M-theory interpretation of the real topological string

We describe the type IIA physical realization of the unoriented topological string introduced by Walcher, describe its M-theory lift, and show that it allows to compute the open and unoriented topological amplitude in terms of one-loop diagram of BPS M2-brane states. This confirms and allows to generalize the conjectured BPS integer expansion of the topological amplitude. The M-theory lift of the orientifold is freely acting on the M-theory circle, so that integer multiplicities are a weighted version of the (equivariant subsector of the) original closed oriented Gopakumar-Vafa invariants. The M-theory lift also provides new perspective on the topological tadpole cancellation conditions. We finally comment on the M-theory version of other unoriented topological strings, and clarify certain misidentifications in earlier discussions in the literature.


Introduction
Topological string theory is a fertile arena of interplay between physics and mathematics. A prominent example is the physics-motivated reformulation of the topological A-model on a threefold X 6 in terms of integer multiplicities of BPS states in the 5d compactification of M-theory on X 6 [1,2], and the corresponding mathematical reformulation of the (in general fractional) Gromov-Witten invariants in terms of the integer Gopakumar-Vafa invariants (see also [3]).
A natural generalization is to consider A-models with different worldsheet topologies. In particular, there is a similar story for the open topological A-model, in which worldsheets are allowed to have boundaries mapped to a lagrangian 3-cycle in X 6 , and which via lift to M-theory admits an open BPS invariant expansion [4]. There has also been substantial work to define unoriented topological A-models, for instance in terms of the so-called real topological strings [5][6][7][8]. The latter was proposed to require a specific open string sector for consistency, and conjectured to admit a BPS-like expansion ansatz, although no physical derivation in terms of M-theory was provided.
In this paper we fill this gap, construct the physical theory corresponding to the real topological string, and show that its M-theory lift reproduces the topological string partition function in terms of certain BPS invariants, which we define and show to be the equivariant subsector of the corresponding closed oriented Gopakumar-Vafa invariants. Along the way, the M-theory picture sheds new light into certain peculiar properties of the topological model, like the so-called tadpole cancellation condition, which requires combining open and unoriented worldsheets in order to produce welldefined amplitudes and integer invariants. Although [5] focused on the quintic and other simple examples (see also [6][7][8]), we keep the discussion general, using these examples only for illustration at concrete points.
The paper is organized as follows. In section 2 we review the Gopakumar In section 5 we describe related systems, by the inclusion of additional brane pairs (section 5.1), or by using other orientifold plane structures (section 5.2); in this respect, we clarify certain misidentifications of the M-theory lifts in the earlier literature on unoriented topological models. Finally, section 6 contains our conclusions. Appendix A reviews the basics of the real topological string, while appendix B discusses the physical couplings computed by the real topological string.

Review of Gopakumar-Vafa expansion
We start with a brief review of the Gopakumar-Vafa interpretation of the closed oriented topological string in terms of BPS states in M-theory [1,2].
The 4d compactification of type IIA on a CY threefold X 6 provides a physical realization of the topological A-model on X 6 , whose genus g partition function F g (t i ), which depends on the Kähler moduli t i , computes the F-term (where the second expression applies for g > 1 only). Here we have used the N = 2 Weyl multiplet, schematically W = F + + θ 2 R + + · · · , with F + , R + being the self-dual components of the graviphoton and curvature 2-form, respectively. These contributions are summed up if we turn on a self-dual graviphoton background in the four noncompact dimensions The sum is given by the total A-model partition function, with coupling λ There is an alternative way to compute this same quantity, by considering the lift of the IIA configuration to M-theory, as follows. We start with the 5d compactification of M-theory on X 6 . There is a set of massive half BPS particle states, given by either the dimensional reduction of 11d graviton multiplets, or by M2-branes wrapped on holomorphic 2-cycles. These states are characterized by their quantum numbers under the 5d little group SU(2) L × SU(2) R . Note that at the classical level, each such particle can have a classical moduli space, but at the quantum level there is only a discrete set of ground states, which provide the BPS particle states we are interested in. For instance, an 11d particle (such as the 11d graviton) has a classical moduli space given by X 6 itself, but quantization leads to wave functions given by the cohomology of X 6 , resulting in a net BPS multiplicity given by χ(X 6 ).
In order to relate to type IIA, we compactify on an S 1 . Corrections to the R 2 term will arise from one-loop diagrams in which the above BPS particles run, in the presence of the graviphoton field, which couples to their SU(2) L quantum numbers. In type IIA language this corresponds to integrating out massive D0-and D2-brane states (and their bound states). In the Schwinger proper time formalism we have Here the sinh 2 factor arises from the 4d kinematics, we have included a sum over KK momenta along the S 1 , the trace is over the Hilbert space H of 5d one-particle BPS states, with central charge Z, and F = 2J L 3 + 2J R 3 . The Hilbert space H of 5d one-particle BPS states from an M2-brane on a genus g holomorphic curve Σ g (in general not the same as the genus of the worldsheet in the type IIA interpretation) in the homology class β is obtained by quantization of zero modes on its worldline. Quantization of the universal Goldstinos contributes to the state transforming as a (half) hypermultiplet, with SU(2) L representation I 1 = 1 2 ⊕ 2(0). There are in general additional zero modes, characterized in terms of the cohomology (2.5) The first factor corresponds to zero modes from the deformation moduli space M g,β of Σ g in X 6 , whose quantization determines the SU(2) R representation. The latter is decoupled from the self-dual graviphoton background, so it only contributes as some extra overall multiplicity in the above trace. 1 The second factor corresponds to zero modes arising from flat connections on the type IIA D2-brane worldvolume gauge field on Σ g . The T 2g should be regarded as the Jacobian of Σ g , Jac Σ g = T 2g . Quantization of these zero modes determines further contributions to the SU(2) L representation of the state as dictated by the SU(2) Lefschetz decomposition of cohomology of T 2g , i.e. with creation, annihilation and number Here k is the Kähler form of the torus, denotes contraction, the bidegree deg is p + q for a (p, q)-form and n is complex dimension.
The SU(2) representation is of the form I g = I ⊗g 1 , where I 1 = 1 2 ⊕ 2(0). For instance, for g = 1 we have a ground state 1 and operators dz and dz, so that the cohomology of T 2 splits as where k ∼ dz ∧ dz. These form the representation I 1 . The argument generalizes straightforwardly to higher genera.
The contribution from a state in the SU(2) L representation I g to the trace is given by (−4) g sinh 2g sλ 2 , so we get where we write Z = β · t. Also, GV g,β are integers describing the multiplicity of BPS states arising from M2-branes on a genus g curve in the class β ∈ H 2 (X 6 ; Z), with the understanding that β = 0 corresponds to 11d graviton states. This multiplicity includes that arising from the SU(2) R representations, in the following sense. Describing the set of BPS states in terms of their SU(2) L × SU(2) R representations

Walcher's real topological string
A prominent example of unoriented topological string is Walcher's real topological string introduced in [5] (see also [6][7][8] [5][6][7][8] have considered cases with H 1 (L; Z) = Z 2 , for instance the quintic, or local CP 2 . Since our description in M-theory is more general, and this condition will only play a role in section 4.3, we keep the description general here as well.
The A-model target space is a Calabi-Yau threefold X 6 , equipped with an antiholomorphic involution σ, whose pointwise fixed set is a lagrangian 3-cycle denoted by L. The model is defined by considering maps of (possibly non-orientable) surfaces In the relation Σ =Σ/Ω, the particular case in which Σ is itself closed oriented and Σ has two connected components is not included.
The topological classification of possibly non-orientable surfaces Σ with boundaries, described as symmetric Riemann surfaces, written (Σ, Ω), generalizes the closed oriented case, through the following classic result: Let us define the (negative of the) Euler characteristic ofΣ/Ω by χ =ĝ − 1. It is useful to separate the worldsheets into three classes, corresponding to having 0, 1 or 2 crosscaps (recall that two crosscaps are equivalent to a Klein handle, namely two holes glued together with an orientation reversal, which in the presence of a third crosscap can be turned into an ordinary handle). This leads to a split of the topological amplitudes into classes, namely: closed oriented surfaces (with amplitude denoted by F (gχ) , with g χ = 1 2 χ + 1 the number of handles), oriented surfaces with h boundaries (with amplitude F (g,h) ), non-orientable surfaces with an odd number of crosscaps (with amplitude R (g,h) ) and non-orientable surfaces with an even number of crosscaps (K (g,h) ).
The Euler characteristic is given by The basic tool used to compute these amplitudes is equivariant localization on the moduli space M of stable maps, following ideas going back to [18] (see also [19], and [20,21] for more recent developments on the formal side). Localization is with respect to a torus action which is compatible with the involution, and leads to a formulation in terms of the diagram techniques of [18].
Ref. [5] finds that, in the example of the quintic or local CP 2 , in order to apply this machinery to unoriented and/or open worldsheets, some constraints, dubbed tadpole cancellation conditions, have to be imposed: as we discuss in appendix A, this is a cancellation between contributions from worldsheets with an unpaired crosscap and worldsheets with boundaries, with one boundary ending on L 'with even degree' (specifically, wrapping the generator of H 1 (L, Z) = Z 2 an even number of times, hence begin topologically trivial). This results in a condition where d ∈ H 2 (X, L; Z) = Z is the relevant homology class. It implies that R-type Mathematically, this condition applies to real codimension one boundary strata in moduli space, in which a given worldsheet piece near L develops a node which 3 Note that we define g such that the negative Euler characteristic is 2g + h − 2, 2g + h − 1, and 2g + h − 2 in the F, R and K cases respectively, i.e. it also accounts for Klein handles.
can be smoothed to yield either a disk or a crosscap. The combined count of these homologically trivial disks and crosscaps leads to cancellation of potentially ill-defined pieces, and produces an invariant count.
Strong evidence for this consistency condition comes from the fact that the invariant numbers thus computed turn out to be all integers. This motivated the proposal of an ansatz reminiscent of a BPS expansion, as a sum over holomorphic embeddings (rather than maps) equivariant with respect to worldsheet parity Ω :Σ →Σ.
If we write the total topological amplitude as In more physical terms, the tadpole cancellation condition means that the background contains a single D-brane wrapped on L, as counted in the covering space. The interpretation in terms of a physical type IIA construction and its lift to M-theory will be discussed in the next section.

Tadpole cancellation, the O4/D4 system and M-theory
It is natural to look for a physical realization of the real topological string in terms of type IIA on the threefold X 6 , quotiented by worldsheet parity times an involution acting antiholomorphically on X 6 . In general, we consider involutions with a fixed point set along the lagrangian 3-cycle L, which therefore supports an orientifold plane.
The total dimension of the orientifold plane depends on the orientifold action in the 4d spacetime, and can correspond to an O6-plane or an O4-plane. The choice of 4d action is not specified in the topological string, but can be guessed as follows.
We expect that the topological tadpole cancellation condition has some translation in the physical theory, as a special property occurring when precisely one D-brane (as counted in the covering space) is placed on top of the orientifold plane. Since the charge of a negatively charged O4-plane is −1 (in units of D4-brane charge in the covering), the configuration with a single D4-brane stuck on top of it is special, because it cancels the RR charge locally (on the other hand, the charge of an O6-plane is −4, and no similarly special property occurs for a single stuck D6-brane).
The presence of a single D4-brane stuck on the O4-plane is not a consistency requirement of the type IIA theory configuration, 4 but rather a condition that we will show leads to a particularly simple M-theory lift, and a simple extension of the Gopakumar-Vafa BPS expansion of topological amplitudes. This nicely dovetails the role played by tadpole cancellation in the real topological string to achieve the appearance of integer invariants.
The M-theory lift of O4-planes with and without D4-branes has been discussed in [22,23]. In particular, a negatively charged O4-plane with no stuck D4-brane, spanning the directions 01234 in 10d Minkowski space M 10 , lifts to M-theory on a Z 2 orbifold , and which also flips the M-theory 3-form, C 3 → −C 3 . The latter action is required to be a symmetry of the M-theory Chern-Simons term, and matches the effect of the type IIA orientifold action on the NSNS 2-form B 2 . Hence, we will classify the M-theory action as 'orientifold' as well.
The construction generalizes to compactification on X 6 , with the orientifold acting holomorphically on X 6 . It produces M-theory on the quotient (M 4 × X 6 )/Z 2 × S 1 , with the Z 2 acting as the antiholomorphic involution σ on X 6 and as on 4d Minkowski space. This system, and its generalization with additional D4-brane pairs (M5-branes in M-theory), is discussed in section 5.2. Here we simply note that the explicit breaking of the SU(2) L symmetry already in the 5d theory makes necessary to make certain assumptions on the structure of BPS multiplets in the theory, obscuring the derivation of the BPS expansion of the amplitude.
The M-theory lift of a negatively charged O4-plane with a stuck D4-brane is however much simpler, and in particular does not suffer from these difficulties. Because of the already mentioned local cancellation of the RR charge, the M-theory lift is a completely smooth space described by a freely acting quotient , as a half-period shift on S 1 (y → y + π for periodicity y y + 2π), and flipping the 3-form C 3 (hence defining an M-theory orientifold).
Concerning the latter, it is important to point out that the negative charge of the 4 Notice that any configuration with additional pairs of D4-branes is continuously connected to it.
See section 5.1 for further discussion. Also, topological A-models related to systems of O4-planes with no stuck D4-brane, and their M-theory interpretation, have appeared in [13], see section 5.2 for further discussion.
O4-plane implies that there is a half-unit NSNS B 2 background on an RP 2 surrounding the O4-plane; consequently, there is a non-trivial half-unit of 3-form background on the corresponding M-theory lift (CP 1 × S 1 )/Z 2 . This will play an important role in the M-theory interpretation of the disk/crosscap tadpole cancellation, see section 4.3.
The construction generalizes to compactification on X 6 , with the orientifold acting holomorphically on X 6 . It produces M-theory on the quotient (M 4 × X 6 × S 1 )/Z 2 , with the Z 2 acting as The geometry is a (Möbius) fiber bundle with base S 1 , fiber M 4 × X 6 , and structure group Z 2 .
As before, it is straightforward to add extra D4-brane pairs away from (or on top of) the O4-plane, since they lift to extra M5-brane pairs in M-theory, see section 5.1.

M-theory BPS expansion of the real topological string
The M-theory configuration allows for a simple Gopakumar-Vafa picture of amplitudes, which should reproduce the real topological string amplitudes. Since the quotient is acting on the M-theory S 1 as a half-shift, its effect is not visible locally on the S 1 . This means that the relevant 5d picture is exactly the same as for the closed oriented setup, c.f. section 2, so the relevant BPS states are counted by the standard Gopakumar-Vafa invariants. When compactifying on S 1 and quotienting by Z 2 , these states run in the loop as usual, with the only (but crucial) difference that they split according to their parity under the M-theory orientifold action. In the Möbius bundle picture, even components of the original N = 2 multiplets will run on S 1 with integer KK momentum, whereas odd components run with half-integer KK momentum.
The split is also in agreement with the reduction of supersymmetry by the orientifold, which only preserves 4 supercharges. Note also that the orientifold is not 4d Poincaré invariant, as Lorentz group is broken as The preserved supersymmetry is not 4d N = 1 SUSY, and in particular it admits BPS particles.

General structure
The states in the Hilbert space H are groundstates in the SUSY quantum mechanics on the moduli space of wrapped M2-branes. In the orientifold model, these BPS states of the 5d theory fall into two broad classes.

Non-invariant states and the closed oriented contribution
Consider a BPS state |A associated to an M2 on a curve Σ g not mapped to itself under the involution σ; there is an image multiplet |A associated to the image curve 5 Σ g . We can now form orientifold even and odd combinations |A ± |A , which run on the S 1 with integer or half-integer KK momentum, respectively. Since each such pair has Z A = Z A and identical multiplet SU(2) L × SU(2) R structure and multiplicities (inherited from the parent theory), we get the following structure: (4. 3) The sinh −2 factor corresponds to 4d kinematics, since the orientifold imposes no restriction on momentum in the directions transverse to the fixed locus. We have also denoted n Σg the possible multiplicity arising from SU (2)  The conclusion is that contributions with even wrapping belong to the sector of non-invariant states, which heuristically describe disconnected curves in the cover and reproduce the closed oriented topological string.

Invariant states and the open and unoriented contributions
The second kind of BPS states correspond to M2-branes wrapped on curvesΣĝ in the cover, mapped to themselves under σ. The overall parity of one such state is determined by the parities of the states in the corresponding SU(2) L and SU(2) R representations.
We introduce the spaces Hĝ ± describing the even/odd pieces of the SU(2) L representation Iĝ for fixedĝ. In what follows, we drop theĝ label to avoid cluttering notation.
We similarly split the equivariant BPS invariant GV ĝ,β (i.e. after removing the pairs of states considered in the previous discussion) into even/odd contributions as Recalling that states with even/odd overall parity have integer/half-integer KK momenta, we have a structure β,ĝ (4.5) In the next to last line, the traces clearly add up to the total trace over the parent N = 2 multiplet and the GV ± add up to the parent BPS invariants, c.f. eq. (4.4).
Noticing also that it corresponds to even wrapping contributions m ∈ 2Z, we realize that this corresponds to a contribution to the closed oriented topological string partition function, c.f. footnote 5. As discussed, it should not be included in the computation leading to the open and unoriented contributions.
The complete expression for the latter is where we have introduced the integers, which we call real BPS invariants, These integer numbersĜV's are those playing the roleGV's in eq. (3.3). Note however that their correct physical interpretation differs from that in [5], where they were rather identified as our GV ĝ,β . Note also that the correct invariants eq. (4.7) are equal mod 2 to the parent GVĝ ,β , proposed in [5], just like the GV ĝ,β .
In eq. (4.6) we have taken into account that these states, being invariant under the orientifold, propagate only in the 2d fixed subspace of the 4d spacetime, resulting in a single power of (2 sinh) in the denominator. This also explains the factor of 2 in the graviphoton coupling relative to eq. (2.4). In the next section we fill the gap of showing the promised equality of the even and odd multiplicities, and compute the trace difference in the last expression.

Jacobian and computation of SU(2) L traces
We must now evaluate the trace over the even/odd components of the Hilbert space of a parent N = 2 BPS multiplet. This is determined by the parity of the corresponding zero modes on the particle worldline. As reviewed in section 2, the traces are nontrivial only over the cohomology of the Jacobian ofΣĝ which determines the SU(2) L representation. We now focus on its parity under the orientifold.
Consider for example the case of I 1 , c.f. eq. (2.7). We introduce the formal split of the trace into traces over H ± where ± denotes orientifold behavior and denotes a formal combination operation, which satisfies 2 = 1 (it corresponds to the (−1) m factor once the wrapping number m has been introduced, c.f. eq. (4.5)). Since the orientifold action is an antiholomorphic involution on the worldsheet, it acts as dz ↔ dz, so eq. (2.7) splits as where, to avoid notational clutter, we have reabsorbed λ into s.
Since the creation and annihilation operators associated to different 1-forms commute, the argument generalizes easily to higher genus, and the trace over a representation Iĝ has the structure where t + g and t − g contain even and odd powers of t − 1 , respectively. For instance, for I 2 we have to trace over and obtain We are now ready to compute the final expression for the BPS expansion

The BPS expansion
Recalling eq. (3.3), the genuine open and unoriented contribution reduces to the odd wrapping number case eq. (4.6). Interestingly, the trace difference can be written (restoring the λ) This has the precise structure to reproduce the conjecture in [5] as in eq. (3.3), with the invariants defined by eq. (4.7). In particular we emphasize the nice matching of exponents of the sinh factors (achieved since for the coveringĝ − 1 = χ) and of the exponential e −Z = q d/2 for a one-modulus X 6 (the factor of 1/2 coming from the volume reduction due to the Z 2 quotient.) The only additional ingredient present in eq. (3.3) is the restriction on the degree, which is related to the conjectured tadpole cancellation condition, and which also admits a natural interpretation from the M-theory picture, as we show in the next section. We simply advance that this restriction applies to examples with H 1 (L; Z) = Z 2 . Our formula above is the general BPS expansion of the real topological string on a general CY threefold.
We anticipate that, once the tadpole cancellation discussed below is enforced, our derivation of eq. (4.15) provides the M-theory interpretation for the integer quantities GV appearing in eq. (3.3) as conjectured in [5]. Therefore the real topological string is

Tadpole cancellation
In

First step: Restriction to even degree
Consider the relative homology exact sequence Since (the embedded image of) a crosscap doesn't intersect the lagrangian L, its class must be in the kernel of the second map, i.e. the image of the first. Thus, every crosscap contributes an even factor to the degree d. For boundaries, the same argument implies that boundaries wrapped on an odd multiple of the non-trivial generator of H 1 (L; Z) = Z 2 contribute to odd degree, while those wrapped on an even multiple of the Z 2 1-cycle contribute to even degree. This restricts the possible cancellations of crosscaps to even degree boundaries.

Second step: Relative signs from background form fields
We now show that there is a relative minus sign between crosscaps and disks associated to the same (necessarily even degree) homology class. As mentioned in section 4.1, the M-theory lift contains a background 3-form C 3 along the 3-cycles (CP 1 × S 1 )/Z 2 , with the Z 2 acting antiholomorphically over CP 1 ; this corresponds to a half-unit of NSNS 2-form flux on any crosscap RP 2 surrounding the O4-plane in the type IIA picture.
In M-theory, the reduction of In contrast, boundaries do not receive such contribution, 6 and therefore there is a relative sign between contributions from curves which fall in the same homology class, but differ in trading a crosscap for a boundary.

Third step: Bijection between crosscaps and boundaries
To complete the argument for the tadpole cancellation condition, one needs to show that there is a one-to-one correspondence between curves which agree except for a replacement of one crosscap by one boundary. The replacement can be regarded as a local operation on the curve, so the correspondence is a bijection between disk and crosscap contributions.
More precisely we want to show that for every homologically trivial disk which develops a node on top of L we can find a crosscap, and viceversa. This is mathematically a nontrivial statement, for which we weren't able to find an explicit construction going beyond the local model of eq. (A.3). Moreover, in higher genera this problem has not 6 Note that odd degree boundaries can receive an extra sign due to a possible Z 2 Wilson line on the D4-brane. This however does not affect the even degree boundaries, which are those canceling against crosscaps. Hence it does not have any effect in the explanation of the tadpole cancellation condition.
been tackled by mathematicians yet. Nonetheless, following [26], we propose a model for gluing boundaries on the moduli space in the genus zero case, which points towards the desired bijection. The original argument applies to holomorphic maps, relevant to Gromov-Witten invariants; we expect similar results for holomorphic embeddings, relevant for Gopakumar-Vafa invariants.
The main point is that integrals over moduli spaces of Riemann surfaces make sense and are independent of the choice of complex structure if the moduli space has a virtually orientable fundamental cycle without real codimension one boundaries (RCOB).
If L = ∅, in order to achieve this, one has to consider together contributions coming roughly speaking from open and unoriented worldsheets, as proposed by [5].
We are interested in elements of RCOB in which a piece of the curve degenerates as two spheres touching at a point q: where f is the holomorphic map, Σ i = CP 1 , the involution exchanges the Σ i and f (q) ∈ L. For real = 0 one can glue Σ into a family of smooth curves, described locally as 7 Σ = (z, w) ∈ C 2 : zw = . leading to either a boundary or a crosscap. The RCOB corresponding to sphere bubbling in the '+' case is the same as the RCOB for the '−' case. By attaching them along their common boundary, we obtain a moduli space whose only RCOB corresponds to disk bubbling. The resulting combined moduli space admits a Kuranishi structure and produces well-defined integrals.

Final step
Using the above arguments, we can now derive eq. (3.1), as follows. First, the tadpole cancellation removes contributions where the number of crosscaps c is odd, so taking where h denotes the number of boundaries. 7 In this equivariant covering picture, we only consider singularities of type (1) as in Definition 3.4 of [27], since the covering does not have boundaries.
Second, the value of d = d i (mod 2) can only get contributions from boundaries and crosscaps (since contributions of pieces of the curve away from L cancel mod 2 from the doubling due to the orientifold image); moreover contributions from crosscaps and even degree boundaries cancel. Hence, the only contributions arise from boundaries with odd terms d i , so clearly d ≡ h (mod 2). We hence recover eq. (3.1).

Adding extra D4-brane pairs
The discussion in the previous sections admits simple generalizations, for instance the addition of N extra D4-brane pairs in the type IIA picture. 8 The two branes in each pair are related by the orientifold projection, but can otherwise be placed at any location, and wrapping general lagrangian 3-cycles in X 6 . For simplicity, we consider them to wrap the O4-plane lagrangian 3-cycle L, and locate them on top of the O4-plane in the spacetime dimensions as well.
In the M-theory lift, we have the same quotient acting as a half shift on the M-theory S 1 (times the antiholomorphic involution of X 6 and the spacetime action , now with the extra D4-brane pairs corresponding to extra M5brane pairs, related by the M-theory orientifold symmetry [22]. Notice that since the orientifold generator is freely acting, there is no singularity, and therefore no problem The sum in m runs only over positive odd integers. The N ± β,R,r are the multiplicities of (even or odd) states from M2-branes on surfaces in the class β, with spin r under the rotational U(1) in the 01 dimensions, 9

Relation to other approaches
In this section we describe the relation of our system with other unoriented A-model topological strings and their physical realization in M-theory.

M-theory lift of the four O4-planes
As discussed in [22,23] there are four kinds of O4-planes in type IIA string theory, with different lifts to M-theory. We describe them in orientifolds of type IIA on X 6 × M 4 , with the geometric part of the orientifold acting as an antiholomorphic involution on X 6 and x 2 , x 3 → −x 2 , −x 3 on the 4d spacetime.
• An O4 − -plane (carrying −1 units of D4-brane charge, as counted in the covering space) with no D4-branes on top. Its lift to M-theory is a geometric orientifold Inclusion of additional D4-brane pairs (in the covering space) corresponds to including additional M5-brane pairs in the M-theory lift.
• An O4 0 -plane, which can be regarded as an O4 − with one stuck D4-brane. We recall that its M-theory lift, exploited in this work, is with the Z 2 including a half-period shift of the S 1 . Additional D4-brane pairs correspond to additional M5-brane pairs, as studied in the previous section.
The M5-branes are stuck because they do not form an orientifold pair, due to a different worldvolume Wilson line [23].
• An O4 + -plane, which can be regarded as an O4 + with an extra RR background field. Its M-theory lift is our M 2 × (R 2 × X 6 × S 1 )/Z 2 geometry, with one stuck M5-brane fixed by the Z 2 action. even locally on the S 1 , i.e. already at the level of the 5d theory, and breaks the SU(2) L symmetry. Therefore the structure of multiplets need not correspond to full SU(2) L multiplets, although this is explicitly assumed in most of these references. Although supported by the appropriate integrality properties derived from the analysis, these extra assumptions obscure the physical derivation of the BPS integrality structures.

The O4
We emphasize again that these properties differ in our system, which corresponds to the lift of the O4 0 -plane. The SU(2) L multiplet structure is directly inherited from the parent theory, and is therefore manifestly present, without extra assumptions.

The O4 + case
The case of the O4 + -plane has been discussed in the literature as a

Conclusions and open issues
In this paper we have discussed the BPS integer expansion of the real topological string in [5], using the M-theory lift of the O4-plane with one stuck D4-brane. Since the geometry is a Z 2 quotient acting freely in the M-theory S 1 , the 5d setup enjoys an enhancement to 8 supercharges and is identical to that in the closed oriented Gopakumar is partially supported by the COST Action MP1210 "The string Theory Universe" under STSM 15772.

A Review of Walcher's real topological string
We now give a short review of A-model localization as in [5]. Take for concreteness Fermat quintic, given by and involution σ : (x 1 : which gives a fixed point lagrangian locus L with RP 3 topology, hence H 1 (L; Z) = Z 2 .

A.1 Tadpole cancellation in the topological string
Requiring a function f to be equivariant implies that fixed points of Ω are mapped to L, but one has to further specify their homology class in H 1 (L; Z) = Z 2 . When that class is trivial, then under deformation of the map it can happen that the boundary is collapsed to a point on L. 10 The local model for this phenomenon is a Veronese-like embedding CP 1 → CP 2 defined by a map (u : v) → (x : y : z) depending on a target space parameter a, The image can be described as the conic xy − a 2 z 2 = 0, and it is invariant under σ if a 2 ∈ R. The singular conic a = 0 admits two different equivariant smoothings, determined by the nature of a: The proposal of [5] to account for this process is to count disks with collapsible boundaries and crosscaps together. Specifically, there is a one-to-one correspondence between even degree maps leading to boundaries and maps leading to crosscaps (which must be of even degree, in order to be compatible with the antiholomorphic involution, as already manifest in the above local example). The tadpole cancellation condition amounts to proposing the combination of these paired diagrams, such that certain cancellations occur. For instance, the amplitude R (g,h) (odd number of crosscaps) vanish (due to the cancellation of the unpaired crosscap with an odd degree boundary). Similarly, for the remaining contributions, in terms of χ and d, the tadpole cancellation imposes the restriction d ≡ h ≡ χ mod 2. (A.5) The first equality follows from the requirement that odd degree contributions only come from boundaries (homologically trivial, i.e. even degree, ones cancel against crosscaps), while the second from the requirement that there be no unpaired crosscaps. where β ∈ H 2 (CP D ; Z) is to be identified with d in this particular case, and n denotes the number of punctures.

A.2 Rules of computation
The next step is to apply Atiyah-Bott localization to the subtorus T 2 ⊂ T 5 compatible with σ. As explained in [18], the fixed loci of the torus action are given by nodal curves, in which any node or any component of non-zero genus is collapsed to one of the fixed points in target space, and any non-contracted rational component is The components of the fixed locus can be represented by a decorated graph Γ and one has well-defined rules for associating a graph to a class of stable maps.
For the case of real maps, one has to be extra careful and require the decoration to be compatible with the action of Ω and σ. For example, consider a fixed edge: if we think of z = w 1 /w 2 , then in eq. (A.8) z → 1/z is compatible (i.e. f is equivariant) with any degree, while z → −1/z requires even degree, because our involution acts on the target space as (x 1 : x 2 : . . .) → (x 2 : x 1 : . . .) i.e. w = x 1 /x 2 → 1/w.
A careful analysis in [5,7,28,29] allows to conclude that the localization formula takes the formG where the following prescriptions are used: (i) the (−1) p(Σ) factor in front of the localization formula is put by hand in order to fix the relative orientation between different components of moduli space; (ii) for any fixed edge of even degree, the homologically trivial disk and crosscap contributions are summed, with a relative sign such that they cancel. This is the above mentioned tadpole cancellation.
Finally, [5] proposes an integer BPS interpretation of the obtained rational num-bersGW: by combining them with the above prescribed signs at fixed χ, one gets integer numbersGV, which are conjectured to reproduce a BPS expansion for the open-unoriented topological amplitudes.

B Physical couplings
An interesting question one can ask is what kind of coupling the topological string is computing in the IIA physical theory. This has a clear answer for the closed oriented sector [30,31], while some proposals have been made for open oriented [4] and unoriented sectors [9]. Here we make a proposal for the analogous expression in our unoriented model.
We have 4 supercharges in 1 + 1 dimensions, and we'd like to find a good splitting of the N = 2 Weyl tensor W ij µν = T ij µν + R µνρλ θ i σ ρλ θ j + · · · , (B.1) where one requires the graviphoton field strength T to acquire a self-dual background, and hence can write where W · v = W αβ σ µν αβ v µν . This has a contribution RTĝ −1 (in the covering picturê g−1 = χ) which in principle can generate (via SUSY) the sinh −1 power in the Schwinger computation, taking into account the fact that the orientifold halves the number of fermion zero modes on the Riemann surface.
It would be interesting to discuss the appearance of this contribution from different topologies at fixed χ. We leave this for future work.