Type IIB flux vacua from G-theory II

We find analytic solutions of type IIB supergravity on geometries that locally take the form $\text{Mink}\times M_4\times \mathbb{C}$ with $M_4$ a generalised complex manifold. The solutions involve the metric, the dilaton, NSNS and RR flux potentials (oriented along the $M_4$) parametrised by functions varying only over $\mathbb{C}$. Under this assumption, the supersymmetry equations are solved using the formalism of pure spinors in terms of a finite number of holomorphic functions. Alternatively, the solutions can be viewed as vacua of maximally supersymmetric supergravity in six dimensions with a set of scalar fields varying holomorphically over $\mathbb{C}$. For a class of solutions characterised by up to five holomorphic functions, we outline how the local solutions can be completed to four-dimensional flux vacua of type IIB theory. A detailed study of this global completion for solutions with two holomorphic functions has been carried out in the companion paper [1]. The fluxes of the global solutions are, as in F-theory, entirely codified in the geometry of an auxiliary $K3$ fibration over $\mathbb{CP}^1$. The results provide a geometric construction of fluxes in F-theory.

Compactifications of string theories with and without flux is a subject of long history, dating back to the seminal papers [2][3][4]. 1 Without fluxes, supersymmetry requires that the internal manifold in type II string compactification is Calabi-Yau, whereas in the presence of fluxes, it must be of generalised Calabi-Yau type. A generalised Calabi-Yau manifold [10] is characterised by the existence of globally defined spinors. Spinor bilinears define polyforms that behave as pure spinors in the generalised tangent space. Supersymmetry is preserved in the four-dimensional theory if the pure spinors satisfy a system of first order differential equations [11]. If the flux also satisfies the relevant Bianchi identities and the internal manifold is compact, a supersymmetric four-dimensional vacuum is obtained.
In the companion paper [1], we present concrete examples of supersymmetric four-dimensional type IIB vacua where all fields can be explicitly written out in an analytic form, even in the presence of fluxes. The solutions are built by gluing local solutions on T 4 × C in a U-duality consistent way.
Such local solutions can be found by starting from non-compact Calabi-Yau geometries, and then applying a sequence of U-duality transformations that rotate the metric into fluxes. In this way, different classes of flux solutions characterised by up to three holomorphic functions are generated.
The aim of this paper is to present more general flux solutions that cannot be related to Calabi-Yau geometries by means of U-dualities. We consider geometries that locally take the form Mink × M 4 × C with M 4 a generalised complex manifold with SU (2) structure. We use an ansatz in which the metric, the dilaton and the type IIB fluxes are parametrised by functions varying over the complex plane, and all form potentials are oriented along M 4 . Under these assumptions, the supersymmetry constraints simplify drastically and can be solved in terms of a finite number of holomorphic functions. We find three classes of solutions with SU (2) structure that we denote A, B and C. The three solutions correspond to different choices of the two angles describing the relative orientations of the two spinors defining the SU (2) structure. The solutions A, B, C in [1] fall into the solution class of that name here for M 4 = T 4 , and correspond to the case where only three of the holomorphic functions characterising the general solutions are non-constant.
The interest in the solutions under study here lies in the fact that they can be given an auxiliary, completely geometric description, following the approach of [12,13] (related ideas have been explored in ). In particular, solutions on T 4 or K3 characterised by n ≤ 5 holomorphic functions can be extended to the whole complex plane (including infinity) away from a finite number of degeneration points. Around these points, the functions undergo non-trivial monodromy transformations in the U-duality group SO(2, n, Z). This group is also the modular group of the space of complex structures 1 For recent reviews (with extensive references) on the subject of supersymmetric and non-supersymmetric flux compactifications of string theory, see [5][6][7][8][9].
of an algebraic K3 surface with Picard number 20 − n. Moreover, the locally holomorphic functions parametrise a coset space that is isomorphic to the complex structure moduli space of this K3. The n holomorphic functions characterising the flux solution can thus be identified with the n holomorphic parameters (periods of the holomorphic two-form) characterising the complex structure of the K3 surface, and the local charges in the flux solution (e.g. branes, orientifold planes) can be read off from the monodromy transformations of the periods around singular points in the base. The presence of singularities (and thus local sources) allow non-trivial flux solutions even when the base C is compactified, in agreement with known no-go theorems [38]. 2 Consequently, we can, in this way, construct four-dimensional flux vacua of type IIB string theory in terms of auxiliary geometries that are fibrations of K3 surfaces over, for example, a two-sphere. This auxiliary description is an extension of F-theory [39], in that it provides a geometric description of fluxes in F-theory compactifications.
The details of this analysis are given in [1] for the case n = 2, 3 and will not be repeated here.
The techniques developed in that paper can also be applied to K3 fibrations with n > 3 complex parameters, and hence to the local solutions of this paper. Since this computation is very technical, it goes beyond the scope of this paper and is left for future work.
The rest of this note is organised as follows. First, in section 2, we give a very brief review of type IIB supergravity, and present the ansatz we will use for the local solutions. In Section 3, we solve the supersymmetry equations and the Bianchi identities (away from local sources) where the internal six-manifold takes the form M 4 × C. We perform the analysis of the supersymmetry equations using the formalism of pure spinors, briefly reviewed in appendix B. We present three classes of solutions, with different flux and metric content, that can be parametrised in terms of a set of holomorphic functions. We also discuss how these different classes of solutions are related by U-duality transformations. Finally, in section 4 we draw some conclusions. Appendix A summarises our conventions, and appendix C rederives one class of local solutions by the more direct approach of solving the Killing spinor equations.

Type IIB supergravity
In this section, we provide a very brief review of type IIB supergravity, in order to clarify our conventions. For more details, we refer the reader to [40] and recent reviews on flux compactifications [5][6][7][8][9]. We also specify the ansatz for the local supersymmetric solutions that will be studied in the next section.

Action and Bianchi identities
In the low-energy supergravity limit, the bosonic field content of type IIB string theory consists of the Neveu-Schwarz-Neveu-Schwarz (NSNS) fields (a metric g, a scalar field called the dilaton φ and a two-form field B) and the Ramond-Ramond (RR) p-form fields C p , where p is 0, 2, 4. This is complemented by the fermionic fields: two gravitinos Ψ A M and two dilatinos λ A , A = 1, 2 of equal chirality.
The action for the bosonic sector is, in the string frame, where g = |det g M N |, H = dB and F n = dC n−1 − H 3 ∧ C n−3 are the NSNS and RR field strengths, respectively. In what follows, we collectively refer to the RR fluxes using a polyform language The fluxes must fulfil the Bianchi identities If sources (NS 5-branes, D p -branes and orientifolds) are present, these will modify the right hand side of these equations.

Killing spinor equations
where Ψ M , and λ are column vectors containing two Majorana-Weyl spinors of the same chirality, ∇ is the standard covariant derivative, n is odd and Γ are the ten-dimensional Dirac matrices (see appendix A for our spinor conventions). The projection matrices P, P n are given by where the Pauli matrices σ i are given in appendix A.
Once the KSE and the Bianchi identities are satisfied in a Minkowski vacuum, it can be shown that all bosonic equations of motion follow [41,42]; as such, the supersymmetric solutions constructed below will satisfy all the constraints required for local type IIB vacua.

The ansatz
In the following section we will solve the KSE and the Bianchi identities corresponding to type IIB supergravity on space-times R 1,3 × M 4 × Σ, with Σ an open subset of C. The torus metric g mn , the dilaton φ, the B-field and C p -fields are assumed to vary over Σ ⊂ C. All the non-trivial fluxes are assumed to be oriented along M 4 .
Let {y 1 , y 2 , y 3 , y 4 } be real coordinates on M 4 and z a complex coordinate on C. In these coordinates, we write the metric and the fluxes as: g mn dy m dy n + e 2D |h(z)| 2 dz dz (2.7) where A, D, g mn , b mn , c mn , c 4 , C 0 and φ are real z-dependent functions.
To cohere with the ansatz for the bosonic fields, the ten-dimensional Killing spinor = 1 , 2 T must decompose into four-and six-dimensional spinors, that we denote ζ a i and η i , respectively. The number of four-dimensional spinors is determined by the number of ten-dimensional spinors and the number n of well-defined internal spinor η i : where ± denotes chiral and anti-chiral components of the spinors, ζ A i− = ζ A * i+ , and η i − = η i * + . We take ζ A i± to be constant spinors and assume that η i vary only along the z-plane, in accordance with our ansatz for the bosonic fields. On SU (2) structure manifolds there are two globally defined spinors η 1 + , which can be written in the form (see Appendix B for details) Different choices of the angles α and θ will lead to different kinds of fluxes and brane sources.

Local supersymmetric solutions
In this section, we present three classes of local supersymmetric type IIB solutions, that all satisfy the Killing spinor equations and source-free Bianchi identities. As reviewed in appendix B.1, the fact that M 4 × Σ allows two well-defined spinors implies that its structure group is reduced to SU (2).
We will use this local SU (2) structure to define pure O(6, 6) spinors, and show that these satisfy the supersymmetry equations, once the supergravity fields on M 4 vary holomorphically over Σ. In appendix C, this result is shown without recourse to the pure spinor language for a class of solutions, named type A in the following, with up to five holomorphic functions.
A six-dimensional manifold of the local form M 4 × Σ has a local SU (2) structure if it admits a set of differential forms, (j, Ω 2 ) on M 4 and K on Σ (cf. appendix B.1). Choosing local holomorphic coordinates (z 1 , z 2 , z), we take for the one-form K and expand the two-forms j, Ω 2 , the NSNS two-form B and the RR potentials C p in a basis of closed forms on M 4 with coefficients that depend on z. The fluxes of these configurations are given by H = dB and F n = dC n−1 − H ∧ C n−1 ; they satisfy the Bianchi identities automatically, and always have one leg along either dz or its complex conjugate.
We take the SU (2) structure on M 4 to be defined by self-dual two-forms We will consider potential forms oriented along M 4 . For a d 4 -closed form χ on M 4 varying only over C, the Hodge dual in six-dimensions can be written as the exterior derivatives on M 4 × C and d 2 , d c 2 their reductions to C. The four-dimensional metric will be computed with the help of formula the Levi-Civita symbol in four dimensions, and c fixed such that I 2 = −1. These equations follow straightforwardly from the corresponding SU (3) structure identities (B.4) and (B.5), cf. Appendix B.

Pure spinor equations
A particularly elegant reformulation of the supersymmetry constraints is found using O(6, 6) pure spinors (or polyforms) Φ 1,2 [11,43]. In terms of these variables the KSE (2.4)-(2.5) translate into a set of first order differential equations with d H χ = dχ − H ∧ χ for any differential form χ, and λ(F ) = F 1 − F 3 + F 5 . In appendix B.1 we review how Φ 1,2 are related to the six-dimensional spinors η i ± : the latter define nowhere vanishing differential forms (j, Ω 2 , K) which in turn specify two nowhere vanishing polyforms Φ ± [44,45] (3.10) By specifying α and θ, we will, in the remainder of this section, find three different types of supersymmetric IIB solutions, that we will label A, B and C. More precisely, we will construct local solutions to the KSE and the Bianchi identities following the ansatz (2.7). In this analysis, we will, from time to time, use the fact that the SU (2) structure defines also an SU (3) structure on the six-dimensional manifold characterised by the forms
In this case the two spinors η A are parallel, but out of phase. 3 The constraints (3.7)-(3.9) become The first two supersymmetry equations lead to where we solve the first two relations by taking with dΩ 3 = dĴ = 0. A sufficient condition to solve the last two constraints is to take B anti-self-dual, since this implies that B wedges to zero with both J and Ω 3 . Since B is a closed z-dependent form on M 4 , this implies that H = dB also wedges to zero with these forms.
The third supersymmetry equation (3.15) then reduces to 4 On the other hand, using (3.3) and the anti-self- where we have used (3.22) and that B is anti-self-dual with respect to * 4 .
As noticed in [13], these equations can be written in the compact form with the holomorphic polyform (3.27) or, in components, where we have used that j ∧ j = 2e 2φ−4A dy 1 ∧ dy 2 ∧ dy 3 ∧ dy 4 . We conclude that a supersymmetric solution is specified by (b − 2 + 2)-holomorphic functions (where b − 2 is the number of globally defined anti-self-dual two-forms on M 4 ) characterising the fluxes, and a choice of warped metric for M 4 .

Example A: 5 holomorphic functions
As an example we can consider M 4 = T 4 with trivial SU (2) structure and K = e D h dz, which give the SU (3) structure forms For this choice 1 2 Ω 2 ∧Ω 2 = j ∧ j implies D = −A and a basis of anti-selfdual two-forms can be taken to be The solution is then parametrised by five holomorphic functions 2 , a = 1, 2, 3 , and can be written as with β i · β j = a β

Solution class B
In the case α = π/2, the two spinors η A are orthogonal. The pure spinors are where the SU (2) structure forms K, j and Ω 2 are defined in (B.6) and we set θ = 0 since this phase can be trivially reabsorbed in the definition of Ω 2 .
The supersymmetry equations (3.7)-(3.9) then require where we use that K ∧K is closed and dχ ∧ K ∧K = 0 for any form χ that is closed on M 4 . The first equation implies that is solved by taking with h(z) and γ(z) holomorphic zero and two-forms respectively. Equation Since B is parallel to j in order to satisfy (3.42), the second constraint is automatic. The first may be solved by with dΩ 2 = 0. The third supersymmetry equation (3.40) decomposes to Using that Ω 2 is self-dual, * 4 Ω 2 = Ω 2 , this is solved by with C 0 a constant and ρ(z) a holomorphic two-form. Writing B = b a χ a , j = j a χ a , γ = γ a χ a and C 2 − C 0 B = −c ReΩ 2 , the solution is specified by the holomorphic functions In the case M 4 = T 4 , after fixing a complex two-formΩ 2 , the j a span a four-dimensional space orthogonal toΩ 2 . The dilaton is fixed by the SU (2) condition j ∧ j = 1 2 Ω 2 ∧Ω 2 . The solution is then specified by five holomorphic functions, one from ρ and four from the γ a 's.

Solution class C
Finally, we consider the case α = 0 and θ = π. Like in case A, the two spinors η A are parallel but now the relative phase is simply a sign. The pure spinors are  On the other hand, from j ∧ j = 1 2 Ω 2 ∧Ω 2 we conclude that Finally the third supersymmetry equation in (3.58) decomposes into (recall that H = 0) holomorphic. We recall that j is a two-form orthogonal to Ω 2 that satisfies d(j ∧ j) = 0. Just as discussed above for solutions of type B, when M 4 = T 4 we can expand j in a four-dimensional basis of two-forms orthogonal to Ω 2 . Writing C 2 = c a χ a we build four holomorphic functions The flux content of general solutions in this class is then characterised by four holomorphic functions.
Additionally, there may be holomorphic functions that parametrise Ω 2 .

Example C: 4 holomorphic functions
As an example of solution in the C-class we choose M 4 = T 4 and The T 4 metric computed from (3.5) and (3.6) is and the solution becomes g mn dy m dy n + e −2A |h(z)| 2 dzdz , (3.72)

Relations between local solutions
In the preceding sections, we presented three types of supersymmetric local solutions to type IIB supergravity. For each class, we displayed an example of solutions on T 4 characterised by 4 holomorphic functions. These three solutions can be related to each other acting with T-and S-dualities.
Under T-duality along a direction y, the metric in the string frame and the NSNS/RR fields transform as [46][47][48]: 7 g yy = 1 g yy , e 2φ = e 2φ g yy , g ym = B ym g yy , B ym = g ym g yy On the other hand, under S-duality for backgrounds with C 0 = 0 is

Conclusions and outlook
In this paper, we presented explicit solutions where the ten-dimensional spacetime takes the local form The reader is referred to the accompanying paper [1], where the details of the flux/geometry dictionary are studied and discussed in great detail.

A Conventions
We use (M, N, P, Q...) to index ten-dimensional quantities, (m, n, p, q...) in the internal six dimensions and (µ, ν, ρ, ..) for the four space-time dimensions. Flat tangent space indices will sometimes be used, and we denote them with a hat:M ,m etc.
Gamma matrices. Ten-dimensional Dirac matrices are denoted Γ M , and six-dimensional ones γ m .
We will choose the latter to be hermitian, γ † m = γm, imaginary and antisymmetric. All Dirac matrices satisfy the Clifford algebra (e.g. in six dimensions {γ m , γ n } = 2g mn ) and totally antisymmetric products of gamma matrices are denoted γ m 1 m 2 ...m k , where e.g.
The chirality operator in d dimensions is given by where we use hatted letters for flat tangent space indices. The eigenvalues of γ d+1 are +1 (-1) for chiral (antichiral) spinors. Thus, a six-dimensional spinor η can be decomposed into chiral and anti-chiral components η ± , where γ 7 η ± = ±η ± (η − = η * + ). Without loss of generality, we will take η ± to have unit norm, η † + η + = η † − η − = 1. The two-dimensional Pauli matrices are given by Differential forms and Hodge duals. We define the components of a differential p-form by The contraction of a q-form with a p-form (p > q) is Our convention for the Hodge star operation * , when acting on a p-form, (A.6) with 1...d = 1, cf. [45,49]. Another very useful relation is the combined identity where A is a p-form, B a q-form, and (d − p) > q. Further relevant identities can be found in [50].
The six-dimensional exterior derivative can be decomposed into holomorphic and antiholomorphic parts, and with local holomorphic coordinates z a , a = 1, 2, 3, we have The 2d Hodge dual satisfies * 2 dz = i dz . If a six-dimensional manifold admits a nowhere vanishing spinor η ± its structure group is reduced to SU (3). Another way to express this constraint is in terms of differential forms. The spinor can also be used to define a set of pure O(6, 6) spinors. In this appendix, we briefly review these different formalisms, in order to pave the way for the analysis of the analysis of the local supersymmetry conditions of type IIB compactifications.
In string compactifications to four dimensions, supersymmetry requires the existence on M 6 of at least one globally defined and nowhere vanishing spinor η. The six-dimensional spinors η ± can be used to build a nowhere vanishing real two-form J and a complex decomposable three-form Ω 3 on the six-dimensional manifold 8 For manifolds of strict SU (3) structure, i.e. those for which η is unique, these are the only nowhere vanishing forms can be defined on M 6 . 9 J and Ω 3 are subject to the constraints The Hodge duals of J and Ω 3 are 8 Gamma matrices are in our conventions imaginary and complex. 9 In particular, there are no globally defined one-forms on M6; for spinors η1,2 of the same chirality, bilinears η † 1 γ m 1 ...m k η2 vanish for odd k, and η † ∓ γ m η± = η T ± γ m η± = 0 follows from the antisymmetry of γ m .
One can show that satisfies I m p I p n = −δ n m for a given normalisation constant c. The matrix I thus defines an almost complex structure [51] (see also sec. 3.1 in [52]). Moreover, the contraction of I with J gives a metric On a Calabi-Yau manifold, J and Ω 3 are closed and I is an integrable complex structure. The Ricciflat Calabi-Yau metric is given by (B.5) once the Kähler form and holomorphic top-form have been correctly identified in the cohomology classes of J and Ω 3 .
If a six-dimensional manifold allows two orthogonal nowhere-vanishing spinors, η and χ, its structure group is further reduced to SU (2). Again, without loss of generality, we take the chiral and antichiral parts of the spinors to have unit norm. The SU (2) structure is characterised by the existence of a nowhere vanishing complex one-form K, a real two-form j, and a complex two-form Ω 2 given by: The SU (2) structure can be embedded into the SU (3) via the relations Using these relations and (B.2), it is straightforward to show the SU (2) structure relations and vice versa these conditions imply that J and Ω 3 given by (B.7) is an SU (3) structure (using the fact that |K| 2 = 2, which follows from the unit norm of χ + ).

B.2 Pure spinors
For manifolds with SU (2) structure, one can write In presence of D-branes the modulus of the two spinor should match 10 , and supersymmetry requires |a| 2 = |b| 2 = e A . We write a = |a|e iθ 1 , b = |b|e iθ 2 . The parameter α interpolates between strict SU (3) (α = 0, parallel spinors) and SU (2) (α = π/2, orthogonal spinors) structures. The O(3) spinors η 1,2 can be used to define two pure O(6, 6) spinors The right hand side of (B.10) can be thought as a polyform via the Clifford map In particular bilinears made out of spinors of the same (different) chirality lead to odd (even) forms.
The various contributions can be written as with θ ± = θ 1 ± θ 2 . The phase θ + can be reabsorbed into the definition of K so we discard this phase and rename θ − = −θ in the main text. By specifying α and θ, we find different supersymmetric solutions, (see Section 3).

C Explicit solution for the Killing spinor
In this appendix, we present explicit local type A solutions of the KSE (2.4)-(2.5) that are parametrised by up to five holomorphic functions. It can be checked that this solution satisfies the equations of motion of type IIB supergravity, and we have done so using Mathematica.
We start from the ansatz and using (C.4) to discard the contribution of the last term in (C.9) one finds ie φ∂ C ny i − τ B ny i Γz n 1 = 0 (C.10) that implies∂ (C mn − τ B mn ) = 0 (C.11) Let us consider now (C.7), The non-trivial components of the spin connection are Finallly taking m = z in (C.7) one finds the differential equation where we use the compact notation β 2 2 = a (β