Open Strings in IIB Orientifold Reductions

We consider type IIB compactifications on a general 4D group manifold with different types of possible spacetime filling O-planes and the corresponding D-branes parallel to them. Once fluxes allowed by the associated orientifold projection are included, a 6D $\mathcal{N}=(1,1)$ gauged supergravity is obtained. In this paper we show how the consistent coupling to dynamical open strings living on the spacetime filling D-branes may be captured by the inclusion of extra vector multiplets and extra embedding tensor deformations on the gauged supergravity side. As a result, the quadratic constraints on the embedding tensor consistently reproduce the source corrected 10D Bianchi identities. Furthermore, the field strength modifications induced by the open string sector could potentially be understood as U-dual versions of the Green-Schwarz terms. Finally, the entire scalar potential of the theory exactly matches the one obtained from reduction of the bulk action plus the source contributions.


Introduction
Extracting consistent low energy effective descriptions from string theory is one the main challenges of theoretical high energy physics.Typically this procedure involves dimensional reduction and (partial) supersymmetry breaking.Depending on the mechanism used in order to realize them, a plethora of viable lower dimensional models arises, with varying amounts of supersymmetry in different dimensions.Low energy effective models obtained in this way are by construction UV consistent and belong to the string landscape.
On the other hand, by adopting a bottom-up approach instead, one could study different lower dimensional (supersymmetric) constructions and assess whether or not they can be consistently coupled with quantum gravity in a UV regime.This is the perspective promoted by the so-called Swampland Program [1,2], which aims at identifying a set of consistency requirements that any effective theory must comply with, in order for it to admit a UV completion.
By restricting oneself to theories enjoying extended supersymmetry, the range of possibilities gets drastically reduced, up to the extent that UV consistency requirements in some instances may be even exhaustively analyzed.This certainly applies to the case of maximal supersymmetry, i.e. 32 supercharges.In 10D, the only consistent maximal supergravities are type IIA and type IIB supergravities and they exactly coincide with the low energy limits of the corresponding superstring theories, respectively.This may be viewed as a prime manifestation of string universality.Now, still within 10D one may consider theories with half-maximal supersymmetry.In such a situation, the aforementioned universality principle was verified in [3] by showing that the only UV consistent half-maximal theories are N = 1 supergravities with gauge groups given by either SO (32) or E 8 × E 8 .Those are indeed the only gauge symmetries that may be ever obtained by considering the low energy limits of heterotic or type I superstring theories.
In the last few decades we have learned a number of things concerning string theories with 16 supercharges and this allowed us to address the string universality issue in dimension lower than 10.By now we can consider it to be fully verified up to dimension 8 [4][5][6].Besides, there have been recent developments even in dimension 7 and 6, the latter both with (2, 0) [7] and (1, 1) [8] supersymmetry, as well as some preliminary studies on D < 6 cases [9][10][11].
Our present work is to be placed within such a context, from which it draws its main motivations.We aim at building a bridge between top-down string theory constructions yielding 6D theories with (1, 1) supersymmetry and the corresponding (gauged) supergravities, which are classified by means of bottom-up based organizing principles 1 .In more concrete terms, the stringy setup's relevant here are compactifications of type I/heterotic strings on T 4 , as well as orientifold reductions of type IIA/IIB on T 4 , or M theory on T 5 .Our interest towards (1,1) supergravity rather than for the (2, 0) one is due to the fact that none of the orientifold projections respecting chiral extended 6D supersymmetry allows to turn on fluxes.This is, on the other hand, consistent with the statement that (2, 0) supergravities do not admit any consistent embedding tensor deformations.Conversely in the non-chiral (1, 1) case, we find a wide range of flux compactifications.
In [17], an analysis of this sort was already presented and all the cases consistent with 6D Lorentz symmetry and (1, 1) supersymmetry were discussed.In each single setup the dictionary was obtained between 10D (11D) fields & fluxes on the one side, and 6D fields & deformations on the other side.The approach used mimics that of [18,19] designed for orientifold reductions down to 4D.Focusing on compactifications over 4d twisted tori, a vacua scan performed with the aid of the 6D gauged supergravity description showed the existence of a wide landscape of Minkowski (Mkw) vacua, but no maximally symmetric backgrounds with non-vanishing cosmological constant appeared.
The aim of this paper is to extend the analysis carried out in [17] to include open string effects such as dynamical brane position moduli and Wilson lines wrapped in internal space, as well as non-Abelian brane gauge groups and non-trivial associated Yang-Mills (YM) flux.While these ingredients are difficult to take into account from a top-down perspective, we show that these are straightforwardly handled from a bottom-up viewpoint, just at the price of including extra vector multiplets within the associated gauged supergravity description.The reason for this is that Lagrangians of half-maximal gauged supergravities (see e.g.[20] for the 4D & 5D cases) are fully determined for a given choice of embedding tensor [21], simply out of imposing consistency and supersymmetry.
At a technical level, the work done in this paper proves the equivalence between the effective scalar potential obtained from type IIB orientifold reductions and that of a suitable gauged supergravity, with the appropriate amount of vector multiplets accounting for both closed and open string excitations.The highly non-trivial result is a full matching between the scalar potential obtained from gauged supergravity and the one arising from reduction of the bulk action plus the contributions coming from the effective actions of the spacetime filling sources.This matching works even in presence of open string effects such as non-Abelian brane gauge groups and non-vanishing YM internal flux.It is worth remarking that such competing effects between closed and open string sectors in some sense require working at finite α .It still remains to be understood whether this set of α effects is also physically reliable, besides being mathematically consistent.
The paper is organized as follows.In Sec. 2 we review some salient features of Op/Dp systems, the associated light dof's, possible gauge groups and consistency requirements.In Sec. 3 we spell out the embedding tensor formulation of 6D N = (1, 1) gauged supergravities coupled to an arbitrary number of vector multiplets.In Sec. 4 we analyze the case of IIB reductions including spacetime filling O5/D5 sources and work out the dictionary between the 6D supergravity side and the type IIB side.A parallel analysis is then carried out in Sec. 5 for O7/D7 sources and later in Sec.6 for O9/D9, i.e. type I reductions.One of the key results of the paper is the discovery of bulk field strength modifications sourced by the open string vector fields, just like in the heterotic case, where this was due to the Green-Schwarz (GS) mechanism [22].Indeed, the modifications derived here could be heuristically understood as U-dual versions of GS terms.Finally, our appendices contain technical support material concerning type IIB supergravity and non-Abelian brane actions, as well as reductions thereof.
2 General Aspects: Op/Dp Systems & Open Strings Within the string theory spectrum Dp-branes appear as higher dimensional spacetime defects representing dynamical boundary conditions for open strings.Such extended objects admit a low energy description in terms of supergravity black brane solutions. 2n particular, a Dp-brane has a positive tension and carries a positive charge µ Dp = T Dp w.r.t. a RR (p + 1)-form field.The corresponding anti-brane Dp will have the same tension, but carry opposite charge.Since the associated supersymmetry projectors are mutually orthogonal, brane configurations involving both Dp's & Dp's at the same time will necessarily be non-supersymmetric.Each Dp-brane has a massless vector multiplet associated with the light open string state attached to it.Its low energy description is given by U(1) maximal SYM in (p+1) dimensions.Now, a set of N Dp-branes which are kept separated from one another at finite distance describes an Abelian U(1) N gauge theory.However though, in the limit where these are made to collide together to form a brane stack, the system undergoes a gauge symmetry enhancement to the non-Abelian gauge group U(N ).In this case, the N 2 generators of U(N ) are in one-to-one correspondence with light strings having each extremum on any D-brane within the stack.We refer to Appendix C for more details concerning non-Abelian brane actions and their relation to non-commutative geometry.
Besides ordinary Dp-branes, more exotic objects are present in the spectrum, i.e. orientifold planes.These objects are the loci of fixed points of a given orientifold Z 2 action Ω Op , which may be defined through where Ω is the worldsheet parity acting on the closed string bulk fields as Ω : where k = 2q + r, with r = 0 (type IIB), or r = 1 (type IIA).The second Z 2 factor σ Op is a spacetime involution flipping the sign of all transverse coordinates Op : Finally, σ F L involves the so-called fermionic number and is given by It turns out that there exist two different types of Op-planes preserving the same supersymmetries as a stack of Dp-branes parallel to them.These are conventionally denoted by Op + & Op − and their tension T Op ± and charge µ Op ± satisfy the following formula which in particular implies that fractional orientifold charges are allowed for p < 4 [24].Despite the fact that Op-planes carry tension and charge, they appear to be completely rigid objects, at least at a perturbative level.When considering a system made out of Dp-branes and parallel Op-planes, in order to fully specify the dynamics, we also need to spell out the orientifold action on the open string states living on the D-branes.This is done by identifying its action on the open string Chan-Paton factors λ.For N Dp's and one Op [25], this reads with The above difference in the orientifold action at the level of the Chan-Paton factors results in different open string SYM gauge groups in presence of an Op + , or an Op − .In the former case we have an USp(2N ) group, while in the latter we have SO(2N ) instead.The corresponding conceptual picture in these two situations can be found in Figure 1.
For the most general system made out of Dp-branes & Op-planes in the absence of fluxes and in flat space, the following tadpole cancellation condition is required for UV-finiteness of the corresponding quantum description which can be written as where N Dp = 2N accounts for the imagine branes as well.It is crucial to remember that the above constraint originates from demanding that string amplitudes be free of divergences and it refers to amplitudes calculated in flat space and in the absence of fluxes.
In our work we will be considering more involved situations where the background fluxes may contribute in several ways to the tadpole constraints for the corresponding spacetime filling sources.We will therefore assume that the relaxed versions of (2.7) that we will be writing in every specific case of our interest play an analogous role in guaranteeing UV-finiteness of string amplitudes.This is however not explicitly shown in our setup's.
If we now go back to purely supersymmetric brane configurations obtained by setting N Dp = 0 in equation (2.7), we find that it is actually possible to have both Op + 's & Op − 's at the same time, as long as the constraint N Op − ≥ N Op + is respected, in such a way that tadpole cancellation is realized without introducing susy breaking sources such as anti-branes.In this situation, tadpole cancellation would simply require adding the following amount of parallel Dp-branes (2.8) In this most general setup, if we furthermore allow for the possibility that these objects be separated into smaller groups from one another, we find that the most general gauge group will be of the form (2.9) The corresponding total number of massless vector fields reads which will precisely coincide with the bare quantity appearing in the lower dimensional supergravity description.In the remainder of the paper, when discussing how these open string gaugings are embedded within the effective lower dimensional gauged supergravity theories, we will no longer specify an explicit form of the YM gauge group, nor specifically discuss concrete brane setup's.We hope to analyze concrete applications of the machinery presented here in the next future.It is perhaps worth making one last general comment before moving to the technical supergravity analysis needed for our present purposes.It concerns the nature of the gauge groups just discussed here.Since in our analysis we will consider situations where the open and the closed string sectors are non-trivially coupled, the effective lower dimensional description will be a gauged supergravity with total gauge group featuring a mixing between G YM & G ISO , where G YM is realized in terms of D-branes and O-planes as we have just seen, while G ISO stems from the isometries of the bulk (internal) geometry.This suggests that the most general form of gauging within our 6D supergravity is expected to be a non-semisimple extension of G YM × G ISO3 .
3 Gauged N = (1, 1) Supergravities in 6D Ungauged N = (1, 1) supergravity stems from dimensional reduction of type I supergravity on a T4 .In this case, the complete set of closed string zero mode excitations is contained in the coupling between the gravity multiplet and four vector multiplets.
Since the goal of this paper is that of using N = (1, 1) supergravities as a tool for studying type IIB orientifold reductions including an excited open string sector, we need to introduce their general formulation featuring the coupling with an arbitrary number of vector multiplets 4 .The (ungauged) theory enjoys the following global symmetry where N is the number of extra vector multiplets.The physical (propagating) dof's of the theory are suitably rearranged into irrep's of the little group and of the global symmetry group as described in Table 1.In particular, the 17 + 4N scalar fields of the theory parametrize the following coset geometry where the scalar coset representative H M N is written in terms of a vielbein V M M as where the local SO(4) × SO(4 + N) index M has been split into (m, m, I), related to its SO(4) timelike , SO(4) spacelike , and SO(N) parts, respectively.The kinetic Lagrangian is given by The deformations of the ungauged theory which are consistent with bosonic symmetry as well as supersymmetry can arranged into the following embedding tensor irrep's where ζ M corresponds to a massive deformation inducing a Stückelberg coupling for the two-form B µν , while the remaining two irreducible pieces are traditional gaugings.
In particular, f M N P purely gauges a subgroup of SO(4, 4 + N), whereas ξ M gauges a combination of the R + Σ generator and generators in the SO(4, 4 + N) part.Now, given a specification of the embedding tensor Θ transforming as in (3.5), the consistency of the deformed theory demands its gauge invariance, which is enforced by imposing the following set of quadratic constraints (QC) which include conditions for the closure of the gauge algebra, i.e. generalized Jacobi identities.In (3.6), contractions are defined by means of the invariant SO(4, 4 + N) metric η M N and its inverse η M N .In what follows, we will perform a lightcone (LC) basis choice within the SO(4, 4) sector, combined with a standard Cartesian basis along the remaining SO(N) directions.The explicit form of η in this case is It is perhaps worth mentioning that this choice of basis precisely matches the one made in [17] within the (4, 4) part, which will represent the closed string sector of our type IIB orientifold compactifications.This choice is justified by the fact that closed string background fluxes have a natural mapping into LC components of the embedding tensor.
Embedding tensor deformations turn out to induce Yukawa-like couplings between scalars and fermions, which are parametrized by the so-called fermionic shift matrices.As a consequence, supersymmetry invariance of the action requires the presence of a scalar potential, which turns out to be quadratic in Θ.Its explicit form in terms of embedding tensor irrep's reads where H M N denotes the inverse of H M N and g is the gauge coupling constant.For simplicity, in the remainder of the paper, we fix g = 2.The four-index antisymmetric object H M N P Q appearing above is instead defined through in terms of the Cartesian vielbein VM M , which is in turn related to the LC one transforming the LC metric into diag(−I 4 , +I 4 , I N ).In [17] all possible orientifold reductions yielding N = (1, 1) theories in six dimensions were studided within the closed string sector.The closed string dynamics turned out to be contained within the theories with only the four (universal) vector multiplets included.In the next sections we will select the type IIB cases of interest and include an excited open string sector.This will require analyzing the 6D supergravity theories in the form presented in this section, i.e. with the inclusion of N extra vector multiplets.Such an extension will allow us to study open string dof's like brane position moduli and/or Wilson line moduli, i.e. axions arising from internal legs of the worldvolume gauge fields.Moreover, we will be able to consider possible physical effects of a non-Abelian worldvolume theory, and/or the presence of worldvolume flux wrapping internal space.

O5/D5 & Open Strings
Let us start by considering the minimal possible spacetime filling O-planes that respect 6D Lorentz symmetry, i.e.O5-planes.These are placed as follows within 10D spacetime, where σ O5 is the orientifold involution, whose action flips the sign of all transverse coordinates.The O5 projection is realized at the level of the 10D supergravity fields by means of the simultaneous action of the aforementioned involution, together with the worldsheet parity operator.Such a procedure yields the correct field content of a half-maximal supergravity in 6D.The resulting details of this projection are collected in Table 2.
The Z 2 parity of all internal components of the different IIB fields in the presence of spacetime filling O5-planes.The allowed ones yield excitable 6D scalar fields.Note that the total amount of resulting scalars correctly gives 17+4N, i.e. the dimension of the supergravity coset (3.2).
In this case, due to presence of O5's and D5's, the reduction Ansatz can be formulated in a SL(4, R) × G YM covariant way.The 10D bulk supergravity Ansatz containing the 17 closed string scalars reads where " + . . ." denotes that we are discarding the terms in the Ansatz that do not contribute to the scalar potential.The scalars ρ, τ represent the volume and dilaton would-be moduli, M mn is an element of SL(4, R)/SO(4) describing deformations of the internal metric, and γ mn is antisymmetric and contains the scalars coming from the R-R two-form C (2) .These modes add up to 17, as they should.The remaining 4N scalars are part of the open-string sector and are denoted as Y Im , where the index I labels the adjoint representation of G YM .On the other hand, h m and f m are constants parametrizing the H (3) and F (1) fluxes within the closed string sector, while in the open-string one we have the possibility of considering a non-Abelian gauge group with structure constants g IJ K .The consistency requirements on the aforementioned flux parameters purely reduce to the Jacobi identity for the Yang-Mills structure constants, Note that the flux tadpole induced by H (3) and F (1) does not have to vanish, since the Bianchi identity for C (2) gets modified by the presence of O5/D5 sources: , where j = Q 5 vol M 4 is the effective current density.In the case at hands, we have that In the last equality we have made use of the relation between the D5 and O5 charges given in (C.15).The sign O5 = ±1 precisely determines the type of O5 ± plane that we are considering. 5The tadpole condition imposes the following condition which must be taken into account when matching the scalar potential of the compactification with that of supergravity, as the latter only knows about the fluxes.The scalar potential of the compactification ignoring the open-string sector was previously computed in [17].Now we build on their results and also take into account open-string effects.The worldvolume action of the D5 branes contains two pieces: the DBI and the WZ actions.The first directly gives a contribution to the scalar potential, while the contribution of the second secretly appears through the bulk scalar potential given in Appendix B. The reason lies in the fact that the WZ action contains couplings between the open-string fields and the R-R potentials which in turn give rise to modified (bulk) field strengths associated to the dual R-R potentials.
Let us consider each contribution separately, first focusing on the one coming from the DBI.The DBI action of the D5 branes is given by (see Appendix C for a detailed description) where is the D5 brane tension.The indices M, N, . . .are worldvolume indices whereas i, j, . . .denote the transverse ones.The matrices M and Q are defined as where E MN = ĜMN + BMN . 6Making use of the above compactification Ansatz, one finds that the matrices M and Q are given by where now the dots mean that we are ignoring terms that do not contribute to the scalar potential and also the ones which are of higher-order in λ.The generators of G YM are denoted by t I , and our conventions are such that [t I , t J ] = i g IJ K t K .Making use of (4.15) and (4.16) in (4.12), we obtain the following contribution to the scalar potential, where 2κ 2 6 = 16πG 6 , being G 6 the six-dimensional Newton's constant.In addition to (4.17), we will have the analogous contribution from the O5.Since the latter is non-dynamical, its contribution merely reduces to tension term (namely, the first one in (4.17)).Hence, the total DBI contribution from both types of sources is where λ5 = (2κ 2 6 N D5 T D5 ) 1/2 λ and g IJK = g IJ L κ LK .Crucially now F m is not pure flux, as it has the second contribution from the open-string scalars.On the contrary, the field strength H mnp is not modified, so it simply reads Finally, all that is left is to take the expression for the bulk scalar potential computed in [17] (see also Appendix B) and replace, according to (4. 19), The resulting expression has to be added to (4.18), which yields the following expression for the full scalar potential, where the bulk and the WZ contributions correspond to the first line, while the rest come from DBI.
Matching with N = (1, 1) gauged supergravity Since the O5 reduction Ansatz respects SL(4, R) × G YM covariance, the fundamental index M of SO(4, 4 + N) is split accordingly where m and m are (anti)fundamental indices of SL(4, R), while I is an adjoint index of G YM .The dictionary between flux parameters and embedding tensor components is summarized in Table 3.With these non-vanishing components of the embedding IIB fluxes Θ components Dictionary Table 3.The embedding tensor/fluxes dictionary for type IIB reductions with spacetime filling O5-planes and D5-branes.
tensor, the general form of the QC (3.6) reduces to the Jacobi identities for the structure constants g IJK , as given in (4.8).
In order to evaluate the general supergravity scalar potential (3.8) in our specific case, we need to identify how the 17 + 4N scalars parametrize Σ and H M N appearing there.This is done by directly expressing Σ and the coset representative V M M as a function of the scalars ρ, τ, M mn , γ mn , Y Im .In particular, we find that and the SO(4, 4 + N) coset element is where By plugging the above parametrization of the scalars and embedding tensor into the general form of the scalar potential (3.8), we find a perfect agreement with (4.22), which was calculated from direct dimensional reduction.

O7/D7 & Open Strings
Let us now consider reductions of type IIB with the inclusion of spacetime filling O7planes.These are placed as follows within ten-dimensional spacetime where σ O7 is the orientifold involution, whose action flips the sign of all transverse coordinates, while leaving the y a internal coordinates invariant.The O7 projection is realized at the level of the ten-dimensional supergravity fields by means of the simultaneous action of the aforementioned involution, together with the fermionic number (−1) F L and the worldsheet parity operator.Such a procedure yields the correct field content of a half-maximal supergravity in six dimensions.The resulting details of this are collected in Table 4.
IIB fields σ O7 (−1) Now, because the embedding of O7-planes breaks internal diffeomorphism covariance, the reduction scheme can only be formulated in a SL(2, R) a × SL(2, R) i × G YM covariant way.The ten-dimensional supergravity Ansatz containing the 17 closed string scalars reads e Φ = ρ τ −2 , (5.2) where again " + . . ." means that we are discarding the terms in the Ansatz that do not contribute to the scalar potential.The scalars ρ and τ still represent the volume and dilaton would-be moduli, σ is a non-universal geometric deformation controlling the relative size between the two-cycle wrapped by the O7 and the transverse one, M ab and M ij are elements of SL(2, R) a(i) /SO(2), and, finally, B ai , C ai , χ, C abij = ab ij ψ are axionic scalars coming from the NS-NS two-form and the R-R forms.These modes again add up to 17, as expected.Since in this case the metric flux is allowed by the O7 involution, we have introduced a parallelization of the internal manifold with torsion given in terms of the Maurer-Cartan one-forms v m = v m m dy m .These satisfy for some constants ω np m , which turn out to be the structure constants of the underlying Lie algebra and therefore fulfill the Jacobi identities as an integrability condition, ω [mn r ω p]r q = 0 . (5.8) The 0-form α = α(y) entering in the reduction Ansatz of C (0) , given in (5.4), introduces the 1-form flux F a via its exterior derivative: with F a = f a .The integrability condition implies that f a η b ab = 0 . (5.10) The 2-form β = 1 2 β ab v a ∧ v b in the reduction Ansatz of B (2) does not give rise to further scalars but rather to fluxes, as it satisfies: where H abi = h i ab denotes the H-flux.The integrability condition of this equation is trivially satisfied.Finally, the 2-form γ = γ ai v a ∧ v i in (5.5) encodes the 3-form flux F abi via its exterior derivative, with F abi = ab f i .
If we now add a stack of N D7 D7-branes parallel to the O7, these will require the existence of N extra vector multiplets, labelled by the index I.In terms of 6D dof's, besides the new vector fields, our resulting gauged supergravity will again have 4N new scalar modes.Half of them, denoted as Y Ii , correspond to the scalar fields living in the N vector supermultiplets, whereas the remaining half come from the internal components of the worldvolume gauge fields A I a .This way, the set Y Ii , A I a exactly parametrizes 4N independent extra scalar modes.The compactification Ansatze for the worldvolume gauge fields and the scalar fields are where v m 0 = v m m | y i =0 dy m , with m = (a, i), are the Maurer-Cartan 1-forms restricted to the worldvolume of the D7 branes7 and σ I is a 1-form satisfying where g IJ K are the Yang-Mills structure constants satisfying the Jacobi identities: In addition, σ I gives rise to the flux F I ab through where The above condition (5.14) on σ I has to be imposed in order to remove undesired dependence on the internal coordinates.Taking a exterior derivative, we can express this condition in terms of the fluxes g I as follows: which tells us that the Killing-Cartan metric, must be degenerate when the fluxes g I are turned on.Consequently, this implies that the Lie algebra cannot be semisimple.As we are going to see in what follows, the constraint (5.17) will arise in supergravity as one of the QC, (5.32).Let us further remark that the integrability condition of (5.16) is automatically satisfied without imposing further constraints on the fluxes g I .Such circumstance is very particular of this case.Taking these considerations into account and the compactification Ansatz, we find that the internal components of the field strength F I are given by Due to the presence of the O7 plane, the Z 2 truncation realized by the product σ O7 (−1) F L Ω projects out, some internal fluxes.Table 5 shows the exhaustive list of fluxes that are projected in by this truncation, where we observe that the fluxes ω ab i , ω ij k , ω ai j are forbidden.As a consequence, the Maurer-Cartan 1-forms v m are required to depend on the internal coordinates y m in a very particular way.This, together with the consistency of the Ansatz when studying the D7 brane effective action imposes the following functional dependence of the twist matrices v m m : . (5.20) The presence of the D7/O7 sources modifies the Bianchi identity of C (0) as follows, , where j is the effective 7-brane current.The above equation involves both the metric flux ω ij a and the F (1) flux, F a .Upon integration over the transverse space, one obtains the following tadpole condition where Q 7 is the total charge, receiving contributions both from the D7 branes and the O7, where O7 = ±1 amounts to considering the presence of O7 ± planes.As a consequence of (5.22), we will obtain a non-vanishing contribution in the scalar potential which is proportional to the effective tension, as we are about to see.
The procedure one has to follow to compute the scalar potential is exactly the same as in D5/O5 case studied in the previous section, so we will skip most of the details here.In order to evaluate the contribution from the DBI, we just need to know the matrices M and Q in (C.3).In the case at hands, these read where (5.27) On the other hand, the bulk contribution was already studied in [17] and has been reviewed in Appendix B. As emphasized in the previous section, we now have to take into account that open strings backreact onto the bulk fields modifying their field strengths as follows, where λ7 ≡ (2κ 2 8 N D7 T D7 ) 1/2 λ and g IJK ≡ g IJ L κ LK .Details on the derivation of these modified field strengths are provided in Appendix C.2.
All these partial results already allow us to compute the scalar potential.It turns out to be given by (5.31) Let us stress three relevant aspects of the potential: (i) as expected from the tadpole condition, the term proportional to the (non-vanishing) effective tension is present, (ii) the last two lines arise from the DBI action of the D7 branes, and (iii) the WZ contributions are entirely encoded in the modified field strengths F (1) and F (3) .8 Matching with N = (1, 1) gauged supergravity where a & ā (i & ī) are (anti)fundamental indices of SL(2, R) a(i) , while I is an adjoint index of G YM .The dictionary between flux parameters and embedding tensor components is summarized in Table 5.

Θ components Dictionary
Table 5.The embedding tensor/fluxes dictionary for type IIB reductions with spacetime filling O7-planes and D7-branes.Note that κ a satisfies (κ a ) i i = 0, and η a parametrizes the partial traces of ω, while still respecting ω mn n = 0, which is required by unimodularity.
With these non-vanishing components of the embedding tensor, the general form of the QC (3.6) reduces to which exactly reproduce the consistency constraints coming from both the Bianchi and Jacobi identities of the corresponding flux background.In particular, the first and last quadratic constraints arise from the Jacobi identities of the structure constants (5.8) associated to the group manifold.
For the evaluation of the general supergravity scalar potential (3.8) in this specific case, we again need to identify how Σ and the coset representative V M M are expressed as a function of the 17 + 4N scalars ρ, τ, σ, M ab , M ij , B ai , C ai , χ, ψ, A I a , Y Ii .We find while where (5.36) Thus, plugging V M M together with the parametrization of the embedding tensor Θ ≡ { f M N P , ξ M , ζ M } in the gauged supergravity potential (3.8), we obtain the same scalar potential as the one calculated from the compactification in (5.31).

O9/D9 & Open Strings
Let us finally study type IIB reductions with spacetime filling O9-planes, i.e. type I reductions.The orientifold planes in this case fill the entire ten-dimensional spacetime: , where σ O9 acts trivially on all the coordinates, due to the absence of transverse directions.In this case the Z 2 action realizing the truncation is purely given by the worldsheet parity operator Ω.The set of resulting scalar modes retained by this operation is shown in Table 6.The spacetime filling O9-planes preserve the diffeomorphism covariance, thus making the reduction scheme to be formulated in a SL(4, R) × G YM covariant way.The ten-dimensional supergravity Ansatz that contains the 17 closed string scalars coincides with the one studied by Kaloper and Myers in [26] except for the fact that they work in the heterotic frame and focus on the Abelian case. 9In terms of the type I fields, our Ansatz reads where λ9 ≡ (2κ 2 10 N D9 T D9 λ 2 ) 1/2 .Once again, the scalars ρ and τ represent the volume and dilaton would-be moduli, M mn is an element of SL(4, R)/SO(4), and C mn parametrizes the axionic scalars coming from the R-R 2-form C (2) .As expected, these modes add up to 17.As before, we have introduced a parallelization of the internal manifold with torsion given in terms of the Maurer-Cartan one-forms v m , which fulfill the equation In addition to these fields, when adding a stack of N D9 D9-branes parallel to the O9, we will require N extra vector multiplets, labelled by the index I to accommodate the full group G YM .In terms of 6D degrees of freedom, besides the new vector fields, our resulting gauged supergravity description will again have 4N new scalar modes arising from the internal components of the worldvolume gauge fields 3) The 1-and 2-form σ I (y) and γ(y) in the Ansatze (6.1) and ( 6.3) give rise to the vector flux F I mn and the 3-form flux F mnp listed in Table 7 via their exterior derivatives [26], The integrability conditions associated to the above equations are, order to compute the contribution to the scalar potential coming from the DBI action (C.3), we just need to know explicitly the matrix M M N (C.4) since the lack of transverse coordinates forbids the existence of Q.The non-trivial components of M M N are M mn = ρ M mn + λ F mn + . . . .(6.9) Instead, the bulk contribution can be extracted from using the results provided in Appendix B, but we need to compute first F mnp .To this aim, one has to bear in mind that the coupling of the sources to C (6) modifies the Bianchi identity of F (3) as in (C.62), which we repeat here for convenience This implies that, locally, the field strength F (3) is given by Restricting to the internal components components and making use of the reduction Ansatz, (6.1) and ( 6.3), we obtain that (6.12) Finally, the complete expression for the scalar potential is, where As a tadpole cannot be generated with the fluxes at our disposal, the condition N D9 + 32 O9 = 0 must be imposed.This implies N D9 = 32 and O9 = −1, so that the total charge and tension vanish.Therefore, the last term in the scalar potential, which is proportional to the effective tension, vanishes as well.

Matching with N = (1, 1) supergravity
The O9 reduction Ansatz preserves SL(4, R) × G YM covariance, in such a way that the fundamental index M of SO(4, 4 + N) is split into where m & m are (anti)fundamental indices of SL(4, R), and I is an adjoint index of G YM .The dictionary between flux parameters and embedding tensor components is summarized in Table 7.

IIB fluxes Θ components Dictionary
. The embedding tensor/fluxes dictionary for type IIB reductions with spacetime filling O9-planes and D9-branes.Note that ω mn p is restricted to satisfy ω mn n = 0, which is required for unimodular gaugings.
With these non-vanishing components of the embedding tensor, the general form of the QC (3.6) reduces to which exactly reproduce the consistency constraints coming from the BI of the corresponding flux background (see Appendix C.3 for further details).
For the evaluation of the general supergravity scalar potential (3.8) in this specific case, we again need to identify how Σ and the coset representative V M M are expressed as a function of the 17 + 4N scalars.In this case, because no fluxes are embedded into ζ M , the scalar potential is entirely written in terms of H M N .We find then Σ = ρ 1/2 , Λ = τ 2 , (6.16)

Concluding Remarks
In this paper we have studied type IIB flux compactifications down to six dimensions with spacetime filling O-planes, D-branes and open strings.Such compactifications turn out to yield 6D N = (1, 1) gauged supergravities.The exact relation between the 10D & the 6D descriptions was studied in [17] within the closed string sector.Now we have been able to generalize this correspondence to the case where open strings are excited.We also included open string effects such as non-Abelian D-brane gauge groups and non-trivial YM flux.Our analysis is very much in the spirit of the one in [28] carried out in the context of compactifications down to four dimensions, the main difference being that our brane gauge groups may be non-Abelian.The dictionary derived here allowed us to understand crucial physical mechanisms like the bulk field strength modification induced by open string effects.At least at a heuristic level, this can be related to the GS mechanism for the heterotic string via a duality chain.Besides this intuition, we were able to explain such a form of the bulk field strengths directly in terms of couplings contained in the WZ brane actions.
The present work sets the ground for interesting developments within the context of compactifications of type IIB string theory with sixteen supercharges.This will first of all, allow us to search for string vacua supported by interactions between the open and the closed string sectors.This moves towards the direction of [29][30][31], where a similar analysis was performed in type IIA strings and conditions for the existence of 4D vacua with mobile D-branes are discussed.Our machinery could be extremely valuable in improving efficiency for vacua searches.
Moreover, the possible existence of certain types of vacua within this setup could shed a light on issues of utmost importance, like the validity of Swampland conjectures or the string universality principle.To this end, it would be very interesting to explore the set of non-supersymmetric vacua of this sort (AdS or dS, even), or to study those consistency conditions for flux backgrounds coming from anomaly cancellation.Finally, if one could find supersymmetric AdS in this class, it would be interesting to study their holographic interpretation.We certainly intend to address all of these issues in the next future.are designed to automatically satisfy the following (modified) Bianchi identities (BI) It is worth mentioning that S IIB in (A.1) is called a pseudoaction because it must be supplemented by the following duality relations that yield the correct number of propagating degrees of freedom and hence allow for an on-shell realization of supersymmetry.By varying (A.1), one obtains the following set of equations of motion for the 10D dilaton Φ, for the form fields, and finally the (trace reversed) Einstein equations (A.12)

B Bulk Reduction
In order to carry out the dimensional reduction of type IIB down to six dimensions, we parametrize the ten-dimensional metric G MN in terms of the six-dimensional one and the moduli describing the four-dimensional internal metric.In particular, by picking the universal moduli ρ and τ singled out, whereas the rest of the moduli are encoded inside g (4) and describe volume preserving deformations of the internal geometry.In addition to that, we introduce local indices m, n as where the matrix M mn parametrizes the coset SL(4, R)/SO(4) and det M = 1.
To obtain the 6D gravity action in the Einstein frame upon compactification, we require the constraint [33] which implies that ρ and τ fix the internal volume and the string coupling.
Let us consider now each of the terms in the type IIB effective action.The determinant of the metric reduces to Then, the reduction of the Einstein term in (A.1) amounts to10 where V ω ≡ −ρ −1 τ −2 R (4) .In case of twisted toroidal compactifications, where v m are the Maurer-Cartan 1-forms, R (4) has the following expression [34]: where M mn is the inverse of M mn and ω mn p are the structure constants entering the Maurer-Cartan equation This, in turn, implies the Jacobi identities as an integrability condition, In addition to this, we will ask the structure constants to fulfill the unimodularity condition ω mn n = 0 for consistency, as we are performing the compactification at the level of the action.
us consider now the scalar potential arising from the H flux. Reducing the corresponding term of the action (A.1) yields so that the contribution consists of V H ≡ 1 12 H mnp H mnp ρ −3 τ −2 and the contraction with the indices is done with the internal metric g (4) .
Regarding the R-R p-forms 11 , their contribution to the scalar potential is . Thus, as the 10D Chern-Simons term does not give any contribution to the potential, the reduced D = 6 theory is given by the following action where the full scalar potential arising from the bulk and the effective tension consists of 12) The kinetic term for the moduli, which span a R + ρ × R + τ × SL(4, R)/SO(4) geometry, is given by

C Non-Abelian Brane Actions and Reductions Thereof
Dp-brane actions on curved backgrounds with fluxes were studied in [35,36].In this appendix we collect some relevant details for evaluating the bosonic effective actions of an Op-plane and a stack of N Dp Dp-branes contributing to the scalar potential upon compactification.We will follow the notation in [37].As opposed to the rest of this paper12 , we denote x M the worldvolume coordinates, whereas the transverse coordinates to the branes are called y i .Therefore, the 10D spacetime coordinates x M split into x M = (x M , y i ).We will work in the static gauge, where the position of each brane reads y Ii = λ Y Ii , with λ ≡ 2π 2 s .We will consider the generators of G YM to live in the fundamental representation of the Lie algebra.Denoted by {t I } I=1,...,N , they satisfy On the other hand, the WZ action reads hat "ˆ" on the fields indicates that they are evaluated at the position of the Dpbranes placed at y i = λY i , which is defined via a Taylor expansion as, for example, . (C.9) With P[• • •] we denote the pullback of the bulk fields over the Dp-brane worldvolume, in such a way the ordinary derivative ∂ M Y i is substituted by the covariant derivative D M Y i : Finally, the symbol ι Y denotes the interior product by a vector Y i , e.g., The bosonic effective action of an Op ( 1 , 2 ) -plane in type II string theory, with where the DBI and WZ terms are Let us note that the theories obtained combining a stack of parallel Dp-branes and a Op ( 1 , 2 ) -plane are not supersymmetric for arbitrary values of 1 and 2 .An intuitive way of seeing this is by noticing that the net force between these objects will not vanish (as expected for a supersymmetric configuration) unless 1 = 2 .Therefore, in the rest of the paper we always assume 131 = 2 ≡ Op , (C. 16) and simply use the notation Op to refer to the Op (±,±) planes which are BPS with respect to the Dp-branes.In addition to this and for the sake of concreteness, we will only discuss the case Dp = +1, corresponding to Dp-branes of possitive charge.When dimensionally reducing upon the entire transverse space, the above action reduces, in its low energy limit, to a maximal SYM theory with gauge groups of C & D type.This is due to the presence of O-planes parallel to the D-branes in our concrete setup.

C.1 Case O5/D5 & Open Strings
Let us compute the contribution of the sources to the scalar potential.Focusing on the scalar sector and making use of eqs.(4.15) and (4.16), we find that the reduction of the DBI action of N D5 coincident D5-branes is given by where the dots mean subleading contributions in λ (or, equivalently, in α ) which cannot be captured by the gauged-supergravity description together with other terms that will not enter the scalar potential.On the other hand, for the O5 planes we get simply the contribution from the tension Hence, the total contribution from the DBI action of the sources amounts to (C.19)This is not the only contribution of the sources to the scalar potential, as the Wess-Zumino terms in the D5-brane action give additional contributions.These, however, have been already included through the modification of the field strength (4.19).Let us discuss this aspect in more detail.The Wess-Zumino terms in the D5-brane action include a coupling to C (8) , Tr P e iλι Y ι Y Ĉ( 8) ∧ e B(2) ∧ e λF + . . ., (C.20) modifies the Bianchi identity of C (0) and consequently the form of the associated field strength.Locally, we now have F (1) = dC (0) + χ (1) for some 1-form χ (1) .The effect of this in our setup is that now the F (1) -flux is no longer constant, since ∆f m is a certain combination of the non-Abelian scalars Y Im .This will result in two additional contributions to the scalar potential coming from the kinetic term of F (1) , namely: By a standard argument, the term in the potential which is linear in ∆f m can be read by evaluating (C.20) using the uncorrected expression for Ĉ(8) . 14Let us do this explicitly in order to show how one can get from a direct calculation the expression of the modified field strength presented in the main text, (4.19).First, we expand the integrand of (C.20) at the relevant order in λ: Now we insert the expression of Ĉ(8) , which is the following 15 Ĉ(8)µ 1 ...µ 6 mn = λ 3 Plugging this in (C.23) yields 16 Comparing this with the term in the scalar potential (C.22) which is linear in ∆f m , we obtain as anticipated in (4.19).

C.2 Case O7/D7 & Open Strings
this section we will explain how to obtain the modified field strengths F (1) and F (3) in (5.29) and (5.30) from the WZ effective actions.We will study the case of F (1) and give similar arguments for F (3) .
Let us firstly consider the contribution of the C (8) potential in the bulk and the WZ actions. 17sing the type IIB democratic formulation (A.1), together with the duality relation F in such a way the source term is straightforwardly determined from the C (8) couplings in the expansion of the WZ action.On the other hand, taking into account that the product σ O7 (−1) ) is, respectively, (+, −, +), only the quantities ( J) ab and ( J) ij will be nonzero.Here, because the flux F a is allowed while F i is projected out, we will focus on ( J) ab .
In this case, we have to consider the following C (8) couplings: where we have used the notation dx The pullback of the 8-form potential is expressed as where, moreover, Ĉ( 8) is Taylor-expanded around the position of the source, Here, C µ 0 •••µ 5 ab is a 10D field, which admits the Kaluza Klein decomposition where ellipses account for other nontrivial terms entering the compactification Ansatz that do not depend on the 8-form potential which will be omitted in this analysis without loss of generality.
All in all, we find that the first term in (C.35) gives the following contribution to ( J) ab : If we multiply both sides of the equation by ij and use the Schouten identity on the kl and ij indices, we have Let us consider now the second term of (C.35).Using the twist matrices (5.20), this term contains the quantity Then, while the first term contributes ( J) ab (we need to use the Schouten identity on the indices [ijk] and take the trace), the latter does not, due to the presence of the longitudinal indices [bc] in the potential.In particular, the full term is given by The last term in (C.35) can be written as Multiplying by ij and using again the Schouten identity, we have where we have omitted terms proportional to D µ .Next, by studying the second term coming from the WZ action (C.32), we observe that the only contribution to ( J) ab arises from where we have used the notation v mn ≡ v m ∧ v n .Precisely, this term cancels the one in (C.43).Finally, let us consider the third and last term in the WZ action (C.32),P[(ι Y ι Y ) 2 Ĉ( 8) ].This consists of the 4th interior product over the vector Y i of a 12form.Because we are dealing with codimension-2 objects, this turns out to be trivially zero.Therefore, according to (C.30), the final expression for ( J) ab is where λ7 ≡ (2κ 2 8 T D7 ) 1/2 λ.Precisely, the compactification Ansatz for C (0) , together with the integrability condition (5.10) allow us to identify ∆ a = F a , in such a way that (5.29) is recovered.
A similar argument applies to the field strength F (3) in (5.30).In this case we need to study the couplings to C (6) in the WZ action, so that we can read off the current ( J) abij , which is defined via this expression:  which is again a quadratic constraint, (6.15).

Figure 1 .
Figure 1.(Left) In the presence of an Op − , a stack of N coincident Dp-branes realizes an SO(2N ) gauge group.Its N (2N − 1) light dof 's can be understood as all open strings with both extrema on one side of the O-plane (type a, N 2 states), plus those with one extremum on each side, with the Chan-Paton rule that i = j (type b, N (N − 1) states).(Right) In the presence of an Op + , we still have open strings of type a & b (these are now N 2 states due to the absence of the i = j rule), and in addition we find strings connecting each D-brane to the O-plane (type c, N states).This yields a total of N (2N + 1) light states realizing USp(2N ).
.18) On the other hand, the Wess-Zumino action contains a coupling between the scalars Y Ii and C(8) , see (C.20).As shown in Appendix C.1, this can be understood through a modified field strength F m of the form, ij , respectively.The matrix C is defined as C mn ≡ γ mn − 1 2 A I m A I n , in terms of

C. 1 )
where κ IJ = g IK L g JL K is the Cartan-Killing metric of G YM .The bosonic worldvolume action describing a stack of N Dp coincident Dp-branes in type II string theory contains two pieces: the Dirac-Born-Infeld (DBI) and the Wess-Zumino (WZ) actions,S Dp = S DBI Dp + S WZ Dp , (C.2)whereS DBI Dp = −T Dp d p+1 x Tr e − Φ −det(M M N ) det(Q i j ) , (C.3) being T Dp = 2π −(p+1) sthe brane tension.The matrices M and Q are defined as Tr P e iλι Y ι Y Ĉ ∧ e B(2) ∧ e λF , (C.6)where µ Dp = Dp T Dp , being Dp = ±1 the charge sign corresponding to the brane and anti-brane cases respectively.The field strength F living on the brane is given byF = dA + iA ∧ A ,

dF ( 1 ) = −2 κ 2 8 N 2 F
D7 µ D7 λ 2 2 • 2! κ II kl Y Ii Y I j ω ai k ω bj l +g JKL η [a A J b] Y Kk Y Li ki v a ∧ v b + . . . .(C.46)because generically its internal part is F (1) = F a v a + F i v i and ω ab i = 0, the only contribution to ( J) ab arises from F a .In particular,dF (1) = − 1 a (ω ij a v i ∧ v j + ω bc a v b ∧ v c ) + . . . .(C.47)Equating the v a ∧ v b components with (C.46) and using the Jacobi identities (5.8), in particular the first equation in(5.32),we obtainκ IJ Y Ii Y Jk jk − g JKL A J a Y Kk Y Li ki + ∆ a , ab ∆ a η b = 0 , (C.48)

Table 1 .
The (64 + 8N) B bosonic dof 's of the theory arranged into irrep's of SO(4) little × G global , the internal global symmetry being the one defined in (3.1).

Table 4 .
The Z 2 parity of all internal components of the different IIB fields in the presence of spacetime filling O7-planes.The allowed ones yield excitable 6D scalar fields.Note that the total amount of resulting scalars correctly gives 17+4N, i.e. the dimension of the supergravity coset (3.2).

Table 6 .
The Z 2 parity of all internal components of the different IIB fields in the presence of spacetime filling O9-planes.The allowed ones yield excitable 6D scalar fields.Note that the total amount of resulting scalars correctly gives 17+4N, i.e. the dimension of the supergravity coset (3.2).