Semi-Flatland

Semi-Flatland

David Vegh and John McGreevy

Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA.

Abstract:

We study perturbative compactifications of Type II string theory that rely on a fibration structure of the extra dimensions à la SYZ. Non-geometric spaces are obtained by using T-dualities as monodromies. These vacua generically preserve supersymmetry in four dimensions, and are U-dual to M-theory on manifolds. Several examples are discussed, some of which admit an asymmetric orbifold description. The massless spectrum is matched to that of the dual M-theory compactification on a Joyce manifold when a comparison is possible. We explore the possibility of twisted reductions where left-moving spacetime fermion number Wilson lines are turned on in the fiber. We also give an explanation from this semiflat viewpoint for the Hanany-Witten brane-creation effect and for the equivalence of the Type IIA orientifold on and Type IIB on .

preprint: MIT-CTP-3960

1 Introduction

A great deal of progress has been made in the study of string compactification using the ten-dimensional supergravity approximation (for a review, see [Douglas:2006es]). However, it has become clear that certain interesting physical features of our world are difficult (if not impossible) to realize when this description is valid. Examples which come to mind include a period of slow-roll inflation [Hertzberg:2007wc, Dimopoulos:2005ac, Grimm:2007hs], certain models of dynamical supersymmetry breaking [Florea:2006si], chiral matter with stabilized moduli [Blumenhagen:2007sm] and parametrically-small perturbatively-stabilized extra dimensions [Douglas:2006es]. This strongly motivates attempts to find descriptions of moduli-stabilized string vacua which transcend the simple geometric description.

One approach to vacua outside the domain of validity of 10d supergravity is to rely only on the 4d gravity description, as in e.g. [Shelton:2005cf, Silverstein:2007ac]. This can be combined with insight into the microscopic ingredients to give a description of much more generic candidate string vacua. A drawback of this approach is that it is difficult to control systematically the interactions between the ingredients. Another promising direction is heterotic constructions, which do not require RR flux and hence are more amenable to a worldsheet treatment [Adams:2006kb, Adams:2007vp]. However, stabilization of the dilaton in these constructions requires non-perturbative physics.

A third technique, which is at an earlier state of development, was implemented in [Hellerman:2002ax], and was inspired by [Greene:1989ya, Vafa:1996xn]. The idea is to build a compactification out of locally ten-dimensional geometric descriptions, glued together by transition functions which include large gauge transformations, such as stringy dualities. This technique is uniquely adapted to construct examples with no global geometric description. In this paper, we build on the work of [Hellerman:2002ax] to give 4d examples.

With S. Hellerman and B. Williams [Hellerman:2002ax], one of us constructed early examples of vacua involving such ‘non-geometric fluxes’. These examples were constructed by compactifying string theory on a flat -torus, and allowing the moduli of this torus to vary over some base manifold. The description of these spaces where the torus fiber is flat is called the semi-flat approximation [Strominger:1996it]. Allowing the torus to degenerate at real codimension two on the base reduces the construction of interesting spaces to a Riemann-Hilbert problem; the relevant data is in the monodromy of the torus around the degenerations [Greene:1989ya]. Generalizing this monodromy group to include not just modular transformations of the torus, but more general discrete gauge symmetries of string theory (generally known as string dualities) allows the construction of vacua of string theory which have no global geometric description [Hellerman:2002ax]. The examples studied in detail in [Hellerman:2002ax] had two-torus fibers, which allowed the use of complex geometry.

A natural explanation of mirror symmetry is provided by the conjecture [Strominger:1996it] that any CY has a description as a three-torus fibration, over a 3-manifold base. In the large complex structure limit, the locus in the base where the torus degenerates is a trivalent graph; the data of the CY is encoded in the monodromies experienced by the fibers in circumnavigating this graph. Further, the edges of the graph carry energy and create a deficit angle – in this description a compact CY is a self-gravitating cosmic string network whose back-reaction compactifies the space around itself. In this paper, our goal is to use this description of ordinary CY manifolds to construct non-geometric vacua, again by enlarging the monodromy group. We find a number of interesting new examples of non-geometric vacua with 4d supersymmetry. In a limit, they have an exact CFT description as asymmetric orbifolds, and hence can be considered ‘blowups’ thereof. We study the spectrum, particularly the massless scalars, and develop some insight into how these vacua fit into the web of known constructions.

We emphasize at the outset two limitations of our analysis. First, the examples constructed so far are special cases which have arbitrarily-weakly-coupled perturbative descriptions and (therefore) unfixed moduli. Our goal is to use them to develop the semiflat techniques in a controllable context. Generalizations with nonzero RR fluxes are naturally incorporated by further enlarging the monodromy group to include large RR gauge transformations, as in F-theory [Vafa:1996xn]. There one can hope that all moduli will be lifted. This is the next step once we have reliable tools for understanding such vacua using the fibration description.

The second limitation is that we have not yet learned to describe configurations where the base of the -fibration is not flat away from the degeneration locus. The examples of SYZ fibrations we construct (analogous to F-theory at constant coupling [Dasgupta:1996ij]) all involve composite degenerations which we do not know how to resolve. The set of rules we find for fitting these composite degenerations into compact examples will be a useful guide to the more difficult general case.

A number of intriguing observations arise in the course of our analysis. One can “geometrize” these non-geometric compactifications by realizing the action of the T-duality group as a geometric action on a fiber. The semi-flat metric on the fiber contains the original metric and the Hodge dual of the B-field. Hence, we are led to study seven-manifolds which are fibrations over a 3d base. They can be embedded into flat compactifications of M-theory down to seven dimensions where the reduced theory has an U-symmetry. U-duality then suggests that may be a manifold since the non-geometric Type IIA configuration can be rotated into a purely geometric solution of maximal supergravity in seven dimensions. Whether or not these solutions can in general be lifted to eleven dimensions is a question for further investigation. In this paper, we study explicit examples of (and Calabi-Yau) manifolds and show that they do provide perturbative non-geometric solutions to Type IIA in ten dimensions through this correspondence. The spectrum of these spaces can be computed by noticing that they admit an asymmetric orbifold description, and it matches that computed from M-theory when a comparison is possible.

The paper is organized as follows. In the next section we review the semiflat approximation to geometric compactification in various dimensions. We describe in detail the semiflat decomposition of an orbifold limit of a Calabi-Yau threefold; this will be used as a starting point for nongeometric generalizations in section four. In section three we describe the effective field theory for type II strings on a flat . We show that the special class of field configurations which participate in -fibrations with perturbative monodromies can alternatively be described in terms of geometric -fibrations. We explain the U-duality map which relates these constructions to M-theory on -fibered -manifolds. In sections four and five we put this information together to construct nongeometric compactifications. In section six we consider generalizations where the fiber theory involves discrete Wilson lines. Hidden after the conclusions are many appendices. Appendix A gives more detail of the reduction on . The purpose of Appendices B–D is to build confidence in and intuition about the semiflat approximation: Appendix B is a check on the relationship between the semiflat approximation and the exact solution which it approximates; Appendix C is a derivation of the Hanany-Witten brane-creation effect using the semiflat limit; Appendix D derives a known duality using the semiflat description. In Appendix E we record asymmetric orbifold descriptions of the nongeometric constructions of section four. In Appendices F through H, we study in detail the massless spectra of many of our constructions, and compare to the spectra of M-theory on the corresponding -manifolds when we can. Appendix I contains templates to help the reader to build these models at home.

2 Semi-flat limit

Since we want to construct non-geometric spaces by means of T-duality, we exhibit the spaces as torus fibrations. We need isometries in the fiber directions in which the dualities act. Hence, we wish to study manifolds in a semi-flat limit where the fields do not depend on the fiber coordinates. This is the realm of the SYZ conjecture [Strominger:1996it]. Mirror symmetry of Calabi-Yau manifolds implies that they have a special Lagrangian fibration. Branes can be wrapped on the fibers in a supersymmetric way and their moduli space is the mirror Calabi-Yau. At tree level, this moduli space is a semi-flat fibration, i.e. the metric has a isometry along the fiber. However, there are world-sheet instanton corrections to this tree-level metric. Such corrections are suppressed (away from singular fibers) in the large volume limit. The mirror Calabi-Yau is then in the large complex structure limit. In this limit the metric is semi-flat and mirror symmetry boils down to T-duality along the fiber directions111It is best to think of the fiber as being very small compared to the size of the base. It is thought that in the large complex structure limit, the total space of the CY collapses to a metric space homeomorphic to which is the base of the fibration (see e.g. [GrossWilson]). .

As a warm-up, we will now briefly review the one-complex-dimensional case of a torus, and the two-dimensional case of stringy cosmic strings [Greene:1989ya]. These sections may be skipped by experts. In Section 2.3, we construct a fibration for a three-dimensional orbifold that will in later sections be modified to a non-geometric compactification.

2.1 One dimension

The simplest example is the flat two-torus. Its complex structure is given by modding out the complex plane by a lattice generated by 1 and (with ). The Kähler structure is where and the area of the torus (again, ).

There is an group acting on the complex modulus . This is a redundancy in defining the lattice. The group action is generated by and . Another group acts on . This is generated by the shift in the B-field and a double T-duality combined with a rotation that is . The fundamental domain for the moduli is shown in Figure 1.

The torus can naturally be regarded as a semi-flat circle fibration over a circle. For special Lagrangian fibers, we choose the real slices in the complex plane. In the large complex structure limit, these fibers are small compared to the base which is along the imaginary axis.

Mirror symmetry exchanges the complex structure with the Kähler structure . This boils down to T-duality along the fiber direction according to the Buscher rules [Buscher:1987sk, Buscher:1987qj]. It maps the large complex structure into large Kähler structure that is .

2.2 Two dimensions

In order to construct semi-flat fibrations in two dimensions, let us consider the dynamics first. Type IIA on a flat two-torus can be described by the effective action in Einstein frame

 S=∫d8x√g(R+∂μτ∂μ¯ττ22+∂μρ∂μ¯ρρ22) (2.1)

where is the complex structure of the torus, and is the Kähler modulus as described earlier. The action is invariant under the perturbative duality group, which acts on and by fractional linear transformations.

Variation with respect to gives

 ∂¯∂τ+2∂τ¯∂τ¯τ−τ=0; (2.2)

and obeys the same equation. Stringy cosmic string solutions to the EOM can be obtained by choosing a complex coordinate on two of the remaining eight dimensions, and taking a holomorphic section of an bundle. Such solutions are not modified by considering the following ansatz for the metric around the string222 By an appropriate coordinate transformation of the base coordinate, this metric can be recast into a symmetric form (see [Strominger:1996it, Loftin:2004qu]).

 ds2=ds2Mink+eψ(z,¯z)dzd¯z+ds2fiber (2.3)

where

 ds2fiber=1τ2(1τ1τ1 |τ|2) (2.4)

The Einstein equation is the Poisson equation,

 ∂¯∂ψ=∂¯∂logτ2 (2.5)

Far away from the strings, the metric of the base goes like [Greene:1989ya]

 ds22D∼|z−N/12dz|2 (2.6)

where is the number of strings. This can be coordinate transformed by to a flat metric with deficit angle.

Solutions and orbifold points. One could in principle write down solutions by means of the -function,

 j(τ)=η(τ)−24(θ81(τ)+θ82(τ)+θ83(τ))3 (2.7)

which maps the and orbifold points to 0 and 1, respectively. The degeneration point gets mapped to . A simple solution would then be

 j(τ)=1z−z0+j0 (2.8)

At infinity, the shape of the fiber is constant, i.e.  and thus this non-compact solution may be glued to any other solution with constant at infinity. However, since covers the entire fundamental domain once, there will be two points in the base where or . Over these points, the fiber is an orbifold of the two-torus. These singular points cannot be resolved in a Ricci-flat way and we can’t use this solution for superstrings.

There is, however, a six-string solution which evades this problem [Greene:1989ya]. It is possible to collect six strings together in a way that approaches a constant value at infinity. can be given implicitly by e.g.

 y2=x(x−1)(x−2)(x−z) (2.9)

There are no orbifold points now because can be written as a holomorphic function over the base. The above equation describes three double degenerations, that is, three strings of tension twice the basic unit. In the limit when the strings are on top of one another, we obtain what is known (according to the Kodaira classification) as a singularity with deficit angle .

The monodromy of the fiber around this singularity is described by

 MD4=(−100 −1) (2.10)

acting on with . This monodromy decomposes into that of six elementary strings which are mutually non-local333For explicit monodromies for the six strings, see [Gaberdiel:1997ud]..

This can be generalized to more than six strings using the Weierstrass equation

 y2=x3+f(z)x+g(z) (2.11)

The modular parameter of the torus is determined by

 j(τ(z))=4f34f3+27g2 (2.12)

Whenever the numerator vanishes, and we are at an orbifold point. We see however that it is a triple root of and no orbifolding of the fiber is necessary. The same applies for the points. The strings are located where that is where the modular discriminant vanishes. Note that the monodromy of the fibers around a smooth point is automatically the identity in such a construction.

Kodaira classification. Degenerations of elliptic fibrations have been classified according to their monodromy by Kodaira. For convenience, we summarize the result in the following table [Bershadsky:1996nh]:

ord(f) ord(g) ord() monodromy singularity
0 none
none

Constructing K3. One can construct a compact example where the fiber experiences 24 degenerations. In the Weierstrass description (2.11), this means that has degree 8, has degree 12, and has degree 24. This is the semi-flat description of a manifold. In a certain limit where we group the strings into four composite singularities, the base is flat and the total space becomes . The base can be obtained by gluing four flat triangles as seen in Figure 3. At each degeneration, the base has deficit angle which adds up to and closes the space into a flat sphere with the curvature concentrated at four points.

As we have seen, in two dimensions the Weierstrass equation solves the problem of orbifold points. In higher dimensions, we don’t have this tool but we can still try to glue patches of spaces in order to get compact solutions. Gluing is especially easy if the base is flat. However, generically this is not the case. Having a look at the Einstein equation (2.2), we see that a flat base can be obtained if is constant. This happens in the case of and singularities. Our discussion in this paper will (unfortunately) be restricted to these singularities.

The cosmic string metric is singular in the above semi-flat description. It must be slightly modified in order to get a smooth Calabi-Yau metric for the total space. This will be discussed in Appendix LABEL:sfvs.

2.3 Three dimensions

In two dimensions, the only smooth compact Calabi-Yau is the surface. In three dimensions, there are many different spaces and therefore the situation is much more complicated. The SYZ conjecture [Strominger:1996it] says that every Calabi-Yau threefold which has a geometric mirror, is a special Lagrangian fibration with possibly degenerate fibers at some points. For the generic case, the base is an . Without the special Lagrangian condition, the conjecture has been well understood in the context of topological mirror symmetry [Gross:1999hc, Tomasiello:2005bp]. There, the degeneration loci form a (real) codimension two subset in the base. A graph is formed by edges and trivalent vertices. The fiber suffers from monodromy around the edges. This is specified by a homomorphism

 M: π1(S3∖Γ)⟶SL(3,Z) (2.13)

There are two types of vertices which contribute to the Euler character of the total space444These positive and negative vertices are also called type (1,2) / type (2,1) [Gross:1999hc] or type III / type II [Ruan1] vertices by different authors. For an existence proof of metric on the vertex, see [Loftin:2004qu].. At the vertices, the topological junction condition relates the monodromies of the edges.

One of the most studied non-trivial Calabi-Yau spaces is the quintic in . However, even the topological description of this example is fairly complicated [Gross:1999hc]. The topological construction contains vertices and edges in the base.

Constructing not only topological, but special Lagrangian SYZ fibrations is a much harder task. In fact, it is expected that away from the semi-flat limit, the real codimension two singular loci in the base get promoted to codimension one singularities, i.e. surfaces in three dimensions. These were termed ribbon graphs [Joyce:2000ru] and their description remains elusive.

A compact orbifold example. In the following, we will describe the singular orbifold in the SYZ fibration picture. One starts with that is a product of three tori with complex coordinates . Without discrete torsion, the orbifold action is generated by the geometric transformations,

 α:(z1,z2,z3)↦(−z1,−z2,z3) (2.14) β:(z1,z2,z3)↦(−z1,z2,−z3) (2.15)

These transformations have unit determinant and thus the resulting space may be resolved into a smooth Calabi-Yau manifold.

In order to obtain a fibration structure, we need to specify the base and the fibers. For the base coordinates, we choose and for the fibers . Under the orbifold action, fibers are transformed into fibers and they don’t mix with the base555It is much harder in the general case to find a fibration that commutes with the group action..

Degeneration loci in the base. The base originally is a . What happens after orbifolding? If we fix, for instance, the coordinate, then the orbifold action locally reduces to (since the other two non-trivial group elements change ). This means that we simply obtain four fixed points in this slice of the base. This is exactly analogous to the example. The fixed points correspond to singularities with a deficit angle of . As we change , we obtain four parallel edges in the base. By keeping instead or fixed, we get perpendicular lines corresponding to conjugate s whose monodromies act on another in the fiber. Altogether, we get lines of degeneration as depicted in Figure 4. These edges meet at (half-)integer points in the base.

Some parts of the base have been identified by the orbifold group. We can take this into account by a folding procedure which we have already seen for . The degeneration loci are the edges of a cube. The volume of this cube is of the volume of the original . The base can be obtained by gluing six pyramids on top of the faces (see Figure 5). The top vertices of these pyramids are the reflection of the center of the cube on the faces and thus the total volume is twice that of the cube. This polyhedron is a Catalan solid666Catalan solids are duals to Archimedean solids which are convex polyhedra composed of two or more types of regular polygons meeting in identical vertices. The dual of the rhombic dodecahedron is the cuboctahedron.: the rhombic dodecahedron. (Note that one can also construct the same base by gluing two separate cubes together along their faces.)

In order to have a compact space, we finally glue the faces of the pyramids to neighboring faces (see the right-hand side of Figure 5). This is analogous to the case of where triangles were glued along their edges (Figure 2).

The topology of the base. The base is an which can be seen as follows777We thank A. Tomasiello for help in proving this.. First fold the three rombi , and , and the corresponding three on the other side of the fundamental domain. Then, we are still left with six rhombi that we need to fold. It is not hard to see that the problem is topologically the same as having a ball with boundary . Twelve triangles cover the and we need to glue them together as depicted in Figure 6. This operation is the same as taking the and identifying its points by an flip. This on the other hand, exhibits the space as an fibration over . The fiber vanishes at the boundary of the disk. This is further equivalent to an fibration over an interval where the fiber vanishes at both endpoints. This space is simply an . The degeneration loci are on the equator of this base and form the edges of the cube.

Edges and vertices. The monodromies of the edges are shown in Figure 7. The letters on the degeneration edges denote the following monodromies:

 x=⎛⎜⎝1000−1000−1⎞⎟⎠y=⎛⎜⎝−1000 1000−1⎞⎟⎠z=⎛⎜⎝−1000−1000 1⎞⎟⎠ (2.16)

This orbifold example contained strings. These are composite edges made out of six “mutually non-local” elementary edges. The edges have deficit angle around them which is where is the deficit angle of the elementary string.

Note that the base is flat. This made it possible to easily glue the fundamental cell to itself yielding a compact space. Since the edges around any vertex meet in a symmetric way, the cancellation of forces is automatic.

There are other spaces that one can describe using edges and the above mentioned composite vertices. Some examples are presented in Section LABEL:examplesec. The strategy is to make a compact space by gluing polyhedra like the above described cubes, then make sure that the dihedral deficit angles are appropriate for the singularity.

2.4 Flat vertices

Codimension two degeneration loci meet at vertices in the base. In the generic case, these are trivalent vertices of elementary strings. Such strings have deficit angle around them measured at infinity. This creates a solid deficit angle around the vertex.

In some cases when composite singularities meet, the base is flat and the vertex is easier to understand. In particular, the total deficit angle arises already in the vicinity of the strings. An example was given in Section 2.3 where composite vertices arise from the “collision” of three singularities (see Figure 5). The singular edges have a deficit angle . The vertex can be constructed by taking an octant of three dimensional space and gluing another octant to it along the boundary walls. The curvature is then concentrated in the axes. The solid angle can be computed as twice the solid angle of an octant. This gives (or a deficit solid angle of ).

In the general (flat) case, a composite vertex may be described by gluing two identical cones (the analogs of octants). Such a cone is shown in Figure 8. Note that the solid angle spanned by three vectors is given by the formula

 θ=α+β+γ−π (2.17)

where , and are the dihedral angles at the edges. This can be used to compute the solid angle around a composite vertex.

The singular edges have a tension which is proportional to the deficit angle around them. This leads to the problem of force balance. In Figure 9, a flat vertex is shown. The two solid lines ( and ) are degeneration loci. The third edge () is pointing towards the reader as indicated by the arrow head. The deficit angle around is shown by the shaded area. In the weak tension limit (where we rescale the deficit angles by a small number), one condition for force balance is that these edges are in a plane. (Otherwise, energy could be decreased by moving the vertex.) This can be generalized for almost flat spaces by ensuring that . This is automatic when we construct the neighborhood of a vertex by gluing two identical cones888In the weak tension limit, the two identical cones almost fill two half-spaces. The slopes of the edges are dictated by the tensions as in [Sen:1997xi]. We leave the proof to the interested reader..

Another problem to be solved is related to the fiber monodromies. These can be described by matrices , and (see Figure 10). The loop around one of the edges (say ) can be smoothly deformed into the union of the other two (). This gives the monodromy condition999Since monodromy matrices do not generically commute, it is important to keep track of the branch cut planes. .

Some composite strings can be easier described than elementary ones because the base metric can be flat around them. Such singularities are , , and with deficit angles , , and , respectively [Greene:1989ya]. Vertices where composite lines meet can also be easily found by studying flat orbifolds. Here we list some of the vertices that will later arise in the examples.

We have already seen the vertex in Section 2.3. If the vertex is located at the origin, then the strings are stretched along the coordinate axes,

 D(1)4:(1,0,0)D(2)4:(0,1,0)D(3)4:(0,0,1) (2.18)

The second example is generated by

 α:(z1,z2,z3) ↦ (−z3,z2,z1) β:(z1,z2,z3) ↦ (z1,−z2,−z3)

It contains different colliding singularities. Their directions are given by

 D(1)4:(1,0,0)D(2)4:(1,0,1)E7:(0,1,0) (2.19)

The group has and subgroups. It is generated by

 α:(z1,z2,z3) ↦ (z2,z3,z1) β:(z1,z2,z3) ↦ (−z1,−z2,z3)

The strings directions are

 D4:(1,0,0)E(1)6:(1,1,1)E(2)6:(1,1,−1) (2.20)

The last example is generated by combining and generators,

 α:(z1,z2,z3) ↦ (z2,z3,z1) β:(z1,z2,z3) ↦ (−z2,z1,z3)

which generate the group. The direction of the strings are the following,

 D4:(1,1,0)E6:(1,1,1)E7:(1,0,0) (2.21)

This is not an exhaustive list; a thorough study based on the finite subgroups of [Fairbairn] would be interesting.

3 Stringy monodromies

In this section, we wish to extend the discussion by including the full perturbative duality group of type II string theory on in the possible set of monodromies. We will find that this duality group can be interpreted as the geometric duality group of an auxiliary . The extra circle is to be distinguished from the M-theory circle but it is related to it by a U-duality transformation.

For simplicity, the Ramond-Ramond field strengths will be turned off. This allows us to use perturbative dualities only. However, in moduli stabilization these fields play an important role. In fact, in the Appendices LABEL:hwapp and LABEL:t5app, we use U-duality [Hull:1994ys] monodromies which act on RR-fields in order to describe two familiar phenomena.

From the worldsheet point of view, string compactifications are expected to be typically non-geometric, since the 2d CFT does not necessarily have a geometric target space. Even though we construct our examples directly based on intuition from supergravity, they will have a worldsheet description as modular invariant asymmetric orbifolds.

For other related works on non-geometric spaces, see [Kumar:1996zx, Hellerman:2002ax, Kaloper:1999yr, Hitchin:2004ut, Gualtieri:2003dx, Grana:2004bg, Kachru:2002sk, Gurrieri:2002wz, Grana:2006kf, Lawrence:2006ma, Dabholkar:2005ve, Hull:2004in, Flournoy:2004vn, Hull:2006va, Shelton:2005cf, Shelton:2006fd, Becker:2006ks] and references therein.

In the following, we study the perturbative duality group in Type IIA string theory compactified on a flat three-torus. We gain intuition by studying the reduced 7d Lagrangian of the supergravity approximation. Finally we discuss how U-duality relates non-geometric compactifications to manifolds in M-theory which will be fruitful when constructing examples in the next section.

3.1 Reduction to seven dimensions

Action and symmetries. Let us consider the bosonic sector of (massless) 10d Type IIA supergravity,

 SIIA=SNS+SR+SCS (3.22)

where

 SNS=12κ210∫d10x√−ge−2ϕ(R+4∂μϕ∂μϕ−12|H3|2) (3.23)
 SR=−14κ210∫d10x√−g(|F2|2+|~F4|2) (3.24)

and the Chern-Simons term is

 SCS=−14κ210∫B∧F4∧F4 (3.25)

with and .

First we set the RR fields to zero101010This can be done consistently since the symmetry forbids a tadpole for any RR field.. This truncates the theory to the NS part which is identical to the IIB action. Compactifying Type IIA on a flat yields the perturbative T-duality group which acts on the coset .

The equivalences of Lie algebras

 so(3,3)≅sl(4) (3.26)
 so(3)⊕so(3)≅su(2)⊕su(2)≅so(4) (3.27)

enable us to realize the T-duality group as an action on . This latter space is simply the moduli space of a flat with constant volume. Therefore, we can think of the T-duality group as the mapping class group of an auxiliary four-torus of unit volume. What is the metric on this in terms of the data of the ? To answer this question, we have to study the Lagrangian.

Reduction to seven dimensions. One obtains the following terms after reduction on [Maharana:1992my] (see Appendix LABEL:redapp for more details and notation)

 S=∫dx√−ge−ϕL (3.28)

with and

 L1 = R+∂μϕ∂μϕ (3.29) L2 = 14(∂μGαβ∂μGαβ−GαβGγδ∂μBαγ∂μBβδ) (3.30) L3 = −14gμρgνλ(GαβF(1)αμνF(1)βρλ+GαβHμναHρλβ) (3.31) L4 = −112HμνρHμνρ (3.32)

The relation of these fields and the ten dimensional fields are presented in Appendix LABEL:redapp. In order to see the symmetry, one introduces the symmetric positive definite matrix

 M=(G−1G−1BBG−1  G−BG−1B)∈SO(3,3) (3.33)

The kinetic terms can be written as the model Lagrangian

 L2=18Tr(∂μM−1∂μM) (3.34)

The other terms in the Lagrangian are also invariant under .

The SL(4) duality symmetry and “N-theory”. Let us now put the bosonic action in a manifestly invariant form (see [Brace:1998xz]). Rewrite as

 L2=18Tr(∂μM−1∂μM)=14Tr(∂μN−1∂μN), (3.35)

where we introduced the symmetric matrix111111This matrix parametrizes the eight complex structure moduli, and one Kähler modulus of .

 N4×4=(detG)−1/2(GG→b→bTG  detG+→bTG→b) (3.36)
 Bij=ϵijkbkbi=12ϵijkBjk. (3.37)

The equality of the Lagrangians can be checked by lengthy algebraic manipulations (or a computer algebra software). We included the Hodge-dualized B-field in the metric as a Kaluza-Klein vector. The inverse of is

 N−1=(detG)−1/2⎛⎝(detG)G−1+→bTb  −→b−→bT  1⎞⎠. (3.38)

Keeping symmetric, the Lagrangian is invariant under the global transformation,

 N(x)↦UTN(x)U,  with U∈SL(4). (3.39)

A useful device for interpreting is the following. Note that we would get the exact same bosonic terms of and , if we were to reduce an eleven dimensional classical theory to seven dimensions. This theory is given by the Einstein-Hilbert action plus a scalar, the “11d dilaton”121212We will denote the extra dimension by . This is not to be confused with the M-theory circle denoted by .

 S=∫d11x√−~ge−ϕ(R(~g)+∂μϕ∂μϕ) (3.40)

This Lagrangian contains no B-field. The description in terms of (3.40) is only useful when and vanish. This means that . Since the size of is constant, its dimensions are not treated on the same footing as the three geometric fiber dimensions. It is similar to the situation in F-theory [Vafa:1996xn], where the area of the is fixed and the Kähler modulus of the torus is not a dynamical parameter.

We have seen that the matrix can be interpreted as a semi-flat metric on a torus fiber. Part of this torus is the original fiber and the overall volume is set to one. The T-duality group acts on in a geometric way. This means that we can hope to study non-geometric compactifications by studying purely geometric ones in higher dimension.

3.2 The perturbative duality group

In the previous section, we have transformed the coset space into via Eq. (3.36). We also would like to see how the discrete T-duality group maps to . We will denote the matrices by , and the matrices by .

Generators of SO(3,3,Z). It was shown in [Schwarz:1998qj] that the following elements generate the whole group

 (3.41)

where , . The first two matrices correspond to a change of basis of the compactification lattice. The last matrix is T-duality along the coordinates. Instead of using directly, we combine double T-duality with a rotation. This gives the matrix

 ˜Q3=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−1 111−11⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (3.42)

Generators of SL(4,Z). In the Appendix of [Brace:1998ku], it was shown that the above matrices have an integral spinor representation and in fact generate the entire . We now list the spinor representations corresponding to these generators131313Note that [Brace:1998ku] uses a different basis for the spinors..

• is mapped to matrices

 W1(n)=⎛⎜ ⎜ ⎜ ⎜⎝1 n231 n31 1 n121⎞⎟ ⎟ ⎟ ⎟⎠ (3.43)

These are the generators corresponding to “T” transformations of various subgroups.

• is mapped to

 ˜W3=⎛⎜ ⎜ ⎜ ⎜⎝11−1 1⎞⎟ ⎟ ⎟ ⎟⎠ (3.44)

This corresponds to a modular “S” transformation. Note that .

• When , the matrix is mapped to the matrix

 W2(R)=(R001) (3.45)

For symmetric matrices it coincides with the prescription of Eq. (3.36).

• The case is more subtle. Even though Type IIA string theory is parity invariant, in the microscopic description reflecting an odd number of coordinates does not give a symmetry by itself. Since this transformation flips the spinor representations , it must be accompanied by an internal symmetry which changes the orientation of the world-sheet and thus exchanges the left-moving and right-moving spinors.

has maximal subgroup and hence has two connected components [Gualtieri:2003dx]. Inversion of an odd number of coordinates is not in the identity component. is the double cover of the connected component of only. We must allow for to obtain , the double cover of the full . Then, the reflections of the , or coordinates have the following representations141414The only non-trivial element in the center of is . This sign may be attached to all the group elements not in the identity component, giving an automorphism of .

 WI7=diag(−1,1,1,1)WI8=% diag(1,−1,1,1)WI9=diag(1,1,−1,1) (3.46)

Upon restriction to , the group is a trivial covering.

Ramond-Ramond fields transform in the spinor representation of the T-duality group151515As discussed in [Brace:1998xz], the fields that have simple transformation properties are .. Therefore they form fundamental multiplets, for instance . We can check the above representation for the coordinate reflections. Reflection of say combined with a flip of the three-form field gives

 (C7,C8,C9,C789)↦(−C7,C8,C9,C789) (3.47)

which is precisely the action of .

3.3 Embedding Sl(2)2 in Sl(4)

In order to get some intuition for the duality group that we discussed in the previous section, we first look at the simpler case of compactifications. In this section we describe how the T-duality group of compactifications can be embedded into the bigger group.

In eight dimensions, the duality group is with the first factor acting on the complex structure of the torus and the second factor acting on where and is the volume of . If we consider a two dimensional base with complex coordinate , then the equations of motion are satisfied if and are holomorphic sections of bundles. Monodromies of around branch points points describe the geometric degenerations of the fibration. Monodromies of , however, correspond to T-dualities and to the semi-flat description of NS5-branes. In particular, if there is a monodromy around a degeneration point in the base, then it implies which describes a unit magnetic charge for the B-field, i.e. an NS5-brane. The monodromy on the other hand is a double T-duality along the combined with a rotation.

Let us denote the two-torus coordinates by . In order to embed this duality group into the of compactifications, we need to further compactify on a “spectator” circle of size . We denote its coordinate by . The metric on () is now

 G3×3=⎛⎜ ⎜⎝g11g12g21g22 L2⎞⎟ ⎟⎠ (3.48)

Then, one can construct the metric on by the prescription of (3.36) which gives

 N=(detg)−1/2⎛⎜ ⎜ ⎜⎝1Lg2×2 LLb LbL(detg+b2)⎞⎟ ⎟ ⎟⎠≡(1LT2×2LR2×2) (3.49)

with

 (3.50)

The -model Lagrangian

 Tr(∂μN−1∂μN)=−2(∂μτ∂μ¯ττ22+∂μρ∂μ¯ρρ22) (3.51)

indeed gives the familiar kinetic terms for the torus moduli (in seven dimensions).

We have seen how the metric and the B-field parametrize the relevant subset of the coset space. The generators of the duality group are also mapped to elements in . We now verify that these images in fact give the transformations that we expect.

• Geometric transformations

These are simply generated by

 T=(1101)⊕\mathbbm12×2andS=(0−110)⊕\mathbbm12×2 (3.52)

They act on by conjugation with the non-trivial part as expected. The determinant of stays the same. The first one is a Dehn-twist and the second one is a rotation.

• Non-geometric transformations

The generators

 T′=\mathbbm12×2⊕(1101)andS′=\mathbbm12×2⊕(0−110) (3.53)

correspond respectively to the shift of the B-field and to a double T-duality on combined with a rotation. The latter one has the monodromy

 M=⎛⎜ ⎜ ⎜ ⎜⎝1 0 000100000−10010⎞⎟ ⎟ ⎟ ⎟⎠ (3.54)

This is basically an exchange of the coordinates and it transforms the submatrix of into its inverse

 R−1=(detg)−1/2(detg+b2−b−b1) (3.55)

After this double T-duality, the (geometric) metric on becomes

 G3×3↦˜G3×3=⎛⎜⎝1detg+b2 g2×200 L2⎞⎟⎠ (3.56)

The B-field transforms as

 b↦~b=−bdetg+b2 (3.57)

The metric on changes, in particular if , then the volume gets inverted. Since we exchanged the coordinates, one might have expected that this affects the metric on . However, we see that it remains the same as it should since it was only a spectator circle.

• Left-moving spacetime fermion number:

This is a global transformation which inverts the sign of the Ramond-Ramond fields. It acts trivially on the vector representation of (which is the antisymmetric tensor of ). It will be important since T-duality squares to . In [Hellerman:2002ax], its representation was determined,

 M(−1)FL=(−100−1)⊕(−100−1)∈SL(2)×SL(2) (3.58)

that is a monodromy combined with a (i.e. a conjugate ). This statement can be proven as follows. Let us define complex coordinates

 zL=x7L+ix8L (3.59)
 zR=x7R+ix8R (3.60)

where and are the left- and right-moving components of the bosonic coordinates. We denote a transformation

 (zL,zR)↦(eθLzL, eθRzR) (3.61)

by . Then,

 θD4=(−π,−π) (3.62)

as it is a reflection of the bosonic coordinates. Moreover, we can use where is a double T-duality with a rotation. We have

 θS=(−π,0)double T-duality+(π2,π2)90∘ rotation=(−π2,π2) (3.63)

from which we obtain

 θD′4=2×θS=(−π,π) (3.64)

Finally,

 θD4+D′4=θD4+θD′4=(−2π,0) (3.65)

which acts trivially on the bosons. However, it inverts the sign of the spinors from left movers which is precisely the action of . Finally, it can be embedded into simply as

 M(−1)FL=diag(−1,−1,−1,−1) (3.66)

3.4 U-duality and G2 manifolds

We have seen that upon compactifying Type IIA on , a torus emerges. We will be eventually interested in compactifications to four dimensions. For vacua without fluxes and T-dualities, the total space of the fibration is a Calabi-Yau threefold. What can we say about the total space of the fibration?

Note that there is an analogous (more general) story in M-theory. Reducing eleven dimensional supergravity on a flat yields a Lagrangian that is symmetric under the U-duality group [Hull:1994ys, Cremmer:1979up, Sezgin:1982gi, Cremmer:1997ct]. By Hodge-dualizing the three-form (), one can define a matrix161616 The relation to F-theory [Vafa:1996xn] can roughly be understood as follows. In the lower right corner of the metric there is a submatrix (with coordinates ). In the ten dimensional language, this matrix contains the dilaton and the three-form which is “mirror” to the axion in Type IIB. Roughly speaking, (conjugate) S-duality acts on this .

 G−1=⎛⎝ωgIJ+1ωXIXJ −1ωXI−1ωXI1ω⎞⎠ (3.67)

which contains the geometric metric on as well. We denote the dimensions171717Note that and are switched. This is because we want to denote the extra M-theory dimension by . We stick to this notation throughout the paper. by , , , , , respectively. The bosonic kinetic terms can be written as a manifestly invariant -model in terms of this metric [Cremmer:1997ct].

We can embed the unit determinant matrix (see Eq. 3.38) into the unit-determinant matrix as follows

 G−1=⎛⎜ ⎜ ⎜⎝δgij+1δbibj 0−1δbi010−1δbi01