Superstring field theory with open and closed strings

We construct Lorentz invariant and gauge invariant 1PI effective action for closed and open superstrings and demonstrate that it satisfies the classical BV master equation. We also construct the quantum master action for this theory satisfying the quantum BV master equation and generalize the construction to unoriented theories. The extra free field needed for the construction of closed superstring field theory plays a crucial role in coupling the closed strings to D-branes and orientifold planes.


Introduction and summary
We now have a consistent formulation of closed superstring field theory (see [1] for a review). Our goal in this paper is to extend this construction to interacting theory of open and closed strings. Such systems arise naturally in the presence of D-branes.
As has been described in [1], full quantum string field theory can be studied at different levels. The basic formulation involves the quantum master action satisfying quantum BV master equation [2][3][4]. However, without giving up any information, one can have various equivalent formulations that are useful for specific studies. One of them is the 1PI effective action, suitable for studying the problem of mass renormalization, vacuum shift and computing S-matrix in the shifted vacuum. Another is the Wilsonian effective action, suitable for studying the dynamics of string states below certain mass scale, but without making any low energy approximation. On the other hand the quantum master action is useful for studying problems that require making all loop momenta integration manifest, e.g. study of unitarity [6] and analyticity [7] properties of the amplitudes. Of these the 1PI effective action has the simplest gauge transformation properties, in that it should be invariant under suitable gauge transformation. As a consequence of this, it satisfies the classical BV master equation [2]. In contrast the quantum BV master action and the Wilsonian effective action are not invariant under any gauge transformation, since the gauge non-invariance of the action needs to compensate for the gauge non-invariance of the path integral measure [2][3][4]. What remains invariant is the combination dµ e 2S where dµ is the integration measure in the space of fields and anti-fields and S is the action [5]. This is ensured by the fact that the action satisfies quantum BV master equation.

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For this reason we begin our study by first constructing the 1PI effective action. To experts in quantum field theory, it may appear strange that we construct the 1PI effective action before writing down the actual action whose off-shell 1PI amplitudes would give the interaction terms of the 1PI effective action. However as explained in [1], since in string theory we already know the formal expressions for on-shell amplitudes (ignoring effects of vacuum shift and mass renormalization), we can first generalize this to construct offshell amplitudes, given as integrals over moduli spaces of punctured Riemann surfaces. By appropriately restricting the region of integration over the moduli spaces, we can construct either 1PI amplitudes or amplitudes corresponding to elementary vertices of the quantum master action. The former requires us to remove certain regions of integration around separating type degenerations in the moduli space, whereas the latter requires us to remove certain regions of integration around all degenerations.
After constructing the 1PI effective action and checking its desired properties, namely gauge invariance and validity of classical BV master equation, we also describe the construction of quantum master action satisfying the quantum BV master equation. This construction only requires a few changes from its 1PI counterpart. We also generalize our results to unoriented string theory, required to describe the interacting theory of closed and open strings in the presence of orientifold planes. Throughout our work, we shall follow the general strategy that has been used in the formulation of bosonic string field theory [8][9][10], but extending this to superstring field theory requires a few additional ingredients that we shall explain.
We now give a summary of our result for the 1PI effective action of superstring field theory of open and closed strings. We define H m,n to be the vector space of GSO even closed string states |s , carrying (left, right) picture numbers (m, n) and satisfying the constraints [8] For A c i ∈ H c and A o i ∈ H o , we define {A c 1 · · · A c N ; A o 1 · · · A o M } to be the off-shell 1PI amplitude, summed over all genera and all number of boundaries, with external closed string 1 These restrictions on the off-shell closed string states are needed to get well-defined off-shell amplitudes, which require us to choose local coordinates at the punctures that are used to insert the vertex operators of the off-shell string states into the correlation function. Such choice of local coordinates at the punctures are possible globally only if we ignore the phase of the local coordinate. The requirement that the vertex operators must be invariant under such phase rotations leads to (1.1).

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states A c 1 , · · · , A c N and external open string states A o 1 , · · · A o M , excluding the one point function of closed string states on the disc from the definition of {A c ; }. In computing this amplitude, we also need to insert in the correlation function appropriate combination of picture changing operators (PCO) and ghosts. For A c ∈ H c , we also define { A c } D to be the disc one point function ofÃ c with appropriate insertion of PCOs and c-ghosts. More detailed definitions of {A c 1 · · · A c N ; A o 1 · · · A o M } and { A c } D have been given in section 2. Note the exclusion of the one point function on the disc from the definition of {A c ; }. If A c is in the RR sector, then the contribution to {A c ; } from the disc does not exist, since the RR vertex operators carry total picture number −1, and therefore it is impossible to satisfy picture number conservation on the disc, requiring total picture number −2, by inserting PCOs which carry positive picture number. For A c belonging to the NSNS sector the disc one point function is non-vanishing, but we shall still exclude its contribution from the definition of {A c ; } for uniformity and define { A c } D to contain contributions from one point function on the disc both for NSNS and RR sector A c . As we shall comment at the end of this section, whether to include the contribution to the disc one point function of NSNS sector closed string states into the definition of {A c ; } or { A c } D is a matter of convention, but for RR states there is no such option. For one point function of closed string states in H c on surfaces with more boundaries/genera, there is no problem with picture number conservation, and we include their contribution in the definition of {A c ; }.
We shall take the closed string states A c i 's and A c to be even elements of the grassmann algebra and the open string states A o i 's and A o i 's to be odd elements of the grassmann algebra. This means that if ζ is an odd c-number element of the grassmann algebra, then we have where in (1.3), A c i , A c , A o i and A o refer to the vertex operators corresponding to the states. 2 Unless mentioned otherwise, all our subsequent equations will assume this assignment of grassmann parities. In this case, {A c 1 · · · A c N ; A o 1 · · · A o M } and { A c } D can be shown to be even elements of the grassmann algebra. Furthermore it follows from the definition given in section 2 that {A c 1 · · · A c N ; A o 1 · · · A o M } is invariant under arbitrary exchanges A c i ↔ A c j and A o k ↔ A o . Note that there is no additional sign even for the grassmann odd vertex operators A o k . More discussion on this can be found in comment 1 at the end of this section. We also define Ref. [10] assigned even grassmann parity to the open string field by taking the open string vacuum |0 to have odd grassmann parity and identified the grassmann parity of the string field as that of the corresponding ket state. In contrast we take the grassmann parity of the string field to that of the corresponding vertex operator.

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Here A| denotes the BPZ conjugate of |A , generated by z → 1/z for closed strings and z → −1/z for open strings. We shall see in (2.3) that [ ] D is related to the boundary state of the D-brane system via appropriate picture changing operation and rescaling. It follows from (1.5), and the fact that and In (1.6), (1.7), G is given by: X andX being holomorphic and anti-holomorphic PCOs. includes multiplicative factors of ±(2πi) −1 .
In the following we shall also need to deal with states of wrong grassmann parityclosed string states which are grassmann odd and open string states which are grassmann even. To derive the relevant relations, we multiply each grassmann odd closed string state and grassmann even open string state by grassmann odd c-numbers so that they acquire standard grassmann parities and therefore obey the standard symmetry properties and other identities described above. We can now bring the grassmann odd c-numbers to the extreme left in both sides of the equations keeping track of the signs picked up during this process. In doing this we follow the convention that a grassmann odd c-number can be passed through { and without any extra sign, -the physical origin of these rules will be described above (2.6). Once this is done, we can remove these c-numbers from both sides of the equations and derive the relevant identities. To avoid confusion, we shall use the convention that string states labelled by roman letters, like this convention, in an expression like Q B A c i , it will be understood that A c i is grassmann even and therefore Q B A c i is grassmann odd. It can be shown that with this prescription, the generalizations of (1.5) remain relatively simple even for states of wrong grassmann parity, provided we use the prescription that [ for [· · · ] c behaves as a grassmann odd object and [ for [· · · ] o behaves as a grassmann even object. In that case, when we replace , the first and the third equations of (1.5) remain the same, and the second equation also remains the same if either A o 0 or the product A c 1 · · · A c N is grassmann even. In case both of these are grassmann odd, we have an extra minus sign on the right hand side of the second equation. To see the necessity of assigning odd grassmann parity for [ in [· · · ] c , let us use the generalization of the first equation of (1.5) to get for a grassmann odd c-number ζ. Now on the left hand side we can bring ζ to the extreme left by picking up a multiplicative factor given by the grassmann parity of A c 0 . On the right hand side we shall get an additional minus sign while moving ζ through c − 0 . Therefore to compensate for this we need the rule: (1.12) There is no such factor for [· · · ] o due to the absence of c 0 factor in the second line of (1.5). Another rule we need to follow is to never move a grassmann odd variable through ] c,o or }. To see this consider and move a grassmann odd c-number ζ through this from left to right. We may conclude that this operation will pick up a minus sign due to three grassmann odd open string vertex operators inserted in between {· · · }. But this may not be the correct result, e.g. the correlation function of three grassmann odd operators on a disc is a grassmann even number. Furthermore, this cannot be compensated by simply assigning an odd grassmann parity to } since the result depends on the number of boundaries. The same rule will be followed for matrix elements like A o |O|B o for any operator O acting on the open string states. Any grassmann odd c-number from inside the matrix element will be taken outside the matrix element from the left. However in this case we could allow grassmann odd c-numbers to be taken out from the right by picking an extra minus sign, i.e. by treating the ket vacuum as grassmann odd.
We are now ready to describe the form of the action. We introduce two sets of grassmann even closed string fields Ψ c ∈ H c and Ψ c ∈ H c and two sets of grassmann odd open string fields Ψ o ∈ H o and Ψ o ∈ H o . The 1PI effective action is given by: where Besides the various identities mentioned earlier, we need another set of identities, known as the 'main identities' [8], to prove gauge invariance of the action. For grassmann even A c , A c i and grassmann odd A o i , these identities take the form: 15) and, in bosonic open-closed string field theory. There we include the contribution of the disc one point function in the definition of {B c ; }. As mentioned below (1.2), this is not an option for RR sector B c in superstring theory due to an obstruction associated with the picture number conservation.

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Note that on the left hand side of (1.16), Q B A c i and Q B A o i are string states of 'wrong grassmann parity', while on the right hand side B c and B o are also of wrong grassmann parity. Therefore the corresponding objects {· · · } will have to be defined by multiplying these wrong parity objects by grassmann odd c-number ζ. It is useful to include this grassmann odd c-number ζ explicitly in the identity so that each term in the identity has only states of correct grassmann parity. This takes the form: (1.17) The (−1) j−1 factor on the left hand side of (1.16) will be generated when we pull ζ through the open string vertex operators A o i to the extreme left. The equations of motion following from the 1PI effective action are given by Now, multiplying the second equation by G and adding to the first equation gives Similarly, multiplying the 4th equation by G and adding to the 3rd equation gives  (1.21) is to add solutions to Q B | Ψ o = 0 to an existing solution. These represent free field degrees of freedom. Furthermore the choice of solutions to these free field equations of motion does not affect the interacting field equations (1.22) and (1.23) since they do not involve Ψ c and Ψ o . Therefore the degrees of freedom associated with Ψ c and Ψ o represent free fields that completely decouple from the interacting part of the theory, and they have no observable signature.
Besides the results reviewed above, this paper also contains the following results: 1. We show in section 4 that the action (1.13) satisfies the classical BV master equation (S, S) = 0 where (F, G) denotes the anti-bracket between two functions F and G of the string field, as defined in (4.7), (4.8).
2. We construct in section 5 the quantum BV master action of superstring field theory of open and closed strings. This action has the same form as (1.13), but with the interaction vertices {· · · } replaced by slightly modified vertices { {· · · } }. Also the main identity satisfied by the new vertices now has additional terms given by the last two terms in (5.5). Due to these additional terms in the vertices the action now satisfies the quantum BV master equation (5.2), with ∆ defined in (5.3).
3. In section 6 we describe how to generalize the construction of the 1PI action and the BV master action to unoriented open closed string field theory. The structure of the action remains the same but the definitions of the interaction terms change. In particular {A c 1 · · · A c N ; A o 1 · · · A o M } now gets additional contribution from non-orientable surfaces, { A c } D gets additional contribution from the projective sphere and there are additional normalization factors in the definitions of these quantities.
We end this section with a few comments.
1. In most formulation of classical open string field theory an associative * -product and its generalization known as the A ∞ algebra plays a significant role [11][12][13][14][15][16][17][18][19][20][21][22]. This is not manifest in the formulation of the quantum action described above. Instead what plays a central role here is the associated Lie algebra and its infinite dimensional generalization -the L ∞ algebra. For example, if we consider Witten's open bosonic string field theory [11], then our definition of [; The price we pay in giving up the A ∞ structure is that we can no longer extract color ordered amplitudes from the theory without digging into the detailed definition of {· · · }. 3 If we want to make manifest the information on color ordering, we need to follow a more elaborate approach described in [10,23,24].

4.
Using the open-closed superstring field theory one can construct gauge invariant 1PI effective actions for theories that are apparently anomalous. For example if we consider type IIB string theory with certain number of space-filling D9-branes, its spectrum will contain, besides the usual closed string fields, additional chiral fermions from the gauge supermultiplet on the D9-branes. This theory is known to suffer from gravitational anomaly [25]. However there is no difficulty in writing down a gauge invariant 1PI effective action for this theory. In the latter description, the inconsistency shows up due to the presence of a term in the action that is linear in the RR 10-form field, encoded in the { Ψ c } D term in (1.13). Due to the presence of this term, the theory does not have a vacuum solution to the equations of motion [27]. However in some cases we may be able to cancel the effect of the { Ψ c } D term, leading to the last term on the left hand side of (1.22), by switching on other background fields contributing to the second term on the left hand side of (1.22). Examples of this kind can be found in compactification of type IIB string theory on Calabi-Yau manifolds where the space-filling D3-brane charge can be cancelled by flux of 3-form fields along the internal 3-cycles of the Calabi-Yau manifold.
5. The action (1.13) contains insertion of c − 0 in several places -in the kinetic term as well as in the definition of { Ψ c } D given in (2.2). This can be traced to the presence of conformal Killing vectors on the associated Riemann surfaces -a sphere with two punctures and a line integral of the BRST current, and a disk with one bulk puncture. We can avoid this by including c − 0 in the definition of Ψ c , declaring H c to be the subspace of states annihilated by c − 0 instead of b − 0 . This will introduce a b − 0 in the Ψ c -Ψ c kinetic term, but this is a more natural operator since the anti-commutator of Q B with b − 0 generates L − 0 . It has in fact been argued in [26] that in this formalism we can take Ψ c and Ψ c to be unconstrained elements in the Hilbert space of closed string states except for the restriction on the picture numbers. This introduces some additional free field degrees of freedom which decouple from the interacting part of the theory. 6. As has been mentioned already, (1.22)  Therefore the question reduces to whether they describe non-trivial elements of the BRST cohomology, since as long as they are BRST trivial one can always find Ψ c and Ψ o satisfying (1. 19) and (1.21). This can be studied separately in the zero momentum sector and the non-zero momentum sector. We shall analyze this question for (1.19), -the analysis of (1.21) will be similar and in fact simpler. In the sector carrying non-zero momentum along the non-compact space-time directions, the contribution from [ ] D can be expressed as 7. Once a consistent superstring field theory for open and closed strings has been formulated, it can be used to systematically study various aspects of string theory that are not easily amenable to the standard world-sheet approach. This includes for example the study of mass renormalization or vacuum shift [1], or studying superstring theory in RR background [28].
8. The superstring field theory of open and closed strings constructed here does not suffer from any ultra-violet divergence. However this theory suffers from all the usual infra-red divergences that a quantum field theory suffers. These have physical origin and need to be dealt with as in a quantum field theory.
9. Superstring field theory action that we write down can be formulated around any background associated with a superconformally invariant world-sheet theory in the NSR formalism. Even for a given background the theory is not unique -it depends on the choice of local coordinates at the punctures and the choice of PCO locations that we have to make in defining the interaction terms of the action. For superstring field theories of closed strings it is known that apparently different string field theories, that one gets by making different choices, are all related by field redefinition.
We expect a similar result to hold for the theory described here, but we have not JHEP01(2020)183 attempted to give a complete proof. Another feature of the theory that we would like to prove is background independence -if we have different backgrounds related by marginal deformation of the world-sheet theory, then the superstring field theories formulated around these different backgrounds should also be related by field redefinition. This analysis is expected to be more complicated than the corresponding analysis for closed superstring field theory, since under marginal deformation of the bulk world-sheet theory we also need to deform the D-brane system appropriately [10]. We hope to return to these problems in the future.

Construction of 1PI vertices and their properties
We shall now describe the construction of the 1PI vertices {A c satisfying the various identities described in section 1. Since the construction proceeds more or less in the same way as in the case of closed superstring field theory reviewed in [1], we shall only emphasize the differences.
We begin by describing our convention for correlation functions of open string vertex operators. If there are N D-branes, not necessarily of the same kind, then a general open string state is described by an N × N matrix, with the i-j matrix element describing the state of an open string whose left end is on the i-th D-brane and the right end is on the j-th D-brane. A correlation function of open and closed string vertex operators on a general Riemann surface with multiple boundaries include taking traces over the N × N Chan-Paton matrices on each boundary of the Riemann surface. These traces will not be written explicitly, but will be understood as part of the definition of the correlation function. We shall follow the standard convention of referring to the locations of the vertex operators on the Riemann surface as punctures, with closed string vertex operators inserted at the bulk punctures and the open string vertex operators inserted at the boundary punctures.
We shall first define { A c } D . We take this to be the one point function of g −1 s c − 0 G A c on the unit disc |z| ≤ 1, with the vertex operator inserted at the center of the disc using local coordinate z e β . Here β is some positive number and 5 where X 0 ,X 0 are zero modes of the PCOs as defined in (1.10). The c − 0 factor is needed due to the presence of conformal Killing vector on the disc, which makes one point function of any operator vanish if the operator is annihilated by b − 0 . The G factor is needed for picture number conservation. Since A c has picture number −2 in the NSNS sector and −3 in the RR sector, GA c has picture number −2 in both sectors. This is the correct picture number for getting a non-vanishing one point function on the disc. The identity (1.15) can now be proved by deforming the integration contour defining Q B to the boundary of the disc. On 5 Since there is no insertion on the boundary, X0 andX0 could be evaluated by taking their defining integration contours on the boundary. Since on the boundary the holomorphic and anti-holomorphic PCOs are equal, X0 andX0 are identical. Therefore we could replace G by X0 orX0.

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the boundary the holomorphic and the anti-holomorphic components of Q B cancel each other. There is no additional contribution from having to pass does not contain a c − 0 and therefore integration over the zero mode c − 0 associated with the conformal Killing vector vanishes.
Given this definition of { A c } D , we can define [ ] D via (1.5). We can also express [ ] D in terms of the boundary state [29][30][31][32] of the D-brane system as follows. Let |B be the boundary state of the D-brane system under consideration so that φ c |c − 0 |B describes the one point function of the closed string vertex operator c − 0 φ c , inserted at the center of the unit disc |z| ≤ 1 in the z coordinate system. This gives Since one point function on the disc is non-zero only for vertex operators of picture number −2, φ c must have picture number −2 for getting non-vanishing φ c |c − 0 |B . On the other hand, φ c |c − 0 |B , being related to a sphere correlation function of φ c and B, is non-zero only when the total picture number of φ c and B add up to −4. Therefore |B has picture number −2. Comparing the last equation of (1.5) and (2.2), we can express [] D ∈ H c as where P denotes projection operator into H c . Its role is to pick the (−1, −1) and (−1/2, −1/2) components of G|B , which otherwise has states in H m,n for all m, n with m + n = −2 in the NSNS sector and m + n = −1 in the RR sector. The construction of {· · · } proceeds as in the case of closed superstring field theory -therefore we shall be brief, emphasizing only the new aspects of this construction. We denote by M g,b,mc,nc,pc,qc,mo,no the moduli space of Riemann surfaces with genus g, b boundaries, m c NSNS punctures, n c NSR punctures, p c RNS punctures, q c RR punctures, m o NS-sector punctures on the boundary and n o R-sector punctures on the boundary, with the understanding that the integration over M g,b,mc,nc,pc,qc,mo,no includes sum over spin structures. It will also be understood that the Ramond punctures carry picture number −1/2, i.e. when we insert a vertex operator at the puncture, it belongs to H c or H o . We denote by P g,b,mc,nc,pc,qc,mo,no a fiber bundle over this moduli space, with the fiber directions specifying the choice of local coordinates at the punctures and also the locations of the PCOs. Note that for Riemann surfaces with boundary, the picture number in the holomorphic and anti-holomorphic sectors are not separately conserved, but only the total picture number is conserved. For this reason, we have specified only the total picture number.
Only for b = 0, we have to have 2g − 2 + m c + n c + (p c + q c )/2 anti-holomorphic PCOs and 2g − 2 + m c + p c + (n c + q c )/2 holomorphic PCOs. Therefore for b = 0 the fiber has ℘ + 1 different branches, with the r-th branch having r holomorphic PCOs and (℘ − r) anti-holomorphic PCOs.

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In order to simplify notation we shall define the formal sum P g,b,N,M ≡ mc,nc,pc,qc,mo,no mc+nc+pc+qc=N,mo+no=M P g,b,mc,nc,pc,qc,mo,no , (2.5) and similarly M g,b,N,M . Following the same procedure reviewed in [1] for closed string theory, given a set of closed string states A c 1 , · · · , A c N and open string states The sign rules of section 1, that tells us that we can take a grassmann odd c-number out of {· · · } from the left, implicitly assumes that in computing the correlation functions that define Ω g,b,N,M p , we insert the vertex operators for the external states first in the order they appear inside {· · · } and then insert all the ghosts and PCOs. Similarly the sign rule that we can move a grassmann odd c-number through without extra sign corresponds to treating the bra vacuum 0| as grassmann even. Ω g,b,N,M p satisfies the useful property: where κ is an appropriate sign factor about which we shall say more later. The identity is derived by deforming the integration contour used in defining Q B away from the vertex operators and making it act on the ghosts / PCO insertions. This generates insertion of stress tensor in the correlation function which in turn has the interpretation of an exterior derivative acting on Ω p−1 . The phase κ could in principle differ from the corresponding result in the closed string case from having to move odd operators through open string vertex operators. It can be determined by careful analysis as in [10], but we shall extract the relevant information using an indirect approach to be described later. A special role will be played by Ω g,b,N,M 6g−6+3b+2N +M since the dimension of the moduli space M g,b,N,M is given by 6g − 6 + 3b + 2N + M .
As in the case of closed (super-)string field theory reviewed in [1], we introduce the notion of a generalized section of P g,b,N,M by extending the notion of a section. A generalized section can be a formal weighted average of many sections -with the understanding that integral over such a generalized section will be given by the weighted average of the integral over the corresponding sections. Unlike a regular section, a generalized section may also contain vertical segments across which the PCO locations jump discontinuously. The JHEP01(2020)183 integral of Ω g,b,N,M 6g−6+3b+2N +M over such vertical segments will have to be defined by adding to the integral over the continuous part of the section some correction terms described in [33,34].
The ability to include vertical segments in the generalized section plays a crucial role in open-closed string field theory -it allows us to choose generalized sections that can jump between the different branches of P g,b,N,M mentioned earlier. This is done by moving one or more PCOs from the bulk to a boundary across a vertical segment [33,34]. Since on the boundary the holomorphic and anti-holomorphic PCOs are identical, we can replace holomorphic PCOs by anti-holomorphic PCOs (or vice versa) and then move them to the desired positions in the bulk across another vertical segment. Such jumps may be necessary in order to ensure that near various boundaries of the moduli space, the generalized section factorizes correctly into the direct product of the sections on the component Riemann surfaces to which the original Riemann surface degenerates.
For defining {· · · } we also need the notion of a section segment where we remove from the base M g,b,N,M certain codimension zero regions and then erect a generalized section on this truncated space. Even though we drop the word generalized for brevity, it should be understood that the section segments we shall be working with refer to generalized section segments, allowing us to take weighted averages and vertical segments. Each such section segment represents a family of punctured Riemann surfaces equipped with a choice of local coordinates at the punctures and the choice of PCO locations. We shall now define two operations on section segments that will be important for us: 1. Sewing. Let us take a pair of section segments, one in P g,b,N,M and the other in P g ,b ,N ,M . Each represents a family of Riemann surfaces equipped with choice of local coordinates at the punctures and PCO locations. We can now construct a new section segment by sewing them at a pair of punctures -one from each section segment. If the sewing is done at a bulk puncture, then this means that we take a Riemann surface from one family and sew one of its punctures to a puncture on the Riemann surface from the other family by making the identification where w 1 and w 2 are the local coordinates at the respective punctures. For sewing at a pair of boundary punctures the analog of (2.7) takes the form In both cases, we also insert a factor of G defined in (1.9) around the origin of the

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2. Hole creation. Another way of producing a new section segment from a given one is to sew a Riemann surface belonging to a section segment to a disc |z| ≤ 1 with one bulk puncture, via the relation

9)
where w 1 is the local coordinate at the bulk puncture on the Riemann surface that is being sewed and w 2 is the local coordinate at the bulk puncture at the center of the disc. We also need to insert a factor of G defined in (2.3) around the origin of the w 1 or w 2 coordinate system. Note the absence of the phase e −iθ even though we sew two closed string punctures -this is related to the presence of a conformal Killing vector on the one punctured disc. We shall take w 2 to be related to the standard coordinate z on the unit disc by w 2 = e β z, where β is the positive constant that appeared in the definition of { A c } D . In that case, hole creation is equivalent to inserting at a bulk puncture on the Riemann surface the state (see (A.16)) 3. Given a set of section segments {R g,b,N,M }, we can generate new section segments from them by repeated application of sewing and hole creation. The demand we make on R g,b,N,M is that the formal sum of all of these section segments produces a full generalized section whose projection to the base covers the full moduli space M g,b,N,M .
As in the case of closed superstring field theory described in [1], we can systematically construct the section segments R g,b,N,M 's satisfying the above requirements as follows. Since the sewing and hole creation operation always increase the dimension of the section segment due to appearance of new parameters s or θ, we shall set up a recursive procedure JHEP01(2020)183 for constructing R g,b,N,M with the dimension of R g,b,N,M as the recursion parameter. We begin by making choices of R 0,0,3,0 , R 0,1,0,3 and R 0,1,1,1 . In each of these cases the moduli space is 0-dimensional, so the only choice we have to make is the choice of local coordinates and PCO locations. 6 We choose the local coordinates w i so that they are related to some natural local coordinate z on these Riemann surfaces by large scaling: w i = e β i z with large |β i |, so that |w i | = 1 describes a small circle around the puncture in the natural coordinates. This is known as adding long stubs to the string vertices [35]. Similarly we choose the parameter β appearing in the definition of { A c } D in (2.3) to be large. In order to make R g,b,N,M symmetric under the exchange of punctures, we may need to average over different choices of local coordinates and / or PCO locations -this is allowed since we only require R g,b,N,M to be a generalized section segment. We now focus on R g,b,N,M 's whose expected dimension is 1, e.g. R 0,1,0,4 , R 0,1,1,2 , R 0,1,2,0 and R 0,2,0,1 . For any such g, b, N, M , we first determine all section segments obtained by sewing or hole creation operation involving zero dimensional R g ,b ,N ,M . As long as the zero dimensional R g ,b ,N ,M have been constructed by adding long stubs, sewing and hole creation of the corresponding section segments will generate one dimensional section segments of P g,b,N,M , whose projection on the base covers only small regions in M g,b,N,M around separating type degenerations, leaving behind large gaps. We now choose one dimensional R g,b,N,M to 'fill the gap' so that together we have a complete generalized section of P g,b,N,M . There is clearly a lot of freedom since the section segments generated by sewing or hole creation of zero dimensional R g ,b ,M ,N only fix the boundaries of one dimensional R g,b,N,M , by requiring them to match the s = 0 boundaries of the sewing operation (2.8) or hole creation operation (2.9). In the interior we can choose local coordinates / PCO locations arbitrarily, subject to the restriction that the local coordinates should carry long stubs, and the PCOs must avoid spurious poles [36] by finite margin [1]. We now repeat the process, generating all the two dimensional section segments by sewing and hole creation of the section segments of lower dimensional R g ,b ,N ,M 's and then choose the two dimensional R g,b,N,M 's by filing the gap. This procedure can be repeated to generate all the R g,b,N,M 's.
Note that allowing R g,b,N,M to be generalized section segments is crucial for this construction. We have already mentioned that making it symmetric under the exchange of punctures may require averaging over different choices of local coordinates and/or PCO locations. Furthermore the boundaries of R g,b,N,M , fixed by sewing or hole creation in lower dimensional R g ,b,M ,N 's, are often generalized sections since they often have insertion of the operator G or G involving average of PCO insertions on a circle.
For the purpose of our analysis we shall not need the explicit form of R g,b,N,M 's. Explicit construction of such R g,b,N,M 's can be done using minimal area metric [8] or hyperbolic metric [37][38][39], but any other choice satisfying the above requirements will also be acceptable for our construction. We now define:

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The proof of the main identity (1.16) can now be given as follows. It is clear from (2.11) that the left hand side of the main identity is given by l.h.s. of (2.6) with p = 6g − 6 + 3b + 2N + M .
(2.12) We can now use (2.6) to express this as  .7) produces the term involving B c on the right hand side of (1.16), the s = 0 boundary of the operation (2.8) produces the term involving B o on the right hand side of (1.16), and the s = 0 boundary of the operation (2.9) produces the term involving {[· · · ]} D on the right hand side of (1.16). The signs can be determined by careful analysis as in [10] since there is no essential difference between bosonic and superstring theories here, the open string fields being grassmann odd in both cases. However we shall determine the signs by an indirect argument. We outline below the general strategy, leaving the detailed analysis to appendix A.
First of all, the fact that using (1.16) we can prove gauge invariance of the action, as shown in section 3, provides an indirect evidence for the correctness of the signs of the terms on the right hand side of (1.16). We can provide a more direct argument for these signs using the factorization property of string amplitudes as follows. Factorization property tells us that near the separating type degeneration, the integrand of a string amplitude breaks into a sum of products of the integrands associated with the component Riemann surfaces into which the original Riemann surface degenerates. This property is needed to ensure that the contribution to the amplitude from the region near a separating type degeneration has the interpretation of a pair of string amplitudes connected by a propagator. This can be used to fix any phase ambiguity in the integrand of a string amplitude by relating it to the product of the phases of amplitudes at lower genus / with lower number of punctures. We shall show in appendix A that if the phases of the amplitudes are fixed this way, then (2.13) gives precisely the right hand side of the main identity (1.16) without any extra sign.
and {Ã c } D are grassmann even can be proved iteratively by starting with an amplitude with no external states (or, for low genus, amplitudes JHEP01(2020)183 with minimal number of external states needed for removing the conformal Killing vectors) and then noting that the addition of a grassmann even closed string state is accompanied by two insertions of b-ghosts in the correlator, while the addition of a grassmann odd open string is accompanied by one insertion of b-ghost. Therefore the grassmann parity of the correlator remains unchanged under these operations. This is also consistent with the fact that in order to get a non-zero result for an amplitude, the total number of grassmann odd component fields (coming from space-time ghosts, fermions etc.) must be even. The identities (1.6)-(1.8) can be proved using (1.5). For example we can express the left hand side of (1.6) as: where in the second step we have used the fact that the both [· · · ] c in (2.14) are grassmann odd. This establishes the symmetry property (1.6). The other identities (1.7), (1.8) can be proven in a similar manner.

Gauge invariance of the 1PI action
In this section we shall prove the invariance of the 1PI action given in (1.13) under the gauge transformation (1.14). The corresponding result for bosonic string field theory of open and closed strings follows from this by setting Ψ = Ψ, Λ = Λ and G = 1. Under the gauge transformation (1.14), the variation in the action (1.13) is given by where we used the identity We shall analyze separately the effect of the transformations generated by Λ c , Λ o , Λ c and Λ o . First we consider the gauge transformation parametrized by Λ c in (1.14). Using Q 2 B = 0

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we see that the variation of the action is given by But this vanishes by the identity (1.15). Similarly, the variation of the action parametrized by Λ o in (1.14) vanishes identically: For the gauge transformation parametrized by Λ c , we have, from (1.14), The first and 3rd terms cancel each other while 4th and 6th terms cancel each other. After using (1.5) we are left with Now, we specialize the identity (1.16) to the following case where ζ is a grassmann odd c-number. This gives After redefining the sums appropriately, we see that the right hand side of the above equation is precisely the gauge transformation δ c S 1P I given in (3.6). This proves gauge

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invariance under the transformation generated by Λ c : Finally, we consider the variation of the action under the gauge transformation parametrized by Λ o . The variation of the action is given by Again, the first and third terms cancel each other and the fourth and sixth terms cancel each other. After using (1.5), we are left with Now, we specialize the identity (1.16) to the following case for some grassmann odd c-number ζ. This gives (3.14) Bringing the grassmann odd parameter ζ to extreme left and summing over M and N after multiplying with {N !(M − 1)!} −1 , we get After redefining the sums and comparing with (3.12), we find This completes the proof of gauge invariance of the 1PI effective action.

JHEP01(2020)183 4 Classical BV master equation for the 1PI action
We shall now show that the 1PI effective action satisfies the classical BV master equation [8,10,40,41]. For this, we first identify the fields and anti-fields of the theory. This is done by dividing the Hilbert spaces as follows The closed string fields are expanded as 7 where ϕ in the exponent for any state |ϕ denotes the grassmann parity of the vertex operator ϕ, taking value 0 for even operators and 1 for odd operators. We define the target space ghost number of the coefficient fields by g = 2 − G where G denotes the worldsheet ghost number of the corresponding basis states. This means that the coefficients (ψ c ) r , ( ψ c ) r have target space ghost numbers ≥ 0 whereas the coefficients (ψ c ) * r and ( ψ c ) * r have target space ghost numbers ≤ −1. In the BV quantization, the (ψ c ) r and ( ψ c ) r will be interpreted as fields whereas (ψ c ) * r and ( ψ c ) * r will be interpreted as anti-fields. The factors of g s in (4.4) ensure that in the action (ψ c ) r , ( ψ c ) r , (ψ c ) * r and ( ψ c ) * r have conventionally normalized kinetic terms.
In a similar way, we expand the open string fields as

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We define the target space ghost number of the coefficient fields by g = 1 − G where G denotes the world-sheet ghost number of the corresponding basis states. This means that the coefficients (ψ o ) r , ( ψ o ) r have target space ghost numbers ≥ 0 whereas the coefficients (ψ o ) * r and ( ψ o ) * r have target space ghost numbers ≤ −1. In the BV quantization, (ψ o ) r and ( ψ o ) r will be interpreted as fields whereas (ψ o ) * r and ( ψ o ) * r will be interpreted as anti-fields. The BV anti-bracket between any two functions F and G of the fields is given by then using (4.4), (4.5), (4.6), one can express the anti-bracket as (4.8) Our goal will be to verify that the action (1.13) satisfies the classical BV master equation: (S 1P I , S 1P I ) = 0 . (4.10)

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This gives, The variation (4.10) can also be written as This gives (4.13) in the convention that the operators Q B and G act on the right even though they are to the right of a bra state. Using (4.8), (1.5), (1.15) and Q 2 B = 0, we now get (4.14) If we specialize the main identity (1.16) to the case we obtain Multiplying by 1 N !M ! and summing over N and M from 0 to ∞, we get Therefore the 1PI effective action satisfies the classical BV master equation. The BV formalism also gives a way to derive the gauge transformation laws (1.14). The classical BV master action -and therefore also the 1PI effective action -is known to be invariant under gauge transformations that transform any function F of the string fields |Ψ c , | Ψ c , |Ψ o and | Ψ o as [5], δF = (F, (S, Λ)) , (4.19) where Λ is any even function of the fields. Choosing 20) we reproduce the gauge transformation laws given in (1.14).

JHEP01(2020)183 5 Quantum BV master action
We can also write down the quantum BV master action for the combined open closed string field theory following the procedure reviewed in [1]. It is given by where the anti-bracket (, ) has been defined in (4.6), and Here ψ r stand for all the fields (ψ c ) r , ( ψ c ) r , (ψ o ) r and ( ψ o ) r and ψ * r stand for all the antifields (ψ c ) * r , ( ψ c ) * r , (ψ o ) * r and ( ψ o ) * r . −(S BV , S BV ) is given by the right hand side of (4.14) with { } replaced by { { } }. Using (5.3) one finds that ∆S is given by: The proof of this follows the same analysis as used in section 2 and appendix A for the proof of (1.16). The last two terms arise due to the fact that 6 Unoriented open-closed string field theory Our construction of the 1PI effective action or BV master action holds for any superconformal field theory that we use to compute the correlation functions of vertex operators that enter the definition of {A c Therefore the same construction is valid for any compactification of type IIA or type IIB superstring field theory involving NSNS background. The construction should also generalize to orientifolds where we have unoriented strings, but there will be a few differences. The main difference will be that the definitions of H c , H c , H o and H o will automatically include projection by the appropriate orientifold operation. Therefore the sewing and hole creation operation will also have this projection operator. For consistency, now we must also include non-orientable Riemann surfaces in the construction of superstring field theory interaction vertex, -if we start with an oriented Riemann surface and sew two of its punctures with the orientifold projection inserted, we shall generate a non-orientable surface. Together, the oriented and JHEP01(2020)183 non-orientable surfaces that will be relevant for us are known as Klein surfaces. Review of the essential results that we shall need can be found in [42][43][44] and has been summarized in appendix B.
Due to the inclusion of non-orientable surfaces, we encounter a few differences from the corresponding results in oriented open-closed string field theory.
1. Off-shell amplitudes now include sum over the moduli spaces of oriented and nonorientable surfaces. The oriented surfaces are as usual characterized by their genus g and the number of boundaries b, while the non-orientable surfaces are characterized by the number of crosscaps c and the number of boundaries b [42], with the Euler character given by b+c−2. Therefore in the definition (2.11) of {· · · }, the contribution from the non-orientable surfaces must be weighted by a factor of (g s ) c+b−2 , whereas the contribution from the oriented surfaces continue to be weighted by (g s ) 2g+b−2 . We shall often express such factors as (g s ) 2g+b+c−2 , with the understanding that oriented surfaces will have c = 0 and non-orientable surfaces will have g = 0.
2. We shall now show that in the definition of {· · · } given in (2.11) we must also include an extra factor of 2 −g−(c+b)/2+M/4 , (6.1) if we normalize Ω g,b,c,N,M in the same way as in the case of oriented string theories, e.g. for c = 0, Ω is defined with the same normalization as for oriented string theory. As we shall explain below, this extra factor (6.1) is needed to ensure that amplitudes factorize correctly near degeneration. 8 There is no such factor for closed strings since the kinetic term is a genus 0 amplitude, and for this there is no additional factor of 1/2.

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b remain unchanged. It is easy to verify that (6.1) changes by a factor of 1/2 under this operation. This again correctly accounts for the factor of 1/2 in the projection operator.
If the original surface was non-orientable, with c crosscaps, then the effect of sewing two of its bulk punctures may be analyzed in a similar manner. We again have the projection operator (1 + W )/2 with 1 corresponding to sewing with a handle and W corresponding to sewing with a cross handle. Both operations increase the number of crosscaps by 2, leaving fixed g(= 0), b and M . Under this (6.1) picks a factors of 1/2, correctly accounting for the 1/2 in the projection operator. It is easy to verify that (6.1) is also compatible with separating type degenerations. For example when we sew two oriented surfaces along bulk punctures, the total numbers for g, c, b and M all remain unchanged and therefore there is no change in (6.1). On the other hand in this case even though sewing introduces a factor of (1 + W )/2, both 1 and W produce the same set of oriented surfaces and therefore there is no factor of 1/2. Similar agreement can be shown for sewing of boundary punctures. A description of different kinds of sewing that can arise in unoriented open closed string field theory can be found in [43] and reviewed in appendix B.
3. Finally, in defining { A c } D we must now include not only one point function of c − o A c on the disc, but also on RP 2 , together with an insertion of G as before. The latter is similar to one point function on the disc, but with the boundary condition on the disc replaced by a crosscap. In keeping with our discussion above we must accompany each crosscap and disc by a factor of 1/ √ 2 by including it in the definition of { A c } D .
4. The choice of local coordinates at the punctures and PCO locations must be compatible with the orientifold projection.

5.
In the presence of one or more crosscaps, only the total picture number is conserved but the holomorphic and anti-holomorphic picture numbers are not separately conserved. Therefore the corresponding P 0,b,c,N,M will have multiple branches. We can jump between the branches by moving the PCOs to the crosscap via vertical segments JHEP01(2020)183 and converting holomorphic PCOs into anti-holomorphic PCOs or vice versa using the boundary condition on the crosscap.
With these few changes, the construction of R g,b,c,N,M for unoriented open-closed string field theory proceeds in the same way as in the case of oriented strings, beginning with zero dimensional R g,b,c,N,M 's. Eqs. (2.11) and (2.3) are generalized to: and , where β b and β c are positive constants and |B and |C are the boundary states for the disc and the crosscap. The form of the action, various identities described in section 1, the gauge transformation laws and the definition of the anti-bracket remains the same. We can also construct quantum BV master action by replacing R g,b,c,N,M 's by R g,b,c,N,M 's that satisfy slightly different constraints as described in section 5.
A Signs of the terms in the 'main identity' In this appendix we shall determine the signs of various terms on the right hand side of the main identity (1.17), following the strategy outlined at the end of section 2. We shall do the analysis iteratively in the dimension of R g,b,N,M (or equivalently M g,b,N,M ) associated with the interaction vertex, which we shall simply refer to as the dimension of the interaction vertex. Therefore we shall assume that (1.17) holds for vertices carrying dimension ≤ K and prove that it holds for vertices of dimension K + 1.
In the following we shall analyze the behaviour of the amplitude near the boundary of the moduli space using the language of string field theory. This may give the impression that our argument is circular, i.e. we use string field theory to prove relations among string field theory interaction vertices. However the factorization property of the string amplitude JHEP01(2020)183 that we use is known to hold independently of string field theory, and tells us that the contribution to an amplitude near a separating type degeneration has the interpretation of the contribution from a Feynman diagram where two different diagrams are connected by an internal line. This has been shown in figure 1. Therefore our argument does not use any essential element of string field theory. We shall also use gauge invariance of the amplitude, that tells us that if in an amplitude we act ζQ B in turn on each external state, the result vanishes. This property of the amplitude also follows from standard world-sheet analysis and does not rely on the existence of an underlying string field theory. Let us consider an amplitude with external closed string states A c 1 , · · · A c n , B c 1 , · · · B c m and external open string states A o 1 , · · · , A o p , B o 1 , · · · , B o q , with ζQ B acting in turn on each external state. Gauge invariance requires this amplitude to vanish. However the contribution to this amplitude from the 1PI vertex does not vanish -rather it is given by the right hand side of the main identity (1.17). 9 Let us first focus on the particular contribution: on the right hand side of (1.17). This is expected to get cancelled against the contribution from the Feynman diagram where a 1PI vertex with the external states A c 1 , · · · , A c n , A o 1 · · · A o p and an internal closed string state and another 1PI vertex with the external states B c 1 , · · · , B c m , B o 1 , · · · B o q and an internal closed string state are joined by a closed string propagator, and ζQ B acts in turn on each of the A c,o i 's and B c,o i 's. Figure 1 shows one of these Feynman diagrams where ζQ B acts on A c 1 . Our strategy will be to evaluate these Feynman diagrams explicitly and use this to test the sign in (A.1).
We shall first compute the contribution to the amplitude from the Feynman diagram when ζQ B acts on A c 1 , as shown in figure 1. Let us denote by |ϕ r the basis states in H c

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and by |φ r the conjugate basis in H c , satisfying so that the closed string field can be expanded as ψ c r |ϕ r . Then the product of the two vertex factors in figure 1 can be expressed as: We shall eventually get a propagator by contracting ψ c r and ψ c s . Therefore ψ c r and ψ c s must have same grassmann parity. Since the closed string field is grassmann even, |ϕ r and |ϕ s will also have the same grassmann parities as ψ c r and ψ c s . We shall denote this grassmann parity by (−1) ϕr . We now pull the ψ c r from the ψ c r |ϕ r factor inside the first vertex and ψ c s from the ψ c s |ϕ s factor inside the second vertex to the left outside the respective {· · · }'s. This does not generate any sign since the A c i 's are grassmann even. ψ c r and ψ c s are now This has grassmann parity (−1) ϕr . Therefore the contraction of ψ c r and ψ c s gives a factor of (−1) ϕr ∆ rs where ∆ rs is the propagator. ∆ rs can be computed as follows. In the kinetic term, we expand Ψ c | as ψ c r ϕ r | and pull ψ c r to the left without picking any sign, and express |Ψ c as ψ c s |ϕ s and pull ψ c s to the left, picking a factor of (−1) ϕr since it has to pass through ϕ r |c − 0 Q B which has the same grassmann parity as ψ c s . Similar operation can be done for fields in H c . Furthermore in the Siegel gauge we can replace Q B by (c 0 L 0 +c 0L0 ). Inverting the kinetic operator we now get a ψ r -ψ s propagator ∆ rs = −g 2 . 10 After cancelling the (−1) ϕr factor in ∆ rs with the (−1) ϕr factor coming from the contraction of ψ r and ψ s across {(ζQ B A c 1 ) · · · A c n ϕ r ; A o 1 · · · A o p }, we arrive at the following expression for the amplitude shown in figure 1: We now use (1.5) to express the last factor as ϕ s |c − 0 |[B c 1 · · · B c m ; B o 1 · · · B o q ] c and then perform the sum over s in (A.4) using the completeness relation to express the product of the last two factors as φ r |c − On the other hand the first {· · · } factor in (A.4) can be expressed as where we have used the fact that [(ζQ B A c 1 ) · · · A c n ; A o 1 · · · A o p ] c is grassmann odd. We can now perform the sum over r in (A.4) using completeness relation and express (A.4) as In computing the propagator, we only use the terms involving QB as kinetic operator and treat the quadratic and linear terms in {· · · } as interaction terms. This allows us to make contact with the worldsheet formulation, although in order to correctly compute the renormalized masses we have to work with the full quadratic term. In [1] the propagator was written as −g 2 s φ s |c − 0 b + 0 G(L + 0 ) −1 |φ r . This can be shown to be equal to −g 2 s (−1) ϕr φ r |c − 0 b + 0 G(L + 0 ) −1 |φ s .

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pull ψ o r and ψ o s to the left, we get a factor of (−1) φr+1 from having to pass ψ o s through φ r |Q B . After replacing Q B by c 0 L 0 in the Siegel gauge and inverting the kinetic operator, we get a ψ o r -ψ o s propagator of the form −g s (−1) φr+1 φ r |b 0 G(L 0 ) −1 |φ s . Cancelling the (−1) φr+1 factor from the propagator with the (−1) φr+1 factor coming from the contraction, we arrive at the expression: The rest of the analysis proceeds almost in the same way as for (A.4), with the [· · · ] c 's replaced by [· · · ] o , c − 0 's removed, b + 0 replaced by b 0 and ϕ r ,φ r replaced by φ r ,φ r . Using the fact that [(ζQ B A c 1 ) · · · A c n ; A o 1 · · · A o p ] o is grassmann even, we can verify that there are no extra signs compared to those that appeared in the earlier analysis. The final result takes the form analogous to (A.6) We now sum over terms where ζQ B acts in turn on all the external states. Each of these terms can be analyzed in the same way, leading to expressions similar to (A.12) with the position of ζQ B shifted. Using the main identity for vertices of lower dimension and throwing away terms that would cancel with Feynman diagrams with two propagators, we can, as in the case of (A.6), express the sum of these terms as the negative of the term where ζQ B acts on b 0 G(L 0 ) −1 , producing just a factor of Gζ. This brings the boundary contribution to the form: This cancels (A.8), confirming that the sign in (A.8) is correct. Finally we turn to the contribution arising on the right hand side of (1.17). The Feynman diagram that cancels it is the sum of the diagrams of the form shown in figure 2, with ζQ B acting on each external state in turn. This amplitude can be analyzed in the same way as (A.3) and is given by an expression similar to (A.4): together with similar terms where ζQ B acts in turn on the other external states. Note that there is no G insertion in the propagator since we have to use the ψ r -ψ s propagator [1]. We now carry out manipulations similar to those for (A.4) to express (A.15) in the form given in (A.6): Therefore we can use the main identity (1.17). Throwing away the terms on the right hand side of (1.17) that would cancel against Feynman diagrams with two propagators, and using BRST invariance of [ ] D , we can express the sum of Feynman diagrams of the type shown in figure 2 as where in the last step we used (1.8). This cancels (A.14), confirming that the sign in (A.14) is correct.

B Review of non-orientable surfaces
In this appendix we shall review some properties of non-orientable surfaces following [42][43][44]. A 2 dimensional surface is non-orientable if we cannot assign an orientation to the surface uniquely, i.e. there exist closed curves such that the tangent bundle, parallel transported along the curve, comes back with opposite orientation. A non-orientable 2 dimensional surface can be described using the 'crosscap' -a disc whose diametrically opposite points have been identified. By attaching an arbitrary number of crosscaps to a sphere with holes, we can generate an arbitrary non-orientable surface in 2 dimensions. Examples of non-orientable surfaces which appear at tree and one loop level in unoriented string theory are real projective plane (RP 2 ), Mobius strip and Klein bottle. The RP 2 is a sphere with one crosscap, the Mobius strip is a disc with one crosscap whereas Klein bottle is a sphere with two crosscaps. Therefore to construct RP 2 , we remove a disc from the sphere and identify the diametrically opposite points of the resulting hole, whereas to construct the Mobius strip, we remove a disc from the interior of the disc and identify the diametrically opposite points of the resulting hole. A Klein bottle can be obtained by removing two discs from the sphere and identifying the diametrically opposite points of each of the resulting hole.

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We now recall some general statements about two dimensional surfaces. Any compact orientable 2 dimensional manifold is topologically equivalent to a sphere with g handles and b boundaries. On the other hand, any compact non-orientable 2 dimensional manifold is topologically equivalent to a sphere with c crosscaps and b boundaries. Surfaces with both handles and crosscaps are redundant, since, in the presence of crosscaps, a handle can be replaced by two crosscaps: handle + crosscap = 3 crosscaps .
(B.1) Therefore for non orientable surfaces, the number of boundaries and crosscaps specify the surface topologically. The Euler number of a general surface, having g handles (or c crosscaps) and b boundaries is given by This, together with the picture numbers carried by the string states, dictates the number of PCOs one needs to insert on the surface. The conformal killing groups of sphere and disc without punctures are SL(2,C) and SL(2,R) respectively. There are 3 complex conformal Killing vectors (CKVs) on the sphere and 3 real CKVs on the disc. Moreover, the volume of these conformal killing groups is infinite. This implies that the 1 and 2 point sphere amplitudes do not give any contribution to the 1PI effective action. 11 Similarly, the 1 and 2 point disc amplitudes for external open strings also do not give any contribution to the 1PI effective action. However, the amplitude of 1 closed string on the disc does not vanish since the resulting surface -disc with one bulk puncture -has only one real CKV, generating a finite volume U(1) group. The conformal killing group of RP 2 is SU(2) which has finite volume. Hence, we can also have non zero 1-point function of closed strings on RP 2 . 11 It has been argued in a recent paper [45] that the two point function on the sphere does not vanish.
This, however, represents the standard forward contribution to the S-matrix present in any quantum field theory, including string field theory, and is needed for unitarity of the theory [6]. Therefore this does not require us to add a new term in the action. The same comment holds for two point function on the disc.

JHEP01(2020)183
The parametrization of the world-sheet with coordinates (σ, τ ) defines an orientation to the world-sheet locally. The orientation can be reversed by making the following transformation Ω : σ → σ = − σ , τ → τ = τ (B.5) The parameter , describing the length of the string in σ coordinates, is usually chosen to be π for open strings and 2π for the closed strings. In complex coordinates z = e τ +iσ , the operator Ω acts as Closed strings : z →z ,z → z , The operator Ω is called world-sheet parity operator. Since acting twice with Ω gives us the original orientation, we have Ω 2 = 1 and hence the eigenvalues of Ω are ±1.
Orientifold operation typically corresponds to taking a projection by the operator Ω possibly accompanied by another symmetry transformation acting on the world-sheet superconformal field theory. It could for example involve reversing the directions of certain space-time coordinates. Let us denote by W this combined operation. We can take W 2 = 1. 12 In the theory obtained by quotienting the original theory by W , we keep only those states that carry W eigenvalue +1. This can be achieved using the projection operator P : Note that the propagator also has GSO projection in all superstring field theories and L 0 =L 0 projection in the closed string sector, but we do not display this explicitly. Let us now review how non-orientable surfaces appear in the unoriented theories. Consider a loop diagram. In the intermediate state, we need to sum over all the states. Since we only want to keep the W = 1 states, the intermediate states must include the projection operator (1 + W )/2. This corresponds to cutting the world-sheet describing the propagation of the intermediate state, inserting the projection operator P and then sewing back the cut-edges. Then the 1 part of the projector corresponds to gluing the cut edges with the same orientation -via the sewing relations (2.7) for closed strings and (2.8) for the open strings. However, the W part of the projector corresponds to first reversing the orientation of one edge and then gluing it with the other edge. The corresponding sewing relations are zw = q ≡ e −s−iθ for closed strings and zw = e −s for open strings. This produces a non-orientable surface even if the original surface was oriented. Note that both, the oriented and the non-orientable surface produced this way, come with weight half.
With this understanding we can now describe the effect of different types of sewing in the unoriented theories. One general result that one can infer from this is that sewing two