ( − / + i √ α ′ k µ (Γ µ ) ˙ αα ηcS α (1 / (cid:17) e ik · X (0) | i , (5.18)which imply that the induced linearized equations of motion are B + i r α ′ ∂ µ A µ = 0 , i∂ µ B − r α ′ ∂ A µ = 0 , Γ µ ∂ µ ˆ λ = 0 , (5.19)in terms of component fields. The first two equations correspond to the Maxwell equation for the gaugefield A µ and the last equation corresponds to the Dirac equation, and hence they are consistent withSYM at the linearized level. 11he spacetime supersymmetry transformations in Eqs. (2.15) and (2.16) are linearized as δ (0) S Φ = S ΞΨ , δ (0) S Ψ = X S η Φ . (5.20)At the massless level, the first transformation is computed as δ (0) S Φ = S ΞΨ = ˆ d k (2 π ) i √ ǫ T Γ µ Cλ ) V µA ( k )(0) | i + η Ω , (5.21)where Ω ≡ ξ S (Ξ − ξ )Ψ is a kind of Ω -gauge transformation in Eq. (2.10). Then, up to Ω -gaugetransformation, the induced linearized transformations of bosonic component fields are obtained: δ (0) S A µ = 1 √ ǫ Γ µ ˆ λ , δ (0) S B = 0 , (5.22)where ˆ ǫ = Cǫ and ¯ˆ ǫ = ˆ ǫ T C . For the second equation of Eq. (5.20), we have calculated as follows: δ (0) S Ψ = X S η Φ = ˆ d k (2 π ) (cid:16)r α ′ k µ A ν ( k ) − k ν A µ ( k ))( ǫ T Γ µν ) ˙ α + (cid:0) B ( k ) + r α ′ k µ A µ ( k ) (cid:1) ǫ ˙ α (cid:17) V ˙ αλ ( k )(0) | i . (5.23)This implies that the induced linearized transformation of the fermionic component field is given by δ (0) S ˆ λ = i r α ′ ∂ µ A ν − ∂ ν A µ )Γ µν ˆ ǫ + ( B + i r α ′ ∂ µ A µ )ˆ ǫ . (5.24)We find that the induced transformations of Eqs. (5.22) and (5.24) are consistent with the conventionalsupersymmetry transformation in the ten-dimensional SYM, up to the equations of motion in Eq. (5.19),at the linearized level. In this paper, we have truncated the string fields in both NS and R sectors in the framework of Kunitomoand Okawa’s SSFT up to the lowest level (massless level) and, by evaluating the action explicitly in termsof the component fields, we have obtained the ten-dimensional SYM action plus an extra O ( A µ ) term. Wehave also investigated a contribution from the massive part in the lowest order in g in the zero-momentumsector and observed that the extra O ( A µ ) cancels in the NS sector and instead an extra O ( λ α ) appearsfrom the R sector, which can be interpreted as α ′ -correction. We have derived the gauge transformationand the spacetime supersymmetry transformation for the massless component fields induced from thoseof string fields at the lowest order. Our explicit calculation implies that the lowest-level truncation ofKunitomo and Okawa’s SSFT action is consistent with the ten-dimensional SYM theory.We have some remaining issues. At the present stage, we have no explicit formula for the realitycondition of the string fields including the R sector. We expect that the Majorana condition for themassless component field in the R sector, ˆ λ † Γ = ˆ λ T C , should be imposed by a consistent reality conditionfor string fields. It would be interesting to perform similar calculations in other SSFTs such as A ∞ -type As noted at the end of Sect. 4, we should include the nonzero-momentum sector in order to discuss O ( α ′ ) -correctioncompletely. Acknowledgments We would like to thank T. Erler, T. Kawano, H. Kunitomo, H. Nakano, and T. Takahashi for valuablediscussions and comments. We would also like to thank the organizers of the conference “Progress inQuantum Field Theory and String Theory II” at Osaka City University and the 22nd Niigata–Yamagatajoint school (YITP-S-17-03) at the National Bandai Youth Friendship Center, where our work was pre-sented. This work was supported in part by JSPS Grant-in-Aid for Young Scientists (B) (JP25800134). A Convention of the spin fields Here, we summarize our convention for explicit computations including the R sector, which is based onthe method developed in Ref. [12]. The worldsheet fermion ψ µ ( µ = 0 , , . . . , can be bosonized using φ a ( a = 1 , , . . . , as i − / ( ψ ∓ ψ ) ≃ e ± φ c ± e , (A.1) − / ( ψ a − ∓ iψ a − ) ≃ e ± φ a c ± e a , a = 2 , , , , (A.2)where the operator product expansion (OPE) among φ a is φ a ( z ) φ b ( w ) ∼ δ a,b log( z − w ) , ( a, b = 1 , . . . , . Involving the bosonized ghost φ ≡ φ , such as φ ( z ) φ ( w ) ∼ − log( z − w ) , the cocycle factor c λ ( λ = P i =1 λ i e i ) is defined by c λ = e iπ P i,j =1 λ i M ij [ ∂φ j ] , [ ∂φ i ] = ˛ dz πi ∂φ i , (A.3)where M ij ( i, j = 1 , , . . . , is given by the matrix M = − − − − − − . (A.4)The GSO ( + ) projection can be expressed as − G , G = X i =1 [ ∂φ i ] . (A.5)13he spin fields with n pic = ± / are expressed as S α ( ± / = e P i =1 A i φ i ± φ c P i =1 A i e i ± e , A i = ± , ( i = 1 , . . . , , Y i =1 A i > , (A.6) S ˙ α ( ± / = e P i =1 ˙ A i φ i ± φ c P i =1 ˙ A i e i ± e , ˙ A i = ± , ( i = 1 , . . . , , Y i =1 ˙ A i < . (A.7)In general, for S λ ≡ e P i =1 λ i φ i c λ with λ = P i =1 λ i e i , the OPE is S λ ( y ) S λ ′ ( z ) = ( y − z ) λ i η ij λ ′ j e λ i φ i ( y )+ λ ′ i φ i ( z ) e iπλ i M ij λ ′ j c λ + λ ′ , (A.8)where η ij = diag(1 , , , , , − . Corresponding to the above convention, we define the Γ -matrix as Γ = − iσ ⊗ ⊗ ⊗ ⊗ , Γ = σ ⊗ ⊗ ⊗ ⊗ , (A.9) Γ = σ ⊗ σ ⊗ ⊗ ⊗ , Γ = − σ ⊗ σ ⊗ ⊗ ⊗ , (A.10) Γ = − σ ⊗ σ ⊗ σ ⊗ ⊗ , Γ = − σ ⊗ σ ⊗ σ ⊗ ⊗ , (A.11) Γ = − σ ⊗ σ ⊗ σ ⊗ σ ⊗ , Γ = σ ⊗ σ ⊗ σ ⊗ σ ⊗ , (A.12) Γ = σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ , Γ = σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ , (A.13)and Γ = Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ = σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ , where σ i ( i = 1 , , is the Pauli matrix,and we take C = e πi σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ , C T = − C, C † = C − . (A.14)Then, we have { Γ µ , Γ ν } = 2 η µν , C − Γ µ C = − Γ µT , (Γ µ ) † = Γ Γ µ Γ . (A.15)In the same way as the linear combination of the bosonization, Eqs. (A.1) and (A.2), we define Γ ± e = i √ ∓ Γ ) = √ σ ∓ ⊗ ⊗ ⊗ ⊗ , (A.16) Γ ± e = 1 √ ∓ i Γ ) = ±√ iσ ⊗ σ ∓ ⊗ ⊗ ⊗ , (A.17) Γ ± e = 1 √ ∓ i Γ ) = −√ σ ⊗ σ ⊗ σ ∓ ⊗ ⊗ , (A.18) Γ ± e = 1 √ ∓ i Γ ) = ∓√ iσ ⊗ σ ⊗ σ ⊗ σ ∓ ⊗ , (A.19) Γ ± e = 1 √ ∓ i Γ ) = √ σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ ∓ , (A.20)where σ + = ( σ + iσ ) , σ − = ( σ − iσ ) , and they can be rewritten as Γ ± e j = ( ± i ) j − √ σ ⊗ ) j − σ ∓ ( ⊗ ) − j j = 1 , , , , . (A.21)14sing the above equations, we find Γ ± e j = ± e j ) α ˙ β (Γ ± e j ) ˙ αβ ! , j = 1 , , , , , (A.22) (Γ ± e j ) α ˙ β = δ ± e j + A, ˙ B √ e ± iπ P k =1 M jk ˙ B k , (Γ ± e j ) ˙ αβ = δ ± e j + ˙ A,B √ e ± iπ P k =1 M jk B k , (A.23) C = C α ˙ β C ˙ αβ ! , (A.24) C α ˙ β = δ A + ˙ B, e − iπ P i,j =1 A i + M ij A j + , A + ≡ ( A i , / , (A.25) C ˙ αβ = − δ ˙ A + B, e − iπ P i,j =1 ˙ A i − M ij ˙ A j − , ˙ A − ≡ ( ˙ A i , − / , (A.26) e iπ P j =1 M j A j = i, e iπ P j =1 M j ˙ A j = − i, (Γ µ C ) αβ = (Γ µ C ) βα , (Γ µ C ) ˙ α ˙ β = (Γ µ C ) ˙ β ˙ α , (A.27)and furthermore, for C − = C − α ˙ β C − αβ ! , C − α ˙ β = − δ A + ˙ B, e iπ P i,j =1 A i + M ij A j + , C − αβ = δ ˙ A + B, e iπ P i,j =1 ˙ A i − M ij ˙ A j − , (A.28) C α ˙ β C − βγ = δ αγ , C ˙ αβ C − β ˙ γ = δ ˙ α ˙ γ , ( C − Γ µ ) αβ = ( C − Γ µ ) βα , ( C − Γ µ ) ˙ α ˙ β = ( C − Γ µ ) ˙ β ˙ α . (A.29)Here, we should note that the correspondence of the spinor index is α ↔ A = ( ± / , ± / , ± / , ± / , ± / 2) = X i =1 A i e i , Y i =1 A i > , (A.30) ˙ α ↔ ˙ A = ( ± / , ± / , ± / , ± / , ± / 2) = X i =1 ˙ A i e i , Y i =1 ˙ A i < , (A.31)for undotted and dotted spinors.For the spin field with n pic = r , S ˆ α ( r ) = e P i =1 ˆ A i φ i + rφ c P i =1 ˆ A i e i + re , α = ( α, ˙ α ) ↔ ˆ A , (A.32)we have the OPE: ψ µ ( y ) S ˆ α ( r ) ( z ) ∼ ( y − z ) − √ µ ) ˆ α ˆ β S ˆ β ( r ) ( z ) . References [1] N. Berkovits and M. Schnabl, “Yang-Mills action from open superstring field theory,” JHEP (2003) 022, arXiv:hep-th/0307019 [hep-th] .[2] N. Berkovits, “SuperPoincare invariant superstring field theory,” Nucl.Phys. B450 (1995) 90–102, arXiv:hep-th/9503099 [hep-th] .[3] H. Kunitomo and Y. Okawa, “Complete action for open superstring field theory,” PTEP (2016) no. 2, 023B01, arXiv:1508.00366 [hep-th] .[4] H. Kunitomo, “Space-time supersymmetry in WZW-like open superstring field theory,” PTEP (2017) no. 4, 043B04, arXiv:1612.08508 [hep-th] .[5] T. Erler, “Superstring Field Theory and the Wess-Zumino-Witten Action,” JHEP (2017) 057, arXiv:1706.02629 [hep-th] . 156] M. Asada, “On a construction of a complete action for open superstring field theory (in Japanese),”Master’s thesis, Niigata University, 2017.[7] T. Erler, Y. Okawa, and T. Takezaki, “Complete Action for Open Superstring Field Theory withCyclic A ∞ Structure,” JHEP (2016) 012, arXiv:1602.02582 [hep-th] .[8] K. Ohmori, “Level expansion analysis in NS superstring field theory revisited,” arXiv:hep-th/0305103 [hep-th] .[9] H. Kunitomo, Y. Okawa, H. Sukeno, and T. Takezaki, “Fermion scattering amplitudes fromgauge-invariant actions for open superstring field theory,” arXiv:1612.00777 [hep-th] .[10] M. Schnabl, “Wedge states in string field theory,” JHEP (2003) 004, arXiv:hep-th/0201095 [hep-th] .[11] M. Schnabl, “Analytic solution for tachyon condensation in open string field theory,” Adv.Theor.Math.Phys. (2006) 433–501, arXiv:hep-th/0511286 [hep-th] .[12] V. A. Kostelecky, O. Lechtenfeld, W. Lerche, S. Samuel, and S. Watamura, “Conformal Techniques,Bosonization and Tree Level String Amplitudes,” Nucl. Phys. B288 (1987) 173–232.[13] J. Scherk and J. H. Schwarz, “Dual Models for Nonhadrons,” Nucl. Phys. B81 (1974) 118–144.[14] A. A. Tseytlin, “Vector Field Effective Action in the Open Superstring Theory,” Nucl. Phys. B276 (1986) 391. [Erratum: Nucl. Phys.B291,876(1987)].[15] A. A. Tseytlin, “On nonAbelian generalization of Born-Infeld action in string theory,” Nucl. Phys. B501 (1997) 41–52, arXiv:hep-th/9701125 [hep-th]arXiv:hep-th/9701125 [hep-th]