Open superstring field theory I: gauge fixing, ghost structure, and propagator

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1 Open superstring field theory I: gauge fixing, ghost structure, and propagator The MIT Faculty has made this article openly available Please share how this access benefits you Your story matters Citation As Published Publisher Kroyter, Michael, Yuji Okawa, Martin Schnabl, Shingo Torii, and Barton Zwiebach Open Superstring Field Theory I: Gauge Fixing, Ghost Structure, and Propagator Journal of High Energy Physics 212, no 3 March 9, Springer-Verlag Version Author's final manuscript Accessed Sun Mar 17 21:52:23 EDT 219 Citable Link Terms of Use Creative Commons Attribution-Noncommercial-Share Alike 3 Detailed Terms

2 arxiv: MIT-CTP-4332 UT-Komaba/12-1 TAUP Open superstring field theory I: gauge fixing, ghost structure, and propagator arxiv: v2 [hep-th] 6 Feb 212 Michael Kroyter, 1 Yuji Okawa, 2 Martin Schnabl, 3 Shingo Torii 2 and Barton Zwiebach 4 1 School of Physics and Astronomy The Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Ramat Aviv, 69978, Israel 2 Institute of Physics, University of Tokyo Komaba, Meguro-ku, Tokyo , Japan 3 Institute of Physics AS CR Na Slovance 2, Prague 8, Czech Republic 4 Center for Theoretical Physics Massachusetts Institute of Technology Cambridge, MA 2139, USA mikroyt@tauacil, okawa@hep1cu-tokyoacjp, schnablmartin@gmailcom, storii@hep1cu-tokyoacjp, zwiebach@mitedu Abstract The WZW form of open superstring field theory has linearized gauge invariances associated with the BRST operator Q and the zero mode η of the picture minus-one fermionic superconformal ghost We discuss gauge fixing of the free theory in a simple class of gauges using the Faddeev- Popov method We find that the world-sheet ghost number of ghost and antighost string fields ranges over all integers, except one, and at any fixed ghost number, only a finite number of picture numbers appear We calculate the propagators in a variety of gauges and determine the field-antifield content and the free master action in the Batalin-Vilkovisky formalism Unlike the case of bosonic string field theory, the resulting master action is not simply related to the original gauge-invariant action by relaxing the constraint on the ghost and picture numbers

3 Contents 1 Introduction and summary 1 2 Gauge fixing of the free theory 6 21 Open bosonic string field theory 6 22 Open superstring field theory 9 23 Comparison with Witten s superstring field theory in Siegel gauge Various gauge-fixing conditions 16 3 Calculation of propagators Propagators for gauge fixing with b and ξ Propagators for gauge fixing with b and d 21 4 Verifying the master equation for the free action 25 5 Conclusions and outlook 31 1 Introduction and summary String field theory is an approach to string theory that aims to address non-perturbative questions that are difficult to study in the context of first quantization Classical solutions that represent changes of the open string background are of particular interest, and considerable progress was made in this subject in the last few years see, for example, [1, 2, 3, 4, 5] A covariant string field theory should satisfy a series of consistency checks The kinetic term, for example, must define the known spectrum of the theory The full action, with the inclusion of interaction terms, has nontrivial gauge invariances It must be possible to gauge fix these symmetries, derive a propagator, and set up a perturbation theory that produces off-shell amplitudes that, on-shell, agree with the amplitudes in the first-quantized theory The purpose of these checks is not necessarily to construct off-shell amplitudes, but rather to test the consistency and understand better the structure of the theory Indeed that was the way it turned out for open bosonic string field theory [6] The Faddeev-Popov quantization of the theory quickly suggested that the full set of required ghost and antighost fields could be obtained by relaxing the ghost number constraint on the classical string field [7, 8, 9] 1 Moreover, the Batalin-Vilkovisky BV quantization approach [1, 11] turned out to be surprisingly effective [12] The full master action for open bosonic string field theory the main object in this quantization scheme is simply the classical action evaluated with the unconstrained string field For the closed bosonic string field theory, the BV master equation was useful in the 1 In this paper we refer to the string field in the gauge-invariant action before gauge fixing as the classical string field, distinguishing it from ghost and antighost fields introduced by gauge fixing 1

4 construction of the full quantum action, since it has a close relation with the constraint that ensures proper covering of the moduli spaces of Riemann surfaces [13] As is the case for open strings, the closed string field theory master action is simply obtained by relaxing the ghost number constraint on the classical string field It is the purpose of this paper to begin a detailed study of gauge fixing of the WZW open superstring field theory [14] using the Faddeev-Popov method and the Batalin-Vilkovisky formalism This theory describes the Neveu-Schwarz sector of open superstrings using the large Hilbert space of the superconformal ghost sector in terms of ξ, η, and φ [15] As opposed to some alternative formulations [16, 17, 18] no world-sheet insertions of picture-changing operators are required and the string field theory action takes the form S = 1 2g 2 e Φ Qe Φ e Φ η e Φ 1 dte tφ t e tφ { e tφ Qe tφ,e tφ η e tφ } 11 Here {A,B} AB + BA, g is the open string coupling constant, η denotes the zero mode of the superconformal ghost field η, and Q denotes the BRST operator These two operators anticommute and square to zero: {Q,η } =, Q 2 = η 2 = 12 The string field Φ is Grassmann even and has both ghost and picture number zero Both Q and η have ghost number one While η carries picture number minus one, Q carries no picture number Products of string fields are defined using the star product in [6], and the BPZ inner product of string fields A and B is denoted by AB or by A B The action is defined by expanding all exponentials in formal Taylor series, and we evaluate the associated correlators recalling that in the large Hilbert space ξzc c 2 cwe 2φy 13 The action can be shown to be invariant under gauge transformations with infinitesimal gauge parameters Λ and Ω: and the equation of motion for the string field is δe Φ = QΛe Φ +e Φ η Ω, 14 η e Φ Qe Φ = 15 In this paper we focus on the linearized theory For notational simplicity we will simply set the open string coupling equal to one: g = 1 To linearized order the action reduces to S given by Using bra and ket notation, the kinetic term can be written as S = 1 2 QΦη Φ 16 S = 1 2 Φ, Qη Φ, 17 2

5 Here we have written Φ = Φ, to emphasize that the classical string field has both ghost number and picture number zero Unless indicated otherwise, we take X g,p to be an object that carries ghost number g and picture number p To this order the equation of motion 15 becomes η QΦ, =, 18 and the gauge transformations 14 become δ Φ, = QΛ + η Ω 19 Let us use ǫ for gauge parameters and rewrite 19 as δ Φ, = Qǫ 1, + η ǫ 1,1, 11 where we have indicated the appropriate ghost and picture numbers in the subscripts Note that both ǫ 1, and ǫ 1,1 are Grassmann odd, both have ghost number minus one, but differ in picture number The gauge invariances 11 have their own gauge invariances We can change ǫ 1, and ǫ 1,1 without changing δ Φ, Indeed, with δ 1 ǫ 1, = Qǫ 2, +η ǫ 2,1, δ 1 ǫ 1,1 = Qǫ 2,1 +η ǫ 2,2, 111 we readily verify that δ 1 δ Φ, =, making use of 12 At this stage we have introduced three gauge parameters, all of ghost number minus two, and with pictures zero, one, and two The above redundant transformations have their own redundancy: δ 2 ǫ 2, = Qǫ 3, +η ǫ 3,1, δ 2 ǫ 2,1 = Qǫ 3,1 +η ǫ 3,2, 112 δ 2 ǫ 2,2 = Qǫ 3,2 +η ǫ 3,3, and this time we verify that δ 2 δ 1 ǫ 1, = δ 2 δ 1 ǫ 1,1 = At step n, in matrix notation, we have δ n ǫ n, ǫ n,1 ǫ n,2 ǫ n,n = Q η Q η Q Q η ǫ n+1, ǫ n+1,1 ǫ n+1,2 ǫ n+1,n ǫ n+1,n The above describes the full structure of redundant symmetries of the theory at linearized level It is the starting point for the BRST quantization of the theory, where we select gauge-fixing conditions and add suitable Faddeev-Popov terms to the action The above gauge parameters turn into ghosts Φ n,p, n 1, p =,1,,n 114 3

6 It follows from the BRST prescription that all of the above ghost fields are Grassmann even, just like the classical string field Φ, 2 Antighosts must be also added The Faddeev-Popov quantization is carried out using a set of gauge conditions that enable us to confirm that the free gauge-fixed action, after elimination of auxiliary fields, coincides with that of Witten s free theory [16] in Siegel gauge The gauge-fixing conditions here are of type b,ξ ;α, meaning that ghosts and antighosts are required to be annihilated by operators made of the zero modes b and ξ with a parameter α See 29 We then turn to the calculation of the propagator of the theory, for which the free gauge-fixed action is sufficient As usual, we add to the action linear couplings that associate unconstrained sources with the classical field, with each ghost, and with each antighost The propagator is then the matrix that defines the quadratic couplings of sources in the action, and it is obtained by solving for all fields in terms of sources using the classical equations of motion We examine this propagator for a few types of gauges In the b,ξ ;α type gauges, the propagator matrix contains the zero mode X = {Q,ξ } of the picture-changing operator and its powers The propagator is quite complicated for α and simplifies somewhat for α =, where it takes the form of matrices of triangular type A more intriguing class of gauges are of type b,d ;α Here d is the zero mode of the operator d = [Q,bξ] InthelanguageofthetwistedN = 2superconformalalgebra[19], d = G isacounterpart of b = G Corresponding to the relation {Q,b } = L, the anticommutation relation {η,d } = L holds In fact, the gauge-fixing conditions b Φ, = d Φ, = were used in the calculation of a four-point amplitude in [2] The propagators in this class of gauges are much simpler than in the b,ξ ;α type gauges and do not involve picture-changing operators They further simplify when α = 1 see 359, 36, and 361 We expect this form of the propagator to be useful in the study of loop amplitudes The gauge structure of the free theory is infinitely reducible In fact, the equations in 113 determine the field/antifield structure of the theory following the usual Batalin-Vilkovisky procedure [1, 11] reviewed in [21, 22, 23] We write the original gauge symmetry of the classical fields φ α schematically as δφ α = R α α 1 ǫ α 1, 115 where sum over repeated indices is implicit and R is possibly field dependent The symmetry is infinitely reducible if there are gauge invariances of gauge invariances at every stage, namely δǫ α 1 = R α 1 1α 2 ǫ α 2 δǫ α 2 = R α 2 2α 3 ǫ α = 2 The spacetime fields in such string fields can be even or odd depending on the Grassmann parity of the CFT basis states 4

7 with the following on-shell relations R αn nα n+1 R α n+1 n+1α n+2 =, for n =,1,2, 117 In this case one introduces fields φ αn with n 1 and antifields φ α n with n such that the BV action reads S = S φ α + n= φ α n R αn nα n+1 φ α n+1 + = S +φ α R α α 1 φ α 1 +φ α 1 R α 1 1α 2 φ α 2 +, 118 where the dots represent terms at least cubic in ghosts and antifields that are needed for a complete solution of the master equation Since all string fields for fields in open superstring field theory are Grassmann even, the R s are Grassmann odd, and since the inner product with 13 needed to form the action couples states of the same Grassmann parity, the string fields for antifields are Grassmann odd 3 The antifield Φ g,p associated with the field Φ g,p is Φ 2 g, 1 p : Φ g,p = Φ 2 g, 1 p 119 This follows from 118 where each term in the sum takes the form φ α n δφ αn, with the gauge parameter replaced by a ghost field of the same ghost and picture number This implies that the inner product with 13 must be able to couple a field to its antifield Since this inner product requires a total ghost number violation of two and a total picture number violation of minus one, the claim in 119 follows The full field/antifield structure of the theory is therefore Φ 2,2 Φ 2,1 Φ 1,1 p Φ 2, Φ 1, Φ, g Φ 2, 1 Φ 3, 1 Φ 4, 1 Φ 3, 2 Φ 4, 2 Φ 4, 3 12 The string fields Φ g,p with g on the left side are the fields, and the Φ g,p with g 2 on the right side are the antifields Note the gap at g = 1 Collecting all the fields in Φ and all antifields in Φ + as g g 1 Φ = Φ g,p, Φ + = Φ g, p, 121 g= p= g=2 p=1 3 In open bosonic string field theory the string fields for fields and those for antifields are of the same odd Grassmann parity 5

8 we can show that the free master action S implied by 118 and by our identification of fields and antifields takes the form: S = 1 2 Φ Qη Φ + Φ + Q+η Φ 122 The master equation {S,S} =, where {, } is the BV antibracket, will be shown to be satisfied 2 Gauge fixing of the free theory In this section we perform gauge fixing of the free open superstring field theory using the Faddeev- Popov method We first review the procedure in the free open bosonic string field theory, and then we extend it to open superstring field theory We also demonstrate that the resulting gauge-fixed action coincides with that of Witten s superstring field theory in Siegel gauge after integrating out auxiliary fields 21 Open bosonic string field theory The gauge-invariant action of the free theory is given by S = 1 2 Ψ 1 Q Ψ 1, 21 where Ψ 1 is the open string field It is Grassmann odd and carries ghost number one, as indicated by the subscript The BRST operator Q is BPZ odd: Q = Q This action is invariant under the following gauge transformation: where ǫ is a Grassmann-even string field of ghost number zero δ ǫ Ψ 1 = Qǫ, 22 The Faddeev-Popov method consists of adding two terms to the gauge-invariant action The first term is given by L GF = λ i F i φ, 23 where F i φ = are the gauge-fixing conditions on the field φ and λ i are the corresponding Lagrange multiplier fields The second term is the Faddeev-Popov term given by L FP = b i c α δ δǫ α δ ǫ F i φ 24 It is obtained from L GF by changing λ i to the antighost fields b i and by changing F i φ to its gauge transformation δ ǫ F i φ with the gauge parameters ǫ α replaced by the ghost fields c α The sum of the two terms L GF + L FP is then BRST exact: L GF + L FP = δ B b i F i φ under the convention δ B b i = λ i Let us apply this procedure to the free theory of open bosonic string field theory and choose the Siegel gauge condition b Ψ 1 = 25 6

9 for gauge fixing Note that b is BPZ even: b = b It is convenient to decompose Ψ 1 into two subsectors according to the zero modes b and c as follows: Ψ 1 = Ψ 1 +c Ψ c 1, 26 where Ψ 1 and Ψc 1 are both annihilated by b The superscript indicates the sector without c and the superscript c indicates the sector with c, although c has been removed in Ψ c 1 Therefore Ψc 1 is Grassmann even and carries ghost number zero, and so the subscript, which is carried over from Ψ 1, does not coincide with the ghost number of Ψ c 1 The operator c we used in the decomposition is BPZ odd: c = c Using this decomposition, the Siegel gauge condition can be stated as Ψ c 1 = 27 The gauge-fixing term S GF implementing this condition can be written as S GF = N c Ψ c 1, 28 where the Lagrange multiplier field N is annihilated by b Note that the insertion of c is necessary for the inner product to be nonvanishing The ghost number of N is two and component fields playing the role of Lagrange multiplier fields have to be Grassmann even, so the string field N is Grassmann even This term can be equivalently written as S GF = N 2 Ψ 1, 29 with the constraint b N 2 = 21 This can be seen by decomposing N 2 before imposing the constraint as N 2 = N2 +c N2 c, 211 where N2 and Nc 2 are annihilated by b The inner product N 2 Ψ 1 is then given by N 2 Ψ 1 = N2 c Ψ c 1 + N2 c c Ψ The constraint b N 2 = eliminates N2 c, and the remaining field N 2 is identified with the Lagrange multiplier field N The string field N 2 is Grassmann even and carries ghost number two Another way to derive S GF is to use the form b Ψ 1 = for the gauge-fixing condition and write S GF = Ñ3 b Ψ We then redefine the Lagrange multiplier as N 2 = b Ñ

10 The resulting field N 2 is subject to the constraint b N 2 = Since {b,c } = 1, any solution N 2 to this constraint can be written as N 2 = {b,c }N 2 = b c N 2 = b Ñ 3 with Ñ3 = c N 2 Therefore, N 2 obtained from the redefinition N 2 = b Ñ 3 is equivalent to N 2 with the constraint b N 2 = The Faddeev-Popov term S FP can be obtained from S GF by changing N 2 to the Grassmann-odd antighost field Ψ 2 of ghost number two and by changing Ψ 1 to its gauge transformation Qǫ with ǫ replaced by the Grassmann-odd ghost field Ψ of ghost number zero We have S FP = Ψ 2 Q Ψ, 215 with the constraint b Ψ 2 =, 216 which is inherited from b N 2 = After integrating out N 2, the total action we obtain is S +S 1 = 1 2 Ψ 1 Q Ψ 1 Ψ 2 Q Ψ, 217 with b Ψ 1 =, b Ψ 2 = 218 This action S +S 1 is invariant under the following gauge transformation: δ ǫ Ψ = Qǫ We can choose b Ψ = 22 for gauge fixing Repeating the same Faddeev-Popov procedure, we obtain S +S 1 +S 2 = 1 2 Ψ 1 Q Ψ 1 Ψ 2 Q Ψ Ψ 3 Q Ψ 1, 221 with b Ψ 1 =, b Ψ 2 =, b Ψ =, b Ψ 3 =, 222 where Ψ 3 has ghost number three and Ψ 1 has ghost number minus one The action S +S 1 +S 2 is invariant under δ ǫ Ψ 1 = Qǫ 2 In this way the gauge-fixing procedure continues, and at the end we obtain S = where S n, 223 n= S = 1 2 Ψ 1 Q Ψ 1, S n = Ψ n+1 Q Ψ n+1 for n with b Ψ n =, n 225 The action S can also be written compactly as S = 1 2 Ψ Q Ψ with Ψ = Ψ n, b Ψ = 226 n= 8

11 22 Open superstring field theory Let us now perform gauge fixing of the free open superstring field theory We denote a string field of ghost number g and picture number p by Φ g,p The gauge-invariant action of the free theory is given by S = 1 2 Φ, Qη Φ,, 227 where η is the zero mode of the superconformal ghost carrying ghost number one and picture number minus one It is therefore BPZ odd: η = η The action S is invariant under the following gauge transformations: δ ǫ Φ, = Qǫ 1, +η ǫ 1,1 228 We can choose ǫ 1,1 appropriately such that the condition ξ Φ, = 229 on Φ, is satisfied The operator ξ we used in the gauge-fixing condition is BPZ even: ξ = ξ Because {η,ξ } = 1, a field Φ, satisfying 229 can be written as Φ, = ξ Φ1, 1, 23 where Φ 1, 1 carrying ghost number one and picture number minus one is in the small Hilbert space, namely, it is annihilated by η The equation of motion Qη Φ, = 231 in the large Hilbert space reduces to Q Φ 1, 1 = 232 Since this is the familiar equation of motion in the small Hilbert space, we know that we can choose the condition b Φ1, 1 = 233 to fix the remaining gauge symmetry This gauge-fixing condition can be stated for the original field Φ, as b Φ, = when ξ Φ, = is imposed The condition b Φ, = can be satisfied by appropriately choosing ǫ 1,, and it is compatible with ξ Φ, = by adjusting ǫ 1,1 To summarize, we can choose as the gauge-fixing conditions on Φ, b Φ, =, ξ Φ, = 234 It is convenient to decompose Φ g,p into four subsectors according to the zero modes b, c, η, and ξ as follows: Φ g,p = Φ g,p +c Φ c g,p +ξ Φ ξ g,p +c ξ Φ cξ g,p, 235 9

12 where Φ g,p, Φc g,p, Φ ξ g,p, and Φcξ g,p are all annihilated by both b and η Note that the subscript g,p is carried over from Φ g,p and does not indicate the ghost and picture numbersof the fields in the subsectors The ghost and picture numbers g,p are g,p for Φ g,p, g 1,p for Φc g,p, g+1,p 1 for Φ ξ g,p, and g,p 1 for Φcξ g,p Using this notation, the gauge-fixing conditions 234 can be stated as Φ, =, Φc, =, Φcξ, = 236 The gauge-fixing term S GF implementing these conditions can be written as S GF = N cξ 2, 1 c ξ Φ, + N ξ 2, 1 c ξ Φ c, + N 2, 1 c ξ Φ cξ,, 237 where the Lagrange multiplier fields N cξ 2, 1, N ξ 2, 1, and N 2, 1 are all annihilated by b and η Note that the insertion of c ξ to each term is necessary for the inner product to be nonvanishing, as can be seen from 13 This term can be equivalently written as S GF = N 2, 1 Φ, 238 with the constraint b ξ N 2, 1 = 239 This can be seen by writing N 2, 1 before imposing the constraint as where N 2, 1, Nc 2, 1, N ξ 2, 1, and Ncξ b ξ N 2, 1 = eliminates N c 2, 1 N 2, 1 = N 2, 1 +c N c 2, 1 +ξ N ξ 2, 1 +c ξ N cξ 2, 1, 24 2, 1 are all annihilated by both b and η The constraint, and the remaining fields Ncξ 2, 1, N ξ 2, 1, and N 2, 1 implement Φ, =, Φc, =, and Φcξ, = Another way to derive S GF is to use the form b Φ, = ξ Φ, = for the gauge-fixing conditions and write We then redefine the Lagrange multiplier as S GF = Ñ3, 1 b Φ, Ñ3, 2 ξ Φ, 241 N 2, 1 = b Ñ 3, 1 +ξ Ñ 3, TheresultingfieldN 2, 1 issubjecttotheconstraintb ξ N 2, 1 =, andthesolutiontotheconstraint can be written as N 2, 1 = b Ñ 3, 1 + ξ Ñ 3, 2 This time, however, Ñ 3, 1 and Ñ3, 2 are not uniquely determined for a given solution Comparing this with the decomposition 24, we find that N 2, 1 is in the part b Ñ3, 1 and c ξ N cξ 2, 1 is in the part ξ Ñ3, 2, but ξ N ξ 2, 1 can be in either part This ambiguity is related to the fact that a part of Ñ 3, 1 and a part of Ñ 3, 2 impose the same constraint Φ c, = More specifically, if we write Ñ3, 1 and Ñ3, 2 as Ñ 3, 1 = c Ñ c 3, 1 +c ξ Ñ cξ 3, 1, Ñ 3, 2 = Ñ 3, 2 +cñc 3,

13 with Ñc 3, 1, Ñcξ 3, 1, Ñ 3, 2, and Ñc 3, 2 all annihilated by both b and η, both Ñcξ 3, 1 and Ñ 3, 2 impose the condition Φ c, = So we should be careful if we use Ñ3, 1 and Ñ3, 2 as Lagrange multiplier fields No such issues arise if we use N 2, 1 with the constraint b ξ N 2, 1 = as the Lagrange multiplier field The Faddeev-Popov term S FP can be obtained from S GF by changing N 2, 1 to the Grassmannodd antighost field Φ 2, 1 and by changing Φ, to its gauge transformations Qǫ 1, +η ǫ 1,1 with ǫ 1, and ǫ 1,1 replaced by the Grassmann-even ghost fields Φ 1, and Φ 1,1, respectively We have S FP = Φ 2, 1 Q Φ 1, +η Φ 1,1 244 with the constraint b ξ Φ 2, 1 =, 245 which is inherited from b ξ N 2, 1 = After integrating out N 2, 1, the total action we obtain is S +S 1 = 1 2 Φ, Qη Φ, + Φ 2, 1 Q Φ 1, +η Φ 1,1 246 with b Φ, =, ξ Φ, =, b ξ Φ 2, 1 = 247 The action S 1 can be written in the following form: S 1 = Φ 2, 1 Q η Φ 1, Φ 1,1 This action S +S 1 is invariant under the following gauge transformations: 248 δ ǫ Φ 1, = Qǫ 2, +η ǫ 2,1, δ ǫ Φ 1,1 = Qǫ 2,1 +η ǫ 2,2, 249 which can also be written as δ ǫ Φ 1, Φ 1,1 Q η = Q η ǫ 2, ǫ 2,1 ǫ 2,2 25 We can choose ǫ 2,1 appropriately such that the condition ξ Φ 1, = 251 on Φ 1, is satisfied Moreover, we can choose ǫ 2,2 appropriately such that the condition ξ Φ 1,1 = 252 on Φ 1,1 is satisfied Then Φ 1, satisfying 251 can be written as 11

14 Φ 1, = ξ Φ, 1, 253 where Φ, 1 is in the small Hilbert space We can then choose the condition b Φ, 1 =, 254 to fix the remaining gauge symmetry This gauge-fixing condition can be stated for the original field Φ 1, as b Φ 1, = when ξ Φ 1, = is imposed The condition b Φ 1, = can be satisfied by appropriately choosing ǫ 2,, and it is compatible with ξ Φ 1, = and ξ Φ 1,1 = by adjusting ǫ 2,1 and ǫ 2,2 To summarize, we can choose as the gauge-fixing conditions on Φ 1, and Φ 1,1 b Φ 1, =, ξ Φ 1, =, ξ Φ 1,1 =, 255 Each of Φ 1, and Φ 1,1 can be decomposed into four subsectors as before so that we have eight subsectors in total It is straightforward to see that the conditions 255 eliminate five of the eight subsectors and three subsectors remain, which match with the three remaining subsectors of Φ 2, 1 after imposing the constraint b ξ Φ 2, 1 = We can thus invert the kinetic term S 1 to obtain the propagator, as we explicitly do in the next section It is also straightforward to see that the five conditions can be implemented by the Lagrange multiplier fields N 3, 1 and N 3, 2 as S GF = N 3, 1 Φ 1, + N 3, 2 Φ 1,1, 256 with the constraints b ξ N 3, 1 =, ξ N 3, 2 = 257 This can be verified by decomposing each of N 3, 1 and N 3, 2 into four subsectors The corresponding Faddeev-Popov term is then given by S FP = Φ 3, 1 Q Φ 2, +η Φ 2,1 + Φ 3, 2 Q Φ 2,1 +η Φ 2,2 258 with b ξ Φ 3, 1 =, ξ Φ 3, 2 = 259 After integrating out N 3, 1 and N 3, 2, the total action we obtain is S +S 1 +S 2 = 1 2 Φ, Qη Φ, + Φ 2, 1 with + Φ 3, 1 Q Φ 2, +η Φ 2,1 Q Φ 1, +η Φ 1,1 + Φ 3, 2 Q Φ 2,1 +η Φ 2,2 26 b Φ, =, ξ Φ, =, b ξ Φ 2, 1 =, b Φ 1, =, ξ Φ 1, =, ξ Φ 1,1 =, b ξ Φ 3, 1 =, ξ Φ 3, 2 =

15 The action S 2 can be written in the following form: S 2 = Φ 3, 1 Φ 3, 2 Q η Q η Φ 2, Φ 2,1 Φ 2,2 262 The action S +S 1 +S 2 is invariant under the following gauge transformations: ǫ Φ 2, Q η 3, δ ǫ Φ 2,1 = Q η ǫ 3,1 ǫ Φ 2,2 Q η 3,2 263 ǫ 3,3 It is straightforward to show that we can impose the conditions b Φ 2, = ξ Φ 2, =, ξ Φ 2,1 =, ξ Φ 2,2 = 264 for gauge fixing In this way the gauge-fixing procedure continues, and at the end we obtain where S n for n 1 is with S = S n = Φn+1, 1 Φ n+1, 2 Φ n+1, n S n, 265 n= Q η Q η Q Q η b Φ n, = ξ Φ n, =, b ξ Φ n+1, 1 =, ξ Φ n,1 Φ n,2 Φ n,n =, ξ Φ n+1, 2 Φ n+1, 3 Φ n+1, n Φ n, Φ n,1 Φ n,n = 23 Comparison with Witten s superstring field theory in Siegel gauge We have seen that string fields of various ghost and picture numbers appear in the process of gauge fixing, and we imposed various conditions on these string fields While those features may look exotic, we will demonstrate that the gauge-fixed action of the free superstring field theory in the Berkovits formulation derived in the preceding subsection describes the conventional physics by showing that it reduces to the gauge-fixed action of the free superstring field theory in the Witten formulation using Siegel gauge after eliminating auxiliary fields 13

16 The gauge-invariant action of Witten s superstring field theory is given by S = 1 2 Ψ 1, 1 Q Ψ 1, 1, 268 where A B is the BPZ inner product of A and B in the small Hilbert space, which is related to A B in the large Hilbert space as A B = 1 A A ξ B up to an overall sign depending on a convention Here 1 A = 1 when A is Grassmann even and 1 A = 1 when A is Grassmann odd Gauge fixing in Siegel gauge is completely parallel to that in the bosonic string, and the gauge-fixed action is given by S = S n, 269 n= where S = 1 2 Ψ 1, 1 Q Ψ 1, 1, Sn = Ψ n+1, 1 Q Ψ n+1, 1 for n 1 27 with b Ψ n, 1 =, n 271 As in the bosonic case, the action S can also be written compactly as S = 1 2 Ψ Q Ψ with Ψ = Ψ n, 1, b Ψ = 272 n= Since Ψ is annihilated by b, we need c from Q for the inner product to be nonvanishing Using {Q,b } = L, we see that the gauge-fixed action reduces to S = 1 2 Ψ c L Ψ 273 Similarly, S n reduces to S = 1 2 Ψ 1, 1 c L Ψ 1, 1, Sn = Ψ n+1, 1 c L Ψ n+1, 1 for n We will compare this with the gauge-fixed action derived in the preceding subsection Let us start with S Under the gauge-fixing conditions 234 the string field Φ, reduces to Then the action S reduces to Φ, = ξ Φ ξ, 275 S = 1 2 Φ ξ, ξ Qη ξ Φ ξ, = 1 2 Φ ξ, ξ Q Φ ξ, = 1 2 Φ ξ, ξ c L Φ ξ, 276 This coincides with S = 1 2 Ψ 1, 1 c L Ψ 1,

17 in Witten s theory under the identification Ψ 1, 1 = Φ ξ, 278 Let us next consider S 1 Under the gauge-fixing conditions 255, Φ 1, and Φ 1,1 reduce to Φ 1, = ξ Φ ξ 1,, Φ 1,1 = ξ Φ ξ 1,1 +c ξ Φ cξ 1,1 279 Then the action S 1 reduces to S 1 = Φ 2, 1 Qξ Φ ξ 1, + Φ 2, 1 η ξ Φ ξ 1,1 + Φ 2, 1 η c ξ Φ cξ 1,1 = Φ 2, 1 Qξ Φ ξ 1, + Φ 2, 1 Φ ξ 1,1 Φ 2, 1 c Φ cξ 1,1 28 The antighost field Φ 2, 1 with the constraint b ξ Φ 2, 1 = can be decomposed as Φ 2, 1 = Φ 2, 1 +ξ Φ ξ 2, 1 +c ξ Φ cξ 2, 1, 281 and the last two terms on the right-hand side of 28 reduce to Φ 2, 1 Φ ξ 1,1 = Φcξ 2, 1 c ξ Φ ξ 1,1, Φ 2, 1 c Φ cξ 1,1 = Φ ξ 2, 1 c ξ Φ cξ 1,1 282 Since Φ ξ 1,1 imposing and Φcξ 1,1 only appear in these terms, these fields act as Lagrange multiplier fields Φ cξ 2, 1 =, After integrating out Φ ξ 1,1 and Φcξ 1,1, the action S 1 therefore reduces to Φ ξ 2, 1 = 283 S 1 = Φ 2, 1 Qξ Φ ξ 1, = Φ 2, 1 c L ξ Φ ξ 1, 284 This coincides with S 1 = Ψ 2, 1 c L Ψ, in Witten s theory under the identification Ψ 2, 1 = Φ 2, 1, Ψ, 1 = Φ ξ 1, 286 We can similarly show that S n with n 1 reduces to S n = Φ n+1, 1 Qξ Φ ξ n, = Φ n+1, 1 c L ξ Φ ξ n, 287 and coincides with S n = Ψ n+1, 1 c L Ψ n+1, in Witten s theory under the identification Ψ n+1, 1 = Φ n+1, 1, Ψ n+1, 1 = Φ ξ n, for n

18 We have thus shown that the gauge-fixed action derived in the preceding subsection coincides with that of Witten s superstring field theory in Siegel gauge after integrating out auxiliary fields While the kinetic term of Witten s superstring field theory is consistent, there are problems in the construction of the cubic interaction term using the picture-changing operator On the other hand, interaction terms can be constructed without using picture-changing operators in the Berkovits formulation We have confirmed that both theories describe the same physics in the free case, and we expect a regular extension to the interacting theory in the Berkovits formulation 24 Various gauge-fixing conditions In subsection 22, we have seen that the completely gauge-fixed action in the WZW-type open superstring field theory is given by the sum 265 of the original action S and all the Faddeev-Popov terms with the gauge-fixing conditions 267 As we will see later in section 4, the action 265 is precisely the solution to the classical master equation in the BV formalism if we identify antighosts with antifields 4 From this point of view S is a universal quantity and different gauge-fixed actions can be obtained simply by imposing different conditions on Φ s In this subsection, we list some gauge-fixing conditions different from 267 For further generalization and for the validity of the gauge-fixing conditions, see [24] Let us first mention a one-parameter extension of 267: b Φ n, = n, ξ Φ n,m +αb Φ n,m+1 = m n 1, ξ Φ n,n = n, b ξ Φ n+1, 1 = n 1, αb Φ n+1, m +ξ Φ n+1, m+1 = 1 m n 1 29 The previous condition corresponds to the case in which the parameter α is zero Unlike 267, the above set of equations includes linear combinations of Φ s Another interesting class of gauge-fixing conditions is obtained when we use the zero mode d of the operator d = [Q,bξ], instead of ξ : b Φ n, = n, d Φ n,m +αb Φ n,m+1 = m n 1, d Φ n,n = n, b d Φ n+1, 1 = n 1, αb Φ n+1, m +d Φ n+1, m+1 = 1 m n In the language of the BV formalism we have chosen a gauge-fixing fermion such that antifields of minimal-sector fields are identified with antighosts In this paper we consider only such gauge-fixing conditions, and thus we will not distinguish antifields and antighosts 16

19 The operator d is identical to the generator G of the twisted N = 2 superconformal algebra investigated by Berkovits and Vafa [19] Because d is a counterpart of b, it seems natural to adopt the symmetric gauge, in which α = 1 In the next section we will calculate propagators, mainly considering the gauge 29 with α =, which is identical to 267, and the gauge 291 with α = 1 3 Calculation of propagators Let us derive propagators under the gauge-fixing conditions proposed in the preceding section For this purpose, we introduce source terms of the form S J = Φ, J 2, 1, 31a S J n = n n Φ n,m J n+2, m 1 + Φn+1, m J n+1,m 1 m= Φ n, = Φ n,n m=1 J n+2, 1 J n+2, n+1 Φn+1, 1 + Φ n+1, n J n 1, J n 1,n 1 n 1, 31b and consider the action S n [J] = S n +S J n n 32 Here J s of positive non-positive ghost number are Grassmann-even Grassmann-odd sources Each source is coupled with a Φ of the same Grassmann parity Note that Φ s are subject to their gaugefixing conditions, but sources are free from any constraints The actions S and S n, for n 1, were defined in 227 and 266, respectively Starting from the action 32, we can calculate propagators as follows First we solve the equations of motion of the Φ s derived from S n [J] in order to find a stationary point Then we put the solution back into S n [J], to obtain a quadratic form of J s, from which propagators can be read off 31 Propagators for gauge fixing with b and ξ Let us first apply the above-mentioned procedure to the gauge 267 To calculate the propagator of Φ, we start from the action S [J] = 1 Φ, Qη Φ 2, + Φ, J 2,

20 Gauge-fixing conditions for the field Φ, are of the form b Φ, = ξ Φ, =, 34 which leads to Φ, = {b,c }{ξ,η }Φ, = b c ξ η Φ, 35 Thus, the equation of motion derived from S [J] is η ξ c b Qη Φ, J 2, 1 = 36 This can be solved easily Using the identity we find that the solution is given by Qη ξ b L = 1 b Q L ξ η + b Q L ξ η, 37 Φ, = ξ b L J 2, 1 38 Note that this solution is consistent with the conditions 34 Evaluating the action 33 for this solution determines the propagator of Φ, : S [J] = 1 2 J 2, 1 ξ b L J2, 1 39 Next, let us consider ghost propagators The action S 1 [J] takes the form S 1 [J] = Φ 2, 1 Q Φ 1, +η Φ 1,1 The gauge-fixing conditions at this step are + Φ 2, 1 J, + Φ 1, J 3, 1 + Φ 1,1 J 3, 2 b Φ 1, = ξ Φ 1, =, 31 ξ Φ 1,1 =, b ξ Φ 2, 1 = 311 Under these conditions, we have Φ 1, = b c ξ η Φ 1,, Φ 1,1 = ξ η Φ 1,1, Φ 2, 1 = {b,c }{ξ,η }Φ 2, 1 = b c +c b ξ η Φ 2, Therefore, the equations of motion are c b +b c η ξ QΦ 1, +η Φ 1,1 +J, =, η ξ c b QΦ2, 1 +J 3, 1 =, η ξ η Φ 2, 1 +J 3, 2 =

21 Let us find a solution compatible with the conditions 311 This can be readily achieved by the use of the zero mode decomposition 235 of Φ s The solution is given by Φ 1, = b ξ η J L,, Φ 1,1 = ξ + b ξ η X J L,, Φ 2, 1 = b L η ξ J 3, 1 + ξ + b L η ξ X J 3, 2, 314 where X is the zero mode of the picture-changing operator X = {Q,ξ} When the equations for Φ 1, and Φ 1,1 hold, the action S 1 [J] reduces to S 1 [J] = Φ 2, 1 J, = J, Φ 2, Substituting the solution 314 into 315, we immediately obtain 1 J3, 1 S 1 [J] = J, b η ξ + J L, 1 1 J3, 2 b η X ξ 316 L On the other hand, when the equation of motion of Φ 2, 1 holds, the action becomes S 1 [J] = Φ 1, J 3, 1 + Φ 1,1 J 3, Needless to say, plugging the solution 314 into 317 gives the same result as in 316 The above expression can be rewritten by using a one-by-two propagator matrix: J S 1 [J] = J, A B 3, 1, 318 J 3, 2 with A 1 L b η ξ, B 1 1 L b η X ξ 319 We emphasize that the propagator includes the zero mode of the picture-changing operator We can continue the calculation in this manner The action S 2 [J] takes the form S 2 [J] = Φ 3, 1 Q Φ 2, +η Φ 2,1 + Φ 3, 1 J 1, + Φ 3, 2 Q Φ 2,1 +η Φ 2,2 + Φ 3, 2 J 1, Φ 2, J 4, 1 + Φ 2,1 J 4, 2 + Φ 2,2 J 4, 3, and the gauge-fixing conditions are given by b Φ 2, = ξ Φ 2, =, ξ Φ 2,1 =, ξ Φ 2,2 =, 321 b ξ Φ 3, 1 =, ξ Φ 3, 2 = 19

22 When the equations of motion for Φ 2,, Φ 2,1, and Φ 2,2 are satisfied, the action S 2 [J] reduces to S 2 [J] = Φ 3, 1 J 1, + Φ 3, 2 J 1,1 = J 1, Φ 3, 1 J 1,1 Φ 3, Substituting into 322 the solution of the equations of motion Φ 3, 1 = AJ 4, 1 +BJ 4, 2 + X BJ 4, 3, Φ 3, 2 = ξ J 4, 3, 323 we obtain S 2 [J] = J 1, J 1,1 A B X B 4, 1 J J ξ 4, J 4, 3 At the next step, the propagator matrix is given by A B X B X 2 B ξ X ξ, 325 ξ and at the n-th step we obtain the n n+1 matrix A B X B X 2 B X n 1 B ξ X ξ X n 2 ξ ξ X n 3 ξ ξ 326 The propagators in the gauge 29 with α can be calculated in the same manner The result, however, is a little complicated To see this, we calculate the first-step ghost propagator Note that since the condition on Φ, does not include α, the propagator of Φ, is independent of the parameter We start with the gauge-fixing conditions below: b Φ 1, =, ξ Φ 1, +αb Φ 1,1 =, ξ Φ 1,1 =, b ξ Φ 2, 1 = This time, the solution to the equations of motion derived from 31 is Φ 1, = Φ 1,1 = Φ 2, 1 = α 1+αL b η ξ + b L ξ η J,, ξ + b α ξ Qb η ξ J 1+αL,, ξ η X + L α b ξ η + b η ξ J 1+αL L 3, 1 + ξ + b η ξ X + L α ξ η b Qξ J 1+αL 3,

23 The action evaluated for the sources is given by S 1 [J] = J, Aα J B 3, 1 α, 329 J 3, 2 with A α α b ξ η + b η ξ, B α 1 b α η X ξ η b Q ξ 33 1+αL L L 1+αL When α =, the expression 329 indeed reduces to the form Propagators for gauge fixing with b and d Thus far we have calculated propagators in the gauge 29, focusing on the α = case These propagators include the zero mode of the picture-changing operator, which originates from the anticommutation relation {Q,ξ } = X 331 If instead of ξ we use an operator whose anticommutator with Q vanishes, we expect that propagators are dramatically simplified This is indeed the case: the operator d, the zero mode of d = [Q,bξ], provides us with simpler propagators It satisfies the following algebraic relations: d 2 = {b,d } =, {Q,d } =, {η,d } = L 332 In this subsection we investigate propagators in the gauge 291, concentrating on the symmetric case α = 1 First we consider Φ,, whose gauge-fixing conditions are b Φ, = d Φ, = 333 InadditiontothesourceJ 2, 1, weintroducethelagrangemultipliersλ 3, 1 andλ 3, 2, andconsider the action S [J]+S λ with The equation of motion is S λ = Φ, b λ 3, 1 + Φ, d λ 3, Qη Φ, +J 2, 1 +b λ 3, 1 +d λ 3, 2 = 335 supplemented by the gauge-fixing conditions 333 We claim that This follows quickly from the identity Φ, = b L d L J 2, Qη b L d L = 1+ d L η + b L Q+ b L d L Qη,

24 acting on J 2, 1 : Qη Φ, = J 2, 1 + d L η J 2, 1 + b L QJ 2, 1 + b L d L Qη J 2, Note that all terms on the right-hand side, except for the first, simply determine the values of the Lagrange multipliers in 335 Such values are not needed in the evaluation of the action since the solution satisfies the gauge-fixing conditions Evaluating the action for this solution gives S [J] = 1 2 For the next step we have the gauge-fixing conditions J2, 1 d L b L J2, b d Φ 2, 1 =, b Φ 1, =, d Φ 1,1 =, d Φ 1, +b Φ 1,1 = 34 We implement the first and last gauge conditions with Lagrange multipliers The relevant action is then S 1 [J]+S1 λ with S1 λ = λ 4, 2 d Φ 1, +b Φ 1,1 + Φ b 2, 1 d λ2, Note that both J, and λ 4, 2 are Grassmann odd The gauge-fixed equations of motion are recall that d and b are BPZ even, while η and Q are BPZ odd QΦ 1, +η Φ 1,1 +J, +b d λ 2, 1 =, c b QΦ 2, 1 +J 3, 1 +d λ 4, 2 =, f d η Φ 2, 1 +J 3, 2 +b λ 4, 2 =, 342 where f is an operator satisfying {d,f } = 1 5 In the last equation one may view J 3, 2 +b λ 4, 2 as a source and solve the equation by writing Φ 2, 1 = d L J 3, 2 d L b λ 4, 2 b L η d L J 3, 1, 343 where the last term has been included with view of the second equation and does not disturb the third due to the η factor it includes Substitution into the second equation with some simplification yields d c b QJ L 3, 2 + d η J L 3, 1 +2d λ 4, 2 = 344 The equation works out if the Lagrange multiplier is given by 5 For a concrete expression of f, see appendix A of [24] λ 4, 2 = 1 2L QJ3, 2 +η J 3,

25 Inserting 345 back in 343, we now have the solution for Φ 2, 1 We find Φ 2, 1 = b J L 3, 1 d J L 3, A small rearrangement yields Φ 2, 1 = 1 2 d b b QJ L L 3, d η J 2L L 3, b + b d η J L L L 3, 1 1 d + d Q b J 2 L L L 3, When the equations for Φ 1, and Φ 1,1 and the gauge-fixing conditions hold, the action is given by Its evaluation immediately gives S 1 [J] = J, 1 2 S 1 [J] = Φ 2, 1 J, = J, Φ2, b + b d J3, 1 η + J, 1 L L L 2 This answer can be rewritten by using a one-by-two propagator matrix: S 1 [J] = J 1 b, 2 L + b d L η L d L + d L Q b L J3, d 2 L + d L Q b J 3, 1 L 35 J 3, 2 When the equation of motion of Φ 2, 1 and the gauge-fixing conditions hold, the action reduces to S 1 [J] = Φ 1, J3, 1 + Φ 1,1 J3, We can thus read the values of the fields, as bras After BPZ conjugation we obtain Φ 1, = 1 b + d b η J 2 L L L,, Φ 1,1 = 1 d + b Q d J 2 L L L, 352 In the next step we have to deal with three fields and two antifields We have the gauge-fixing conditions b Φ 3, 1 +d Φ 3, 2 =, b Φ 2, =, d Φ 2, +b Φ 2,1 =, d Φ 2,1 +b Φ 2,2 =, d Φ 2,2 = The relevant action is S 2 [J]+S λ 2 with S λ 2 = λ 5, 2 d Φ 2, +b Φ 2,1 λ 5, 3 d Φ 2,1 +b Φ 2,2 λ, d Φ3, 2 +b Φ3,

26 The equations of motion obtained by varying the fields, c b QΦ 3, 1 +d λ 5, 2 +J 4, 1 =, QΦ 3, 2 +η Φ 3, 1 +b λ 5, 2 +d λ 5, 3 +J 4, 2 =, f d η Φ 3, 2 +b λ 5, 3 +J 4, 3 =, 355 are simpler to solve than those obtained by varying the antifields By solving the equations for the antifields we can determine the Lagrange multipliers: The antifields are then given by λ 5, 2 = 1 2L QJ4, 2 +η J 4, 1, λ 5, 3 = 1 2L QJ4, 3 +η J 4, Φ 3, 1 = 1 2 b + b d η J L L L 4, d L J 4, 2, Φ 3, 2 = 1 J 2L 4, b d L + d L Q b L J 4, Thus the action takes the form S 2 [J] = 1 b J 1, J 1,1 2 L + b d L η 1 d L 2 L 1 b 1 2 L 2 d L + d L Q b L J 4, 1 J 4, J 4, 3 The full pattern is now clear The full action S[J] written in terms of propagators and bilinear in sources takes the form S[J] = 1 J2, 1 d b J 2 L L 2, 1 + S n+1 [J], 359 where S n+1 [J] is the term coupling the sources of the n+1 antifields at ghost number n+2, to the sources of the n+2 fields at ghost number n+1: S n+1 [J] = n= J n, J n,1 J n,n P n+1,n+2 J n+3, 1 J n+3, 2 J n+3, n+2 36 Here the propagator matrix P n+1,n+2 has n + 1 rows and n + 2 columns Its general form is the 24

27 extension of our results in 35 and 358: P n+1,n+2 = 1 2 b L + b d L η d L L b L d L b L d L b L d L b L d L + d L Q b L n 361 We can readily obtain the result with a general α 1 as well The propagator matrices are given by P 1,2 = P b P d, P2,3 = P b d α+1l αb α+1l P d, 362a d P b α+1l αb d α+1l α+1l αb α+1l P n+1,n+2 = d α+1l αb d α+1l α+1l αb α+1l P d n, 362b with P b = α+ η d b, P d = L α+1l 1+α Qb L d α+1l 363 Note that the case in which α = 1 is exceptional: the propagators diverge, which means that gauge fixing is not complete See [24] for details When α =, the above propagators correspond to those obtained from 326 by the replacement ξ d L, X Verifying the master equation for the free action Dynamical fields in string field theory are component fields The BV formalism is defined in terms of these component fields, but it is convenient to recast it in terms of string fields In this section we 25

28 present the BV formalism of open superstring field theory in terms of string fields We then show that the free action 122 satisfies the master equation The classical master equation is given by {S,S} =, 41 and the antibracket is defined by {A,B} = k R A φ k L B φ k RA L B φ, 42 k φ k where φ k forms a complete basis of fields and φ k are the associated antifields The Grassmann parity of a field can be arbitrary but the corresponding antifield has the opposite parity In our case the fields are the component fields of Φ and the antifields are the component fields of Φ +, with Φ ± defined in 121 With a slight abuse of language we will call Φ the string field and Φ + the string antifield Let us expand Φ and Φ + in terms of their component fields f and a with indices g, p, and r as follows: String field even String antifield odd Φ = g,p Φ + = g,p + r f r g,pφ r g,p, a r g,p Φr g,p r 43a 43b We took f and a from the initials of fields and antifields For each pair g,p of the world-sheet ghost number g and picture number p, we chose a complete basis of states Φ r g,p labelled by r such that 6 Φ r g,p Φr g,p = δ g+g,2δ p+p, 1δ r,r g 44 Since the Grassmann parity of Φ r g,p is 1g, we have Φ r g,p Φ r g,p = 1g δ g+g,2δ p+p, 1δ r,r g 2, 45 which follows from A B = 1 AB B A, 46 with 1 gg = 1 g2 g = 1 g2 = 1 g Here and in what follows a string field in the exponent of 1 representsitsgrassmannparity: itismod2foragrassmann-evenstringfieldand1mod2for a Grassmann-odd string field While the states Φ r g,p carry ghost and picture numbers, the component fields fg,p r and a r g,p do no carry these numbers and their subscripts g and p refer to the states that multiply them We also introduced the lattice defined by the collection of pairs g,p that appear in Φ and the lattice + defined by the collection of pairs g,p that appear in Φ + 6 We do not need to consider states with g = 1, since they do not appear in the expansion 43 26

29 As we mentioned in the introduction, fg,p r and a r 2 g, 1 p should be paired in the BV formalism: Field-antifield pairing: f r g,p a r 2 g, 1 p 47 The Grassmann parity of f r g,p is 1 g and that of a r g,p is 1 g The Grassmann parity of a r 2 g, 1 p is indeed opposite to that of f r g,p, which is paired with a r 2 g, 1 p It then follows that Φ is Grassmann even and Φ + is Grassmann odd Note that a r g,p and Φr g,p in Φ + commute, while f r g,p Φr g,p = 1 g Φ r g,p fr g,p in Φ The antibracket 42 is thus defined by {A,B} = g,p R A r f r g,p L B a r 2 g, 1 p R A a r 2 g, 1 p L B fg,p r 48 Our goal is to rewrite this antibracket 48 directly in terms of string fields and string antifields In previous sections we used the notation AB or A B for the BPZ inner product of string fields A and B When more than two string fields are involved, it is convenient to introduce the integration symbol as follows: 7 A B = AB = A B 49 The relation 46 is generalized to A 1 A 2 A n = 1 A 1A 2 ++A n A 2 A n A 1 41 The BPZ inner products of states in the basis 44 and 45 are translated into Φ r g,p Φ r g,p = δ g+g,2δ p+p, 1δ r,r g, 411a Φ r g,p Φ r g,p = 1g δ g+g,2δ p+p, 1δ r,r g 2 411b We are interested in evaluating {A,B} where A and B depend on fields and antifields only through Φ ± Let us first consider {Φ,Φ + } This takes value in a tensor product of two Hilbert spaces of the string field We therefore introduce a space number label and write it as {Φ 1 } We see that only the first term on the right-hand side of 48 contributes and find,φ2 + {Φ 1,Φ2 + } = g,p r R Φ 1 f r g,p L Φ 2 + a r 2 g, 1 p = g,p 1 g Φ r1 g,p Φr2 2 g, 1 p, 412 r where the expansion 43 was used and the sign factor 1 g came from the right derivative that must go through the state Φ r g,p to get to the component field fr g,p An important property of {Φ1,Φ2 + } is that it acts as the projector P + to the subspace defined by the lattice + in the following sense: X 1 1 {Φ 1,Φ2 + } = P + X 2, We use the symbol to denote the star product in this section 1 27

30 where the subscripts attached to the integration symbol and the star product represent the space number label To see this, insert 412 into the left-hand side of 413 X g Φ r1 g,p Φ r2 2 g, 1 p = 1 g X 1 1 Φ r1 r2 g,p Φ 2 g, 1 p, g,p r g,p r and expand X 1 in a complete basis of ghost and picture numbers, 1 X 1 = g,p = X r 1 g,p Φr g,p 415 r In 414 this expression is contracted with states in the subspace defined by Therefore, in 415 only states in the subspace defined by + give nonvanishing contributions and the right-hand side of 414 becomes Xg r,p g,p + r g,p r Using the second equation in 411, we find Xg r,p g,p + r g,p r 1 g r Φ 1 g,p 1 Φ r1 r2 g,p Φ 2 g, 1 p g 1 g δ r,r δ g+g,2δ p+p, 1Φ r2 2 g, 1 p = 1 g,p + X r 2 g,p Φr g,p r 417 The right-hand side is the string field X 1 copied into the state space 2, with a projection to the subspace defined by + We have thus shown the relation 413 Similarly, one can prove {Φ 1,Φ2 2 + } 2 X 2 = P X 1, 418 where P is the projector to the subspace defined by We can also evaluate {Φ 1 {Φ 1 +,Φ2 } = 1 g,p X 1 1 {Φ 1 +,Φ2 } = P X 2, r +,Φ2 } to obtain Φ r1 2 g, 1 p Φr2 g,p, 419 {Φ 1 +,Φ2 2 } 2 X 2 = P + X 1 42 Let us next consider {A,Φ + } where A is given by an integral of a product of Φ + and Φ It is useful to define variational derivatives δ RA δφ and δ RA δφ + by δ R A δa = δφ + δ RA δφ δφ δφ + We also define δ LA δφ and δ LA δφ + by δa = δφ δ LA +δφ + δ LA, 422 δφ δφ + 28

31 which are related to δ RA δφ and δ RA δφ + as follows: δ R A = 1 Aδ LA, δφ + δφ + δ R A = δ LA δφ δφ 423a 423b It is important to note that δ RA δφ and δ LA δφ are string fields in the subspace defined by + since they are contracted with the variation δφ in the subspace defined by Similarly, δ RA δφ + and δ LA δφ + are string fields in the subspace defined by since they are contracted with the variation δφ + in the subspace defined by + These can be expressed as follows: P + δ R A δφ = δ RA δφ, P + δ L A δφ = δ LA δφ, P δ R A δφ + = δ RA δφ +, P δ L A δφ + = δ LA δφ We can now write We then obtain R A f r g,p {A,Φ 2 + } = = = g,p δr A δφ RΦ f r g,p r R A L Φ 2 + fg,p r a r 2 g, 1 p P + δ R A δφ 2 = δr A δφ 2, = 1 g δr A δφ Φ r g,p 425 = 1 δr A 1 1 {Φ 1 δφ,φ2 + } 426 where we used 412, 413, and 424 Deleting the space number label, we can write the relation as follows: Similarly, one can derive {A,Φ + } = δ RA δφ 427 as well as the relations R A a r g,p L A f r g,p L A a r g,p δr A = RΦ + δr A δφ + a r = Φ r g,p g,p δφ, + L Φ = fg,p r δ LA = Φ r g,p δφ δ LA, δφ L Φ + = δ LA = Φ r g,p δ LA, δφ + δφ + a r g,p {A,Φ } = δ RA δφ +, {Φ +,A} = δ LA δφ, {Φ,A} = δ LA δφ a 429b 429c 29

32 We can see, using 423, that these relations are consistent with the familiar property of the antibracket: {A,B} = 1 A+B+AB {B,A} 43 Finally, let us consider {A,B} where both A and B are integrals of products made of Φ + and Φ In this case, we begin with 48 and use 425, 428, 412, and 419 to show that δr A 1 {A,B} = 1 {Φ 1 δφ,φ2 + } δl B 2 δr A {Φ 1 + δφ + δφ,φ2 } δl B δφ Using 413 or 418, 42, and 424, we obtain δ R A {A,B} = δ LB δ RA δ LB 432 δφ δφ + δφ + δφ This is the final expression of the antibracket The expression 48 in terms of component fields has now been written in terms of the string field and the string antifield Since Let us evaluate the antibracket {S,S} for the free action 122: S = 1 2 Φ Qη Φ +Φ + Q+η Φ 433 we find δ R S = P + Qη Φ +Q+η Φ +, δφ 1 2 {S,S} = δr S δ LS = δφ δφ + δ L S = P Q+η Φ, 434 δφ + P + Qη Φ +Q+η Φ + Q+η Φ 435 Here we dropped the projector P because it is automatically enforced by the other projector P + through the BPZ contraction Since the only string field in Φ such that Qη Φ is in the subspace defined by + is Φ, and the action of Q or η takes string fields in the subspace defined by + to string fields in the subspace defined by +, we have 1 2 {S,S} = P + = Qη Φ +Q+η Φ + Q+η Φ Qη Φ, +Q+η Φ + Q+η Φ 436 Using 12, we conclude that the antibracket {S, S} vanishes for the free action 433 3