Hidden gauge reducibility of superstring field theory and Batalin-Vilkovisky master action
aa r X i v : . [ h e p - t h ] F e b Hidden gauge reducibility of superstring field theoryand Batalin-Vilkovisky master action
Hiroaki Matsunaga ∗ Institute of Physics of the Czech Academy of Sciences,Na Slovance 2, Prague 8, Czech Republic
Abstract
In this paper, we show that there exists a hidden gauge reducibility in superstring fieldtheory based on the small dynamical string field Ψ ∈ H βγ whose gauge variation is alsosmall δ Ψ ∈ H βγ . It requires additional ghost–antighost fields in the gauge fixed or quantumgauge theory, and thus changes the Batalin-Vilkovisky master action, which implies thatadditional propagating degrees of freedom appear in the loop superstring amplitudes via thegauge choice of the field theory. We present that the resultant master action can take adifferent and enlarged form, and that there exist canonical transformations getting it backto the canonical form. On the basis of the Batalin-Vilkovisky formalism, we obtain severalexact results and clarify this underlying gauge structure of superstring field theory. Contents
A On the string fields–antifields 31
A.1 String antibracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.2 Constraints on BV spectrums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ∗ Email: [email protected]
Introduction
Gauge theory is a theory whose dynamical variables are redundant, in which we should takegauge degrees of freedom into account and clarify its gauge reducibility [1–4]. Iff all gaugetransformations are independent, it is called irreducible, which is the simplest gauge theory. Inother cases, reducible gauge theory, a kind of the ( g + 1)-st gauge invariance for the g -th gaugeinvariance arises and there exists a hierarchy of gauge invariances. If g -th gauge transformationsare independent, then the theory is called g -th order reducible. Infinitely reducible
It is known that (super-)string field theory is an infinitely reducible gauge theory, and thusit necessitates a set of infinite number of ghost–antighost fields for the quantization. Theseghost–antighost string fields never appear in the action without gauge fixing, but propagatein the loop amplitudes. One can find what kind of and how many ghosts are necessitated bystudying the gauge reducibility of the kinetic term, namely, free theory. In many cases, thekinetic operator of (super-)string field theory is the BRST operator Q of the world-sheet theory,which gives the on-shell condition Q Ψ = 0 of a string field Ψ. Since Q is nilpotent, it has thegauge invariance under δ Ψ =
Q λ . However, clearly, there exists the gauge transformation forthe gauge transformation δ λ = Q λ − , where we write λ − g for the g -th gauge parameter field.Likewise, we find the g -th gauge variation δ g λ − g = Qλ − g preserving ( g − Q Ψ = 0 δ Ψ =
Q λ δ g λ − g = Q λ − g ( g > { λ − g } g ≥ = { λ , λ − , ..., λ − g , ... } appears in the analysis ofthe gauge reducibility, the theory needs corresponding ghost fields { Ψ − g } g ≥ . The pair of thestring field and ghost fields { Ψ , Ψ − g } g ≥ requires its antighost fields part { Ψ ∗ , (Ψ − g ) ∗ } g ≥ . Thesefields appear in gauge fixed theory and quantum calculations. Hence, in the Batalin-Vilkoviskyformalism [1, 2], the minimal set of fields–antifields is given by { Ψ , Ψ ∗ , Ψ − g , (Ψ − g ) ∗ } g> :Fields Ψ , { Ψ − g } g> Antifields (Ψ ) ∗ ≡ Ψ ∗ , { (Ψ − g ) ∗ } g> One promised way to achieve gauge fixed or quantum gauge theory is to construct a Batalin-Vilkovisky master action S bv based on the minimal set of fields–antifields. Usually, it is a toughwork to find S bv for given gauge theory: In many cases, S bv will need nontrivial ghost–antighostterms in addition to the original action S and be given by a highly complicated form. However,in (super-)string field theory, we often encounter an interesting situation: The master action S bv takes the same form as the original action S except that it includes all fields–antifields. Ingeneral, an action S [Ψ] for interacting (super-)string field theory takes the following form S [Ψ] = kinetic term z }| { K (Ψ , Q Ψ) + X n ≥ n -vertex z }| { V n (Ψ , ..., Ψ) + X g X n ≥ g -loop correction z }| { V g,n (Ψ , ..., Ψ | {z } n ) . (1.1)We know that in several types of (super-)string field theory, such as [5–14], one can obtain itsclassical (and quantum) Batalin-Vilkovisky master action S bv without changing the form of the It is equivalent to clarify the existence condition of the propagator in given gauge theory. S bv = S [ ψ ] where ψ ≡ Ψ + P g< Ψ g + P g ≤ (Ψ g ) ∗ . Free theory gives amore trivial example. Let us consider the kinetic term K = K (Ψ , Q Ψ) of [9–15], K (cid:0) Ψ , Q Ψ (cid:1) = 12 (cid:10)(cid:10) Ψ , Q Ψ (cid:11)(cid:11) . Since this is a free theory, its master action K bv is obtained by adding Lagrange-multiplier-likeghost–antighost terms fixing the above reducibility. Thus K bv takes the following form K bv = 12 (cid:10)(cid:10) Ψ , Q Ψ (cid:11)(cid:11) + X g> (cid:10)(cid:10) (Ψ g ) ∗ , Q Ψ − g (cid:11)(cid:11) = K (cid:0) ψ, Q ψ (cid:1) . Why can we obtain the master action S bv [ ψ ] by just relaxing the ghost number constraint ofthe original action S [Ψ]? This naive question is the first motivation of this paper. While onemay think that it is provided by the nilpotency of Q in the case of free theory (or homotopyalgebraical relations satisfied by the set of vertices { V g,n } g,n for interacting theory), it seemsthat there is an additional reason related to its geometrical interpretation. Recall that the(super-)string BRST operator Q works as the exterior derivative on the moduli space M g,n of(super-)Riemann surfaces Σ g,n with g -genus and n -punctures. For example, see [16–19].
Geometrical restriction
It is well-known that the on-shell g -loop n -point amplitude of (super-)strings are described by theintegration over the (super-)moduli space M g,n . String field theory gives its off-shell extensionas a gauge theory, and there are several ways to construct gauge invariant actions reproducingthis on-shell property. However, as we know, a straightforward but powerful way of constructingfield theory is to regard it as one of the off-shell defining properties: Consider a Feynman graphdecomposition of the (super-)moduli space M g,n = V g,n ∪ R (1) g,n ∪ · · · ∪ R (3 g − n ) g,n , find n -foldmultilinear functions ω g,n of string fields which are pull back from the volume forms of M g,n ,and define all vertices V g,n of superstring field theory by V g,n ≡ Z V g,n ω g,n . Then, one can prove that Q and the resultant vertices { V g,n } g,n satisfy the (loop) A ∞ /L ∞ relations, which are key algebraic relations providing a simple Batalin-Vilkovisky procedure. Ifthese { Q, V g,n } g,n give a unique gauge generator of theory, we can obtain S bv = S [ ψ ].A string field Ψ should be correspond to a set of world-sheet vertex operators inserted intothe coordinate patch around each puncture of Σ g,n . In this set up, it would be simple and naturalto use a string field living on the small Hilbert space H βγ because this geometrical interpretationarises from the properties of bc - and βγ -ghost systems. Roughly, the gauge invariance of this There is auxiliary kinetic term K aux [Ψ , e Ψ] for [10, 11], and K bv takes the form of K bv = K [ ψ ] + K aux [ ψ, e ψ ]. Off course, for superstrings, NS and R punctures should be distinguished: n = ( n NS | n R ). Here, R ( I ) g,n denotes the region covered by all g -loop n -point graphs including I -propagators: V g,n ≡ R (0) g,n . By taking advantage of this old well-known fact, we can algebraically construct string tree vertices satisfying A ∞ /L ∞ relations without references to these geometrical aspects [12–14]. Interestingly, this purely algebraicprescription, the homotopy algebraic formulation, also reproduces the same on-shell tree amplitudes [20]. However,while these vertices satisfy the A ∞ /L ∞ relations by construction, it has remained unclear whether they give pull-backs of differential forms on {M g,n } g,n . Recently, this point was clarified for lower vertices [15]. {M g,n } g,n of the (super-)moduli spaces. Thus, if we want to keep this geometrical interpretation of the gauge invarianceoff-shell, it would be reasonable to restrict not only the string field Ψ but also its gauge variation δ Ψ onto the small Hilbert space H βγ . Using Ψ ∈ H βγ satisfying δ Ψ ∈ H βγ as a dynamical stringfield, we can obtain an off-shell gauge theory based on these.On the basis of the minimal set of fields–antifields ψ which also satisfies ψ ∈ H βγ , one canobtain the master action S bv for this type of gauge theory S [Ψ] by just relaxing ghost numberconstraint S bv = S [ ψ ], which would be a well-established fact. Unrestriction
However, there is an implicit but too strong assumption in the derivation of these master ac-tions: All fields–antifields are restricted onto the small Hilbert space H βγ . There exists a slightmismatch between the statements “the gauge variation is small δ Ψ ∈ H βγ ” and “the gaugeparameter field is small λ ∈ H βγ ”, which becomes significant for higher gauge parameters. Inthe case of free theory, we take the gauge variation δ Ψ =
Q λ . Apparently, to be δ Ψ ∈ H βγ , thegauge parameter λ must belong to H βγ except for Q -exact terms, and one can find that otherhigher gauge parameter fields λ − g do not have to be small. In other words, although we startfrom the small dynamical string field Ψ ∈ H βγ satisfying δ Ψ ∈ H βγ , the first gauge parameter λ can protrude from H βγ as long as BRST exact, λ ∈ H βγ ⊕ Q H ξηφ , and higher gauge parameters λ − g are no longer restricted, λ − g ∈ H ξηφ . We call this unrestricted state space H ξηφ as the largeHilbert space [21]. Hence, using the above minimal set of fields–antifields ψ satisfying ψ ∈ H βγ is too restrictive, which is indeed sufficient but not necessary. As a gauge theory, it correspondsto fix some higher gauge symmetry, which we see in the next section.What happens if we unrestrict these constraints on these gauge parameters? Clearly, theaction S [Ψ] and its (first) gauge invariance do not change. However, we can consider moreenlarged gauge transformations for the gauge transformations, the large gauge symmetry of the small theory, and find quite different gauge hierarchy: The gauge reducibility of superstringfield theory is drastically changed. As we will see, it necessitates additional ghost and antighostfields in the set of fields–antifields, which will be labeled by relaxed world-sheet picture numbers.As a result, we find that additional propagating degrees of freedom appear in loop amplitudes ofsuperstring field theory. Our second motivation is to clarify these.Since the set of fields–antifields is enlarged, the resultant Batalin-Vilkovisky master action S bv can take some different forms. In particular, one cannot obtain unique S bv by just relaxingthe ghost number constraint unlike well-established cases. It would be a natural consequencefrom relaxing geometry-inspired constraints on the fields–antifields. However, at the same time,it would imply the existence of a larger class of consistent master actions for superstring fieldtheory, which is our third motivation. Interestingly, this unrestricted S bv and its behaviourunder canonical transformations recall the WZW-like formulation [22–31]. Organization of the article
In section 2, we give an analysis of the large gauge symmetry of superstring field theory basedon the small
Hilbert space. On the basis of the Batalin-Vilkovisky formalism, we derive cor-responding master action, which takes a slightly different form from usual one. We study this3nlarged master action and show that canonical transformations bridge the gap from the well-established analysis. In section 3, we clarify the hidden gauge reducibility and the underlying large gauge structure of superstring field theory based on the small Hilbert space. In section 4,we see how these additional fields–antifields appear in the canonical form of the master action.Then, we give master actions for interacting theories based on the set of fields–antifields withoutgeometry-inspired restrictions. We end with remarks on the relation between the hidden gaugereducibility, the large class of the master action, and the general WZW-like formulation.
The gauge structure of superstring field theory is infinitely reducible. We know that the g -thgauge parameter λ − g has its gauge invariance δλ − g = Q λ − g and it is preserved under the( g + 1)-st gauge transformation λ − g → λ − g + δλ − g with δλ − g = Qλ − ( g +1) . However, there is animplicit assumption: As well as a dynamical string field Ψ and its gauge variation δ Ψ, all gaugeparameter fields must belong to the small Hilbert space, { λ − g } g ⊂ H βγ . Note that using thezero mode η ≡ η of η ( z )-current of the ξηφ -system, one can express this restriction as λ − g ∈ H βγ ⇐⇒ η λ − g = 0 , ( g ∈ N ) . We often call this restriction onto the small Hilbert space H βγ as the η -constraint. If and onlyif we impose this constraint on not only the string field but also all gauge parameter fields, thetheory has this type of the gauge reducibility.The above gauge reducibility structure is drastically changed by relaxing the η -constraint onthe gauge transformations for the gauge transformations, keeping on the action and its gaugeinvariance. To see it explicitly at the level of the Batalin-Vilkovisky master action, it is usefulto switch from the small BPZ inner product hh A, B ii to the large BPZ inner product h A, B i : S [Ψ] = 12 (cid:10) ξ Ψ , Q Ψ (cid:11) ≡ (cid:10)(cid:10) Ψ , Q Ψ (cid:11)(cid:11) . (2.1)Here, ξ is a homotopy operator for η , namely, η ξ + ξ η = 1 in the large Hilbert space H ξηφ . Since2 δS = −h δ Ψ , ( Qξ − ξQ )Ψ i , we find the gauge invariance δ Ψ =
Q λ provided that η Ψ = 0 and η ( δ Ψ) = 0 . It implies that λ must satisfy η λ = Q ω with some state ω : To be gauge invariant,the constraint η λ = 0 (and η λ − g = 0 for g ∈ N ) is sufficient but not necessary. In this section,we see what happens if we relax ηλ = 0 under η Ψ = 0 and η ( δ Ψ) = 0.Note that one can apply the analysis which we present below to every types of superstringfields Ψ ∈ H βγ . The mismatch of their Grassmann parities based on world-sheet ghost numbersis resolved by using the grading based on the appropriately suspended degrees. For example,see [7, 13] or appendix A. However, for simplicity, one can regard g, p of Ψ g,p as the ghost andpicture number labels of open superstring fields in the rest of this section. Unrestricted gauge parameter
Let us consider the kinetic term of [8–15], namely the free theory (2.1). Now, we impose the η -constraints on the dynamical string field Ψ and its gauge variation δ Ψ only, η Ψ = 0 , η ( δ Ψ) = 0 . (2.2)4ecause of these constraints, any assignment of ξ in the action (2.1) can be permissible, and wecan rewrite the variation of (2.1) as δS = − (cid:10) δ Ψ , Q ξ Ψ (cid:11) . We write Ψ , − ≡ Ψ . The p -label ofΨ g,p denotes its world-sheet picture number. In this set up, the action is invariant under thegauge transformation δ Ψ , − = Q λ , − . (2.3)Now, a gauge parameter string field λ , − ≡ λ has to keep (2.2), but there is no other restrictions.We consider to enlarge the state space of λ , − from H βγ to H β ⊕ Q H ξηφ . Clearly, it keeps theform of the gauge transformation (2.3) and the constraint δ Ψ , − ∈ H βγ because all enlargedcomponents of λ , − have no new-contributions into (2.3). Then, because of η ( Qλ , − ) = 0 ,there exists a gauge parameter λ , − such that η λ , − + Q λ , − = 0 . (2.4a)It gives a weaker constraint on λ , − rather than η λ , − = 0 . Here, λ , − is an auxiliary gaugeparameter string field which belongs to H βγ ⊕ Q H ξηφ . It is not the end of story: This weakenconstraint (2.4a) implies η ( Qλ , − ) = 0 , which provides the constraint also on λ , − . Again,using an auxiliary gauge parameter λ , − , we have to impose η λ , − + Q λ , − = 0 . As a result,we find that for the gauge transformation (2.3) keeping the η -constraint (2.2), there exists afamily of infinite number of the auxiliary gauge parameter string fields { λ , − p } p satisfying η λ , − p + Q λ , − ( p +1) = 0 , ( p ∈ N ) . (2.4b)We therefore find that in the theory based on (2.2), while the dynamical string field and itsgauge variation must belong to the small Hilbert space, its gauge parameter and auxiliary fieldscan protrude from the small Hilbert space as long as they are BRST exact:Ψ , − ∈ H βγ , λ , − p ∈ H βγ ⊕ Q H ξηφ . Gauge reducibility with constraints
The relations (2.4a) and (2.4b) will give the first class constraints on the set of fields–antifields[42]. On the basis of the gauge transformation (2.3) and gauge parameter fields living in H βγ ⊕ Q H ξηφ , we study the gauge reducibility of superstring field theory. Note that there is no positive-picture gauge parameter fields λ , + p for p ∈ N in the above set up. We consider the gaugetransformations δ for the gauge transformation (2.3), which must preserve (2.2), δ ( δ Ψ , − ) = 0 , δ ( η λ , − p + Q λ , − ( p +1) ) = 0 , ( p ∈ N ) . For type II theory, we should write it as Ψ g,p, ˜ p and consider the left- and right-moving picture numbers This is our implicit assumption in section 2. If we remove it, the space H β ⊕ Q H ξηφ does not give the largestextension. Then, we can further enlarge it to Ker[ Qη ], in which the same story goes by replacing H β ⊕ Q H ξηφ with Ker[ Qη ]. We discuss it again in section 3. The author would like to thank Ted Erler. One may be possible to introduce additional gauge parameter field λ , and consider a similar tower ofauxiliary fields satisfying η Ψ , + Q Ψ , − = 0; see section 3. It would correspond to switch the roles of Q and η in the following analysis. Since Q - and η -complexes are exact in the large Hilbert space H ξηφ , one could introducea homotopy operator for Q and consider the switched version in the completely same way.
5n other words, for p ∈ N , we want to specify δ λ , − p such that Q ( δ λ , − ) = 0 and η ( δ λ , − p ) + Q ( δ λ , − ( p +1) ) = 0 . We find the first higher gauge transformations, δ λ , − = Q λ − , − , δ λ , − ( p +1) = η λ − , − p + Q λ − , − ( p +1) . Next, we consider the gauge transformations δ preserving these, δ ( δ λ , − ) = Q ( δ λ − , − ) = 0and η ( δ λ − , − p ) + Q ( δ λ − , − ( p +1) ) = 0 for p ∈ N . We find that again, they are given by δ λ − , − = Q λ − , − and δ λ − , − ( p +1) = η λ − , − p + Q λ − , − ( p +1) for p ∈ N . Likewise, for p ∈ N ,we obtain the g -th gauge transformations δ g λ − g, − = Q λ − ( g +1) , − , δ g λ − g, − ( p +1) = η λ − ( g +1) , − p + Q λ − ( g +1) , − ( p +1) , (2.5)which preserves the ( g − δ g (cid:0) δ g − λ − ( g − , − (cid:1) = Q (cid:0) δ g λ − g, − (cid:1) = 0 ,δ g (cid:0) δ g − λ − ( g − , − − p (cid:1) = η (cid:0) δ g λ − g, − p (cid:1) + Q (cid:0) δ g λ − g, − − p (cid:1) = 0 . Hence, as well as the theory based on the restriction λ − g ∈ H βγ , our unrestricted theory basedon δ Ψ ∈ H βγ has an infinitely reducible gauge structure. However, in our case, the gaugehierarchy takes a more enlarged form. As we will see, this change of the gauge reducibilityyields the different spectrum of fields–antifields and Batalin-Vilkovisky master action. The above analysis of the gauge reducibility tells us that in addition to the string field Ψ and the(first) gauge parameter λ , an infinite tower of higher gauge parameters { λ − g, − p } g,p> appearsin the theory. Hence, we introduce the set of fields–antifields as follows.Fields : Ψ , − , { Ψ , − − p } p ∈ N , { Ψ − , − − p } p ∈ N , . . . , { Ψ − g, − − p } p ∈ N , . . . Antifields : (Ψ , − ) ∗ | {z } Φ ∗ , , { (Ψ , − − p ) ∗ } p ∈ N | {z } { Φ ∗ ,p } p ∈ N , { (Ψ − , − − p ) ∗ } p ∈ N | {z } { Φ ∗ ,p } p ∈ N , . . . , { (Ψ − g, − − p ) ∗ } p ∈ N | {z } { Φ ∗ g,p } p ∈ N , . . . This is a non-minimal set of fields–antifields and there exist the first class constraints corre-sponding to (2.4a) and (2.4b) [42, 43]. For Neveu-Schwarz (NS) open string fields, the fieldΨ − g, − − p ≡ A g Z − g, − − p consists of a set of space-time fields A g whose space-time ghost num-ber is g and a set of CFT basis Z − g, − − p whose world-sheet ghost and picture numbers are1 − g and − − p . When we use the large BPZ inner product to construct the master action, the corresponding antifields (Ψ − g, − − p ) ∗ consists of a set of space-time fields A − ( g +1) whosespace-time ghost number is − ( g + 1) and a set of CFT basis Z g,p whose world-sheet ghost andpicture numbers are 1 + g and p , namely, (Ψ − g, − − p ) ∗ ≡ A − ( g +1) Z g,p . Thus, for g, p ≥ ∗ g,p for the antifield of Ψ − g, − − p as followsΦ ∗ g,p ≡ (Ψ − g, − − p ) ∗ . One may wonder if it changes the properties of the master action which are valid when we the small BPZinner product, such as K bv = K ( ψ, Qψ ). But it is not the case. The use of the large BPZ inner product providesjust a framework to describe the modification or enlargement, and it is mainly caused by the BV spectrum andconstraints on fields–antifields, which we will see in section 3.2. See also appendix A. − ) Ψ = ( − ) Φ+1 . In this paper, g ofΨ − g, − − p or Φ ∗ g,p denotes the g -th reducibility, and p of that indicates the p -decreasing fromthe natural picture number of considering string fields. Because of (2.4a), the subset of fields { Ψ , − , Ψ , − p } p> must satisfy the constraint equations η Ψ , − = 0 , η Ψ , − p + Q Ψ , − − p = 0 . (2.6)There is no constraint on the other fields or antifields. We write G for the gauge generator ofthe theory, namely, G ≡ Q in this case and G ≡ {
Q, V g,n } g,n for the interacting theory discussedin section 3. We therefore consider the following BV spectrum with constraints (2.6)Ψ , − ∈ H βγ , { Ψ , − p } p> ⊂ H βγ ⊕ G H ξηφ , { Ψ − g, − p , Φ ∗ g,p − } g,p> ⊂ H ξηφ . (2.7)We construct a BV master action based on (2.7) and (2.6) and see its properties in the rest ofthis section. As we will see, the constraints (2.6), or simply η Ψ , − = Q η Ψ , − = 0, work welland play an important role in the BV master equation. Antifield number expansion
We derive the master action S bv [Ψ , Φ ∗ ] on the basis of the antifield expansion, S bv [Ψ , Φ ∗ ] = S (0) [Ψ] + ∞ X a =1 S ( a ) [Ψ , Φ ∗ ] . (2.8)Here, S ( a ) denotes the antifield number a part of the master action S bv . The antifield numberis additive and assigned to antifields only: the a -th antifield Φ ∗ a,p has antifield number a , forwhich we write afn[Φ ∗ a,p ] = a . For simplicity, we define the antifield number of the field Ψ g,p by 0, namely afn[Ψ g,p ] = 0. Every functions F = F [Ψ , Φ ∗ ] of fields–antifields { Ψ , Φ ∗ } have theantifield number which is equivalent to the sum of inputs antifield numbers. Therefore, notethat the derivative with respect to the antifield Φ ∗ a,p must decrease antifield number a . The BPZinner product does not have the antifield number: afn[ h A, B i ] = afn[ A ] + afn[ B ].As the initial condition of the master action, the antifield number 0 part S (0) is given by theoriginal action S [Ψ] itself, S (0) [Ψ] ≡ S [Ψ] = 12 (cid:10) ξ Ψ , − , Q Ψ , − (cid:11) . Let us consider the antifield expansion of the master equation { S bv , S bv } = P ∞ g =0 { S bv , S bv }| ( g ) ,where { , } denotes the antibracket; see appendix A. The antifield number g part is given by n S bv [Ψ , Φ ∗ ] , S bv [Ψ , Φ ∗ ] o(cid:12)(cid:12)(cid:12) ( g ) = ∞ X p =0 (cid:28) ∂ r S ( g ) ∂ Ψ − g, − − p , ∂ r S (1+ g ) ∂ Φ ∗ g,p (cid:29) . First, we specify the antifield number 1 part S (1) of S bv , which has to satisfy the antifield number0 part of the master equation with S (0) : n S (0) + S (1) , S (0) + S (1) o(cid:12)(cid:12)(cid:12) (0) = X p (cid:28) ∂ r S (0) ∂ Ψ , − , ∂ r S (1) ∂ Φ ∗ ,p (cid:29) = 0 . a, b ∈ R satisfying a + b = 0 , we obtain ∂ r S (0) ∂ Ψ , − = 1 a + b (cid:0) a ξ Q − b Q ξ (cid:1) ( η ξ Ψ , − ) = 1 a + b (cid:0) a ξ Q − b Q ξ (cid:1) Ψ , − . To solve the equation { S (0) + S (1) , S (0) + S (1) }| (0) = 0 , we set ∂ l S (1) ∂ Φ ∗ , = Q Ψ , − . = ηξ ( Q Ψ , − ) . One may think this is a natural and unique choice. However, note that the constraint relation(2.7) is crucial to satisfy the master equation using this right derivative, unlike usual cases. Wefind an antifield number 1 part of the solution, S (1) [Ψ , Φ ∗ ] = (cid:10) Φ ∗ , , Q Ψ , − (cid:11) . (2.9a)In this case, unlike well-established analysis based on the restriction { Ψ g , (Ψ g ) ∗ } g ⊂ H βγ , theprojector ηξ does not work as the identity on fields, which leads another expression. This S (1) provides the right derivative with respect to the ghost string field Ψ , − , ∂ r S (1) ∂ Ψ , − = Q Φ ∗ , . In this expression of S (1) , clearly, other additional ghost derivatives vanish. Note however thatthe recursive relation η Ψ , − p + Q Ψ , − − p = 0 on Ψ , − p = ( η ξ + ξ η )Ψ , − p implies Q Ψ , − = Q (cid:0) η ξ Ψ , − (cid:1) − Q X Ψ , − = Q η ξ (cid:0) Ψ , − − X Ψ , − (cid:1) + Q X Ψ , − ...= Q ( η ξ Ψ , − ) + Q ∞ X p =1 ( − ) p X p ( η ξ Ψ , − − p ) . Here, X ≡ Q ξ + ξ Q changes the picture number. We obtain alternative expression of theleft derivative with respect to Φ ∗ , . In this alternative expression, a projector ηξ is insertedin front of the states, which resolves the ambiguity of ξ -assignments. Note that because ofthe constraints (2.6), all ghost fields Q Ψ , − p ( p ∈ N ) have this expression. Hence, S (1) [Ψ , Φ ∗ ]potentially includes all auxiliary fields Ψ , − − p , and we can rewrite (2.9a) as follows, S (1) [Ψ , Φ ∗ ] = (cid:10) Φ ∗ , , Q ( ηξ Ψ , − ) (cid:11) + ∞ X p =1 ( − ) p (cid:10) Φ ∗ , , Q X p ( ηξ Ψ , − − p ) (cid:11) . (2.9b)Interestingly, each term individually vanishes in the master equation. Thus, if one prefers, onecould set different coefficients for each term at this level. It suggests that it may be possible touse X explicitly for solving the master equation, which we discuss later. This expression of S (1) provides the right derivatives with respect to each ghost string field { Ψ , − p } p ∈ N , ∂ r S (1) ∂ Ψ , − − p = − ( − ) p ξ Q X p ( η Φ ∗ , ) . (2.10)8ote that η and ξ are inserted into the Ψ , − -derivative in this expression.Before solving the next order equation, let us consider about X -including terms of (2.9b) .When we impose the rigid small-space constraint Ψ , − = ηξ (Ψ , − ) on the ghost field, theseadditional ghost terms should vanish and (2.9b) reduces to (2.9a) with Ψ , − ∈ H βγ . If ghostfield Ψ , − is exactly small, the consistent Ψ , − -ghost derivative should be a ξ -exact state inthe large BPZ inner product. It implies that the antifield Φ ∗ , satisfies Q Φ ∗ , = ξη ( Q Φ ∗ , )in S (1) . This constraint on Φ ∗ , , the antifield for the string field Ψ , − , yields X η Q Φ ∗ , = 0 , which kills (2.10) and additional terms appearing in (2.9b) .One can also see the consistency of two expression (2.9a) and (2.9b) via direct computationsof the next order BV master equation. We consider the antifield number 2 part S (2) , which hasto satisfy the antifield number 1 part of the master equation,12 n S bv [Ψ , Φ ∗ ] , S bv [Ψ , Φ ∗ ] o(cid:12)(cid:12)(cid:12) (1) = X p (cid:28) ∂ r S (1) ∂ Ψ − , − − p , ∂ l S (2) ∂ Φ ∗ ,p (cid:29) = 0 . From the ghost derivatives of S (1) , we find that left derivatives should be given by ∂ l S (2) ∂ Φ ∗ , = Q Ψ − , − , ∂ l S (2) ∂ Φ ∗ , p = η Ψ − , − − p + Q Ψ − , − − p . While we quickly find that these satisfy the master equation if we use (2.9a) , on the basis of(2.9b) , we obtain the following pieces of the master equation, (cid:28) ∂ r S (1) ∂ Ψ , − , ∂ r S (2) ∂ Φ ∗ , (cid:29) = D ξ Q η Φ ∗ , , Q Ψ − , − E = D η Q X Φ ∗ , , Ψ − , − E , ∞ X p =1 (cid:28) ∂ r S (1) ∂ Ψ , − − p , ∂ r S (2) ∂ Φ ∗ ,p (cid:29) = ∞ X p =1 ( − ) p D Q η X p Φ ∗ , , Ψ − , − p + X Ψ − , − − p E . By summing up all p -labels, these satisfy the master equation12 (cid:8) S bv , S bv (cid:9)(cid:12)(cid:12) (1) = ∞ X p =0 ( − ) p D ( Q η + η Q ) X p Φ ∗ , , Ψ − , − − p E = 0 . Hence two expression are consistent, and at this step, we obtain the following solution S (2) [Ψ , Φ ∗ ] = (cid:10) Φ ∗ , , Q Ψ − , − (cid:11) + ∞ X p =1 (cid:10) Φ ∗ ,p , η Ψ − , − p + Q Ψ − , − − p (cid:11) . Its ghost derivatives are given by ∂ r S (2) ∂ Ψ − , − − p = η Φ ∗ , p + Q Φ ∗ ,p .
9o satisfy the next order master equation, { S bv , S bv }| (2) = 0 , we have to set the antighostderivatives of S (3) as follows ∂ r S (3) ∂ Φ ∗ , = Q Ψ − , − , ∂ r S (3) ∂ Φ ∗ ,p = η Ψ − , − p + Q Ψ − , − − p , which gives the antifield number 3 part of the solution S (3) in the similar form as S (2) . Likewise,we find that the antifield number g part of S bv is given by S ( g ) [Ψ , Φ ∗ ] = (cid:10) Φ ∗ g, , Q Ψ − g, − (cid:11) + ∞ X p =1 (cid:10) Φ ∗ g,p , η Ψ − g, − p + Q Ψ − g, − − p (cid:11) . Note that this type of the antifield number g part S ( g ) works as Lagrange-multiplier-like ghost–antighost term fixing the higher gauge symmetries (2.5). To see how constraints (2.7) work in the master equation, we introduce a set of Lagrangemultipliers L = {L , , L ,p } p ∈ N . Note that L g,p has space-time ghost number − g , world-sheetghost number g , and picture number p . By summing up all antifield number S ( g ) , we get S bv [Ψ , Φ ∗ ; L ] = 12 (cid:10) ξ Ψ , − , Q Ψ , − (cid:11) + ∞ X g =1 (cid:10) Φ ∗ g − , , Q Ψ − g, − (cid:11) + ∞ X g =1 ∞ X p =1 (cid:10) Φ ∗ g,p , η Ψ − g, − p + Q Ψ − g, − ( p +1) (cid:11) + (cid:10) L , , η Ψ , − (cid:11) + ∞ X p =1 (cid:10) L ,p , η Ψ , − p + Q Ψ , − ( p +1) (cid:11) . (2.11a)After integrating out the Lagrange multipliers L of (2.11a), we obtain the BV master action, S bv [Ψ , Φ ∗ ] = Z D [ L ] S bv [Ψ , Φ ∗ ; L ] . (2.11b)We derived the master action using the antifield number expansion. But off course, because ofthe free theory, one can apply the BRST formalism as its gauge fixing procedure and find S bv by the guess from it as [32–34]. Master equation
Let δ BV Ψ and δ BV Φ ∗ be BV-BRST transformations of fields and antifields respectively. Wecheck that our master action S bv [Ψ , Φ ∗ ] satisfies the master equation n S bv [Ψ , Φ ∗ ] , S bv [Ψ , Φ ∗ ] o = 0 . (2.12)Then, the Batalin-Vilkovisky formalism implies that BRST transformations are given by δ BV Ψ − g, − − p = ∂ l S bv [Ψ , Φ ∗ ] ∂ Φ ∗ g,p = n Ψ − g, − − p , S bv [Ψ , Φ ∗ ] o ,δ BV Φ ∗ g,p = ∂ l S bv [Ψ , Φ ∗ ] ∂ Ψ − g, − − p = n Φ ∗ g,p , S bv [Ψ , Φ ∗ ] o .
10e write S bv [ L ] ≡ S bv [Ψ , Φ ∗ ; L ] for brevity. In the previous section, we found that thederivatives with respect to the first pair of field–antifield { Ψ , − , Φ ∗ , } are given by ∂ r S bv [ L ] ∂ Ψ , − = a ξ Q − b Q ξa + b Ψ , − + η L , , ∂ r S bv [ L ] ∂ Φ ∗ , = Q Ψ , − . where a, b ∈ R are real parameters satisfying a + b = 0 . For the primary ghosts { Ψ − g, − } g andauxiliary ghosts { Ψ − g, − − p } g,p , we obtained their right derivatives as ∂ r S bv [ L ] ∂ Ψ − g, − − p = η Φ ∗ g, p + Q Φ ∗ g,p + δ ,g (cid:0) η L g,p +1 + Q L g,p (cid:1) , for given g ∈ N and p ∈ { } ∪ N . For the primary antighosts { Φ ∗ g, } g and auxiliary antighosts { Φ ∗ g,p } g,p with any fixed g, p ∈ N , their left derivatives are given by ∂ l S bv [ L ] ∂ Φ ∗ g, = Q Ψ − g, − , ∂ l S bv [ L ] ∂ Φ ∗ g,p = η Ψ − g, − p + Q Ψ − g, − ( p +1) . Using these, we find that up to the terms including Lagrange multipliers, at each level of the g -label, the master equation holds after the sum over the p -label:12 (cid:8) S bv [ L ] , S BV [ L ] (cid:9) = ∞ X g =0 (cid:28) ∂ r S BV [ L ] ∂ Ψ − g, − , ∂ l S BV [ L ] ∂ Φ ∗ g, (cid:29) + ∞ X g =1 ∞ X p =1 (cid:28) ∂ r S BV [ L ] ∂ Ψ − g, − − p , ∂ l S BV [ L ] ∂ Φ ∗ g,p (cid:29) = (cid:28) ∂ r S bv [ L ] ∂ Ψ , − , ∂ l S bv [ L ] ∂ Φ ∗ , (cid:29) + ∞ X g =0 ∞ X p =1 D(cid:0) Φ ∗ g,p + L ,p (cid:1) , ( Q η + η Q )Ψ − g, − p E . We would like to emphasise that the second term vanishes without requiring higher constraintequations on fields, which may reduce (2.6) and give Ψ ,p< − ∈ H ξηφ . As a result, we obtain12 (cid:8) S bv [ L ] , S bv [ L ] (cid:9) = (cid:10) Q Ψ , − , ξ Q Ψ , − (cid:11) + (cid:10) L , , η Q Ψ , − (cid:11) , which clearly reduces to zero when η Ψ , − = 0 and η Q Ψ , − = 0 hold. While we introduced L , to impose the η -constraint on the string field Ψ , − ∈ H βγ , in the master equation, it alsoworks to impose the η -constraint on the gauge variation δ BV Ψ , − ∈ H βγ . It completes a proofthat the action (2.11b) satisfies the master equation (2.12) and that the master action S bv [Ψ , Φ ∗ ]is invariant under the BRST transformations δ BV Ψ − g, − = Q Ψ − g, − , δ BV Ψ − g, − − p = η Ψ − g − , − p + Q Ψ − g − , − − p ,δ BV Φ ∗ , = Q ξ Ψ , − , δ BV Φ ∗ g,p = η Φ ∗ g,p +1 + Q Φ ∗ g,p . We will see that the master action (2.11b) reduces to that of exactly small theory. Roughly,by integrating out the additional antifields { Φ g,p } p> of (2.11b), all ghost fields { Ψ − g. − p } g,p are restricted on the subspace Σ satisfying the constraint equations η Ψ − g, − p + Q Ψ − g, − p = 0 .Then, Q Ψ − g, − = ηξ ( Q Ψ − g, − ) because of η ( Q Ψ − g, − p ) = 0 , and we find S bv [Ψ − , Φ ∗ + ] (cid:12)(cid:12) Σ = 12 (cid:10) ξ Ψ , − , Q Ψ , − (cid:11) + ∞ X g =1 (cid:10) Φ ∗ g, , ηξ ( Q Ψ − g, − ) (cid:11) = 12 D ξ (Ψ − + η Φ ∗ + ) , Q (Ψ − + η Φ ∗ + ) E , − ≡ Ψ , − + P ∞ g =1 Ψ − g, − and Φ ∗ + ≡ Φ ∗ , + P ∞ g =1 Φ ∗ g, . This is the small BV masteraction based on the large BPZ inner product, exactly small fields Ψ − ∈ H βγ , and unrestrictedantifields Φ ∗ + ∈ H ξηφ . Hence, by identifying η Φ ∗ + with the antifield Ψ ∗ s of the exactly smalltheory, Ψ ∗ s ∼ = η Φ ∗ + , or by imposing constraints ξ Φ ∗ + = 0, which is equivalent to restrict theminimal set of fields–antifields onto H βγ , it reduces to the small BV master action based onthe small BPZ inner product and { Ψ − , Ψ ∗ s } ⊂ H βγ . See appendix B for the small theory. Interms of the Batalin-Vilkovisky formalism, it implies that there exists a (partially) gauge fixingfermion which reduces (2.11b) to the small master action, which we explain. Gauge fixing fermion
Using two types of trivial pairs of BV fields { C g, , N g +1 , } g and { A g,p , L g +1 ,p } g,p , we add anauxiliary term, a trivial solution of the master equation S aux [ L, N ; A ∗ , C ∗ ] = ∞ X g = −∞ (cid:10) C ∗ − g, − , N g, (cid:11) + ∞ X g =1 ∞ X p =1 (cid:10) A ∗ − g, − − p , L g,p (cid:11) , into the master action (2.11b). Here, C ∗ − g, − and A ∗ − g, − − p are antifields for C g, and A g,p re-spectively. We also introduce BV fields { Ψ † g, − } g , which will be identified with the antifields ofthe small master action. Let us consider the following gauge fixing fermion Γ[ ψ ] = Γ[Ψ , A, C ; Ψ † ]consisting of this non-minimal set of fields,Γ[ ψ ] = ∞ X g =1 h(cid:10) C − g, , η Ψ † g, − (cid:11) + (cid:10) ξ Ψ † g, − + η C g, , Ψ − g, − (cid:11) + ∞ X p =1 (cid:10) A g,p , Ψ − g, − − p (cid:11)i . It gives the following ghost field derivatives ∂ Γ[ ψ ] ∂ Ψ − g, − = ξ Ψ † g, − + η C g, , ∂ Γ[ ψ ] ∂ Ψ − g, − p = A g,p − ,∂ Γ[ ψ ] ∂C g> , = η Ψ − g, − , ∂ Γ[ ψ ] ∂C g< , = η Ψ † g, − , ∂ Γ[ ψ ] ∂A g,p = Ψ − g, − − p . On this gauge fixing fermion, we have Φ ∗ ≡ ∂ Ψ Γ and S [ ψ ; ψ ∗ ] | Γ = S [ ψ ; ∂ ψ Γ]. Therefore, byintegrating out { L, N } , we obtain the small BV master action S bv [Ψ − , Ψ † + ] = Z D [ L ] D [ N ] (cid:16) S bv [Ψ , ∂ Ψ Γ] + S aux [ L, N ; ∂ A Γ , ∂ C Γ] (cid:17) = 12 D ξ (Ψ − + Ψ † + ) , Q (Ψ − + Ψ † + ) E , where Ψ − ≡ P ∞ g =0 Ψ − g, − and Ψ † + ≡ P ∞ g =0 Ψ † g, − . Note that η Ψ − g, − = η Ψ † g, − = 0 and { Ψ g,p } p< = 0 because of N - and L -integrations.Therefore, the well-established BV master action based on the geometry-inspired constraintsΨ − ∈ H βγ kills the large gauge symmetries of higher ghost fields and is equivalent to a partiallygauge-fixed version of the gauge theory without restrictions. It is known that the Batalin-Vilkovisky master action is unique up to adding trivial pairs andcanonical transformations if it is proper. In this section, we discuss three important types of12anonical transformations. In particular, we show that there exist a canonical transformationwhich rotates only higher ghost-fields–antifields and transforms the master action (2.11b) into S bv [Ψ , Φ ∗ ] = 12 (cid:10) ξ Ψ , − , Q Ψ , − (cid:11) + ∞ X g =0 ∞ X p =0 (cid:10) Φ ∗ g,p , Q Ψ − g, − − p (cid:11) , (2.13)where we used Φ ,p> = 0 for brevity and { Ψ , − , Ψ − g, − − p , Φ ∗ , , Φ ∗ g,p } g,p is the set of fields–antifields. Although it has the same form as the master action based on the geometry-inspiredconstraints { Ψ g , (Ψ g ) ∗ } g ⊂ H βγ , it includes additional propagating ghost–antighost fields. Wecall (2 .
13) as the canonical form, and (2 . b ) as the large form. On the explicit X -insertions Recall that because of the constraints (2.6), all ghost fields Q Ψ , − p ( p ∈ N ) have anotherexpression, and Q X p Ψ , − − p satisfies the antifield number 0 part of the master equation for any p ∈ N . Hence, for example, we could start from S (1) [Ψ , Φ ∗ ] = (cid:10) Φ ∗ , , Q Ψ , − (cid:11) + ∞ X p =1 ( − ) p (cid:10) Φ ∗ , , Q X p Ψ , − − p (cid:11) . If one prefers, one could use different coefficients for each term. Using (2.7), this S (1) providesthe right derivatives with respect to ghost string fields { Ψ , − p } p ∈ N , ∂ r S (1) ∂ Ψ , − − p = ( − ) p X p Q Φ ∗ , + ( − ) p +1 X p +1 η Φ ∗ , . (2.14)This S (1) also satisfies { S bv , S bv }| (1) = 0 via the same mechanism as we found, ∞ X p =0 (cid:28) ∂ r S (1) ∂ Ψ , − − p , ∂ r S (2) ∂ Φ ∗ ,p (cid:29) = ∞ X p =0 ( − ) p D ( Q η + η Q ) X p +1 Φ ∗ , , b Ψ − , − − p E = 0 , and it leads the same type of the master action as (2.11b). These master actions will be relatedto each other via a canonical transformation. For fixed g, p ≥ b Ψ − g, − − p ≡ Ψ − g, − − p + ∞ X q =1 ( − ) q X q Ψ − g, − − p − q , (2.15a)which gives the field relation between new and old pairs of the fields–antifields. In this notation,we can rewrite the above S (1) and its antighost derivative as S (1) = (cid:10)b Φ ∗ , , Q b Ψ , − (cid:11) , ∂ l S (1) ∂ b Φ ∗ , = Q b Ψ , − . Here, we write b Φ ∗ g,p for the antifield corresponding to the field b Ψ − g, − − p . Then, from it ghostderivatives or (2.14), we find ∂ l S (2) ∂ b Φ ∗ , = Q b Ψ − , − , ∂ l S (2) ∂ b Φ ∗ , p = η b Ψ − , − − p + Q b Ψ − , − − p . Thus, one could take a short-cut to the reduction presented in the previous section by finding an appropriategauge-fixed basis killing additional fields–antifields of (2.13). ∗ g,p and b Φ ∗ g,p are related byΦ ∗ g,p = b Φ ∗ g,p + p X q =1 ( − ) q X q b Φ ∗ g,p − q . (2.15b)Therefore, ambiguity of the form of S bv coming up from using explicit X -insertions can beabsorbed by canonical transformations. Switching transformation
Interestingly, there exist canonical transformations switching the roles of η and Q . We considerthe following generating function R [Ψ , b Φ ∗ ] of the canonical transformation R [Ψ , b Φ ∗ ] = ∞ X g =0 (cid:10)b Φ ∗ g, , Ψ − g, − (cid:11) + ∞ X g =1 ∞ X p =1 (cid:10)b Φ ∗ g,p , η ξ Ψ − g, − − p − ξ Q Ψ − g, − − p (cid:11) . Apparently, it leaves pairs of field-antifield labeled by p = 0 invariant. The new fields { b Ψ , b Φ ∗ } and old fields { Ψ , Φ ∗ } are related by b Ψ − g, − − p ≡ ∂ l R [Ψ , b Φ ∗ ] ∂ b Φ ∗ g,p , Φ ∗ g,p ≡ ∂ r R [Ψ , b Φ ∗ ] ∂ Ψ − g, − − p . Since the first ghost fields satisfy (2.6), we find b Ψ , − − p = ( ηξ + ξη )Ψ , − − p . Thus, R [Ψ , b Φ ∗ ]generates identity transformation not only for the p = 0 subset { Ψ − g, − , Φ ∗ g, } g , but also thefirst ghost-fields–antifields { Ψ , − − p , Φ ∗ ,p } p . For generic g, p ≥
1, it gives b Ψ − g, − − p = η ξ Ψ − g, − − p − ξ Q Ψ − g, − − p , Φ ∗ g,p = ξ η b Φ ∗ g,p − Q ξ b Φ ∗ g,p − . Using these, for the higher ghost-fields–antifields, one can quickly find (cid:10) Φ ∗ g, p , Q Ψ − g, − − p (cid:11) = (cid:10) ξ η b Φ ∗ g, p , Q Ψ − g, − − p (cid:11) = (cid:10) b Φ ∗ g, p , η (cid:0) η ξ Ψ − g, − − p − ξ Q Ψ − g, − − p (cid:1)(cid:11) . By summing up all g, p ≥
0, we can transform the higher ghost-fields–antifields terms of themaster action (2.13) into ∞ X g =1 ∞ X p =1 (cid:10) Φ ∗ g,p , Q Ψ − g, − − p (cid:11) = − ∞ X g =1 ∞ X p =1 (cid:10)b Φ ∗ g,p , η b Ψ − g, − p (cid:11) . The minus sign of the right hand side would be natural from the point of view of the η - Q switching relation appearing in WZW-like superstring field theory. If one prefer, one can absorbethis sign by redefining the fields–antifields or by appropriate canonical transformations. Whilewe introduced the canonical transformation leaving the p = 0 subset { Ψ − g, − , Φ ∗ g, } g , onecan consider canonical transformations switching all pair of the fields–antifields similarly. For example, one can perform I g,p [Ψ , b Φ ∗ ] = i h b Φ ∗ g,p , Ψ − g, − − p i or more trivial transformations. he canonical form and the large form After the above canonical transformation switching η - and Q -terms, we consider to take backthe R [Ψ , b Φ ∗ ]-transformed master action via the following canonical transformation W [Ψ , b Φ ∗ ] = (cid:10)b Φ ∗ , , Ψ , − (cid:11) + ∞ X g =0 ∞ X p =0 (cid:10)b Φ ∗ g,p , Ψ − g, − − p (cid:11) + ∞ X g =1 ∞ X p =1 (cid:10)b Φ ∗ g,p , ξ Q Ψ − g, − − p (cid:11) . Two minimal sets of fields–antifields { Ψ , Φ } and { b Ψ , b Φ ∗ } are related by b Ψ − g, − − p ≡ ∂ l W [Ψ , b Φ ∗ ] ∂ b Φ ∗ g,p , Φ ∗ g,p ≡ ∂ r W [Ψ , b Φ ∗ ] ∂ Ψ − g, − − p . By construction, W [Ψ , b Φ ∗ ] acts as the identity on the p = 0 subset { Ψ − g, − , Φ ∗ g, } g and onthe first ghost-fields–antifields { Ψ , − − p , Φ ∗ ,p } p . For other g, p >
0, it generates b Ψ − g, − − p = Ψ − g, − − p + ξ Q Ψ − g, − − p , Φ ∗ g,p = b Φ ∗ g,p + Q ξ b Φ ∗ g,p − . Therefore, via W [Ψ , b Φ ∗ ], the higher ghost-fields–antifields terms are transformed as (cid:10)b Φ ∗ g,p , η b Ψ − g, − p (cid:11) = (cid:10)b Φ ∗ g,p , η Ψ − g, − p + ηξQ Ψ − g, − − p (cid:11) = (cid:10)b Φ ∗ g,p , Q Ψ − g, − − p (cid:11) + (cid:10)b Φ ∗ g,p , η Ψ − g, − p + ξ Q η Ψ − g, − − p (cid:11) . By summing up all g, p ≥ ∞ X g =1 ∞ X p =1 (cid:10)b Φ ∗ g,p , η b Ψ − g, − p (cid:11) = ∞ X g =1 ∞ X p =1 h(cid:10) Φ ∗ g,p , Q Ψ − g, − − p (cid:11) + (cid:10) Φ ∗ g,p , η Ψ − g, − p (cid:11)i . Hence, there exists a canonical transformation between the large form of the master action(2.11b) and the canonical form of the master action (2.13). Note that this canonical transfor-mation does not change the dynamical string field, its antifield, and string fields of the firstghost-fields–antifields: It rotates string fields of the additional ghost-fields–antifields only.The use of the large Hilbert space H ξηφ enable us to consider various forms of S bv and canon-ical transformations drastically changing S bv . It would make quantization of large superstringfield theory based on the WZW-like formulation highly complicated problem. In section 2, the hidden gauge symmetries arising from δ Ψ , − = Q λ , − were revealed byenlarging the space of the gauge parameter λ , − from H βγ to H βγ ⊕ Q H ξηφ (or Ker[ Qη ] ).What is the origin of these large gauge symmetries in the small theory?—we clarify it in thissection. Recall that the variation of the action (2.1) is given by δS [Ψ] = − (cid:10) δ Ψ , − , Q ξ Ψ , − (cid:11) + 12 (cid:10) δ Ψ , − , X Ψ , − (cid:11) , See [44] for the BV formalism in the large Hilbert space: Several classical BV master actions were obtained. While λ , − ∈ ( η ⊕ Q ) H ξηφ has no new-contributions into δ Ψ , − = Q λ , − , however, λ , − ∈ Ker[ Qη ] does.We consider Ker (cid:2) η (cid:3) ∪ Ker (cid:2) Q (cid:3) in this paper, but one can consider Ker (cid:2) Qη (cid:3) instead of it in the same way. X = ξ Q + Q ξ . The second term always vanishes because of δ Ψ ∈ H βγ . It gives theon-shell condition Q Ψ , − = 0, which is invariant under δ Ψ , − = Q λ , − with η ( Q λ , − ) =0. However, the condition η ( δ Ψ) = η ( Qλ , − ) = 0 implies that there exists ω , such that Qλ , − = η ω , . It provides another expression of this gauge transformation δ Ψ = η ω , with Q ( η ω , ) = 0. If we permit to use this ambiguous expression, it yields more enlarged gaugehierarchy, which we first explain in this section. Alternatively, we can identify λ , − ≡ η Λ − , by using a large gauge parameter Λ − , ∈ H ξηφ . As we explain, it gives unambiguous expressionof the hidden gauge reducibility and clarifies the origin of the large gauge symmetry. We write 2 µ , − ≡ λ , − and 2 µ , ≡ ω , for brevity. Note that δS [Ψ] = 0 holds if δ Ψ ∈ Ker[ η ] ∩ Ker[ Q ] = ( η H ξηφ ) ∩ ( Q H ξηφ ) . We therefore find that the action is invariant under the gauge transformations δ Ψ , − = Q µ , − + η µ , . (3.1)These two gauge parameters { µ , − k } k =0 , belong to the kernel of η or Q , µ , − , µ , ∈ Ker (cid:2) η (cid:3) ∪ Ker (cid:2) Q (cid:3) , (3.2)and they are not independent Q µ , − = η µ , . The state Q µ , − or the state η µ , lives in thesubspace Ker[ η ] ∩ Ker[ Q ] because of µ , − , µ , ∈ Ker (cid:2) Q (cid:3) ∪ Ker (cid:2) η (cid:3) .To see the relation of µ , − and µ , explicitly, it may be helpful to recall that Ker[ η ] ∪ Ker[ Q ] ⊂ Ker[ Qη ] holds. Let us consider two states V , , V , − ∈ Ker[ Qη ] / (Im[ Q ] ∪ Im[ η ]). Weconsider µ , − , µ , ∈ Ker (cid:2)
Q η (cid:3) instead of (3.2), which reduce to (3.2) by setting V , = V , − = 0below. We also introduce three states b µ − , − , b µ − , , b µ − , ∈ H ξηφ living in the large Hilbertspace. Since these µ , − and µ , live in Ker[ Qη ], we can write them as follows, µ , − = V , − + η b µ − , + Q b µ − , − ,µ , = V , − Q b µ − , − η b µ − , . By construction, V , − and V , must satisfy Q V , − = η V , . We thus write V , − ≡ η V , = Q V , − , which gives V , − = Q − V , − and V , = ξ V , − in the above expression. Note that V , − ∈ Ker[ η ] ∩ Ker[ Q ]. Using these V , − and b µ − , , we find Q µ , − = V , − + Q η b µ − , = η µ , . (3.3a)The hidden gauge reducibility of superstring field theory based on the small Hilbert spaceis essentially provided by the large gauge variation δ b µ − , = Q b µ − , + η b µ − , preserving (3.3a),where b µ − , , b µ − , , and b µ − , all live in the large Hilbert space H ξηφ .16 uxiliary gauge parameters The (first) gauge parameters µ , − and µ , live in Ker (cid:2) η (cid:3) ∪ Ker (cid:2) Q (cid:3) . It implies that the state η µ , − lives in Ker[ Q ] = Q H ξηφ and the state Q µ , lives in Ker[ η ] = H βγ ≡ η H ξηφ . We findauxiliary gauge parameters µ , − and µ , − such that η µ , − ∈ Ker[ Q ] ⇐⇒ η µ , − + Q µ , − = 0 ,Q µ , ∈ Ker[ η ] ⇐⇒ Q µ , + η µ , = 0 . However, note that these µ , − and µ , − must satisfy η Qµ − = 0 and η Qµ , − = 0. Hence,these auxiliary parameters µ , − and µ , live in the same subspace as µ , − and µ , , µ , − , µ , ∈ Ker (cid:2) η (cid:3) ∪ Ker (cid:2) Q (cid:3) . Likewise, we find a set of auxiliary gauge parameters { µ ,p } p = − , ⊂ Ker (cid:2) η (cid:3) ∪ Ker (cid:2) Q (cid:3) such that η µ ,p +1 + Q µ ,p = 0 , ( p = − . (3.3b)The (first) gauge parameters { µ , − , µ , } and all auxiliary gauge parameters { µ ,p } p = − , belongto the kernel of η or Q , (cid:8) µ ,p (cid:9) p ∈ Z ⊂ Ker (cid:2) η (cid:3) ∪ Ker (cid:2) Q (cid:3) . Note that there gauge parameters are dependent each other through (3.3a) and (3.3b).
Higher gauge transformations
The above gauge transformation (3.1) is completely equivalent to δ Ψ , − = Q λ , − since it isjust a redefinition of the gauge parameters. However, as we see, all higher gauge parameters(and those of auxiliary gauge parameters) appearing in its gauge reducibility are independenteach other unlike { µ ,p } p ∈ Z ⊂ Ker (cid:2) η (cid:3) ∪ Ker (cid:2) Q (cid:3) , which would be an interesting point.We find the gauge transformations preserving the first gauge transformation (3.1) δµ , − = Q µ − , − + η µ − , ,δµ , = Q µ − , + η µ − , , and the gauge transformations of auxiliary gauge parameter fields δµ ,p = Q µ − ,p + η µ − ,p +1 , ( p = − , , where all gauge parameters { µ − ,p } p ∈ Z belong to the large Hilbert space: { µ − ,p } p ∈ Z ⊂ H ξηφ .The new ingredients are higher gauge parameters µ − ,p labeled by positive p , and the p -labelruns over all integer numbers. They are invariant under the third gauge transformations δµ − , − = Q µ − , − + η µ − , ,δµ − , = Q µ − , + η µ − , ,δµ − , = Q µ − , + η µ − , , δµ − ,p = Q µ − ,p + η µ − ,p +1 , ( p = − , , . Likewise, we find the ( g + 1)-st gauge transformations preserving g -th gauge transformations δµ − g,p = Q µ − g,p + η µ − g,p +1 , ( p = − , , , . . . , g − , (3.4a)and the g -th gauge transformations preserving ( g − δµ − g,p = Q µ − g,p + η µ − g,p +1 , ( p < − , g − < p ) . (3.4b)Note that all higher gauge parameters { µ − g,p } g> , − ≤ p ≤ g − and those of auxiliary gauge param-eters { µ − g,p } g> ,p< − ,g ≤ p are independent and live in the large Hilbert space: { µ − g,p } g> ,p ∈ Z ⊂ H ξηφ . Nonminimal set with constraints and free master action
The above analysis of the gauge reducibility implies that the set of gauge parameters { µ g,p } g< ,p ∈ Z appears in superstring field theory based on Ψ , δ Ψ ∈ H βγ . Hence, the set of fields–antifields isgiven byΨ , − ∈ H βγ , (cid:8) Ψ ,p (cid:9) p ∈ Z ⊂ Ker (cid:2) η (cid:3) ∪ Ker (cid:2) Q (cid:3) , (cid:8) Ψ − − g,p , (Ψ − g,p ) ∗ (cid:9) g ≥ ,p ∈ Z ⊂ H ξηφ . (3.5)We write (Ψ g,p ) ∗ for the antifield of Ψ g,p , whose ghost and picture numbers is determined viathe BPZ inner product of the theory as h (Ψ g,p ) ∗ , Ψ g,p i 6 = 0. Note that the dynamical string fieldΨ , − ∈ Ker[ η ] must satisfy the constraint η Ψ , − = 0 , (3.6a)and the first ghost fields Ψ ,p ∈ Ker (cid:2) η (cid:3) ∪ Ker (cid:2) Q (cid:3) must satisfy the constraints Q Ψ , − = η Ψ , , Q Ψ ,p + η Ψ ,p +1 = 0 , ( p = − . (3.6b)There is no constraint on the other fields and antifields: They live in the large Hilbert space.The large form of the master action is given by the same form as (2.11b) S bv [Ψ , Ψ ∗ ] = 12 (cid:10) ξ Ψ , − , Q Ψ , − (cid:11) + ∞ X g =0 ∞ X p = −∞ (cid:10) (Ψ − g,p ) ∗ , Q Ψ − g,p + η Ψ − g, p (cid:11) (3.7)except for that the p -label runs over all integer numbers. It closely resembles to that of WZW-like theory [32–35]. One can find that it also reduces to the canonical form via canonicaltransformations as we proved in section 2. We can construct the canonical form of the masteraction for interacting theory in the same way as section 2. Again, the property Ψ , − , Ψ , ∈ Ker (cid:2) η (cid:3) ∪ Ker (cid:2) Q (cid:3) ( ⊂ Ker[ Qη ]) is crucial for the master equation. .2 The hidden gauge reducibility without constraints We found that large gauge symmetries are hidden in superstring field theory based on the smallHilbert space. However, because of the first class constraints (3.3a-b), these large symmetries(3.1) and (3.4a-b) are expressed in redundant way. We give a non-redundant expression of theselarge gauge symmetries and clarify the hidden gauge reducibility without using constraints.Let Λ − , be a string field of gauge parameters which lines in the large Hilbert space. Thegauge transformation free gauge transformation is written as follows δ Ψ , − = Q η Λ − , . (3.8)This is the origin of large gauge symmetries arising from the small theory. Clearly, this Λ − , equals to the half of gauge parameters appearing in the large theory [33, 44]. This gauge trans-formation (3.8) is invariant under the following gauge transformations δ ( δ Ψ) = 0 , δ Λ − , = Q Λ − , + η Λ − , , where Λ − − g,p denotes a higher gauge parameter. Note that these Λ − − g,p have the oppositeGrassmann parity to µ − g,p of (3.4a-b). Likewise, we find the higher gauge transformations δ g +1 ( δ g Λ − g,p ) = 0 , δ g Λ − g,p = Q Λ − − g,p + η Λ − − g,p +1 . (3.9)The p -label of the higher gauge parameter Λ − g,p runs from 0 to g −
1, and thus, the g = p lineof Λ − g,p does not appear in these large gauge symmetries of the small theory. The minimal set and free master action
Since these gauge parameters are Grassmann even unlike Ψ ≡ Ψ , − , we write Φ − − g,p for thestring field of ghosts corresponding to Λ − − g,p . We write (Ψ , − ) ∗ for the antifield of Ψ and(Φ − − g,p ) ∗ for the antifield of Φ − − g,p respectively. These are defined by h (Φ α ) ∗ , Φ α i = 1 andtheir Grassmann parities satisfy ( − ) Ψ = ( − ) Φ+1 , ( − ) Ψ ∗ = ( − ) Ψ+1 , and ( − ) Φ ∗ = ( − ) Φ+1 . Bycounting the hidden gauge reducibility (3.9), we find the minimal set of fields–antifieldsΨ , − ∈ H βγ , (cid:8) Φ − − g,p , (Ψ , − ) ∗ , (Φ − − g,p ) ∗ (cid:12)(cid:12) ≤ g , ≤ p ≤ g (cid:9) ⊂ H ξηφ . (3.10)Note that there is no constraint. The large gauge parameter (3.8) enables us to obtain theminimal set of fields–antifields for the small theory with large gauge symmetries.We find that for the free theory, a proper BV master action is given by S bv = 12 (cid:10) ξ Ψ , Q Ψ (cid:11) + (cid:10) (Ψ) ∗ , Q η Φ − , (cid:11) + ∞ X g =0 g X p =0 D (Φ − − g,p ) ∗ , Q Φ − − g,p + η Φ − − g,p +1 E . (3.11)Clearly, this is nothing but a partially gauge-fixed version of the free master action for Berkovitstheory [33, 44]. The origin of the large gauge symmetries of the small theory is the very trivialembedding of the small theory into the large Hilbert space (2.1), which seems to be trivial atthe classical level but gives such results at the level of the master action.While one can apply some useful techniques developed in the previous section to the masteraction (3.7) based on the nonminimal set with constraint, the master action (3.11) based on theminimal set (3.10) necessitates the BV formalism in the large Hilbert space [44]. We thus focuson the former and give a recipe for interacting theories in the rest of this paper.19 BV master action for interacting theory
In the previous sections, we showed that there is a hidden gauge reducibility in superstringfield theory based on the small dynamical string field Ψ ∈ H βγ whose gauge variation is alsosmall δ Ψ ∈ H βγ . It requires additional propagating ghost–antighost fields in the gauge fixedor quantum gauge theory, and thus changes the set of BV fields–antifields. While the resultantmaster action can takes different and enlarged forms, there exist canonical transformationsgetting it back to the canonical form. In this section, on the basis of these results, we presentmaster actions for several types of interacting superstring field theories. In the set of BV fields–antifields (2.7), while the subspace of BV fields Ψ g,p is restricted bythe constraints, there is no restriction on the subspace of BV antifields Φ ∗ − g, − − p ≡ (Ψ g,p ) ∗ .It enables us to have various patterns of the master action and its canonical transformations.However, when we take various canonical transformations into account and focus on the canonicalform of the master action S bv , these unrestricted antifields Φ ∗ ∈ H ξηφ all appear in the form of η Φ ∗ ∈ H βγ in S bv . Then, as we show, the master equation holds in a simple manner.We write Ψ − for the sum of all fields and Φ ∗ + for the sum of all antifields. When the minimalset of fields–antifields is given by (2.7), these Ψ and Φ ∗ take the following forms,Ψ − ≡ Ψ , − + ∞ X g =0 h Ψ − g, − + ∞ X p =1 Ψ − g, − − p i , Φ ∗ + ≡ Φ ∗ , + ∞ X g =0 h Φ ∗ g, + ∞ X p =0 Φ ∗ g,p i . We set ϕ ≡ ξ Ψ − + Φ ∗ + using these Ψ − and Φ ∗ + . When the original action S [Ψ] is given by (1.1),we can obtain the canonical form of the master action S bv using this ϕ by just replacing Ψ ofthe original action S [Ψ] with η ϕ as S bv = S [ ηϕ ], namely, S bv = K (cid:0) η ϕ, Q η ϕ (cid:1) + X n ≥ V n (cid:0) η ϕ, . . . , η ϕ (cid:1) + X g X n ≥ V g,n (cid:0) η ϕ, . . . , η ϕ | {z } n (cid:1) . (4.1)By construction, it quickly satisfies the classical master equation { S bv , S bv } = 0 or the quantummaster equation { S bv , S bv } = ~ ∆ S bv as the same manner as well-established theory basedon the geometry-inspired restrictions. In particular, BV-BRST transformations of Ψ − and Φ ∗ + are orthogonally split: δ BV Ψ − is η -exact and δ BV Φ ∗ + is ξ -exact, which kills extra higher gaugesymmetries. They are natural consequences of that we considered the canonical form.Therefore, additional ghost–antighost string fields arising from the hidden gauge reducibilitycertainly propagate and contribute in the loop amplitudes of superstrings. Contribution of eachghost–antighost term will be changed via a gauge choice and canonical transformations. Wethus expect that as usual gauge field theory, there exist appropriate gauge and suitable formof the master action for considering situations. For this purpose, the large class of canonicaltransformations and the large form of S bv should be clarified. However, unfortunately, it remainsunclear yet. We would like to emphasis that the above canonical form of S bv will be canonical-transformed one from this unknown but large form of S bv . See also [44] for new results.20 xample: classical master action for open superstrings In the rest of this subsection, we show it explicitly by taking open superstring field theory as anexample. We consider the NS action for [8, 11, 12] or the NS + R action for [14], S [Ψ] = 12 (cid:10) ξ Ψ , − , Q Ψ , − (cid:11) + X n> n + 1 (cid:10) ξ Ψ , − , M n (cid:0) n z }| { Ψ , − , . . . , Ψ , − (cid:1)(cid:11) , (4.2)where M n denotes the classical open superstring vertices { V g,n } g =0 , of (4.1). The string field Ψand its gauge variation must satisfy (2.2). Then, S bv = S [ ηϕ ] gives a solution of { S bv , S bv } = 0.Using coalgebraic notation (See [13] for example.), we can express S bv as S bv [Ψ − , Φ ∗ + ] = Z dt D ξ Ψ − + Φ ∗ + , π M − t η ( ξ Ψ − + Φ ∗ + ) E . (4.3)Here, t ∈ [0 ,
1] is a real parameter. It derivatives are given by the following forms, ∂ r S bv [Ψ − , Φ ∗ + ] ∂ Ψ − g, − − p = π ξ M − η ϕ (cid:12)(cid:12)(cid:12)(cid:12) g,p , ∂ l S bv [Ψ − , Φ ∗ + ] ∂ Φ ∗ g,p = π M − η ϕ (cid:12)(cid:12)(cid:12)(cid:12) − g, − − p . These are orthogonally split. In particular, the A ∞ vertices M acts on H βγ because all fields–antifields appear in the form of η ϕ = ξ Ψ − + Φ ∗ + , which permits any ξ -assignment in the classicalmaster equation. Because of η = η ξ η and η M + M η = 0, we find (cid:8) S bv , S bv (cid:9) = (cid:28) π M − η ( ξ Ψ − + Φ ∗ + ) , π ξ M − η ( ξ Ψ − + Φ ∗ + ) (cid:29) = 0 . The classical master action (4.3) can be derived by induction based on the antifield expansion.
On-shell gauge reducibility and BV spectrum
Let us check that (4.2) gives the same BV spectrum as (2.7) before considering the constructionof (4.3). The action (4.2) has the gauge invariance under δ Ψ , − = π M λ , − − Ψ , − ≡ Q Λ , − + ∞ X n =1 X cyclic M n +1 (cid:0) n z }| { Ψ , − , . . . , Ψ , − , λ , − (cid:1) , (4.4)where λ g,p is the coderivation inserting the state λ g,p . It is well-known that this gauge symmetryis on-shell infinitely reducible . Off-shell, there is no gauge reducibility and the gauge invariancenecessitates λ , − ∈ H βγ , or equivalently η λ , − = 0 . (4.5a)We often write δ Ψ , − = G λ , − for (4.4), and call G (or M ) as the gauge generator; this G becomes nilpotent operator on-shell, which yields the on-shell gauge reducibility. We find thaton-shell, the gauge parameter field Λ , − can protrude from the constraint (4.5a) as long as λ , − = η ξ λ , − + ∞ X n =1 n X k =1 M n (cid:0) n − k z }| { Ψ , − , . . ., ξ λ , − , k − z }| { Ψ , − , . . . (cid:1) , Then, as the inner product, we have to use that of [14]. Note that g and p of Ψ − g, − − p denote g -th reducibilityand p -decreasing from the natural picture number of considering string fields respectively. λ , − is an on-shell auxiliary gauge parameter fields. As well as (2.4a), it leads a familyof the on-shell auxiliary gauge parameter fields { λ , − − p } p> satisfying the constraint η Λ − p + π M − Ψ , − ⊗ λ , − − p ⊗ − Ψ , − = 0 . (4.5b)Hence, on-shell, we obtain { λ , − p } p> ⊂ H βγ ⊕ G H ξηφ with δ Ψ , − ≡ G λ , − again.Next, let us consider the on-shell gauge transformations δ for the gauge transformation(4.4), which must preserve (4.5b): For p ≥
0, they satisfy π M ( δ λ , − ) 11 − Ψ , − = 0 , η ( δ λ , − p ) + π M ( δ λ , − − p ) 11 − Ψ , − = 0 . Clearly, these yield the on-shell infinite gauge reducibility, and we find the g -th gauge transfor-mations δ g which preserve the ( g − δ g ( δ g − λ − g, − − p ) = 0, δ g λ − g, − = π M λ − g, − − Ψ , − , δ λ − g, − − p = η λ − , − p + π M λ − , − − p − Ψ , − . (4.6)Therefore, as we found in the previous section, in addition to Ψ , − and λ , − , an infinite towerof higher on-shell gauge parameters (4.6) appear in the interacting theory. Hence, the set offields–antifields is given by the same BV spectrum as (2.7). General form of the antifield number expansion
We construct a BV master action which consists of the BV spectrum (2.7). Again, we considerthe antifield number expansions of the master action (2.8) and the master equation, (cid:8) S bv , S bv (cid:9) = ∞ X a =0 (cid:8) S bv , S bv (cid:9)(cid:12)(cid:12) ( a ) . In this case, the initial condition of the BV master action, S (0) [ ψ ] ≡ S [Ψ], is given by (4.2). Forthis purpose, we derive the explicit form of the antifield number a part of the master equation { S bv , S bv } = 0 . Then, we find that only the perturbative solutions { S ( n ) } a +1 n =0 up to the antifieldnumber ( a + 1) appear in the antifield number a part of the master equation, (cid:8) S bv , S bv (cid:9)(cid:12)(cid:12) ( a ) = (cid:8) S (0) + · · · + S ( a +1) , S (0) + · · · + S ( a +1) (cid:9) , which is one of powerful properties of the antifield expansion in string field theory. When weconsider the BV master action S BV [Ψ , Φ ∗ ] based on (2.7), by the assignment of the antifieldnumber and space-time ghost number, each S ( a ) must satisfy the following relations ∂S (1+ a ) ∂ Φ ∗ g,p = ∂S ( a ) ∂ Ψ − g, − − p = 0 , ( g > a ) . They completely determine the explicit form of the antifield number expansion of the masterequation. Note that, by construction, afn[ S ( a ) ] = a holds. Because of afn[Φ ∗ g,p ] = 1 + g andafn[Ψ − g, − − p ] = 0 by definition, the antifield number of devatives are assigned asafn (cid:20) ∂S (1+ a ) ∂ Φ ∗ g,p (cid:21) = a − g , afn (cid:20) ∂S ( a ) ∂ Ψ − g, − − p (cid:21) = a .
22e thus obtain the following relation in the antifield number expansion, ∞ X a =0 (cid:8) S bv , S bv (cid:9)(cid:12)(cid:12) ( a ) = ∞ X a =0 X t,s X g,p (cid:28) ∂ r S ( t ) ∂ Ψ − g, − − p , ∂ l S ( s ) ∂ Φ ∗ g,p (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) t + s − g − a . The antifield number a terms are given by (cid:8) S bv , S bv (cid:9)(cid:12)(cid:12) ( a ) = a X s =0 s X g =0 ∞ X p =0 (cid:28) ∂ r S ( a − [ s − g ]) ∂ Ψ − g, − − p , ∂ l S (1+ s ) ∂ Φ ∗ g,p (cid:29) , (4.7)and we find that { S ( n ) } n>a +1 do not appear in { S bv , S bv }| ( a ) = 0 . Regarding ambiguity of ξ -assignments and lowest solution We start with the initial condition S (0) ≡ S [Ψ] given by (4.2). First, we would like to specifythe lowest solution S (1) = S (1) [Ψ , Φ ∗ ] which has to satisfy12 n S bv [Ψ , Φ ∗ ] , S bv [Ψ , Φ ∗ ] o(cid:12)(cid:12)(cid:12) (0) = 12 n S (0) + S (1) , S (0) + S (1) o = (cid:28) ∂S (0) ∂ Ψ , − , ∂S (1) ∂ Φ ∗ , (cid:29) = 0 . To find an appropriate S (1) , we have to determine these derivatives. Note that because ofthe constraint η Ψ , − = 0, the action S (0) [Ψ] = S [Ψ] permits any ξ -assignment: Using realparameters t , . . . , t n ∈ R satisfying t + · · · + t n = 1 , we can express it as S (0) [Ψ] = ∞ X n =1 n + 1 h t A n, [Ψ] + t A n, [Ψ] + · · · + t n A n,n [Ψ] i , where A n, and A n,k for k = 1 , . . . , n are defined by their ξ -assignments in M n , A n, [Ψ] ≡ (cid:10) Ψ , − , ξ M n (cid:0) n z }| { Ψ , − , . . . , Ψ , − (cid:1)(cid:11) ,A n,k [Ψ] ≡ − (cid:10) Ψ , − , M n (cid:0) k − z }| { Ψ , − , . . . , Ψ , − , ξ Ψ , − , n − k z }| { Ψ , − , . . . , Ψ , − (cid:1)(cid:11) . Hence, one can choose the right derivative of S (0) as − (cid:28) ∂ r S (0) [Ψ] ∂ Ψ , − , Ψ , − (cid:29) = t A [Ψ] + t A [Ψ] + · · · + t n A n [Ψ] . This ambiguity may enable us to construct various forms of S bv or give some hint to obtain alarge class of consistent master actions. However, we consider the simplest case in this paper.Since ηA [Ψ] = · · · = ηA n [Ψ] holds by acting η on these, we find an unambiguous expression η ∂ r S (0) [Ψ] ∂ Ψ , − = π M − Ψ , − = ∞ X n =1 M n (cid:0) n z }| { Ψ , − , . . . , Ψ , − (cid:1) . Thus if we set the following left derivative of S (1) , ∂ l S (1) ∂ Φ ∗ , = − π M ( η ξ Ψ , − ) 11 − Ψ , − ,
23t always satisfies the antifield number (0) part of the master equation with any ∂S (0) ∂ Ψ , − . Here, η ξ Ψ , − denotes the coderivation inserting η ξ Ψ , − into the tensor algebra of Ψ ≡ Ψ , − .Then, we obtain the lowest solution S (1) for { S (0) + S (1) , S (0) + S (1) } = 0 as S (1) = D Φ ∗ , , π M ( η ξ Ψ , − ) 11 − Ψ , − E . In general, the antifield number expansion does not uniquely determine the form of the masteraction because one can consider various canonical transformations at each order: It would bean interesting to consider all possible transformations of S (0) + · · · + S ( a ) . However, we considerthis simplest form of S (1) , the same form as well-established theory, and we would like to focuson the canonical form in this paper, which makes analysis very simple. Inductive construction of the simplest solution
In the rest, we consider the construction of the master action which takes the canonical form.Thus, we omit the projector η ξ in front of the coderivation η ξ Ψ g,p and write Ψ g,p for brevity.At this step, we have the following right derivatives ∂ r S (1) ∂ Ψ , − = π M Φ ∗ , − Ψ , − + η -exact ,∂ r S (1) ∂ Ψ , − = π M Φ ∗ , Ψ , − − Ψ , − + η -exact . We set ∂ r S (1) ∂ Ψ , − p = 0 for p >
1. Thus, to solve the antifield number 1 part of the equation, (cid:8) S BV , S BV (cid:9)(cid:12)(cid:12) (1) = ∞ X p =0 (cid:28) ∂ r S (1) ∂ Ψ , − − p , ∂ l S (2) ∂ Φ ∗ ,p (cid:29) + (cid:28) ∂ r S (0) ∂ Ψ , − , ∂ l S (2) ∂ Φ ∗ , (cid:29) + (cid:28) ∂ r S (1) ∂ Ψ , − , ∂ l S (1) ∂ Φ ∗ , (cid:29) , we need the following terms12 D M Φ ∗ , , M Ψ , − Ψ , − E + 12 D M Φ ∗ , Ψ , − Ψ , − , M E . These terms should be provided by the inner product of the above field derivatives and the nextantifield derivatives. Therefore, the left derivatives of S (2) should be ∂ l S (2)0 ∂ Φ ∗ ,p = π M h Ψ − , − − p + p X q =0 Ψ , − − q Ψ , − − ( p − q ) i − Ψ , − ,∂ l S (2)0 ∂ Φ ∗ , = π M ( η Φ ∗ , ) h Ψ − , − + 12 ( Ψ , − ) i − Ψ , − . We thus find that the antifield number 2 part is given by S (2) = ∞ X p =0 (cid:28) Φ ∗ ,p , π M (cid:20) Ψ − , − − p + 12 ( Ψ , − ) + p X q =0 Ψ , − − q Ψ , − − ( p − q ) (cid:21) − Ψ , − (cid:29) + 12 (cid:28) η Φ ∗ , , π MΦ ∗ , h Ψ − , − + 12 ( Ψ , − ) i − Ψ , − (cid:29) . n ≡ Ψ , − + n − X g =0 h Ψ − g, − + ∞ X p =1 Ψ − g, − − p i , Φ ∗ n ≡ Φ ∗ , + n − X g =0 h Φ ∗ g, + ∞ X p =0 Φ ∗ g,p i , we can obtain the following expression for n = 2 , S (0) + · · · + S ( n ) = Z dt D ξ (Ψ n + η Φ ∗ n − ) , π M − t η ( ξ Ψ n + Φ ∗ n − ) E . (4.8)Using this S (2) , one can determine the derivatives of S (3) . For example, we have the p = 0right derivatives ∂ r S (2)0 ∂ Ψ − , − = π M h Φ ∗ , + 12 ( η Φ ∗ , ) Φ ∗ , i − Ψ , − + η -exact ,∂ r S (2)0 ∂ Ψ − , − = π M h Φ ∗ , + 12 ( η Φ ∗ , ) Φ ∗ , i Ψ , − − Ψ , − + η -exact ,∂ r S (2)0 ∂ Ψ , − = π M h Φ ∗ , + 12 ( η Φ ∗ , ) Φ ∗ , i(cid:0) Ψ − , − + 12 ( Ψ , − ) (cid:1) − Ψ , − + η -exact . Note that although it seems that the second terms are enough to satisfy the lower masterequation, the first terms of these derivative are necessitated to fix the on-shell gauge reducibilityof S (0) , which satisfy the lower master equation individualy. The antifield number 2 part is (cid:8) S bv , S bv (cid:9)(cid:12)(cid:12) (2) = ∞ X p =0 (cid:20)(cid:28) ∂ r S (2) ∂ Ψ − , − − p , ∂ l S (3) ∂ Φ ∗ ,p (cid:29) + (cid:28) ∂ r S (1) ∂ Ψ , − − p , ∂ l S (3) ∂ Φ ∗ ,p (cid:29)(cid:21) + (cid:28) ∂ r S (0) ∂ Ψ , − , ∂ l S (3) ∂ Φ ∗ , (cid:29) + ∞ X p =0 (cid:28) ∂ r S (2) ∂ Ψ , − − p , ∂ l S (2) ∂ Φ ∗ ,p (cid:29) + (cid:28) ∂ r S (1) ∂ Ψ , − , ∂ l S (2) ∂ Φ ∗ , (cid:29) + (cid:28) ∂ r S (2) ∂ Ψ , − , ∂ l S (1) ∂ Φ ∗ , (cid:29) . Therefore, for example, we find that the p = 0 slice of the second line requires (cid:28) π M h Φ ∗ , + 12 ( η Φ ∗ , ) Φ ∗ , i Ψ , − − Ψ , − , π M h Ψ − , − + 12 ( Ψ , − ) i − Ψ , − (cid:29) + (cid:28) π MΦ ∗ , Ψ , − − Ψ , − , π M ( η Φ ∗ , ) h Ψ − , − + 12 ( Ψ , − ) i − Ψ , − (cid:29) + (cid:28) π M h Φ ∗ , + 12 ( η Φ ∗ , ) Φ ∗ , i(cid:0) Ψ − , − + 12 ( Ψ , − ) (cid:1) − Ψ , − , π MΨ , − − Ψ , − (cid:29) . It implies that the p = 0 left derivatives of S (3) should be ∂ l S (3) ∂ Φ ∗ , = π M h Ψ − , − + Ψ − , − Ψ , − + 13! ( Ψ , − ) i − Ψ , − ∂ l S (3) ∂ Φ ∗ , = π M ( η Φ ∗ , ) h Ψ − , − + Ψ − , − Ψ , − + 13! ( Ψ , − ) i − Ψ , − ∂ l S (3) ∂ Φ ∗ , = π M (cid:0) η Φ ∗ , + 12 ( η Φ ∗ , ) (cid:1)h Ψ − , − + Ψ − , − Ψ , − + 13! ( Ψ , − ) i − Ψ , − . In the same manner, one can obtain the p > Ψ − g, − p and Φ ∗ g,p in the above p = 0 ones. These derivatives determine S (3) , andthis S (3) gives (4.8) for n = 3 . Inductively, one can prove that S ( n +1) satisfies (4.8) for n + 1when S ( n ) satisfying it. Hence, we obtain the canonical form of the classical master action S bv as the n → ∞ limit of (4.8). 25 .2 BV master actions for superstring field theories We found that by adding higher p -labeled fields–antifields into the known minimal set, by con-sidering the sum of all fields Ψ − and antifields Φ ∗ + , and by setting ϕ ≡ ξ Ψ − + Φ ∗ + , the canonicalform of the master action S bv including these additional propagating fields–antifields is obtainedby just replacing Ψ of the known action S [Ψ] with ηϕ , namely, S bv = S [ ηϕ ]. Open superstring field theories
We know classical master actions S [ e Ψ] for geometrical open superstring field theories [11], andtheir homotopy algebraic versions [12,14]. From the analysis of the gauge reducibility of [12,14],we find that the minimal set of fields–antifields can be enlarged as (cid:8) Ψ NS1 − g, − − p , Φ ∗ NS1+ g,p ≡ (Ψ NS1 − g, − − p ) ∗ ; Ψ R1 − g, − − p , Φ ∗ R1+ g, + p ≡ (Ψ R1 − g, − − p ) ∗ (cid:9) g,p ≥ . Note that except for Ψ
NS1 , − , Ψ R1 , − ∈ H βγ and { Ψ NS / R0 , − p } ⊂ H βγ ⊕G H ξηφ , the other fields–antifieldsare unrestricted { Ψ , Φ ∗ } ⊂ H ξηφ . Therefore, by setting ϕ ≡ ξ Ψ − + Φ ∗ + whereΨ − ≡ Ψ NS1 , − + Ψ R1 , − + ∞ X g =0 ∞ X p =0 h Ψ NS − g, − p + Ψ R − g, − p i , Φ ∗ + ≡ Φ ∗ NS1 , + Φ ∗ R1 , + ∞ X g =0 ∞ X p =0 h Φ ∗ NS1+ g,p + Φ ∗ R1+ g, + p i , we can quickly obtain the canonical form of the master action S bv by just replacing e Ψ of theaction S [ e Ψ] given in [14] with ηϕ , namely S bv = S [ ηϕ ]. (Note that there is the Y -insertion inthe inner product of R states.) For [11], we consider the set of fields–antifields (cid:8) Ψ NS1 − g, − − p , Φ ∗ NS1+ g,p ; Ψ R1 − g, − − p , Φ ∗ R1+ g, − + p ≡ (Ψ R1 − g, − − p ) ∗ , e Ψ R1 − g, − − p , ( e Ψ R1 − g, − − p ) ∗ (cid:9) g,p ≥ . We set ϕ ≡ ξ Ψ − + Φ ∗ + and e ψ ≡ ξ e Ψ − + e Ψ ∗ + whereΨ − ≡ Ψ NS1 , − + Ψ R1 , − + ∞ X g =0 ∞ X p =0 h Ψ NS − g, − p + Ψ R − g, − p i , e Ψ − ≡ e Ψ R1 , − + ∞ X g =0 ∞ X p =0 e Ψ R1 − g, − − p , Φ ∗ + ≡ Φ ∗ NS1 , + Φ ∗ R1 , − + ∞ X g =0 ∞ X p =0 h Φ ∗ NS1+ g,p + Φ ∗ R1+ g, − + p i , e Ψ ∗ + ≡ ( e Ψ R1 , − ) ∗ + ∞ X g =0 ∞ X p =0 ( e Ψ R1 − g, − − p ) ∗ . Then, the enlarged BV master action S bv is obtained by just replacing (Ψ , ψ ) and φ of S [(Ψ , ψ ); φ ]given in [11] with ηϕ and η e ψ respectively: S bv = S [ ηϕ ; η e ψ ]. Type II and Heterotic theories
We have quantum master actions S q [Ψ , e Ψ] for type II and heterotic superstring field theories[9, 10], and their classical and homotopy algebraic versions [13]. We consider the quantummaster action S q of [10]. Then, the analysis of its gauge gauge reducibility implies that one canintroduce the additional fields–antifields (cid:8) Ψ NS − g, − − p , Φ ∗ NS2+ g,p ; Ψ R − g, − − p , Φ ∗ R2+ g, − + p , e Ψ R − g, − − p , ( e Ψ R − g, − − p ) ∗ (cid:9) g,p ≥ . − and Φ ∗ + be the sums of all fields and antifields without “tilde”, respectively, and let e Ψ − and e Ψ ∗ + be the sums of all fields and antifields with“tilde”, respectively. Using these, we set ϕ ≡ ξ Ψ − + Φ ∗ + and e ψ ≡ ξ e Ψ − + e Ψ ∗ + . Then, the quantum BV master action S bv is obtained byjust replacing Ψ and e Ψ of S q [Ψ , e Ψ] given in [10] with ηϕ and η e ψ respectively; S bv = S q [ ηϕ, η e ψ ]. In this paper, we showed that there exists a hidden gauge reducibility in superstring field theorybased on Ψ , δ Ψ ∈ H βγ . It necessitates additional propagating ghost–antighost fields in thegauge fixed or quantum gauge theory, and thus changes the set of fields–antifields. In terms of agauge theory, it corresponds to hundle higher gauge symmetry which is fixed or ignored so far.We proved that the resultant master action can takes a different and enlarged form, and thatcanonical transformations fills their gap.We also checked that these additional fields–antifields can be put into the master actions forthe interacting theories. Hence, these additional propagating degrees of freedom indeed appearin loop amplitudes of superstring field theory. It is known that we sometime encounter singularsituations, such as spurious poles, in usual loop calculations [36, 37]. Our analysis of the gaugestructure implies that one can include additional contributions for loops via the gauge choice .Thus, it will be an interesting question whether one can control such singularities appearing inthe loop superstring amplitudes via the gauge invariance of the field theory.Since the set of fields–antifields is enlarged, one cannot obtain unique S bv by just relaxing theghost number constraint, unlike usual cases. It implies the existence of a larger class of consistentBatalin-Vilkovisky master actions for superstring field theory. We presented it explicitly for freetheory, and gave not all but several exact results for interacting theory. Interestingly, it remindus the WZW-like formulation of superstring field theory [22–31].What is the origin of the hidden gauge reducibility and these additional loop-propagatingdegrees? It will be the ambiguity appearing in the expressions of the gauge transformations (forgauge transformations) in superstring field theory based on Ψ , δ Ψ ∈ H βγ . In section 3.1, weconsidered another gauge parameter ω , ∈ Ker[ η ] ∪ Ker[ Q ] such that Q λ , − = η ω , , whichprovides another expression δ Ψ , − = η ω , of the gauge transformation δ Ψ , − = Q λ , − with λ , − ∈ Ker[ η ] ∪ Ker[ Q ] . Although it is just a redefinition of gauge parameters because of δ Ψ , − ∈ H βγ , as we showed, the ambiguous expression of the (first) gauge transformation δ Ψ , − = Q µ , − + η µ , provides larger and independent set of higher gauge parameters. Inparticular, the ( g + 1)-th gauge transformations take the form of δ g µ − g,p = Q µ g,p + η µ g,p +1 as Berkovits theory [32–35], and the p -label can run over all integer numbers. It requires manyadditional fields–antifields { Ψ g,p , (Ψ g,p ) ∗ } g ≤ ,p ∈ Z into the BV spectrum. As we showed in section3.2, if we take a gauge variation δ Ψ =
Q η Λ − , using a large gauge parameter Λ − , ∈ H ξηφ , theexpression of the large gauge invariance is no longer ambiguous. Then, additional fields-antifields { Φ − − g,p , (Φ − − g,p ) ∗ } ≤ p ≤ g are nothing but those of Berkovits theory. But it requires the BVformalism in the large Hilbert space [44]. In this sense, the additional fields–antifields arise fromambiguous expressions of the gauge invariances of superstring field theory based on the smallHilbert space, which is a result of the very trivial embedding (2.1) into the large Hilbert space.27his underling gauge structure of superstring field theory based on the small Hilbert spaceresembles that of the WZW-like formulation. Recently, it was shown that superstring fieldtheory based on H βγ can be embedded into the WZW-like formulation unless taking their gaugereducibility into account [30, 31]. In several example, it is known that classical actions based on H βγ can be obtain from WZW-like ones via field redefinitions reducing gauge symmetries [38–40],which anticipates corresponding canonical transformations. We may expect some exact relationsof these at the level of their master actions. We end this paper with some remarks about it. The general WZW-like formulation is a purely algebraic generalisation of the geometrical frame-work explained in section 1, in which the linear η -constraints on the states are extend to anonlinear C -constraints based on the homotopy algebra C whose linear part is η [30, 31]. Forsimplicity, we take open superstring field theory as an example. Let ( C , V ) be a mutually com-mutative pair of A ∞ algebras: C is some nonlinear extension of η and V is the string vertices { Q, V g,n } g,n given in section 1. Then, using a dynamical string field ϕ of the theory, we considera solution A C [ ϕ ] of the Maurer-Cartan equation for C , π C − A C [ ϕ ] = 0 . This A C [ ϕ ] is a functional of the dynamical string field ϕ . Note that Ψ ∈ H βγ satisfy η Ψ = 0and gives a trivial example for the case of C = η . Note also that when we take C = η − m and V = Q ( m is Witten’s star product), it reduces to the Berkovits theory [22].When D is a derivation operator for C , or more generally, an A ∞ product D commutingwith C , one can define a functional A D [ ϕ ] such that( − ) D D − A C [ ϕ ] = C − A C [ ϕ ] ⊗ A D [ ϕ ] ⊗ − A C [ ϕ ] . It is a generalisation of the relation ∂ t ( e tφ de − tφ ) = d ( e tφ ∂ t e − tφ ) + [ e tφ de − tφ , e tφ ∂ t e − tφ ] satisfiedby a pure-gauge state e − tφ ( d e tφ ) of Chern-Simons theory ( d is the exterior derivative and theproduct is the wedge product.). For a real parameter t ∈ [0 , ∂ t worksas a derivation for C . The variation δ of the field also satisfies the Liebniz rule for C . Thus,one can take D = ∂ t or D = δ for example. Because of mutual commutativity, one can also take D = V . Using these, the general WZW-like action S wzw is given by S wzw [ ϕ ] = Z dt D A ∂ t [ tϕ ] , π V − A C [ tϕ ] E . (5.1)This S wzw [ ϕ ] gives a gauge invariant action for any A ∞ pairs ( C , V ). One can quickly get a proofby omitting one of the constraints in [30]. See also [31] for detailed and pedagogical explanationsabout the general WZW-like action. The gauge transformations are A δ [ ϕ ] = π (cid:16)(cid:2) C , Λ C (cid:3) + (cid:2) V , Λ (cid:3)(cid:17) − A C [ ϕ ] . Here, Λ C and Λ are gauge parameters. See [31] or section 7 of [41] for the coalgebraic notation.In general, field redefinitions b U drastically change the string vertices V in highly nontrivial28anner. In terms of the A ∞ pairs, it just gives a (weak) A ∞ morphism between two A ∞ pairs, b U : ( C , V ) → ( C ′ , V ′ ) , which preserves the solutions of the Maurer-Cartan equations but maychange the forms of the above functionals. Hence, this S wzw [ ϕ ] is covariant under string fieldredefinitions. Therefore, as a gauge field theory, S wzw [ ϕ ] and its Batalin-Vilkovisky masteraction S BV will capture very general properties of superstrings, which is also supported by ourresults obtained in section 2.Unfortunately, we do not have enough understandings about the most general form of S BV yet. In the rest, we consider a slightly generalised version of our analysis, the same pair ( η , V )but large fields ϕ ∈ H ξηφ , which is the second simplest but first nontrivial example of S wzw [ ϕ ]. Example: Large A ∞ open superstring field theory We write Φ , for a dynamical string field, which lives in the state space of ξηφ -system, the largeHilbert space H ξηφ . Thus, there is no constraint on the string field Φ , . In this set up, we find A C [Φ] = η Φ , , and the classical action is given by S A [Φ] = 12 (cid:10) Φ , , Q η Φ , (cid:11) + X n> n + 1 (cid:10) ξ η Φ , , M n (cid:0) n z }| { η Φ , , . . . , η Φ , (cid:1)(cid:11) , which takes the same form as (4.2) except for the dynamical string field, S A [Φ] = S [ η Φ]. Thistheory has large gauge invariances generated by two gauge generators: δ Φ , = η Λ − , + X cyclic ∞ X n =0 M n +1 (cid:0) n z }| { η Φ , , . . . , η Φ , , Λ − , (cid:1) , where we write Λ − g,p for a gauge parameter field living in H ξηγ . Since its kinetic term is that ofthe Berkovits theory, it is infinitely reducible gauge theory [32, 33]. The gauge invariance of thekinetic term is given by δ Φ , = η Λ − , + Q Λ − , . Using the nilpotency of ( Q ) = ( η ) = 0 andthe graded commutation relation η Q + Q η = 0, we find the g -th gauge transformations for the( g − δ g Λ − g,p = η Λ − g − ,p +1 + Q Λ − g − ,p satisfying δ g ( δ g − Λ − g,p ) = 0for 0 ≤ p ≤ g . Since these gauge parameter fields turn into ghosts { Φ g,p } g ≤− , ≤ p ≤| g | and theylead antighosts, the minimal set of the fields–antifields is given by (cid:8) Φ − g,p , Φ ∗ g, − − p (cid:9) ≤ g, ≤ p ≤ g ⊂ H ξηφ . (5.2)Note that there is no constraints on the BV spectrum. The free master action takes the sameform as (2.11b) except for the BV spectrum. See [32–35] for details. Large master action for interacting theory
As we showed in section 3, we can construct the master action in the canonical form, which willbe a canonical-transformed version of the unclear original form of S BV . Unfortunately, we donot have clear understanding about the most general form of S BV . However, in this case, onecan find a more enlarged form of S BV . See also [44]. We set ϕ ≡ X g ≥ g X p =0 h Φ − g,p + Φ ∗ g, − − p i . For type II theory, see [30]. One can consider heterotic theory by omitting one of the constraint L ∞ algebrasin type II theory or by replacing the pair of A ∞ algebras of open superstring theory with that of L ∞ algebras. ϕ is given by using the same symbols as (5.2), these fields–antifields { Φ − g,p , (Φ − g,p ) ∗ } ≤ p ≤ g should be regarded as some canonical transformed ones from (5.2). Let ǫ be an operator counting the grading of the state: ǫ Φ = ( − ) ǫ [Φ] Φ = ( − ) | Φ | Φ. Then, we findthat the following S BV satisfies the master equation, S BV [ ϕ ] = Z dt D ϕ − ǫ η ϕ , π ( η + M ) 11 − t ϕ − t ǫ η ϕ E = 12 (cid:10) ϕ , η ϕ (cid:11) + ∞ X n =1 n + 1 D ϕ − ǫ η ϕ , M n (cid:0) n z }| { ϕ + ǫ η ϕ , . . . , ϕ + ǫ η ϕ (cid:1) E . Since the space-time ghost number of S BV [ ϕ ] equals to zero, for which we write s ( S BV [ ϕ ]) = 0,its total degree is also zero: ǫ ( S BV [ ϕ ]) = 0 . The variation of S BV is given by δS bv [ ϕ ] = D δϕ , π (cid:0) η + M − η M (cid:1) − ϕ − ǫ η ϕ E = (cid:10) δϕ , η ϕ (cid:11) + ∞ X n =1 D δϕ , M n (cid:0) n z }| { ϕ + ǫ η ϕ , . . . (cid:1) − η M n (cid:0) n z }| { ϕ + ǫ η ϕ , . . . (cid:1) E . We thus find that the gauge invariance of S BV [ ϕ ], or the BV-BRST transformations of the BVmaster action, is given by δϕ = η ϕ + π (cid:2) M − η M (cid:3) − ϕ − ǫ η ϕ . We write ( s ; g, p ) of | s ; g,p for the projection onto the space-time ghost number s world-sheet ghostnumber g , and picture number p state. Then, one can express the BV-BRST transformations δ BV Φ − g,p and δ BV Φ ∗ g, − p as follows ∂ l S BV [ ϕ ] ∂ Φ ∗ g, − p = η Φ − g,p + ∞ X n =1 h M n (cid:0) n z }| { ϕ + η ϕ , . . . (cid:1) − η M n (cid:0) n z }| { ϕ + η ϕ , . . . (cid:1)i g ;1 − g,p − ,∂ r S BV [ ϕ ] ∂ Φ − g,p = η Φ ∗ g, − p + ∞ X n =1 h M n (cid:0) n z }| { ϕ + η ϕ , . . . (cid:1) − η M n (cid:0) n z }| { ϕ + η ϕ , . . . (cid:1)i − g ;2+ g, − − p . These derivatives provide the classical BV master equation. Recall that A ∞ relations imply ∞ X i =1 ∞ X j =1 D M i ( A, ..., A ) , M j ( A, . . . , A ) E = ∞ X n =1 n − X l =0 D M l +1 ( A, ..., A ) , M n − l ( A, . . . , A ) E = ∞ X n =1 n + 1 n − X l =0 (cid:2) ( l + 1) + ( n − l ) (cid:3)D M l +1 ( A, ..., A ) , M n − l ( A, . . . , A ) E = ∞ X n =1 n + 1 n − X l =0 h l X k =0 D A, M l +1 ( k z }| { A, ..., M n − l ( A, . . . , A ) , l − k z }| { ..., A ) Ei = 0 . Because of these A ∞ relations and ( η ) = 0 , we quickly find12 n S BV [ ϕ ] , S BV [ ϕ ] o = X m D η ϕ, M m ( ϕ + ǫ η ϕ, ... ) E + X n D M n ( ϕ + ǫ η ϕ, ... ) , η ϕ E = 0 , η and M . Note that the antibracket { S BV , S BV } has space-time ghost number one, s ( { S BV , S BV } ) = 1 , and thus its total degree is one, ǫ ( { S BV , S BV } ) = 1 .Hence, in this case, (cid:10) A, B (cid:11) = − (cid:10) B, A (cid:11) holds for any states
A, B satisfying s ( A ) + s ( B ) = 1 .Off course, one may be able to construct a more enlarged form of the master action in-cluding not only η but also many M . We would like to emphasise that a natural perturbativeconstruction based on the antifield number expansion anticipates such a larger solution. Thus,the above S BV will be also canonical-transformed one. These unknown but interesting feature ofthe gauge invariances may be understood by canonical transformations as we shown in section2. We expect that our results gives a first step to obtain clear insights into the gauge structureand field theoretical properties of superstrings. See [44] for other types of master actions. Acknowledgements
The author would like to thank Theodore Erler, Hiroshi Kunitomo, Martin Schnabl. The authoralso thank Hiroshige Kajiura and Jim Stasheff. This research has been supported by the GrantAgency of the Czech Republic, under the grant P201/12/G028.The author thanks Nathan Berkovits for email.
A On the string fields–antifields
In this appendix A.1, after explaining some properties of the BPZ inner product and its basis,we give a string field representation of the BV antibracket. In appendix A.2, we first give the freeBV master action for the exactly small theory. Then, we show how the BV spectrum is changedby relaxing the η -constraints on higher ghost-fields–antifields of the exactlly small theory. A.1 String antibracket
Let { Ψ g,p , (Ψ g,p ) ∗ } g,p ⊂ H ξηφ be the minimal set of fields–antifields in superstring field theory.While the g -label of Ψ g,p denotes that it corresponds to the g -th ghost fields, the p -label of Ψ g,p distinguishes their difference at the same g -label. Then, the antibracket { F, G } of two functions F = F [Ψ , Ψ ∗ ] and G = G [Ψ , Ψ ∗ ] can be presented by (cid:8) F, G (cid:9) = X g X p (cid:20)(cid:28) ∂ r F∂ Ψ g,p , ∂ l G∂ Ψ ∗ g,p (cid:29) − (cid:28) ∂ r F∂ Ψ ∗ g,p , ∂ l G∂ Ψ g,p (cid:29)(cid:21) , (A.1)where h A, B i is the large BPZ inner product of two string fields A and B . One can quickly derivethis string field representation of the BV antibracket (A.1) by direct computations of usual BVantibracket under two assumptions about string fields and their functions. Graded symplectic BPZ product
The large BPZ inner product h A, B i bpz takes nonzero value if and only if the sum g of theinputs’ world-sheet ghost numbers and the sum p of the inputs’ picture numbers equals to the31ppropriate value : ( g, p ) = (2 , −
1) for open strings, ( g, p ) = (4 , −
1) for heterotic strings, and( g, p, ˜ p ) = (3 , − , −
1) for type II strings. Its Grassmann parity depends on the theory. In thispaper, we introduce an appropriate grading, so-called degree, and consider suspended versions,in which string fields have degree 0 and the string vertices have degree 1 (see [7, 13].). For opensuperstring field theory based on Ψ ∈ H βγ , the degree of states are defined by “space-time ghostnumber + world-ghost number − ∈ H βγ ,the degree of states are defined by “space-time ghost number + world-ghost number − h A, B i is given by (cid:10) A, B (cid:11) ≡ ( − ) G ( A ) (cid:10) A, B (cid:11) bpz , where G ( A ) denotes the Grassmann parity of the state A . Then, the large BPZ inner productbecomes symplectic, all BV fields Ψ ∈ H βγ have degree 0, and all antifields (Ψ) ∗ ∈ H ξηφ havedegree −
1. We write ǫ [Ψ] for the (total) grading of Ψ, its (total) degree.In string field theory, string fields { Ψ g,p } g,p consist of a set of space-time fields { A g,p } g,p and a set of world-sheet basis {Z g,p } g,p , for which we write Ψ s ; g,p ≡ A s,p |Z g,p i . As well as theworld-sheet basis Z g,p , the space-time field A s,p also has its grading ǫ [ A s,p ] which is equal to itsspace-time ghost number s [ A s,p ] = s . We thus find ǫ [Ψ s ; g,p ] = ǫ [ A s,p ] + ǫ [ Z g,p ] and (cid:10) Ψ s ; g,p , Ψ t ; h,q (cid:11) = − ( − ) ǫ [Ψ s ; g,p ] ǫ [Ψ t ; h,q ] (cid:10) Ψ t ; h,q , Ψ s ; g,p (cid:11) . We write ǫ [ ω ] for the grading of the symplectic BPZ product: ǫ [ h A, B i ] = ǫ [ ω ] + ǫ [ A ] + ǫ [ B ]. Inthis paper, our computations are based on the following defining relations (cid:10) A s Z g,p , A t Z h,q (cid:11) = ( − ) ǫ [ A t ] ǫ [ Z h,q ] (cid:10) A s Z g,p , Z h,q (cid:11) A t = ( − ) ǫ [ ω ] ǫ [ A s ] A s (cid:10) Z g,p , A t Z h,q (cid:11) . Batalin-Vilkovisky antibracket
We write { A g,p , ( A g,p ) ∗ } g,p for the minimal set of space-time fields–antifields in usual gaugefield theory. The g -label of A g,p denotes that A g,p corresponds to the g -th gauge reducibility,namely, its space-time ghost number, for which we write s [ A g,p ] = g . Then, by construction,corresponding antighost ( A g,p ) ∗ has space-time ghost number − ( g + 1), for which we write s [( A g,p ) ∗ ] = − ( g + 1). Let F = F [ A, A ∗ ] and G = G [ A, A ∗ ] be functions of these space-timefields–antifields. In the Batalin-Vilkovisky formalism, using these space-time ghost–antighostfields, the BV antibracket { F, G } is given by { F, G } = X g X p (cid:18) ∂ r F∂A g,p ∂ l G∂A ∗ g,p − ∂ r F∂A ∗ g,p ∂ l G∂A g,p (cid:19) . (A.2)The antibracket has space-time ghost number +1 and s [ { F, G } ] = s [ F ] + s [ G ] + 1 holds. Recallthat computations of the antibracket is based on the following expression of the variation, δF [ A, A ∗ ] = X g,p (cid:20) ∂ r F∂A g,p δA g,p + ∂ r F∂A ∗ g,p δA ∗ g,p (cid:21) = X g,p (cid:20) δA g,p ∂ l F∂A g,p + δA ∗ g,p ∂ l F∂A ∗ g,p (cid:21) . We consider the ( c − ˜ c )-inserted one for closed string field theory, in which all string fields Ψ satisfy( b − ˜ b )Ψ = 0 and ( L − ˜ L )Ψ = 0. tring field representation We would like to obtain a string field representation of the BV antibracket. Let λ − g,p be the g -th gauge parameter fields, which consists of sets of space-time gauge parameter fields { B g,p } p and world-sheet basis {Z − g,p } p . Then, its ghost field Ψ − g,p is obtained by replacing B g,p of λ g,p by corresponding g -th space-time ghost field A g,p , namely, Ψ − g,p = A g,p |Z − g,p i . Hence,the field Ψ − g,p has space-time ghost number g , world-sheet ghost number 1 − g , and Grassmannparity 1, for which we write s [Ψ g,p ] = g , gh[Ψ g,p ] = 1 − g , and G [Ψ g,p ] = 1 respectively. Then,the antifield (Ψ − g,p ) ∗ consists of sets of the g -th space-time ‘antighost’ fields { A ∗ g,p } p havingspace-time ghost number − ( g +1) and the world-sheet basis {Z ∗ − g,p } p satisfying hZ g,p , Z ∗ g,p i 6 = 0.Therefore, as we saw in section 2, we have Ψ ∗ − g,p = A ∗ g |Z g, i for open superstring field theory.Note that all fields { Ψ g,p } g,p have degree zero and all antifields { Ψ ∗ g,p } g,p have degree −
1, forwhich we write ǫ [Ψ g,p ] = 0 and ǫ [Ψ ∗ g,p ] = −
1. Note also that s [(Ψ g,p ) ∗ ] = − ( g + 1).Let F = F [Ψ , Ψ ∗ ] be a function of given minimal set of fields–antifields { Ψ g,p , (Ψ g,p ) ∗ } g,p .In string field theory, any C -value function of string fields is written by using the BPZ innerproduct. Then, we assume that the variations of fields–antifields are given by δ Ψ g,p ≡ δA g,p |Z g,p i , δ Ψ ∗ g,p ≡ δA ∗ g,p |Z ∗ g,p i . Likewise, we assume that the variation of any C -value functional F [Ψ , Ψ ∗ ] is given by δF [Ψ , Ψ ∗ ] ≡ X g,p (cid:20)D δ Ψ g,p , ∂ l F∂ Ψ g,p E + D δ Ψ ∗ g,p , ∂ l F∂ Ψ ∗ g,p E(cid:21) = X g,p (cid:20)D ∂ r F∂ψ g,p , δ Ψ g,p E + D ∂ r F∂ Ψ ∗ g,p , δ Ψ ∗ g,p E(cid:21) . On the basis of these string field representations of the variations, the BV antibracket (A.2) iswritten into its string field representation (A.1). The string field derivatives are defined in thesame manner. Then, we have to pay attention to the grading of the inner product.
A.2 Constraints on BV spectrums
We consider to relax the restrictions on the gauge parameters step by step. Then, the BVspectrums are enlarged, and the master actions has larger gauge invariances. For brevity, in thisappendix, we consider open superstring fields as an example.
The exactly small theory
First, let us consider to restrict the dynamical string field Ψ , − and the gauge parameter field λ , − onto the small Hilbert space H βγ . Using the small BPZ inner product, the gauge invariantaction is given by S s = 12 (cid:10)(cid:10) Ψ , − , Q Ψ , − (cid:11)(cid:11) , η Ψ , − = 0 . (A.3)Because of the constraints, this theory is gauge invariant under the (first) gauge transformation δ Ψ , − = Q λ , − , η λ , − = 0 . It yields the gauge reducibility δ n +1 ( δ n λ − n, − ) = 0 under the n -th gauge transformation for the( n − δ n λ − n, − = Q λ − ( n +1) , − . , − and λ , − must satisfy the constraint equations η Ψ , − = 0 and η λ , − = 0 ,the other higher gauge parameters { λ − g, − } g> do not have to satisfy any constraint equation.However, when we use the small BPZ inner product to obtain the BV master action, we haveto impose the same constraint equations on the all higher gauge parameters, η λ − n, − = 0 , ( n ∈ N ) . Then, the set of gauge parameters { λ g, − } g ≤ appears in the theory. It implies that theminimal set of fields–antifields is given by (cid:8) Ψ , − , Ψ − g, − (cid:9) g ≥ ⊂ H βγ , (cid:8) Ψ ∗ g, − ≡ (Ψ − g, − ) ∗ (cid:9) g ≥ ⊂ H βγ . We write (Ψ g,p ) ∗ for the antifield of Ψ g,p , whose ghost and picture numbers is determined viathe BPZ inner product of the theory as hh (Ψ g,p ) ∗ , Ψ g,p ii 6 = 0. We thus write Ψ ∗ g, − for theantifield of (Ψ − g, − ) ∗ , namely Ψ ∗ g, − ≡ (Ψ − g, − − p ) ∗ . These fields and antifields must satisfythe constraint equations η Ψ − g, − = 0 , η Ψ ∗ g, − = 0 , ( g ∈ { } ∪ N ) . Note that for each field Ψ or each antifield Ψ ∗ , the sum of the space-time and world-sheet ghostnumbers is just 1. In other words, the minimal set consists of the degree 0 states: ǫ [Ψ − g, − ] = 0and ǫ [Ψ ∗ g, − ] = 0 . Using this minimal set, one can construct the master action, S bv , s [Ψ , Ψ ∗ ] = 12 (cid:10)(cid:10) Ψ , − , Q Ψ , − (cid:11)(cid:11) + ∞ X g =1 (cid:10)(cid:10) Ψ ∗ g − , − , Q Ψ − g, − (cid:11)(cid:11) . Since ∂ r S bv , s [Ψ , Ψ ∗ ] ∂ Ψ − g, − = Q Ψ ∗ g, − and ∂ r S bv , s [Ψ , Ψ ∗ ] ∂ Ψ ∗ g, − = Q Ψ − g, − , it solves the master equation12 (cid:8) S bv , s , S bv , s (cid:9) = ∞ X g =0 (cid:10)(cid:10) Q Ψ ∗ g, − , Q Ψ − g, − (cid:11)(cid:11) = 0 . Very trivial embedding into the large Hilbert space
We can re-express the action (A.3) using the large BPZ inner product S l = 12 (cid:10) ξ Ψ , − , Q Ψ , − (cid:11) η Ψ , − = 0 . Iff we impose η λ , − = 0, it has the same gauge invariance δ Ψ , − = Q λ , − with η λ , − =0 . As well as λ , − , the other higher gauge parameters { λ − g } g> do not have to satisfy anyconstraint equation. However, if we restrict these on H βγ , namely η λ − n, − = 0 for n ≥
0, thistheory has the same gauge reducibility δ n +1 ( δ n λ − n ) = 0 under the n -th gauge transformationfor the ( n − δ n λ − n = Q λ − ( n +1) .If we consider to construct the master action S bv , l based on the large BPZ inner product, onecan slightly enlarge the minimal set. All antifields (Ψ g, − ) ∗ can live in the large Hilbert space H ξηφ because of h (Ψ g, − ) ∗ , Ψ g, − i 6 = 0. Then, the minimal set of fields–antifields is given by (cid:8) Ψ , − , Ψ − g, − (cid:9) g ≥ ⊂ H βγ , (cid:8) Φ ∗ g, ≡ (Ψ − g, − ) ∗ (cid:9) g ≥ ⊂ H ξηφ .
34e write Φ ∗ g, for the antifield of Ψ − g, − , namely Φ ∗ g, ≡ (Ψ − g, − ) ∗ . In this case, whileall fields { Ψ − g } g ≥ = { Ψ , − , Ψ , − , ..., Ψ − g, − , ... } must satisfy the constraint equations, η Ψ − g, − = 0 , ( g ≥ , there is no constraint equations on the antifields. As we will see, essentially, the above constraintsare too strong, and weaker constraints, η Q Ψ − g, − = 0 for g ≥
0, are sufficient for the masterequation. One can find that the master action is given by S bv , l [Ψ , Φ ∗ ; L ] = 12 (cid:10) ξ Ψ , − , Q Ψ , − (cid:11) + ∞ X g =1 (cid:10) Φ ∗ g, , Q Ψ − g, − (cid:11) + ∞ X g =0 (cid:10) L g, , η Ψ − g, − (cid:11) , where we introduced Lagrange multipliers {L g, } g ≥ to see the role of the constraints. Since ∂ r S bv , l ∂ Ψ , − = ξ Q Ψ , − + η L , , ∂ r S bv , l ∂ Ψ − g, − = Q Φ ∗ g, + η L g, , and ∂ r S bv , l ∂ Φ ∗ g, = Q Ψ − g, − , after integratingout all Lagrange multipliers, we find12 (cid:8) S bv , l , S bv , l (cid:9) = (cid:10) Q Ψ , − , ξ Q Ψ , − (cid:11) + X g ≥ (cid:10) Q Ψ − g, − , η L g, (cid:11) = 0 . We write Ψ / Φ ∗ for the sum of all fields/antifields. After inposing the constraints, we can rewritethe master action into the following form S bv , l [Ψ , Φ ∗ ] = 12 (cid:10) ξ (Ψ + η Φ ∗ ) , Q (Ψ + η Φ ∗ ) (cid:11) = 12 (cid:10) η ( ξ Ψ + Φ ∗ ) , Q ( ξ Ψ + Φ ∗ ) (cid:11) . Therefore, one can reduce it to S bv , s [Ψ , Ψ ∗ ] by using gauge-fixing fermion providing η Φ ∗ ≡ Ψ ∗ ,or one may be able to regard it as some reduced version of larger master action. Removing restrictions
What happens if we relax the above constraints on higher gauge parameters { λ − g, − } g ? Letus consider to remove the constraints on the ( g ≥ n ) higher gauge parameters. Namely, weintroduce two types of gauge parameters: { λ − g, − } n − g =0 ⊂ H βγ and { Λ − g, − } ∞ g = n ⊂ H ξηφ . Then, n gauge parameters λ − g, − satisfy the constraints η λ − g, − = 0 , ( g = 1 , . . . , n ) , and infinite number of gauge parameters { Λ − g, − } ∞ g = n are constraint free. Because of η λ − g, − =0 , we have δ g λ − g, − = Q λ − g, − , ( g = 1 , . . . , n − , which preserves the ( n − δ n λ − n, − , the following gauge reducibility arises δ n λ − n, − = Q Λ − n, − , δ g Λ − g, − = Q Λ − g, − , ( g > n ) . However, if we require δ n λ − n, − ∈ H βγ as λ − n, − ∈ H βγ , we obtain the slightly different gaugereducibility. First, the n -th gauge parameter Λ − n, − cannot live in H ξηφ and must belong to35er[ Qη ] (or Ker[ η ] ∪ Ker[ Q ]). We thus write µ − n, − ≡ Λ − n, − . Because of µ − n, − ∈ Ker[ Qη ],there exist auxiliary gauge parameters { µ − n,p } p ∈ Z \{− } ⊂ Ker[ Qη ] such that Q µ − n, − = η µ − n, , η µ − n,p + Q µ − n,p − = 0 , ( p ≤ − , ≤ p ) . These give the constraint equations on the n -th (auxiliary) gauge parameters { µ − n,p } p ∈ Z , andthus these gauge parameters are dependent each other. Using them, we find δ n λ − n, − = Q µ − n, − + η µ − n, and the following gauge reducibility δ g +1 µ − g,p = Q µ − − g,p + η µ − − g,p +1 , ( g ≥ n, p ∈ Z ) . If we also impose δ n +1 µ − n,p ∈ Ker[ Qη ] as µ − n,p ∈ Ker[ Qη ], the next gauge parameters { µ − n − ,p } p must live in Ker[ Qη ]. Then, a half part of the set of fields–antifields is given by (cid:8) Ψ − g, − (cid:9) ng =0 ⊂ H βγ , (cid:8) Ψ − g,p (cid:9) g ≥ n,p ∈ Z ⊂ Ker[ Qη ] . These fields must satisfy the constraint equations η Ψ − g, − = 0 ( g < n ) , Q Ψ − n, − = η Ψ − n, , η Ψ − g,p + Q Ψ − g,p − = 0 (otherwise) . If we do not restrict δ n +1 µ − n,p , the higher gauge parameters { µ − g,p } g>n,p ∈ Z belong to the largeHilbert space H ξηφ . Then, a half part of the set of fields–antifields is given by (cid:8) Ψ − g, − (cid:9) ng =0 ⊂ H βγ , (cid:8) (Ψ − n,p ) ∗ (cid:9) p ∈ Z ⊂ Ker[ Qη ] , (cid:8) (Ψ − g,p ) ∗ (cid:9) g>n,p ∈ Z ⊂ H ξηφ . In this case, only the fields Ψ g,p labeled by g ≥ − n satisfy the constraint equations η Ψ − g, − = 0 ( g < n ) , Q Ψ − n, − = η Ψ − n, , η Ψ − n,p + Q Ψ − n,p − = 0 ( p ∈ Z ) , and the other fields are constraint free. In either cases, there is no constraints on the other halfof the set of fields–antifields, (cid:8) (Ψ − g, − ) ∗ (cid:9) ng =0 ⊂ H ξηφ , (cid:8) (Ψ − g,p ) ∗ (cid:9) g ≥ ,p ∈ Z ⊂ H ξηφ . The master action is given by the same form as the large form given in section 2 except for thatthe g - and p -labels run over the appropriate regions. References [1] I. A. Batalin and G. A. Vilkovisky, “Gauge Algebra and Quantization,” Phys. Lett. B (1981)27.[2] I. A. Batalin and G. A. Vilkovisky, “Quantization of Gauge Theories with Linearly DependentGenerators,” Phys. Rev. D (1983) 2567 [Erratum-ibid. D (1984) 508].[3] M. Henneaux, “Lectures on the Antifield-BRST Formalism for Gauge Theories,” Nucl. Phys. Proc.Suppl. (1990) 47.[4] J. Gomis, J. Paris and S. Samuel, “Antibracket, antifields and gauge theory quantization,” Phys.Rept. (1995) 1 [hep-th/9412228].
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