Construction of a Gauge-Invariant Action for Type II Superstring Field Theory
aa r X i v : . [ h e p - t h ] M a y Construction of a Gauge-Invariant Actionfor Type II Superstring Field Theory
Hiroaki Matsunaga ∗ Institute of Physics, University of TokyoKomaba, Meguro-ku, Tokyo 153-8902, Japan
UT-Komaba 13-04
Abstract
We construct a gauge-invariant action for covariant type II string field theory in the NS-NS sector. Our construction is based on the large Hilbert space description and Zwiebach’sstring products are used. First, we rewrite the action for bosonic string field theory into anew form where a state in the kernel of the generator of the gauge transformation appearsexplicitly. Then we use the same strategy and write down our type II action, where aprojector onto the small Hilbert space plays an important role. We present lower-orderterms up to quartic order and show that three-point amplitudes are reproduced correctly. ∗ E-mail: [email protected] ontents L ∞ -algebras . . . . . . . . . . . . . . . . . . . . . . . . 42.3 A New Action and the Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . 52.4 Another Calculation and Extra/BRST-exact Terms . . . . . . . . . . . . . . . . . 7 Q G and the State Q G Ψ . . . . . . . . . . . . . . . . . . . . . . . . 144.3 The Classical Action and the Gauge Invariance . . . . . . . . . . . . . . . . . . . 154.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 L ∞ -algebras and Closed String Field Theory 23B A Parallel Structure of Actions 27C Open String Field Theory 28 Introduction
String field theory is one possible approach to understanding nonperturbative aspects of stringtheory [1–16]. In the bosonic theory, there exist two known Lorentz covariant theories: openstring field theory [3] and closed string field theory [11]. Open string field theory is described bythe Chern-Simons-like action [3]. By contrast, closed string field theory necessitates an infinitenumber of fundamental vertices to cover the moduli space of Riemann surfaces and the actionbecomes a nonpolynomial form [7, 8]. However, underlying gauge structures of both theoriesare essentially equivalent: there exists some nilpotent homotopy algebra which is related toporoperties of each moduli space. In free theory, using each BRST operator, we can constructthe action for each string field theory and verify its gauge invariance in the same way. Hence,the underlying gauge structure of each free theory is described by the BRST-complex. Addinginteraction terms requires us to extend or modify this framework. It is known that in fullinteraction theory, A ∞ -algebras for open strings, which includes differential graded algebras assubalgebras, and L ∞ -algebras for closed strings, which are extentions of Lie algebras equippedwith a derivation, appear respectively. Using the nilpotency of A ∞ / L ∞ -algebras, we can easilycheck the gauge invariance, the equation of motion, and so on irrespective of the apparentcomplexity of the action. In this sense, we have good understandings of bosonic theory.By contrast, in superstring field theory, these geometrical and/or algebraical understandingsare very little known. The formulation of superstring field theory has been in process and wehave not constructed even a complete action until now. In addition to it, the situation assumesnew aspects. In the NSR formalism of superstrings [29,31], there exist ghost and picture numberanomalies, which are one of difficulties in the formulaton of field theory: these anomalies make itdifficult to construct supersymmetric theory as a naive extension of bosonic theory. To solve thisproblem, we often use the local insertion of picture changing operators but this leads to anotherproblem: the collision of these operators causes a divergence. We would like to obtain thecomplete formulation of field theory of superstrings. In the construction of the concrete action,we have used two descriptions: the small Hilbert space description, which is based on superghosts( β, γ ), and the Large Hilbert space description, which is based on bosonized superghosts ( ξ, η, φ ).The difference is whether state space does not include the zero-mode of ξ or does. In the smallHilbert space description [17–19], the action always needs local insertions of picture changingoperators and these operators causes difficulties: divergences, nontrivial kernels, and so on. Bycontrast, in the large Hilbert space description, the action does not always need local insertionsof picture changing operators. However, it is not easy to construct the R-sector action. So far,limited to the NS sector, we know two full actions of superstring fields without picture changingoperators: the action of open strings [20] and that of heterotic strings [21]. In this paper, usingthe large Hilbert space description, we propose a construction of a classical action for type IIstring field theory in the NS-NS sector. In fact, if we change the way of interaction from [3], we cannot write down the cubic action [4]. Then,although it is open string field theory, the action has higher order vertices and its gauge structure is governed by A ∞ -algebras, which are yielded from properties of disk moduli space [37, 38]. There exist some proposals of making R-sector theory. See, for example, [22–24].
1n bosonic theory, from a geometrical point of view, the gauge structure is governed by L ∞ -algebras. We expect that in the NS-NS sector, the type II action has such properties like thebosonic action. (Of course, it is not clear yet what controls the algebraic poroperty of type IItheory including all sectors. Recently, there has been a progress [41].) Therefore, we proposethe following action, whose gauge structure is governed by L ∞ -algebras: S = Z dt h ∂ t Ψ( t ) , Q G ( t ) Ψ( t ) i η . (1.1)It is constructed from type II string fields Ψ( t ) and a nilpotent operator Q G , which consists oftype II string fields, the BRST operator, and the extension of Zwiebach’s closed string products.As the inner product, we use the BPZ inner product with η -currents insertion: h A, B i η := h η +0 η − A, B i . The variation of this action, discussed in detail later, is given by the followingform: δS = Z dt ∂∂t h δ Ψ( t ) , Q G ( t ) Ψ( t ) i η . (1.2)Thus it has a simple gauge invariance: δ Ψ = Q G Λ, which is yielded from Q G = 0, and all infor-mation about the gauge structure described by L ∞ -algebras are encoded into Q G ’s nilpotency.In this paper, we call this operator Q G the BRST operator around G ( t ), which is deeply relatedto a pure gauge solution in bosonic theory. We would like to mention that our construction isbased on algebraic properties of closed string fields, so we do not touch the geometrical under-standing of superstrings, such as the correspondence of the full action and the decomposition ofthe moduli space of super-Riemann surfaces as the case of bosonic theory.This article is organized as follows. In section 2, first, we briefly review the related resultsof bosonic string field theory [11]. Then we rewrite the action into more suggestive form, whichis easy to understand the gauge structure and its supersymmetric extension. This is one of newresults. In section 3, we review the construction of a pure gauge solution of bosonic theory [10].We pick up heterotic theory as an example and show the construction of the heterotic actionusing a (formal) bosonic pure gauge solution [21]. At the end of this section, we put commentson a pure gauge solution of heterotic theory and the procedure of constructing type II theory.This section is devoted to present an idea of the construction of our type II action and doesnot include material which is necessary for reading other sections. After that, in section 4 andsction 5, we propose the concrete form of the type II full action and see its properties: thegauge invariance and the equation of motion and so on. In paricular, we would like to show thecorrespondence of cohomology [28], the lower order action, and its gauge invariance through theperturbative expansion. We also check that it reproduces correct three point amplitudes whichare correspond to the result of the first quantization theory [29–31] and estimate higher orderamplitudes. At the top of section 2 and section 3, we pick up some formulae, which are necessaryto follow practical calculations. In appendix A, we summarize homotopy algebras [32–40], whichare used in this paper. In appendix B, we present simple calculus of free theory using our newform of the action. It will be helpful for understanding this article because the same calculusgoes in interacting theory. In Appendix C, we consider open string field theory from the pointof view of our construction. In this paper, any type closed string fields A all are imposed on thelevel matching condition L − A = 0 and the subsidiary condition b − A = 0.2 Basic Facts of Closed String Field Theory
In this section, we review some results of bosonic theory [11] and discuss several properties whichare related to our construction of type II theory. In particular, we rewrite Zwiebach’s actioninto another equivalent action which is rather clear to see the gauge structure: the generator ofthe gauge transformation, the correspondence between free and full theories, and so on.Relations between the BPZ Inner Product and String Products: h A, B i = ( − ( A +1)( B +1) h B, A i , (2.3) h QA, B i = ( − A h A, QB i , (2.4) h [ A, B ] , C i = ( − ( A + B ) h A, [ B, C ] i . (2.5) L ∞ -identities: ( l ≥ k ≥
2, and σ is the sign of splittings.) X l + k = n σ ( i l , i k ) (cid:2) A i , . . . , A i l , [ A j , . . . , A j k ] (cid:3) = 0 . (2.6)Stokes’s Theorem of Three Point Vertices: h QA, [ B, C ] i + ( − A h A, [ QB, C ] i + ( − A + B h A, [ B, QC ] i = 0 . (2.7) The action of bosonic closed string field theory is described by closed string fields Ψ, the BPZinner product, string products, and the BRST operator of closed strings Q . In particular, thereis an infinite set of string products, all of which are graded-commutative. For example, thelowest product satisfies [ A, B ] = ( − AB [ B, A ], where string states in the exponent representtheir Grassmann property, 0 (mod 2) for Grassmann even states and 1 (mod 2) for Grassmannodd states. This means, in closed string field theory, we need an infinite set of fundamentalvertices [7, 8]. The BPZ inner product is defined by BPZ conjugation as h A, B i := h A | c − | B i ,where h A | is the BPZ conjugate of | A i , and c − = ( c − ¯ c ). It is nondegenerate on closed stringHilbert space. Using these components, the action of bosonic closed string field theory is givenby S B := 12 h Ψ , Q Ψ i + X n ≥ κ n ( n + 2)! h Ψ , [Ψ n , Ψ] i , (2.8)where Ψ is a bosonic closed string field which carries ghost number 2, and [Ψ n − , Ψ] is a closedstring n -product which carries ghost number 3 + P ni =1 (gh(Ψ i ) −
2) [11, 15]. Here, the symbolgh(Ψ i ) means the ghost number of Ψ i . The variation of this action is given by δS B = ∞ X n =0 κ n ( n + 1)! h δ Ψ , [Ψ n , Ψ] i . (2.9)Therefore, we obtain the following equation of motion: F (Ψ) ≡ Q Ψ + X n ≥ κ n ( n + 1)! [Ψ n , Ψ] = 0 . (2.10) We use a compact notation: [Ψ n ] := [Ψ , . . . , Ψ]. In paticular, [Ψ ] := 0 and [Ψ , Ψ] := [Ψ] = Q Ψ. δ Ψ = ∞ X n =0 κ n n ! [Ψ n , Λ] . (2.11)The reason is, we can take resummention and use the following simple L ∞ -identities: X l + k = n n ! l ! k ! [Ψ l , [Ψ k ]] = 0 ( l, k ≥ . (2.12)Note that the inner product has ciclicity and closed string products are symmetric. So we obtain δS B = 0 under such δ Ψ, which is the gauge symmetry of closed string field theory. L ∞ -algebras Let us consider the shift: Ψ → Ψ + Ψ ′ . If S B (Ψ) is invarinat under Ψ → Ψ + δ Λ, then clealy S B (Ψ + Ψ ′ ) is invariant under Ψ + Ψ ′ → Ψ + Ψ ′ + δ (Ψ + Ψ ′ ), and as a consequence it isinvariant under Ψ ′ → δ (Ψ + Ψ ′ ). This is the gauge invariance of the shifted action [9, 11]. S B (Ψ + Ψ ′ ) = ∞ X n =0 κ n − n ! n X m =0 n ! m !( n − m )! { Ψ ′ m , Ψ n − m } = 1 κ ∞ X n =0 κ n n ! ∞ X m =0 κ m m ! { Ψ ′ n , Ψ m } (2.13)Here, we use multilinear functions: { B , B , . . . , B n } ≡ h B , [ B , . . . , B n ] i . We can define a newset of string products, denoted by the lowwer index of Ψ [ B , . . . , B n ] Ψ = ∞ X m =0 κ m m ! [ B , . . . , B n , Ψ m ] , (2.14)which are related to new multilinear functions as { B , B , . . . , B n } Ψ ≡ h B , [ B , . . . , B n ] Ψ i .Using the definition of the lowwer-indexed ones, the shifted action simply reads S B (Ψ + Ψ ′ ) ≡ S ′ B (Ψ ′ ) = ∞ X n =0 κ n − n ! { Ψ ′ n } Ψ . (2.15)Here, the first two terms are not equal to zero for general Ψ : { Ψ ′ } = ∞ X m =2 κ m m ! { Ψ m } ≡ κ S B (Ψ ) , { Ψ ′ } = ∞ X m =1 κ m m ! { Ψ ′ , Ψ m } = h Ψ ′ , κ F (Ψ ) i , (2.16)where F (Ψ ) is equation of motion. Thus the shifted action, reads S B (Ψ + Ψ ′ ) ≡ S ′ B (Ψ ′ ) = S (Ψ ) + h Ψ ′ , F (Ψ ) i + . . . . (2.17)If and only if Ψ satisfied the equation of motion: F (Ψ ) = 0, the term linear Ψ ′ would havevanished. [11]. The shifted action is invariant under the transformation: δ ′ Ψ ′ = δ (Ψ + Ψ ′ ) = ∞ X n =0 κ n n ! [(Ψ + Ψ ′ ) n , Λ] = ∞ X n =0 κ n n ! [Ψ ′ n , Λ] Ψ , (2.18)4hich is just the same form of the ordinary one. The first term is the new BRST-like operator,we call it the redefined BRST operator around Ψ :[Λ] Ψ ≡ Q Ψ Λ := ∞ X n =0 κ n n ! [Λ , Ψ n ] = Q Λ + ∞ X n =1 [Λ , Ψ n ] . (2.19)It does not have nilpotency for general Ψ except for the case of classical solution: F (Ψ ) = 0. In the previous subsection, we reviewed the shifting structure of string field theory and considerednew string products around constant string fields. However, algebraically, we are able to definenew string products around an arbitrary string field φ as (cid:2) A , . . . , A m (cid:3) φ := X n κ n n ! (cid:2) φ n , A , . . . , A m (cid:3) . (2.20)There is a L ∞ -morphism between original products [ A , . . . , A n ] and new ones [ A , . . . , A m ] φ .These new operators Q φ ≡ [ · ] φ and [ A , . . . , A m ] φ have almost the same properties as those ofold ones Q and [ A , . . . , A n ]. (See [11, 40], etc.) For instance, the nilpotency of Q becomes Q φ ( Q φ Ψ) = − κ [ F ( φ ) , Ψ ] φ , (2.21)where F ( φ ) is the equation of motion of bosonic closed string field theory. The most importantfact is that these form a new L ∞ -algebra when φ satisfies the equation of motion.Using these operators, we can rewrite the gauge transformation of bosonic string field theory δ Ψ = P κ n n ! [Ψ n , Λ] into the simple form δ Ψ = Q Ψ Λ. As well as we identify the BRST operator Q with the generator of the gauge transformation in free theory, we can regard this Q Ψ as thegenerator of the gauge transformation in full theory. We introduce the following notation: Q [ a ]Ψ := Q + X n ( a · κ ) n n ! (cid:2) Ψ n , (cid:3) . (2.22)The upper index [ a ] on Q Ψ means we consider closed string products of the coupling constant a times κ . Then, Q [ a ]Ψ gives a deformation of the gauge structure: Q [ a ]Ψ connects Q [0]Ψ = Q with Q [1]Ψ = Q Ψ , where a ∈ [0 , Q [ a ]Ψ from a = 0 to a = 1, we can define aMaurer-Cartan operator Q ′ Ψ as follows, Q ′ Ψ := Z da Q [ a ]Ψ = ∞ X n =0 κ n ( n + 1)! (cid:2) Ψ n , (cid:3) , (2.23)which is deeply related to the action for string field theory, disucussed in the rest of this sub-section. Note that the operator Q ′ Ψ maps a string field Ψ to a state F (Ψ) and the equation ofmotion is given by F (Ψ) = Q ′ Ψ Ψ = 0.Properties of the State Q ′ Ψ ΨBefore rewriting the action into a new form, we would like to see two useful properties, whichare necessary to our calculus. The state Q ′ Ψ Ψ has the following properties: Q Ψ ( Q ′ Ψ Ψ) = 0 , (2.24)( − X X ( Q ′ Ψ Ψ) = Q Ψ ( X Ψ) , (2.25)5here X is a derivation which satisfies the relation [[ X, Q ]] = 0. The first line implies that thestate Q ′ Ψ Ψ belongs to the kernel of the generator Q Ψ . It is a result from L ∞ -identities for anarbitrary string field ψ whose ghost number is two: Q ψ Q ′ ψ ψ = X k + l = n κ n k ! · l ! (cid:2) ψ k , [ ψ l ] (cid:3) = 0 . (2.26)The second line means the X -derivative state X ( Q ′ Ψ Ψ) becomes the Q Ψ -exact state. It is aresult from the derivation propertiy of X for string products ( − X X [ A n ] = n [ A n − , XA ]: X ( Q ′ Ψ Ψ) = X ∞ X n =0 κ n ( n + 1)! (cid:2) Ψ n +1 (cid:3) (2.27)= ( − X ∞ X n =0 κ n n ! (cid:2) Ψ n , X Ψ (cid:3) = ( − X Q Ψ ( X Ψ) . (2.28)A New Action and the Gauge InvarianceUsing these operators, we can rewrite the action S B = P κ n ( n +2)! h Ψ , [Ψ n , Ψ] i into a suggestiveform. First, introducing real parameters t ∈ [0 ,
1] and a ∈ [0 , S B = Z dt h ∂ t ( t Ψ) , ∞ X n =0 κ n ( n + 1)! (cid:2) ( t Ψ) n , ( t Ψ) (cid:3) i = Z dt Z da h ∂ t ( t Ψ) , ∞ X n =0 ( aκ ) n n ! [( t Ψ) n , ( t Ψ)] i . (2.29)Then, using (2.23), we obtain the following action: S B = Z dt Z da h ∂ t ( t Ψ) , Q [ a ] t Ψ ( t Ψ) i = Z dt h ∂ t ( t Ψ) , Q ′ t Ψ ( t Ψ) i . (2.30)Or more formally, using Ψ( t ) which satisfies Ψ(0) = 0 and Ψ(1) = Ψ, the action S B becomes S B = Z dt h ∂ t Ψ( t ) , Q ′ Ψ( t ) Ψ( t ) i . (2.31)This action has of course the same properties as the original action: the equation of motion Q ′ Ψ Ψ = 0, the gauge invariance δ Ψ = Q Ψ Λ, and so on. Let us check these properties.Using (2.4), (2.3), and (2.25) for X = δ and X = ∂ t , the following relation holds: h ∂ t Ψ , δ (cid:0) Q ′ Ψ Ψ (cid:1) i = h ∂ t Ψ , Q Ψ δ Ψ i = h Q Ψ ∂ t Ψ , δ Ψ i = h δ Ψ , Q Ψ ∂ t Ψ i = h δ Ψ , ∂ t (cid:0) Q ′ Ψ Ψ (cid:1) i . (2.32)Thus we can quickly calculate the variation δS as follows: δS B = Z dt (cid:16) h δ (cid:0) ∂ t Ψ( t ) (cid:1) , Q ′ Ψ( t ) Ψ( t ) i + h ∂ t Ψ( t ) , δ (cid:0) Q ′ Ψ( t ) Ψ( t ) (cid:1) i (cid:17) = Z dt (cid:16) h ∂ t (cid:0) δ Ψ( t ) (cid:1) , Q ′ Ψ( t ) Ψ( t ) i + h δ Ψ( t ) , ∂ t (cid:0) Q ′ Ψ( t ) Ψ( t ) (cid:1) i (cid:17) = Z dt ∂ t h δ Ψ( t ) , Q ′ Ψ( t ) Ψ( t ) i = h δ Ψ , Q ′ Ψ Ψ i . (2.33) In this paper, we always use the single bracket [Ψ n ] for string products, we therefore use the double bracketfor the graded commutator: [[ A, B ]] ≡ AB − ( − AB BA . L ∞ -identities (2.24), we obtain the expected gauge invariance δ Ψ = Q Ψ Λ and theequation of motion Q ′ Ψ Ψ = 0. We would like to emphasize that the variation of this type actionis realized by exchanging the place of δ and ∂ t : δS B = Z dt δ h ∂ t Ψ( t ) , Q ′ Ψ( t ) Ψ( t ) i = Z dt ∂ t h δ Ψ( t ) , Q ′ Ψ( t ) Ψ( t ) i . (2.34)The statement that the variation is realized by exchanging the place of δ and ∂ t is not limetedto the bosonic theory and it is correct in our type II theory. The result of (2.34) also impliesthat using this representation, we are able to treat the full action S B = R dt h ∂ Ψ , Q ′ Ψ Ψ i as thefree action S = R dt h ∂ Ψ , Q ′ Ψ i algebraically. (Cf. Appendix B) In the previous subsction, we obtain δS B from the relation h ∂ t Ψ , δ (cid:0) Q ′ Ψ Ψ (cid:1) i = h δ Ψ , ∂ t (cid:0) Q ′ Ψ Ψ (cid:1) i which is a result from (2.25). To compare with the calculation of type II theory, it is helpfullto give a second look at this calculation: not using the property of the state Q ′ Ψ Ψ but of theBRST-like operator Q [ a ]Ψ for a derivation X satisfying [[ X, Q ]] = 0, namely,( − ) X X ( Q [ a ]Ψ Ψ) = Q [ a ]Ψ ( X Ψ) + aκ (cid:2) X Ψ , Ψ (cid:3) [ a ]Ψ . (2.35)This relation is an a -integrand of the relation (2.25). The upper index [ a ] on [ A, B ] Ψ means weconsider string products of the coupling constant a times κ . The relation (2.35) implies that fora derivation X satisfying [[ X, Q ]] = 0, the X -derivative state X ( Q Ψ Ψ) becomes the BRST-exactstate Q Ψ ( X Ψ) plus extra terms κ [ X Ψ , Ψ] Ψ . Using (2.35) for X = δ and X = ∂ t , we obtain thefollowing calculation, which also goes in type II theory: δS B = Z dt Z da (cid:16) h δ∂ t Ψ , Q [ a ]Ψ Ψ i + h ∂ t Ψ , δ ( Q [ a ]Ψ Ψ) i (cid:17) = Z dt Z da (cid:16) h ∂ t δ Ψ , Q [ a ]Ψ Ψ i + h ∂ t Ψ , Q [ a ]Ψ δ Ψ i + h ∂ t Ψ , aκ [ δ Ψ , Ψ] [ a ]Ψ i (cid:17) = Z dt Z da (cid:16) h ∂ t δ Ψ , Q [ a ]Ψ Ψ i + h δ Ψ , Q [ a ]Ψ ∂ t Ψ i + h ∂ t Ψ , aκ [ δ Ψ , Ψ] [ a ]Ψ i (cid:17) = Z dt Z da (cid:16) h ∂ t δ Ψ , Q [ a ]Ψ Ψ i + h δ Ψ , ∂ t Q [ a ]Ψ Ψ i (cid:17) + Z dt Z da (cid:16) h ∂ t Ψ , aκ [ δ Ψ , Ψ] [ a ]Ψ i − h δ Ψ , aκ [ ∂ t Ψ , Ψ] [ a ]Ψ i (cid:17) = Z dt Z da ∂ t h δ Ψ , Q [ a ]Ψ Ψ i = h δ Ψ , Q ′ Ψ Ψ i . (2.36)Although there appear extra terms of κ [ X Ψ , Ψ] Ψ , in above calculation, using the cyclicity of theBPZ inner product, those of X = δ and X = ∂ t cancel each other for general δ Ψ.We would like to mention that for δ Ψ = Q Ψ Λ, each extra term h ∂ t Ψ , [Ψ , Q Ψ Λ] Ψ i all vanishesitself on the mass shell because of R da h Q Ψ Λ , [Ψ , ∂ Ψ] Ψ i ∝ h Λ , [ F (Ψ) , ∂ Ψ] Ψ i . It is a result of (2.21)and/or essentially equivalent to the decoupling mechanism of BRST-exact terms, discussed inthe rest of this subsection.Gauge Transformation as a Shift of an Expanding Point7e can regard the gauge transformation δ Ψ = Q Ψ Λ of the action S B as one kind of aninfinitesimal shift of the expanding point of Q Ψ , which is gauge equivalent to zero: Z dt Z da h ∂ t Ψ , Q [ a ]Ψ Ψ i −→ Z dt Z da h ∂ t Ψ , Q [ a ]Ψ+ δ Ψ Ψ i . (2.37)The series of shifted action consists of various terms which have BRST-exact states. We wouldlike to emphasize that those terms like h A, [ B, QC ] i , which often appear in the series of shiftedactions, are always vanish from the decoupling mechanism of BRST-exact terms. For example,we know h ∂ t ψ, [ ψ, Qλ ] i = 0 from h [ ∂ t ψ, ψ ] , Qλ i = h [ Q∂ t ψ, ψ ] + [ ∂ t ψ, Qψ ] , λ i , (2.5), and (2.7).When Ψ satisfies the equation of motion F (Ψ) = 0, there exist an isomorphism of L ∞ -algebras between original one ( H , Q, [ · ]) and new one ( H , Q Ψ , [ · ] Ψ ). Thus, above extra termsin (2.36) all vanish respectively on the mass shell. In this section, we construct a pure gauge solution of bosonic string field theory and see its rolein considering the supersymmetric extension of bosonic theory. As a simple example, we pickup heterotic string field theory [21]. A pure gauge solution as a functional of superstring fieldswill also play an impotant role in the construction of type II theory. This section is devoted topresent an idea and does not include material which is necessary for the following sections.Action of η for the BPZ Inner Product and String Products: h η A, B i = ( − A h A, η B i , (3.38) η [ A , . . . , A n ] + n X i =1 ( − ( A + ··· + A i − ) [ A , . . . , η A i , . . . , A n ] = 0 . (3.39) We can always construct a formal solution of bosonic closed string field theory G ( A ) from closedstring products and parameter fields A which carries ghost number 1 as follows: G ( A ) := Z dτ Q G ( τA ) A. (3.40)Using L ∞ -identities, we can check quickly that the state G ( t ) = G ( A ( t )) satisfies the equationof motion F ( G ) = 0 in boconic closed string field theory: F ( G ( A ( t ))) ≡ QG ( A ( t )) + ∞ X n =1 k n ( n + 1)! [ G ( A ( t )) n , G ( A ( t ))]= Z da (cid:16) Q [ a ] G ( A ( t )) G ( A ( t )) (cid:17) = 0 . (3.41)In this sence, we call G ( A ( t )) as a pure gauge solution [10, 36, 38–40]. It is equivalent to theresult of [21]: G ( A ) is the solution of the following differential equation ∂ τ G ( τ A ) = Q G A := QA + ∞ X n =1 κ n n ! [ G ( τ A ) n , A ] (3.42)8ith the initial condition G (0) = 0. This equation means that such a pure gauge solution G ( A )can be built by successive infinitisimal gauge transformations path-dependently. The path isdecided by choosing τ -dependence of A ( t ). As in the case of other gauge theories, we also choosea straight line connects 0 and A , and parametrize the path linearly as τ A ( t ) with 0 ≤ τ ≤ τ = 1, the first few terms of G ( A ) are given by G ( A ) = QA + κ (cid:2) A, QA (cid:3) + κ (cid:2) A, QA, QA (cid:3) + κ (cid:2) A, [ A, QA ] (cid:3) + O ( κ ) . (3.43)An Important Property of a Classical SolutionThe pure gauge solution has an important property: the state G ( A ) is mapped into the Q G -exact state by a derivation X satisfying [[ X, Q ]] = 0. In supersymmetric theory, we will usethis property for practical calculations as XQ ′ Ψ Ψ = ( − X Q Ψ X Ψ in the previous section. Letus prove it quickly [21]. This property is equivalent to the following statement: suppose thata pure gauge solution G ( A ) and a derivation X which satisfies [[ X, Q ]] = 0 are given, then, wecan always construct the state H ( A, X ) which satisfies the following relation, XG ( A ) = ( − X Q G H ( A, X ) . (3.44)We call this state H ( A, X ) “a related field”. For instance, in bosonic closed string field theory,variation δ and derivation ∂ t satisfy the above conditions of X , so there exist H ( A, δA ) and H ( A, ∂ t A ). We can obtain such H ( A, X ) as the solution of the following differential equation: ∂ τ H ( A, XA ; τ ) = XA + κ [ A, H ( A, XA ; τ )] G . (3.45)Then we can solve this in power of κ as follows: H ( A, XA ) = XA + κ (cid:2) A, XA (cid:3) + κ (cid:2) A, QA, XA (cid:3) + κ (cid:2) A, [ A, XA ] (cid:3) + O ( κ ) . (3.46) We reviewed the procedure of constructing a formal pure gauge solution G ( t ) from parameterfields A ( t ) which carries ghost number (gh[ G ] − S are determinded by the natural correspondence to the vertex operator of strings of thefirst quantization theory. The difference between the large and small Hilbert space is whether The reason becomes clear when we define the auxiliary field H ( τ ) as H ( τ ) := Q G ( τA ) H ( A, X ; τ ) − ( − X XG ( τ A ). Then the result which we expect is equivalent to H ( τ = 1) = 0. If our differencial equationis satisfied, we obtain ∂ τ H = κ [ A, H ] G . Using the initial condition H ( τ = 0) = 0, it gives H = 0 for any τ .
9e consider the zero mode of ξ ( z ) or not. Thus we usually use the following identification [20]of the small space string field Φ S and the large space string field Φ L :Φ S ∼ = η Φ L . (3.47)Here, η is the zero mode of η ( z ), which carries ghost number 1 and picture number −
1. Inparticular, it is a current which has conformal weight 1 like the BRST operator. In the restof this section, as an example of the supersymmetric extension, we pick up the construction ofheterotic string field theory [21] and see the role of the pure gauge solution. (cf. Appendix C.)Heterotic String Field TheoryIn heterotic string field theory [21], the small space string field V S carries ghost number 2 andpicture number −
1. Therefore, in the large space description, the heterotic string field V carriesghost number 1 and picture number 0. It carries the same ghost number as above parameterfields A ( t ), so we can built a pure gauge solution G ( t ) by V ( t ). Then the action is given by S H = Z dt h η ∂ t ( tV ) , G ( tV ) i , (3.48)where we choose t -dependence linearly V ( t ) = tV . This is the Wess-Zumino-Witten-like action.Or more formally, this action can be written for general t -dependent V ( t ) which satisfies V (0) = 0and V (1) = V , where Ψ Q := G ( V ( t )) and Ψ X := H ( V ( t ) , X ), as follows: S H = Z dt h η Ψ t , Ψ Q i . (3.49)Note that for X = ( ∂ t , δ, η ), related fields Ψ X satisfy the following relations ∂ t Ψ Q = Q Ψ Q Ψ t , δ Ψ Q = Q Ψ Q Ψ δ , η Ψ Q = − Q Ψ Q Ψ η . (3.50)It is just the same result as that of open superstrings. The variation of this action is give by δS H = Z dt ∂∂t h η Ψ δ , Ψ Q i . (3.51)So we obtain the equation of motion η Ψ Q = 0 and the gauge invariance underΨ δ = Q Ψ Q Λ (0 , + η Ω (0 , , (3.52)where Λ (0 , and Ω (0 , are gauge parameters with respect to Q and η .We would like to mention that the ξ -decomposition of pure gauge solution Ψ Q is given byΨ Q ≡ ˆ ψ Q + ξ ˆ ψ Q = ˆ ψ Q + ξ ( − Q Ψ Q Ψ η ) . (3.53)Using Ψ η := H ( V, η V ) = η V + . . . , we obtain the following form of the action for V ( t ) = tV : S H = Z dt h η ∂ t V, Q Ψ Q V i + . . . . (3.54)10 .3 Pure Gauge Solutions of Heterotic String Field Theory To construct type II theory, it is not necessary to see contents of this subsection, but helpfull.As a pure gauge solution of bosonic theory G ( A ) can always be constructed from parameterfields A ( t ) which carry ghost number gh( G ) −
1, we can also construct a pure gauge solution ofheterotic theory A (Φ) from parameter fields Φ( t ) which carry ghost number gh( A ) − Q in the large Hilbert space. The ξ -zero modedecomposition of Ψ is given by Ψ = ˆ ψ + ξ ˆ φ . Here, ˆ ψ and ˆ φ live in the small Hilbert space. Q Ψ = Q ( ˆ ψ + ξ ˆ φ )= (cid:16) Q ˆ ψ + X ˆ φ (cid:17) + ξ (cid:16) − Q ˆ φ (cid:17) , (3.55)where X := [[ Q, ξ ]]. Of course, the BRST operator Q is nilpotent on the large Hilbert space: Q Ψ = Q (cid:16) Q ˆ ψ + X ˆ φ (cid:17) + Q (cid:16) − ξ Q ˆ φ (cid:17) = (cid:16) Q ˆ ψ + Q X ˆ φ (cid:17) + (cid:16) −X Q ˆ φ + ξ Q ˆ φ (cid:17) . (3.56)So we notice the suggestive fact about the nilpotency of the BRST operator: Q ( P Q Ψ) = 0,where P is the projector onto the small Hilbert space, which satisfies P + P ⊥ = and P = P .We know that the operator Q is a linear operator on the small Hilbert spase Q : H S → H S . So ifthe state ˆ ψ lives in the Kernel of η , the state Q ˆ ψ also lives in the same space. By contrucs, thecomponent ξ ˆ φ ∈ H L is mapped to elements of both components: X ˆ φ ∈ H S and − ξ ( Q ˆ φ ) ∈ H L .Classical Solutions of Heterotic Theory and Type II String FieldsThe equation of motion of heterotic theory is η Ψ Q = 0. It is helpfull for considering typeII theory to construct a pure gauge solution of heterotic theory. Constructing a pure gaugesolution A of heterotic theory is equivalent to finding Ψ Q which lives in the small Hilbertspace. Using gauge parameter fields of heterotic theory Φ, we can make such a field A quickly: A (Φ) := P ⊥ ( Q G Φ). The state A ( t ) lives in the large Hilbert space and G ( A ) belongs to thesmall Hilbert space because P ⊥ = 1 − P and Q G = 0. Thus it is a pure gauge solution.The state space of closed strings is composed from the tensor product of two sectors, those ofright mover and left mover [29, 31]. Heterotic string theory is the supersymmetric theory whosechiral sector is supersymmetric and the other is bosonic. There exist other supersymmetrictheories that each sector has supersymmetry, so called type II theories. Through the analysisof this section, we know that a pure gauge solution plays an impotant role in supersymmetricextension and we notice the possibility that as we can always construct a formal pure gaugesolution of bosonic string field theory as a functional of heterotic string fields [21], we wouldbe able to construct it as a functional of type II string fields similarly: identification of gaugeparameter fields of heterotic theory and type II string fields. In the next section, we demonstratethe construction of type II string field theory based on this perspective.11 Type II String Field Theory
Bosonic string field theory possesses the gauge symmetry generated by Q Ψ B . The generator Q Ψ B has the kernel, which is an origin of the gauge symmetry: the state Q ′ Ψ B Ψ B . This stateconsists of bosonic string fields Ψ B and string products and satisfies Q Ψ B Q ′ Ψ B Ψ B = 0 becauseof L ∞ -identities. Then a gauge invariant action is constructed as follows S B = Z dt h ∂ t Ψ B ( t ) , Q ′ Ψ B Ψ B ( t ) i . (4.57)We expect that the NS-NS-sector type II action also has such a gauge structure and assumethat the generator is geven by Q G , where G = G (Ψ) is a formal pure gauge solution of bosonictheory and carries ghost number 2, which is a functional of type II string fields Ψ. The generator Q G has the kernel: the state Q G (Ψ) Ψ. This state consists of type II string fields Ψ and stringproducts and satisfies Q G Q G (Ψ) Ψ = 0 because of L ∞ -identities. Therefore, using this state, wepropose the following classical action, which possesses the gauge symmetry generated by Q G : S = Z dt h ∂ t Ψ( t ) , Q G ( t ) Ψ( t ) i η . (4.58)It is constructed from string fields Ψ( t ), pure gauge solutions of bosonic theory G ( t ) = G (Ψ( t ))and the BRST-like nilpotent operator Q G . As the inner product, we use the BPZ inner productwith η -currents insertion: h A, B i η := h η +0 η − A, B i . They are discussed in subsection 4.1 and 4.2.In this section, we give a concrete form of this action by constructing G = G (Ψ) as a functionalof type II string fields Ψ and using properties of the state Q G Ψ which are shown in subsection4.2, the equation of motion and the gauge invariance of this action are presented.
Type II String Fields ΨWe write a type II string field as Ψ and suppose it belongs to the large Hilbert space. Sincewe assume that η +0 η − Ψ corresponds to the type II closed string vertex operator in superstringtheory, the type II string field Ψ carries ghost number 0 and picture number 0. Note that upper-indexed η -currents are given by η ± := η ± ¯ η and satisfy η +0 η − Ψ = η ¯ η Ψ. Let us consider apath Ψ( t ) which has the starting point 0 and the end point Ψ on the state space of cloesd stringswhere t ∈ [0 ,
1] is a real parameter. Note that we can consider any t -dependent path as long asΨ( t ) satisfies Ψ(0) = 0 and Ψ(1) = Ψ. Using this Ψ( t ), we write down the type II action.In the rest of this subsection, we give one concrete construction of a pure geuge solution G and show an important property which general pure gauge solutions have.A Pure Gauge Solution G (Ψ) as a Functional of Type II String Fields ΨWe would like to construct a pure gauge solution G = G (Ψ) concretely, which is a functionalof type II string fields Ψ. To this purpose, using Zwiebach’s string products, we define G ( τ ) asthe solution of the following differential equation with the initial condition G (0) = 0: ∂ τ G ( τ ) = ∞ X n =0 ∞ X m =0 κ n + m n ! m ! (cid:2) G ( τ ) n , P − [ G ( τ ) m , Ψ] (cid:3) ≡ Q G P − Q G Ψ , (4.59)12here P − is the projector onto Im[ η − ], κ is the coupling constant, and τ ∈ [0 ,
1] is a real param-eter. We can always obtain the solution G ( τ ) by the formal Taylor series G ( τ ) = P ∞ n =0 τ n n ! ∂ nτ G (0).This solution G ( τ ) satisfies the equation of motion in bosonic theory for any τ . It is a resultfrom the uniqueness of the solution of the following first-order differential equation: ∂ τ F ( G ( τ )) = Q∂ τ G ( τ ) + ∞ X n =1 κ n ( n + 1)! ∂ τ [ G ( τ ) n , G ( τ )] (4.60)= Q G ∂ τ G ( τ ) = Q G ( P − Q G Ψ) = κ (cid:2) P − Q G Ψ , F ( G ( τ )) (cid:3) G . (4.61)Using the initial condition F ( G (0)) = 0, we obtain F ( G ( τ )) = 0 for any τ , which is the solutionof ∂ τ F = κ [ P − Q G Ψ , F ]. Thus G ( τ ) is a pure gauge solution, and we define G as G := G (1).The first few terms of G are calculated as follows: G (Ψ) = Q P − Q Ψ + κ (cid:16)(cid:2) Q P − Q Ψ , P − Q Ψ (cid:3) + Q P − (cid:2) Q P − Q Ψ , Ψ (cid:3)(cid:17) (4.62)+ κ (cid:16)(cid:2)(cid:0) Q P − Q Ψ (cid:1) , P − Q Ψ (cid:3) + (cid:2) [ Q P − Q Ψ , P − Q Ψ] + Q P − [ Q P − Q Ψ , Ψ] , P − Q Ψ (cid:3) + Q P − (cid:16)(cid:2)(cid:0) Q P − Q Ψ (cid:1) , Ψ (cid:3) + (cid:2) [ Q P − Q Ψ , P − Q Ψ] + Q P − [ Q P − Q Ψ , Ψ] , Ψ (cid:3)(cid:17)(cid:17) + . . . . We succeeded to construct a pure gauge solution G as a functinal of type II string fields Ψ.Using it, we introduce a function G ( t ) ≡ G (Ψ( t )), which is of course a pure gauge solution forany t . Note that the t -dependence of G ( t ) is determineded by that of Ψ( t ) and we can take itarbitrary as long as Ψ( t ) satisfies Ψ(0) = 0 and Ψ(1) = Ψ.A Important Property of Pure Gauge Solutions and Variation δ Ψ = Q G ΛThe pure gauge solution has an important property: the state G (Ψ) is mapped into the Q G -exact state by a derivation X satisfying [[ X, Q ]] = 0. In other words, suppose the state G (Ψ)satisfying F ( G (Ψ)) = 0 and some derivation X satisfying [[ X, Q ]] = 0 are given, then there existsa state Λ X (Ψ) which satisfies the following relation: (We also call this Λ X “a related field”.) X G (Ψ) = ( − X Q G Λ X (Ψ) . (4.63)This Λ X ( τ ) is also a functional of Ψ. In paticular, Λ X (0) = 0 and we define Λ X as Λ X := Λ X (1).As well as the case in the previous section, we can construct this Λ X ( τ ) concreatly as thesolution of the first-order differential equation ∂ τ Λ X ( τ ) = X P − Q G Ψ + κ [ P − Q G Ψ , Λ X ( τ )] G withthe initial condition Λ X (0) = 0. The most interesting one is the related field Λ δ associated withthe variation δ Ψ = Q G Λ. In the rest of this subsection, we see useful properties of Λ X ( τ ), whichis helpfull to our calculus. Suppose this derivation X commutes with the projector P − , then wecan calcurate ∂ τ Λ X ( τ ) as ∂ τ Λ X = ( − X P − (cid:16) Q G X Ψ + κ [ X G , Ψ] G (cid:17) + κ [ P − Q G Ψ , Λ X ] G . (4.64)For δ Ψ = Q G Λ, it becomes ∂ τ Λ δ ( τ ) = κ (cid:0) P − [ Q G Λ δ ( τ ) , Ψ] G + [ P − Q G Ψ , Λ δ ] G (cid:1) . Thus we obtain δ G ( τ ) = Q G Λ δ ( τ ) = 0 under δ Ψ = Q G Λ because Λ δ ( τ ) = 0 for any τ when Q G δ Ψ = 0. Combining with P + , which is the projector onto Im[ η +0 ], we can define the projector onto the closed stringsmall Hilbert space as P + P − , namely, P − P + : H L ⊗ H L → H S ⊗ H S . Of course, P + / − satisfies ( P + / − ) = P + / − . .2 The Generator Q G and the State Q G Ψ In the previous section, we constructed G ( t ) = G (Ψ( t )) as a functional of type II string fieldsΨ, which carries the same ghost number as a bosonic string field. Therefore we can define theBRST-like operator Q G ( t ) as an operator which depends on type II string fields Ψ( t ) throughthe functional G = G (Ψ), which will become the generator of the gauge transformation of ourtype II action. The generator Q G has the kernel: the state Q G ( t ) Ψ( t ) ≡ Q Ψ( t ) + ∞ X n =1 κ n n ! (cid:2) G ( t ) n , Ψ( t ) (cid:3) , (4.65)which satisfies Q G Q G (Ψ) Ψ = 0 at t = 0. In particular, the state Q G Ψ has the following properties: Q G ( Q G Ψ) = 0 , (4.66)( − X X ( Q G Ψ) = Q G ( X Ψ) + κ [ Q G Λ X , Ψ] G , (4.67) h Ψ , X ( Q G Ψ) i = ( − X h Φ , Q G ( X Ψ) i , (4.68)where X is a derivation which satisfies [[ X, Q ]] = 0 and Φ is an arbitraly string field. Theseproperties are used in later calculations, so let us prove them. The first line means the nilpotencyof the generator Q G . Since G (Ψ) is a pure guage solution, using (2.6), we can check it as follows: Q G Q G Ψ = X n,m X σ κ n + m n !( m + 1)! (cid:2) G n , σ (cid:0) [ G m , Ψ] (cid:1)(cid:3) = − κ (cid:2) Ψ , F ( G ) (cid:3) G = 0 , (4.69)where σ ([ A , . . . , A n ]) is a permutation of the order of [ A , . . . , A n ] and the sum of σ runs overall possible permutations. The second line implies the state Q G Ψ is mapped into the Q G -exactstate plus extra terms by a derivation X and the third line guarantees that these extra terms allvanish in the BPZ inner product. Using the property (4.63), we can prove the property (4.67)as same as (2.35). Note that extra terms all are the form h Φ , [Ψ , Q G Λ X ] G i and it is sufficient forthe proof of (4.68) to give h Φ , [Ψ , Q G Λ] G i = 0. The proof becomes understandable by separatingtwo cases: whether X is even or odd. When X is even, namely, ( − X = 1, the Grassmannparity of Φ becomes even. Then, by (2.5), (2.4), (2.6), and (2.3), the following calculation goes: h Φ , [Ψ , Q G Λ X ] G i = h [Φ , Ψ] G , Q G Λ X i = h [ Q G Φ , Ψ] G , Λ X i + h [Φ , Q G Ψ] G , Λ X i (4.70)= h [ Q G Φ , Λ X ] G , Ψ i + h [ Q G Ψ , Λ X ] G , Φ i = h Q G Φ , [Λ X , Ψ] G i + h Φ , [ Q G Ψ , Λ X ] G i . We therefore obtain h Φ , [Ψ , Q G Ψ X ] G i = 0 because of (2.7) and h Φ , [Ψ , Q G Λ X ] G i = 12 (cid:16) h Q G Φ , [Ψ , Ψ X ] G i + h Φ , [ Q G Ψ , Λ X ] G i + h Φ , [Ψ , Q G Λ X ] G i (cid:17) . (4.71)For odd X , the parity of Φ becomes odd and we can check (4.68) similarly. This is equivalentto the decoupling mechanism of BRST-exact states in bosonic string field theory. We would like to mention that in our construction, although the operator Q G anticommutewith η − , it dose not anticommute with η +0 as [[ Q G , η +0 ]] = κ [ η +0 G , ] G . However, using η +0 G = Or more simply, these extra terms of δ G = Q G Λ δ and η G = − Q G Λ η are equivalent to the gauge-trivail shiftof Q G ’s expanding point: Q G Q G + Q G λ . Thus we can expect that the result is not affected by these terms. Q G Λ η and (4.68), this commutator becomes an element of the kernel of cyclic L ∞ -algebras.As a result, the operator Q G and η +0 anticommute in the BPZ inner product. We therefore treatthe classical solution G as it belongs to the small Hilbert space in the BPZ inner product. Thus,for example, we obtain h η − ∂ t Ψ , η +0 (cid:0) Q G Ψ (cid:1) i = −h η − ∂ t Ψ , Q G (cid:0) η +0 Ψ (cid:1) i . This mechanism admits usto use a simple inner product in which Q G works as a derivation: the BPZ inner product with η -currents insertion h A, B i η := h η +0 η − A, B i . It is nonzero if and only if the total ghost andpicture number of the states in the inner product are equal to 1 and 0 respectively. h A, Q G B i η = ( − A h Q G A, B i η . (4.72) Using operators defined in the previous subsection, we propose the following NS-NS-sector actionfor type II string field theory in the large Hilbert space description: S = Z dt h ∂ t Ψ( t ) , Q G ( t ) Ψ( t ) i η ≡ Z dt h η +0 η − ∂ t Ψ( t ) , Q G ( t ) Ψ( t ) i . (4.73)The t -dependence of G ( t ) = G (Ψ( t )) is determinded by that of Ψ( t ), and we can take Ψ( t ) asan arbitrary function of t ∈ [0 ,
1] as long as it satisfies Ψ(0) = 0 and Ψ(1) = Ψ. In practicalcalculations, we often take t -dependence linearly Ψ( t ) = t Ψ for simplicity. This action is almostthe same form as the bosonic action. In fact, using (4.68) for X = δ, ∂ t and (4.72), we can carryout calculations as same as bosonic theory. Note that the following calculation goes: h ∂ t Ψ , δ ( Q G Ψ) i η = h ∂ t Ψ , Q G ( δ Ψ) i η = h Q G ( δ Ψ) , ∂ t Ψ i η = h δ Ψ , Q G ( ∂ t Ψ) i η = h δ Ψ , ∂ t ( Q G Ψ) i η . (4.74)Hence, the variation of the action is given by: δS = Z dt (cid:16) h ∂ t ( δ Ψ( t )) , Q G ( t ) Ψ( t ) i η + h ∂ t Ψ( t ) , δ ( Q G ( t ) Ψ( t )) i η (cid:17) = Z dt (cid:16) h ∂ t ( δ Ψ( t )) , Q G ( t ) Ψ( t ) i η + h δ Ψ( t ) , ∂ t ( Q G ( t ) Ψ( t )) i η (cid:17) = Z dt ∂ t h δ Ψ( t ) , Q G ( t ) Ψ( t ) i η = h δ Ψ , Q G Ψ i η . (4.75)As a result, we obtain the gauge invariance under the gauge transformation δ Ψ = Q G Λ. As anelement of the large Hilbert space, the gauge transformation can be writen down as follows: δ Ψ = Q G Λ + η +0 Ω + + η − Ω − . (4.76)Here, Λ, which carries ghost number − ± , which carries ghostnumber − Q G and η ± .The equation of motion of our type II theory is given by Q G η +0 η − Ψ = η +0 η − Q G Ψ = 0 . (4.77)In particular, on the mass shell, we obtain η ± G (Ψ) = 0, which is the result of the equation ofmotion. We can obtain all interaction terms from this action. In the next section, we calculatethe expansion with respect to κ and see that it gives correct three point amplitudes.15 .4 Summary We proposed the following type II full action in the large Hilbert space description, S = Z dt h η +0 η − ∂ t Ψ( t ) , Q G ( t ) Ψ( t ) i . (4.78)In this theory, Q G and η ± commute each other in the BPZ inner product and the nonlineartransformations with respect to η ± Ω ± do not appear. Therefore, we can include η ± into thedefinition of the innner product as h A, B i η := h η +0 η − A, B i . Of course, the inner product h A, B i η is degenerate in the large Hilbert space H L : all of the small Hilbert space components areidentified with null states. Thus we consider the equivalence relation A ∼ A + η +0 Ω + + η − Ω − ,and introduce the quotient space H η := H L / Ker[ η ± ]. In this quotient space, the inner product h A, B i η becomes nondegenerate. This situation is similar to the c − -insertion (considering b − -vanishing states) in bosonic closed string field theory. Using the quotient space H η as the statespace, we can write down the action as follows: S = Z dt h ∂ t Ψ( t ) , Q G Ψ( t ) i η . (4.79)Here, we regard Ψ( t ) as an element of H η . Then this action is invariant under the gaugetransformation δ Ψ = Q G Λ. The equation of motion becomes Q G Ψ = 0. Note that while thesymmetric prorerties of string products also hold, the naive cyclicity of the inner product is lostin this discription. (It is natural because G (Ψ) is equivalent to 0 in H η .)This representation of the type II action will give us an one-to-one correspondence to thesmall Hilbert space description, so there is a possibility that we can completely write down thetheory in the small Hilbert space description, but it is not yet clear. In the next section, we willsee some results which are yielded from this action. In this section, we study some properties of type II string field theory which was proposed inthe last section. In paticular, we would like to check whether our action can reproduce correctamplitudes expected from the first quantized theory.
In the last section, we proposed a concrete form of the type II full action S via bosonic puregauge solutions as functionals of type II string fields. The full action S produces the free term S and every interaction term S , S , . . . in each order of κ through the following expansion: S = ∞ X n =2 κ n − S n . (5.80)In this subsection, we pick up the kinetic term and see details: the gauge transformation andthe cohomology, or the physical state space. In the next subsection, we will see the propertiesof lower-order terms. 16 type II string field Ψ satisfies the level matching condition L − Ψ = 0 ( L − := L − ¯ L ) asother closed string fields. The free type II action in the NS-NS sector is given by S = Z dt h ∂ t ( t Ψ) , Q ( t Ψ) i η = 12 h η +0 η − Ψ , Q Ψ i , (5.81)where Q is the BRST operator of type II superstrings and we define η ± := η ± ¯ η . The innerproduct is defined as h A, B i = h A | c − | B i . Here, h A | is the BPZ conjugation of | A i . This innerproduct is nondegenerate just on Ker[ b − ]. The normalization of correlators in the full conformalfield theory for a flat space-time background is given by h ξ ( w ) ¯ ξ ( ¯ w ) e − φ ( w ) e − φ ( ¯ w ) c ( z )¯ c (¯ z ) c ( z )¯ c (¯ z ) c ( z )¯ c (¯ z ) e ip µ X µ ( z, ¯ z ) i (5.82)= 2(2 π ) D δ D ( p ) | z − z | | z − z | | z − z | , where D = 10 is the critical dimension of space-time. Since we use the c − -inserted inner product,the total ghost and picture number in the action must be ( g, p ) = (3 , − g, p ) of Ψ are ( g, p ) = (0 , ≡ Ψ (0 , .This free action is invariant under the gauge transformation Ψ Ψ + Q Λ in H η . In thelarge Hilbert space description, it is equivalent to the following gauge transformation: δ Ψ = Q Λ + η +0 Ω + + η − Ω − , (5.83)where Λ = Λ ( − , , Ω + = Ω +( − , , and Ω − = Ω − ( − , all are gauge parameters. Then the equationof motion of free theory is given as follows, Qη +0 η − Ψ = 0 (Ψ ∈ H L ) . (5.84)Using the identification η +0 η − Ψ = Ψ S , it is equivalent to the ordinary small Hilbert space one: Q Ψ S = 0 (Ψ S ∈ H S ) , (5.85)where Ψ S is the type II string field in the small Hilbert space description and ξ ± := ( ξ ± ¯ ξ ). Inthe quotient space H η , choosing a representative element, we can write Ψ ∈ H η as Ψ eqv . = [ ξ − ξ +0 ˆ ψ ]where ˆ ψ (2 , − ∈ Ker[ η ± ]. So the equation of motion becomes Q Ψ eqv . = h ξ − ξ +0 Q ˆ ψ (2 , − i = 0 (Ψ ∈ H η ) (5.86)and we obtain Q ˆ ψ (2 , − = 0, which is equivalent to above two equations. They all give thecondition of the first quantization. It implies that physical state spaces which are described inthe large/small Hilbert space H L / H S and the quotient space H η are equavalent to each other,Ker[ Q : H η → H η ]Im[ Q : H η → H η ] ∼ = Ker[ η +0 η − Q : H L → H L ]Im[ Q, η ± : H L → H L ] ∼ = Ker[ Q : H S → H S ]Im[ Q : H S → H S ] . (5.87) In the last section, the type II full action was given by the following form: S = Z dt h ∂ t Ψ( t ) , Q G Ψ( t ) i η ≡ Z dt ∞ X n =0 κ n n ! h ∂ t Ψ( t ) , [ G ( t ) n , Ψ( t )] i η . (5.88)17t would reproduce all interaction terms of type II string field theory. In this subsection, we seethe result yielded from the lower-order terms S + κS + κ S + O ( κ ). Recall that the puregauge solution G ( t ) is expanded with respect to the coupling constant κ as follows: G (Ψ) = Q P − Q Ψ + κ (cid:16)(cid:2) Q P − Q Ψ , P − Q Ψ (cid:3) + Q P − (cid:2) Q P − Q Ψ , Ψ (cid:3)(cid:17) (5.89)+ κ (cid:16)(cid:2)(cid:0) Q P − Q Ψ (cid:1) , P − Q Ψ (cid:3) + (cid:2) [ Q P − Q Ψ , P − Q Ψ] + Q P − [ Q P − Q Ψ , Ψ] , P − Q Ψ (cid:3) + Q P − (cid:16)(cid:2)(cid:0) Q P − Q Ψ (cid:1) , Ψ (cid:3) + (cid:2) [ Q P − Q Ψ , P − Q Ψ] + Q P − [ Q P − Q Ψ , Ψ] , Ψ (cid:3)(cid:17)(cid:17) + . . . . Therefore we obtain the following lower-order perturbative action: S = Z dt (cid:16) h ∂ t Ψ( t ) , Q Ψ( t ) + κ G ( t ) , Ψ( t )] + κ
3! [ G ( t ) , Ψ( t )] i η + . . . (cid:17) = Z dt h ∂ t Ψ , Q Ψ i η + κ Z dt h Ψ , [ Q P − Q Ψ , Ψ] i η + κ Z dt h ∂ t Ψ , [( Q P Q Ψ) , Ψ] i η + κ Z dt h Ψ , (cid:2) [ Q P − Q Ψ , P − Q Ψ] + Q P − [ Q P − Q Ψ , Ψ] , Ψ (cid:3) i η + . . . . (5.90)Then, the variation of the action becomes the following form: δS = Z dt ∂ t h δ Ψ( t ) , Q G ( t ) Ψ( t ) i η = h δ Ψ , Q G Ψ i η = h δ Ψ , Q Ψ i η + κ h δ Ψ , [ Q P − Q Ψ , Ψ] i η + κ h δ Ψ , (cid:2) ( Q P − Q Ψ) , Ψ (cid:3) i η + κ h δ Ψ , (cid:2) [ Q P − Q Ψ , P − Q Ψ] + Q P − [ Q P − Q Ψ , Ψ] , Ψ (cid:3) i η + . . . . (5.91)The gauge transformation is therefore given as follows: δ Ψ = Q Λ + κ (cid:2) Q P − Q Ψ , Λ (cid:3) + κ (cid:2) ( Q P − Q Ψ) , Λ (cid:3) + κ (cid:16)(cid:2) [ Q P − Q Ψ , P − Q Ψ] + Q P − [ Q P − Q Ψ , Ψ] , Λ (cid:3)(cid:17) + . . . . (5.92)Let us check the gauge invariance of the action perturbatively. We can write it down by thesum of the following all lines which are equal to zero as follows: κ : h Q Λ , Q Ψ i η = 0 κ : h Q Λ , (cid:2) Q P − Q Ψ , Ψ (cid:3) i η + h (cid:2) Q P − Q Ψ , Λ (cid:3) , Q Ψ i η = 0 κ : h Q Λ , (cid:2) ( Q P − Q Ψ) , Ψ (cid:3) + (cid:2) [ Q P − Q Ψ , P − Q Ψ] + Q P − [ Q P − Q Ψ , Ψ] , Ψ (cid:3) i η + h [( Q P − Q Ψ) , Λ] + (cid:2) [ Q P − Q Ψ , P − Q Ψ] + Q P − [ Q P − Q Ψ , Ψ] , Λ (cid:3) , Q Ψ i η + h [ Q P − Q Ψ , Λ] , [ Q P − Q Ψ , Ψ] i η + h [ Q P − Q Ψ , Λ] , [ Q P − Q Ψ , Ψ] i η = 0 κ n : ... (5.93)Vanishing of the first line is equivalent to the nilpotency of the BRST operator: Q = 0, thatof the second line means the derivation propertiy: Q [ A, B ] + [
QA, B ] + ( − A [ A, QB ] = 0,18nd that of the third line implies the lowest-order (homotopy-) Jacobi identity: ∆ Q [ A, B, C ] +∆ J [ A, B, C ] = 0 where∆ Q [ A, B, C ] = Q [ A, B, C ] + [
QA, B, C ] + ( − A [ A, QB, C ] + ( − A + B [ A, B, QC ] , (5.94)∆ J [ A, B, C ] = [ A, [ B, C ]] + ( − A ( B + C ) [ B, [ C, A ]] + ( − C ( A + B ) [ C, [ A, B ]] . (5.95)We can also read higher-order relations from vanishing of higher-order terms similarly: ∆ Q [ A , . . . , A n ]+∆ J [ A , . . . , A n ] = 0, where ∆ Q/ J [ A , . . . , A n ] means the violation of the derivation/Jacobi prop-erty of the n -product [ A , . . . , A n ]. Let us consider three point amplitudes. Recall that the κ -order action is given by S = Z dt h ∂ t Ψ( t ) , [Ψ( t ) , Q P − Q Ψ( t )] i η = 13 h Ψ , [Ψ , Q P − Q Ψ] i η , (5.96)where Ψ( t ) = t Ψ. This cubic term of the action reduces to the expected correlator for physicalstates. Writing Ψ = ξ +0 ξ − ˆΨ (2 , − where Q ˆΨ (2 , − = 0 and P − = η − ξ − , we obtain Q P − Q Ψ = Qη − ξ − Qξ +0 ξ − ˆΨ (2 , − = Q X − ξ +0 ˆΨ (2 , − e.o.m. = X +0 X − ˆΨ (2 , − . (5.97)Note that the last equal sign holds on the mass shell. Hence, we see that S = 13 h ˆΨ , [ ξ +0 ξ − ˆΨ , X +0 X − ˆΨ] i ∼ = 13 hh ˆΨ , ˆΨ , X +0 X − ˆΨ ii . (5.98)Here, the symbol hh A, B, C ii denotes the correlator. This is the expected result.Pure Gauge Solutions in the CorrelatorIn general, there is the correspondence between correlation functions in the small Hilbertspace hh A , . . . , A n ii and those in the large Hilbert space, where φ and ¯ φ are bosonized su-perghosts: hh A , . . . , A n ii = h ξ ¯ ξe − φ e − φ A . . . A n i . (5.99)This implies that the correlator selects one ξ +0 ξ − -component and other small space components.For example, suppose { A i } ni =1 belong to the small Hilbert space, and Φ/Ψ can be ξ ± -decomposedas Φ = ˆ φ + ξ +0 ξ − ˆ φ ξ /Ψ = ˆ ψ + ξ +0 ξ − ˆ ψ ξ , then the correlation function of these elements is givenby the following form: h η +0 η − Φ , [Ψ , A , . . . , A n ] i = h ˆ φ ξ , [ ξ +0 ξ − ˆ ψ ξ , A , . . . , A n ] i . (5.100)Our type II action was given by this form: S = Z dt h ∂ t Ψ , Q G Ψ i η = Z dt ∞ X n =0 κ n n ! h η +0 η − Ψ , [ G ( t Ψ) n , t Ψ] i . (5.101)19n the next subsection, we estimate higher-point amplitudes. To this purpose, in the rest ofthis subsection, we see the property of G in the correlator. Note that using (4.63) for η ± and η +0 η − G (Ψ) = 0, we obtain the ξ ± -decomposition of G (Ψ): G (Ψ) = ˆ ψ Q − ξ +0 (cid:0) Q G Λ η + (cid:1) − ξ − (cid:0) Q G Λ η − (cid:1) . (5.102)In our construction, it becomes G (Ψ) = ˆ ψ Q − ξ +0 Q G Λ η because η − G (Ψ) = 0. Then we noticethat ξ ± -components of G (Ψ) are given by BRST-exact terms. As a result, it is expected thatthe value of the correlation function depends only on the small space component of G (Ψ). Wetherefore regard the pure gauge solution G (Ψ) as an element of the small Hilbert space: G (Ψ) ∼ = P + P − G (Ψ) = ˆ ψ Q , (5.103)although it is constructed from type II string fields Ψ which belong to the large Hilbert space.Since there are one η +0 and one η − in our large Hilbert space action, the value of correlationfunctions in our theory will be determinded by the large Hilbert space component of Ψ and thesmall Hilbert space component of G (Ψ). Let us consider amplitudes, which are expected to correspond to the result of the first quatizationtheory. To this purpose, the small Hilbert space description is rather reasonable because a smallstring field Ψ S naturaly corresponds to the vertex operator of closed strings. We thereforeconsider the reduction to the small Hilbert space, and seek fundamental vertices which produceFeynman graphs of string interaction in the small Hilbert space. We would like to expect thatthere exists the correspondence Ψ S ≡ η +0 η − Ψ between the small string field Ψ S and the largestring field Ψ as the free case. To see it explicitly, let us consider the zero mode decomposition:Ψ = ˆ ψ ◦ + ξ +0 ˆ ψ + + ξ − ˆ ψ − + ξ +0 ξ − ˆ ψ (5.104)where ˆ ψ ◦ , ˆ ψ ± , and ˆ ψ belong to Ker[ η ± ], namely, the small Hilbert space H S . Then the largecomponent ˆ ψ has ghost number 2 and picture nummber −
2. (We often write it explicitly asˆ ψ ≡ ˆ ψ (2 , − .) Therefore, we identify the small string field Ψ S with this ˆ ψ up to a sign factor:Ψ S ⇐⇒ − ˆ ψ (2 , − . (5.105)It is equivalent to impose on Ψ the following (partial) fixing conditions: ξ +0 Ψ = ξ − Ψ = 0 . (5.106)We assume that the other elements ˆ ψ ◦ and ˆ ψ ± correspond to auxiliary fields as well as thecorrespondence in free theory. We thus consentrate on the part which is related to the elementˆ ψ (2 , − . The pure gauge solution, which is a functional of type II string fields, is given by G (Ψ) = Q P − Q Ψ + κ (cid:16) [ Q P − Q Ψ , P − Q Ψ] + Q P − [ Q P − Q Ψ , Ψ] (cid:17) + . . . . (5.107)20e can calculate the ξ ± -decomposition of Q P − Q Ψ, P − Q Ψ, and Q P − Ψ as follows: Q P − Q Ψ ∼ = P + Q P − Q Ψ = P + Q P − Q (cid:16) ˆ ψ ◦ + ξ +0 ˆ ψ + + ξ − ˆ ψ − + ξ +0 ξ − ˆ ψ (cid:17) = P + Q (cid:16) Q ˆ ψ ◦ + Qξ +0 ˆ ψ + + X − ˆ ψ − − X − ξ +0 ˆ ψ (cid:17) = Q X +0 ˆ ψ + + Q X − ˆ ψ − − X +0 X − ˆ ψ, (5.108) P − Q Ψ ∼ = P + P − Q Ψ = P + P − Q (cid:16) ˆ ψ ◦ + ξ +0 ˆ ψ + + ξ − ˆ ψ − + ξ +0 ξ − ˆ ψ (cid:17) = Q ˆ ψ ◦ + X +0 ˆ ψ + + X − ˆ ψ − , (5.109) Q P − Ψ ∼ = P + Q P − Ψ = Q ˆ ψ ◦ + X +0 ˆ ψ + . (5.110)Suppose auxiliary fields are integrated out and BRST-exact terms are dropped, then we obtain Q P − Ψ = P − Q Ψ = 0 , Q P − Q Ψ = −X +0 X − ˆ ψ . (5.111)Thus we regard G (Ψ) ∼ = −X +0 X − ˆ ψ in the small Hilbert space description. Using the identifica-tion Ψ S ∼ = − ˆ ψ (2 , − , we can write down F ( G (Ψ)) = F (cid:0) X +0 X − Ψ S (cid:1) = 0 as follows: Q (cid:0) X +0 X − Ψ S (cid:1) + ∞ X n =1 κ n ( n + 1)! [ (cid:0) X +0 X − Ψ S (cid:1) n , X +0 X − Ψ S ] ∼ = 0 . (5.112)Then we can estimate the following part of the small Hilbert space action, which is expected tocontribute to the value of correlation functions: e S = 12 hh Ψ S , Q Ψ S ii + ∞ X n =1 X σ κ n ( n + 2)! hh Ψ S , σ (cid:0)(cid:2) ( X +0 X − Ψ S ) n , Ψ S (cid:3)(cid:1) ii , (5.113)where σ is a permutation and the sum of σ runs over all possible permutations. Here, hh A, B ii is the inner product in the small Hilbert space. It is nonzero if and only if the pair of the totalghost and picture nummber is equal to ( g, p ) = (5 , n -point vertices e S n of κ e S = P κ n e S n from the large space action κ S = P κ n S n . First, we consider the three point vertex. In this case, we can read it quickly as S = 13 h Ψ , [Ψ , Q P − Q Ψ] i η −→ e S = 13 { Ψ S , Ψ S , X +0 X − Ψ S } S (5.114)where { A, B, C } S := hh A, [ B, C ] ii . It is an expected one. Next, we considr higher-point vertices.Recall that the κ n -order large space action is given by S n = 1 n ! X σ h Ψ , σ (cid:16)(cid:2) ( Q P − Q Ψ) n − , Ψ (cid:3) + · · · + (cid:2) Q P − [ . . . [Ψ , Q P − Q Ψ]] , Ψ (cid:3)(cid:17) i η . (5.115)From the correspondence discussed above, we can estimate higher-point vertices similarly: e S n = 1 n ! X σ { Ψ S , σ (cid:0) Ψ S , ( X +0 X − Ψ S ) n − (cid:1) } S . (5.116)This is a result what we expected from the correspondence to the first quantization theory.Since we use bosonic closed string products, the single covering of the moduli space of Riemannsurfaces is realized automatically. The problem of the divergence with respect to the local picturechanging operators would be also resolved by the nilpotency of homotopy algebras.21 Summary and Discussion
In this paper, using algebraic properties of closed strings, we gave a concrete construction of typeII string field theory. It is based on the large Hilbert space description like [20, 21], so there areno local insertions of picture changing operators. In the construction of supersymmetric theory,a f ormal pure gauge solution of bosonic theory has played a nonnegligible role. A pure gaugesolution of bosonic string field theory is always constructed from gauge parameter fields [10]. Inthe large Hilbert space description, we have constructed superstring field theories by identifyingsuperstring fields with these gauge parameter fields, so it is with our construction. Makinga pure gauge solution as a functional of type II string fields, we construct the full action fortype II string field theory. We would like to note that our construction is based on algebraicproperties of closed string fields, L ∞ -algebras. It is a result from that we use the extension ofZwiebach’s closed string products [11]. (Recall that the action for closed string field theory whoseFeynman graphs reproduce a single covering of the moduli space has the geometrical verticessatisfying the algebraic relations of the BV-master equation, L ∞ -algebras and their quantumextensions [34, 35, 38].) Thus, in this paper, we did not touch the geometrical understandingof superstrings like the correspondence of the full action and the decomposition of the modulispace of super-Riemann surfaces as the case of bosonic theory. In the rest of this section, wewould like to disucuss related aspects of this type II action and put some comments.Comparison with Bosonic TheoryWe can rewrite the bosonic action into R dt h ∂ Ψ , Q ′ Ψ Ψ i , which consits of the Maurer-Cartanoperator Q ′ Ψ and has the gauge invariance generated by the BRST operator Q Ψ . On the otherhand, we write down the type II action by R dt h ∂ Ψ , Q G Ψ i η , which consists of the BRST operator Q G and has the gauge invariance generated by the BRST operator Q G . In bosonic theory, wecan construct the Maurer-Cartan operator Q ′ Ψ as the linear combination of the BRST operators Q [ a ]Ψ . We regard this Q [ a ]Ψ as a deformation of the gauge structure because Q [0]Ψ generates thegauge transformation of free theory and Q [1]Ψ generates that of full theory. In type II theory, wecan introduce Q [ a ] G by the replacement of κ → aκ and define a Maurer-Cartan (like) operator Q ′G by the integration of a from 0 to 1. Using this operator, we can write down the another action as S ′ = R dt h ∂ Ψ , Q ′G Ψ i η , which gives Q ′G Ψ = 0 as the equation of motion. Then the perturbativeexpansion of κ S ′ = P κ n S ′ n gives the almost same result as our action κ S = P κ n S n . Thedifference is the coefficients of n -point vertices: ( n − S ′ n = S n . Note that this S ′ is not invariantunder the transformation generated by Q G (but Q [ a ] G ). Pure Gauge Solution and L ∞ -structureIn our construction, we used the method which is based only on the properties of nilpotenthomotopy algebras, so this framework is available for considering other gauge theories as longas the gauge invariance is governed by nilpotent homotopy algebras. In particular, consideringa shift by a pure gauge solution G , we wrote down the type II action. However, we would liketo note that there exist some ambiguities in constructing a pure gauge solution. For example, Of course, there is some possibilities that the gauge transformatin is realized by the nontrivial form asBerkovits theory: the element generating gauge symmetry is not δ Ψ but some combination of Ψ like δ ( e Φ ).
22e can use some another projector like P + , or modify the defining equation of the pure gaugesolution as ∂ τ G = Q G P − Q Ψ, and so on. Or as well as open string field theory, in closed stringfield theory, we may be able to construct a pure gauge solution without using a differential(or integral) equation which we used. In this article, we chose simplest one which realizes δ G (Ψ) = Q G Λ δ = 0 under the gauge transformation because we would Q G ’s expanding point G like to belong the equivalent class of the gauge transformation.Algebraic structure of Type II String Field TheoryThe gauge symmetry of string field theory is infinitly reducible. Thus it is not clear whetherwe can obtain a gauge fixed action as a simple extension of the original action [25–27]. Since thetype II action consists of string fields, the BRST operator Q , a projector P − , and closed stringproducts, we can expect that guage fixing is carrid out by relaxing the constraint of the ghostnumber of string fields as the case of bosonic theory. However, to prove it exactly, we need toconstruct vertices for type II fields (not for bosonic fields) from the decomposition of the modulispace of super-Riemann surfaces, which would naturally lead to (extended) homotopy algebrasof type II fields. We need these geometrical understandings of superstring fields for quantization,constructing theory which includes R -sector, considering the problem of background indepen-dence, other nonperturbative effects, and so on. It is rather not clear in the case of the LargeHilbert space description. In particular, in the large Hilbert space description, we know thecyclicity of a graded symplectic form and the sign of products of homotopy algebras are notwell-suited naively, so we also need to find a good algebraic structure of supersting fields on thispoint. Recently, there is a progress in this geometrical point of view [41]. They gave an outlineto construct type II vertices and the quantum BV-master action geometrically in the SmallHilbert space description. In quantum bosonic closed string field theory, the vertices satisfy therelations of a loop homotopy Lie algebra, whose classical part gives L ∞ -algebras. In [41], theyintroduced N = 1 loop homotopy Lie algebras by considering the operad which deeply relatesto type II string field theory. Ackonledgements:
This is an extended work of my master’s thesis, so I would like to thank Yuji Okawa andMitsuhiro Kato, my supervisors, for fruitful discussions, suggestive comments, and their kind-ness. I am also grateful to the members of our Komaba particle theory group, in particular,Yuki Iimori, Shota Komatsu, and Shingo Torii for useful discussions and comments, and theirkindness. I would also like to thank the organizers of the conference ‘String Field Theory andRelated Aspects: SFT 2012’ hosted by the Israel Institute for Advanced Studies. A L ∞ -algebras and Closed String Field Theory A representation of the L ∞ -operad on a fixed graded vector space H is an L ∞ -algebra ( H , L )[34,35,39,40]. In string field theory, we regard the string state space, Hilbert space of conformalfield theory, as a graded vector space H through the identification of the world sheet ghostnumber and H ’s grading [12, 14–16, 34–38]. In Appendix A, we give a short review of the L ∞ -algebra and its role in closed string field theory without using operads.23effinition. (un-shuffle)By a ( k, l )-unshuffle of A , . . . , A n with n = k + l is meant a permutation σ such that for i < j ≤ k , we have σ ( i ) < σ ( j ) and similarly for k < i < j ≤ k + l . We denote the subgroupof ( k, l )-unshuffle in S k + l by S k,l and by S k + l = n , the union of the subgroup S k,l with k + l = n .Similary, a ( k , . . . , k i )-unsuffle means a permutation σ ∈ S n with n = k + · · · + k i such thatthe order is preserved within each block of length k , . . . , k i . The subgroup of S n consisting ofall such unshuffle we denote by S k ,...,k i .The Koszul Sign of PermutationBy decomposing permutations as a product of transpositions, there is then defined the signof a permutation of n graded elements: A , . . . , A n , e.g for any σ ∈ S n , the permutation of n graded elements, is defined by σ ( A , . . . , A n ) = ( − σ ( A σ (1) σ, . . . , A σ ( n ) ) . (A.117)Since we will have many fomulas with such indices and their permutations, we will use thenotion I := ( i , . . . , i n ) and A I := A i ⊗ · · · ⊗ A i n . Then, for any σ ∈ S n , we use σ ( I ) to denote( σ ( i ) , . . . , σ ( i n )) and hence A σ ( I ) = A σ ( i ) ⊗ · · · ⊗ A σ ( i n ) .Deffinition. ( L ∞ -algebra)Let H be a graded vector space and suppose that a collection of degree (3 − k ) gradedsymmetric linear maps L := { L k : H ⊗ k → H} k ≥ is given. The pair ( H , L ) is called L ∞ -algebraif the maps satisfy the following relations: X σ ∈ S k + l = n ( − σ L l +1 ( L k ( A σ (1) , . . . , A σ ( k ) ) , A σ ( k +1) , . . . , A σ ( n ) ) = 0 (A.118)for n ≥
1. A weak L ∞ -algebra consists of a collection of degree (3 − k ) graded symmetric linearmaps m := { m k : H ⊗ k → H} k ≥ satisfying the same relation but for n ≥ k, l ≥ L ∞ -morphism)For two L ∞ -algebras ( H , L ) and ( H ′ , L ′ ), suppose that there exists a collection of degreepreserving graded symmetric multi-linear maps F := { f k : H ⊗ k → H ′ } k ≥ where f is a mapfrom C to a degree zere subvector space of H . F is called an L ∞ -morphism if it satisfies thefollowing relations X σ ∈ S k + l = n ( − σ f l ( L k ⊗ ⊗ l )( A σ ( I ) ) = X σ ∈ S k ··· + kj = n ( − σ j ! L ′ j ( f k ⊗ · · · ⊗ f k j )( A σ ( I ) ) (A.119)When ( H , L ) and ( H ′ , L ′ ) are weak L ∞ -algebra, then a weak L ∞ -morphism consists of multi-linear maps { f k } k ≥ satisfying the above condition and f ◦ L = P k k ! L ′ k ( f , . . . , f ).The Maurer-Cartan EquationIn an L ∞ -algebra ( H , L ), the Maurer-Cartan equation for A ∈ H is given by F ( A ) := X k ≥ k ! L k ( A, . . . , A ) = 0 . (A.120)We donote the set of the solutions of the Maurer-Cartan equation as MC ( H , L ).24iven two elements G , G ∈ MC ( H , L ) are called gauge equivalent if there exists a piecewisesmooth path G ( τ ), τ ∈ [0 ,
1] such that ddτ G ( τ ) = X k ≥ k ! L k +1 (Λ( τ ) , G ( τ ) , . . . , G ( τ )) (A.121)for a degree deg[ G ( τ )] − τ ) where G (0) = G and G (1) = G . Gauge transforma-tion preserves MC ( H , L ) and defines the equivalence relation. Then a quotient space by thisequivalence relation: M ( H , L ) := MC ( H , L ) / ∼ gives the moduli space of the deformations.Null forms of the Maurer-Cartan operatorFor A ∈ MC ( H , L ) and B ∈ H satisfying deg[ B ] = deg[ A ] −
1, there exist special element: L A ( B ) := X k ≥ k ! L k +1 ( A, . . . , A, B ) . (A.122)It gives a null state in the Maurer-Cartan operator, or conversely, the image of the Maurer-Cartan perator is included in the kernel of this operator: L A ( F ( A )) = 0.Symplectic StructureSkew-symmetric bilinear map ω : H ⊗ H →
C is called constant symplectic structure whenit has fixed integer degree | ω | ∈ Z and is non-degenerate. Namely, ω ( A, B ) = − ( − AB ω ( B, A ) (A.123)for any
A, B ∈ H and degree | ω | implies that ω ( A, B ) = 0 except for | A | + | B | + | ω | = 0.Cyclic L ∞ -algebrasSuppose that an L ∞ -algebra ( H , L ) is equipped with constant symplectic structure ω : H ⊗ H →
C. For L = { L k : H ⊗ k → H} , let us define the multi-linear maps by V n +1 ( A , A , . . . A n ) := ω ( A , L n ( A , . . . , A n )) . (A.124)The degree of V n +1 is | ω | + 1. An L ∞ -algebra equipped with constant symplectic structure( H , L , ω ) is called cyclic L ∞ -algebra if V n +1 is graded symmetric with respect to any permutationof H ⊗ ( n +1) : V n +1 ( A , . . . , A n ) = ( − σ V n +1 ( A σ (0) , . . . , A σ ( n ) ) , σ ∈ S n +1 . (A.125)In string field theory, we sometimes use the Stokes’s Teorem: X V X σ ∈ S n +1 ( − σ V n +1 ( L ( A σ (0) ) , A σ (1) , . . . , ( A σ ( n ) )) = 0 . (A.126)If the Stokes’s Theorem holds in cyclic L ∞ -algebra, then L ∞ -identities splits into two groups: L ( L n ( A , . . . , A n )) + X σ ∈ S n ( − σ L n ( L ( A σ (1) ) , A σ (2) , . . . , ( A σ ( n ) )) = 0 , (A.127) X σ ∈ S k + l = n ( − σ L l +1 ( L k ( A σ (1) , . . . , A σ ( k ) ) , A σ ( k +1) , . . . , A σ ( n ) ) = 0 , (A.128)25here l ≥ k ≥
2. Then L becomes the derivation for all products { L k } k ≥ . It is naturallyrelized in considering scattering amplitudes of multi-strings in closed string field theory. L ∞ -algebras and Cyclicity in Closed Sting Field TheoryIn string field theory, these mathematical objects naturally corespond to physical ones. Weoften use L ∞ -identities by the following form: ( l ≥ k ≥ σ is the sign of splittings) X l + k = n σ ( i l , i k )[ A i , . . . , A i l , [ A j , . . . , A j k ]] = 0 . (A.129)The sum runs over all different splittings of the set { , . . . , n } into a first group { i , . . . , i l } anda second group { j , . . . , j k } , where l ≥ k ≥
2. Two splittings are the same if the corre-sponding first groups contain the same set of integers regardless of their order. The sign factor σ ( i l , j k ) is defined to be the sign picked up when one rearranges the sequence { Q, A , A , . . . , A n } into the sequence { A i , . . . , A i l , Q, A j , . . . , A j k } taking into account the Grassmann property.The gauge structure of closed string theory is governed by L ∞ -algebras. In particular, theMaurer-Cartan equation corresponds to the equation of motion F (Ψ) = 0 and the null form L Ψ (Λ) corresponds to the gauge transformation δ Ψ = Q Ψ Λ. The differential equation whichdefines a pure gauge solution comes from above equation which decides the gauge equivalence ofMaurer-Cartan elements. In addition to these, the symplectic form ω ( A, B ) is given by the BPZinner product h A | c − | B i of conformal field theory. The cyclic L ∞ -structure {V n } n correspondsto the set of fundamental vertices of string field theory. Furthermore, in string field theory, thereexists Stokes’s Theorem on the moduli space of Riemann surfaces. Therefore, closed string fieldtheory is described by cyclic L ∞ -algebras equipped with Stokes’s Theorem.Stokes’s Theorem on Moduli Space in Closed String Field TheoryThe closed string products are constructed form the decomposition of the moduli space ofRiemann surfaces [11]. We know the fact that the BRST operator acts on the moduli sapce M n as an exterior product acting on the cotangent bundle of (2 n − n -point correlation functionΩ A ...A n of string field theory corresponds to the top form Ω [0] on the moduli space M n . As aresult, we obtain Z M n +1 Ω [0]( P Q ) A ...A n = Z M n +1 d Ω [ − A ...A n = Z ∂ M n +1 Ω [ − A ...A n = 0 (A.130)where M n +1 is the moduli space of the ( n + 1)-punctured sphere and Ω [ − r ] is the volume formof such a space whose dimension is (dim[ M n +1 ] − r ). By definition, ( n + 1)-point correlationfunctions A( A , . . . , A n ) are given through the integral of the top form over such a moduli space M n +1 . X graph n X i =0 ( − A + ··· + A i − h A , [ A , . . . , QA i , . . . A n ] i = Z M n +1 Ω [0]( P Q ) A ...A n (A.131)The summention runs over decomposition of the moduli space M n ≡ V n ⊕ R n ⊕· · ·⊕ R n − n , where V n gives the n -point fundamental vertex and R in represents the region of n -point graph vertices26onstructed from the possible combination of lower vertecies and propergators. Therefore, X graph (cid:16) h QA , [ A , . . . , A n ] i + n X i =1 ( − A + A + ··· + A i − h A , [ A , . . . , QA i , . . . , A n ] i (cid:17) = 0 . (A.132)Note that in the case of the three point vertex, amplitudes are obtained by the fundamentalvertex only, namely, M ≡ V . Then BRST-exact terms decouple. B A Parallel Structure of Actions
Free Theory of Bosonic Closed StringsWe already know the free action for bosonic closed string field theory: S = h Ψ , Q Ψ i . Thenthe generator of the gauge transformation is given by Q . The state Q ′ Ψ which belongs to thekernel of this generator Q is given by Q ′ Ψ := Q Ψ because of the nilpotency Q = 0. Therefore,in free theory, we can regard the Maurer-Cartan operator Q ′ as the BRST operator Q itself.Using this Q ′ , we can rewrite the free action of bosonic string field theory as follows, S = Z dt h ∂ t Ψ( t ) , Q ′ Ψ( t ) i , (B.133)where Ψ( t ) satisfies Ψ(0) = 0 and Ψ(1) = Ψ for t ∈ [0 ,
1] and Q ′ Ψ ≡ Q Ψ. For a given derivation X satisfying [[ Q, X ]] = 0, the state Q ′ Ψ satisfies XQ ′ Ψ = XQ Ψ = ( − X QX Ψ. Thus we caneasily calculate the variation as follows: δS = Z dt (cid:16) h ∂δ Ψ( t ) , Q ′ Ψ( t ) i + h ∂ Ψ( t ) , Qδ Ψ( t ) i (cid:17) = Z dt D ∂ t δ Ψ( t ) , Q ′ Ψ( t ) i + h Q∂ t Ψ( t ) , δ Ψ( t ) i (cid:17) = Z dt ∂ t h δ Ψ( t ) , Q ′ Ψ( t ) i = h δ Ψ , Q ′ Ψ i . (B.134)In full theory, the generator of the gauge transformation Q Ψ does not have the nilpotency:we obtain Q = 0 if and only if Ψ satisfies the equation of motion Q ′ Ψ Ψ = 0. For a generalstring field Ψ, the state which belongs to the kernel of Q Ψ is given by Q ′ Ψ Ψ from L ∞ -identities: Q Ψ Q ′ Ψ Ψ = 0. Thus Q ′ Ψ = Q Ψ in full theory.The Difference for Bosonic and Type II TheoriesIn contrast to calculations (2.36) in bosonic theory, [ δ Ψ , ∂ t Ψ] [ a ]Ψ and [ ∂ t Ψ , δ Ψ] [ a ]Ψ in (2.36) arereplaced by [ δ Ψ , ∂ t G ] G and [ ∂ t Ψ , δ G ] G in type II theory. However, these terms do not cancel eachother but vanish respectively. If the following relation holds, there would exist cancellation[ ∂ t Ψ( t ) , δ G ( t )] G ? = [ δ Ψ( t ) , ∂ t G ( t )] G . (B.135)This relation is trivial for the case like analytic functions, graded differential algebras, and soon because of the associativity of the product. A typical example is a analytic function f [ z ( t )]: δf [ z ( t )] · ∂ t z ( t ) = ∂ t f [ z ( t )] · δz ( t ) . (B.136)27nfortunately, this relation does not hold for a general element of L ∞ -algebras. However,considering the property of the BPZ inner product, we notice that such extra elements belongto the kernel of cyclic L ∞ -algebras. The decoupling mechanism of BRST-exact states works.There is a slight difference between bosonic action and our type II action and it is a resultfrom that we use a naive extension of Zwiebach’s string products which is constructed fromthe decomposition of bosonic moduli space and carry appropriate ghost numbers for bosonicstring fields. We expect that after constructing type II vertices as the decomposition of themuduli space and identifying some underlying (homotopy) algebra, using such vertices for typeII theory, we will be able to write down the action which have exactly parallel structure. C Open String Field Theory
Open string field theory [3] is described by the Chern-Simons-like action S cs = − g (cid:18) h Φ , Q Φ i + g h Φ , Φ ∗ Φ i (cid:19) . (C.137)Thus the equation of motion is given by Q Φ+ Φ ∗ Φ = 0 and the gauge invariance is generated by Q +[[ g Φ , ]]. Here, [[ A, B ]] means the graded commutater: [[
A, B ]] := A ∗ B − ( − AB B ∗ A . In thistheory, the gauge structure is governed by differential graded algebras, which are the special caseof A ∞ algebras [16, 37, 38]. We can also rewrite this action into our form: S = R dt h ∂ t Φ , Q Ψ Φ i .Let us define the BRST operator around A with coupling constant ag and the Maurer-Cartanoperator Q ′ A as follws: Q [ a ] A Φ := Q Φ + ag [[ A, Φ]] , Q ′ A Φ := Z da Q [ a ] A Φ = Q Φ + g A, Φ]] . (C.138)Then we can rewrite this Charn-Simons-like action into the following form: S cs = − g (cid:18) h Φ , Q Φ i + g h Φ , [[Φ , Φ]] i (cid:19) = − g Z dt h ∂ t ( t Φ) , Q ( t Φ) + g t Φ) , ( t Φ)]] i = − g Z dt h ∂ t Φ( t ) , Q ′ Φ( t ) Φ( t ) i , (C.139)where t ∈ [0 ,
1] is a real parameter. At the last line, we introduce Φ( t ) which satisfies Φ(0) = 0and Φ(1) = Φ. Then we notice that the equation of motion is give by the Maurer-Cartanoperator: Q ′ Φ Φ = 0 and the gauge invariance is generated by Q Φ . (Of course, when A satisfiesthe equation of motion in open string field theory, Q A becomes nilpotent operator.)Open Superstring Field TheoryRecall that using a gauge parameter field of bosonic theory λ whose ghost number is 0,we can construct a pure gauge solution of boconic theory by A Q := e − λ ( Qe λ ). If there exista derivation X which satisfies [[ Q, X ]] = 0, a related field A X := e − λ ( Xe λ ) which satisfies( − X XA Q = Q A Q A X ≡ QA X + g [[ A Q , A X ]] appears. In the large Hilbert space description,28eplacing these parameter fields λ by superstring fields Φ( t ) which satisfy Φ(0) = 0 and Φ(1) = Φ,we obtain the following action of Berkovits’ theory [20]: S wzw = Z dt h η A ∂ t ( t ) , A Q ( t ) i Φ( t )= t Φ = Z dt h η ∂ t ( t Φ) , A Q ( t ) i . (C.140)The equation of motion is given by η A Q = 0 and this action is invariant under A δ = Q A Q Λ. Wewould like to mention that for linear t -dependent Φ( t ), introducing a real parameter a ∈ [0 , S wzw = Z dt h η ∂ t Φ( t ) , Q ′ A Q ( t ) Φ( t ) i , (C.141)where Q ′ A Q is defined by the a -integration of Q [ a ] A Q from 0 to 1. Note that now Q [ a ] A Q is given by Q [ a ] A Q ( t ) = Qφ + [[ A [ a ] Q ( t ) , ]] , A [ a ] Q ( t ) ≡ e − ag Φ( t ) (cid:16) Qe ag Φ( t ) (cid:17) . (C.142)The equivalence of (C.140) and (C.141) is provided by the relation A Q ( t ) = Q ′ A Q ( t ) Φ( t ) forΦ( t ) = t Φ. Let us check this relation. Using Φ = g ˜Φ, we can rewrite A Q into Q ′ A Q Φ as follows: A Q ( t ) = e − t Φ (cid:0) Qe t Φ (cid:1) = tQ Φ + t Q Φ ∗ Φ − Φ ∗ Q Φ) + t
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