Contents

IPhT-T17/057


The bosonic string on string-size tori


from double field theory


Yago Cagnacci, Mariana Graña, Sergio Iguri and Carmen Nuñez


Instituto de Astronomía y Física del Espacio (CONICET-UBA) and

Departamento de Física, FCEN, Universidad de Buenos Aires

C.C. 67 - Suc. 28, 1428 Buenos Aires, Argentina

Institut de Physique Théorique, CEA/ Saclay

91191 Gif-sur-Yvette Cedex, France



Abstract: We construct the effective action for toroidal compactifications of bosonic string theory from generalized Scherk-Schwarz reductions of double field theory. The enhanced gauge symmetry arising at special points in moduli space is incorporated into this framework by promoting the duality group of -tori compactifications to , being the dimension of the enhanced gauge group, which allows to account for the full massless sector of the theory. We show that the effective action reproduces the right masses of scalar and vector fields when moving sligthly away from the points of maximal symmetry enhancement. The neighborhood of the enhancement points in moduli space can be neatly explored by spontaneous symmetry breaking. We generically discuss toroidal compactifications of arbitrary dimensions and maximally enhanced gauge groups, and then inspect more closely the example of at the point, which is the simplest setup containing all the non-trivialities of the generic case. We show that the entire moduli space can be described in a unified way by considering compactifications on higher dimensional tori.

December 29, 2018

1 Introduction

The low energy limit of string theory compactifications on string-size manifolds cannot be obtained by usual Kaluza-Klein reductions of ten-dimensional supergravity, since these do not incorporate the light modes originating from strings or branes wrapping cycles of the internal manifold. At special points in moduli space, some of these modes become massless, and there is an enhacement of symmetry promoting the (typically Abelian) gauge groups appearing in Kaluza Klein compactifications to non-Abelian groups. For toroidal compactifications of the heterotic string, this phenomenon has been beautifully described by Narain [1]. In compactifications of the bosonic string on -dimensional tori, the symmetry of the Kaluza-Klein reduction gets enhanced to at special points in moduli space, where are simply-laced groups111For simplicity we consider the cases of equal groups on the left and on the right, i.e. , but the construction can be generalized to different groups of equal rank . of rank and dimension , and there are massless scalars transforming in the adjoint representation of the left and right symmetry groups. These special points correspond to tori whose radii are of order one in string units, and the states that provide the enhancement have non-zero winding number besides non-zero momentum in some torus directions.

Capturing winding states within a field theory requires new ingredients beyond those of ordinary Kaluza-Klein compactifications. In the double field theory (DFT) framework [2], these include the introduction of a T-dual coordinate to every torus direction, which is the Fourier dual of the corresponding winding mode (for reviews see [4]). DFT is therefore a field theory formulated on a double torus incorporating the T-duality symmetry of the bosonic string on a -torus. Consistency of the theory requires constraints. Although the most general form of these constraints is unclear, a sufficient but not necesary constraint is the so-called section condition or strong constraint, that restricts the fields to depend on a maximally isotropic subspace with respect to the inner product, such as the original -torus or its T-dual one, and the original field theory is recovered but now formulated in an -covariant way. Understanding to what extent this constraint can be violated while keeping a consistent theory remains an open question, about which there are a few answers in particular setups.

The setup relevant to this paper is that of generalized Scherk-Schwarz reductions of double field theory [5, 6] on generalized parallelizable manifolds [7], namely manifolds for which there is a globally defined generalized frame on the double space, such that the C-bracket algebra (the generalization of the Lie algebra that is needed in double field theory) on the frame gives rise to (generalized) structure constants, which are on the one hand trully constant, and on the other should satisfy Jacobi identities. As in standard Scherk-Schwarz reductions [8], the only allowed dependence on the internal (double) coordinates is through this frame. Even though the dependence of the frame on internal coordinates might violate the strong constraint, it was shown in [9, 10] that the theory is consistent at the classical level, as long as the structure constants satisfy the Jacobi identities.

A generalized Scherk-Schwarz reduction of double field theory on a double circle gives the low energy action for compactifications of the bosonic string on a circle, and the procedure implemented in [11] allows to describe the string theory features described above when the radius is close to the self-dual one. In this paper we extend the results of [11], and show that the low-energy action for compactifications of the bosonic string on any -torus in a region of moduli space close to a point of symmetry enhancement to a group can be obtained from double field theory. To that aim, we consider double field theory on the double torus (of dimension ), and build an structure on it (where denotes the number of external directions and is the dimension of ), given by a generalized metric. The generalized metric, or rather the generalized vielbein for it, is of the Scherk-Schwarz form, namely it is the product of a piece that depends on the external coordinates, and involves the vector and scalar fields of the reduced theory that are massless at the enhancement point, and a piece depending on the internal, doubled, coordinates. The internal piece is such that the C-bracket algebra gives rise to the symmetry. Plugging this generalized metric in the double field theory action and following the generalized Scherk-Schwarz reduction of [5, 6], we obtain an action that exactly reproduces the string theory three-point functions at the point of symmetry enhancement. Furthermore, we show how the process of symmetry breaking by Higgsing in the effective action gives the exact string theory masses for the vector and scalar fields close to the enhancement point in moduli space, up to second order in deviations from this point. The Higgsing process amounts to giving vacuum expectation values to the scalars along the Cartan directions of the group , and we show that these vevs are precisely given by departure of the metric and B-field on the torus away from their values at the enhancement point.

We provide the explicit expression for the generalized vielbein. For , the piece of the vielbein that depends on the internal coordinates is a straightforward extension of the one corresponding to that was constructed in [11]. The algebra is obtained from the C-bracket of a block-diagonal frame made of -blocks, where each block involves the vertex operators of the corresponding ladder currents. Geometrically, this translates into a 2-dimensional fibration of the directions corresponding to positive and negative roots over the Cartan direction, given by the corresponding circle coordinate . The fibration has trivial monodromy. For groups that have additionally non-simple roots, we show that the bracket can be deformed in a way that preserves the covariance. The deformation accounts for the cocycle factors that are necessary in the vertex representation of the current algebra, and then we can reproduce the algebra with a generalized vielbein that depends on coordinates only. An alternative generalized frame can be constructed from the formulation of DFT on group manifolds [12, 14], in which it depends on coordinates. The question whether there exists a vielbein depending strictly on coordinates that gives rise to the algebra under the usual C-bracket, when has at least one non-simple root, remains open.

A very interesting question is whether there is a description of the full moduli space, namely a formulation that includes all the states that are massless at any point in the moduli space. We show that such a description requires considering a higher-dimensional torus at a point of maximal enhancement. For and , one gets the effective action at any point in moduli space by considering one of the points of maximal enhancement on a torus of one higher dimension ( and respectively), and combining the process of spontaneous symmetry breaking together with a decompactification limit. For , a description that includes the whole moduli space requires considering enhancement points at an even larger torus, namely a . We explain how this process works dimension by dimension. Note, though, that the action obtained this way does not correspond to a low energy action, since states that are massless at one point in moduli space get string-order masses at another point.

The paper is organized as follows. In section 2 we review toroidal compactifications of bosonic string theory. We consider the covariant formulation of compactifications on with constant background metric and antisymmetric 2-form fields and the basics of T-duality. The enhancement of the gauge symmetry at special points in moduli space is discussed in general for and details are provided for the case. The basic features of DFT and generalized Scherk-Schwarz compactifications are reviewed in section 3. Using this framework, we construct the effective action of bosonic string theory compactified on in the vecinity of a point of symmetry enhancement in section 4. In particular, we show that a deformation of the C-bracket involving the cocycle factors of the vertex algebra allows to reproduce the structure constants of the enhanced symmetry algebra. In section 5 we check that the construction reproduces the string theory results when moving slightly away from that point. A higher dimensional formulation that allows to accommodate all maximal enhancement points in a single approach is presented in section 6. Finally, an overview and conclusions are given in section 7. Three appendices collect the necessary definitions and notation used in the main text. Basic notions of simply laced Lie algebras and Lie groups are reviewed in Appendix A, some basic facts about cocycles are contained in Appendix B and the explicit discussion of symmetry breaking on is the subject of Appendix C.

2 Toroidal compactification of the bosonic string

In this section we recall the main features of toroidal compactifications of the bosonic string. We first discuss the generic case and then we concentrate on the example. For a more complete review see [15].

2.1 Compactifications on

Consider the bosonic string propagating in a background manifold that is a product of a dimensional space-time times an internal torus with a constant background metric

(2.1)

and antisymmetric two-form field , . For simplicity we take the dilaton to be zero. The set of vectors define a basis in the compactification lattice such that the target space is the -dimensional torus .

The contribution from the internal sector to the world-sheet action is

(2.2)

The metric and the -field are dimensionless222We will write explicit factors of later in the text when they are needed for clarification., the world-sheet metric has been gauge fixed to () and the internal string coordinate fields satisfy

(2.3)

where are the winding numbers and

(2.4)

with

(2.5)

the dots standing for the oscillators contribution.

The periodicity of the wavefunction requires quantization of the canonical momentum333The unusual factor is due to the use of Euclidean world-sheet metric.

(2.6)

and (2.3) implies the quantisation condition

(2.7)

These equations give

(2.8a)
(2.8b)

The vectors constitute the canonical basis for the dual lattice , i.e. , and thus they obey

(2.9)

The pairs transform as vectors under and they expand the -dimensional momentum lattice . From (2.8) one sees they satisfy

(2.10)

and therefore they form an even Lorentzian lattice. In addition, self-duality follows from modular invariance [1, 16].

The space of inequivalent lattices and inequivalent backgrounds reduces to

(2.11)

where is the T-duality group (we give more details about it in the next section), and the factor accounts for the world-sheet parity , a symmetry acting on the background as .

2.2 covariant formulation

The mass of the states and the level matching condition are respectively given by

(2.12a)
(2.12b)

These can be written in terms of the momentum and winding numbers using an -covariant language by introducing the vector and the invariant metric

(2.13)

as well as the “generalized metric” of the -dimensional torus, given by the matrix

(2.14)

The mass formula (2.12a) and the level matching condition (2.12b) then read

(2.15a)
(2.15b)

respectively.

Note that both the mass formula and the level matching condition are invariant under the T-duality group acting as

(2.16)

The group is generated by integer theta-parameter shifts, associated with the addition of an antisymmetric integer matrix to the antisymmetric -field,

(2.17)

lattice basis changes

(2.18)

and factorized dualities, which are generalizations of the circle duality, of the form

(2.19)

where is a matrix with all zeros except for a one at the component.

Notice the particular role played by the element viewed as a sequence of factorized dualities in all tori directions, i.e.

(2.20)

Its action on the generalized metric is

(2.21)

and, together with the transformation which accounts for the exchange , it generalizes the duality of the circle compactification. These transformations define the dual coordinate fields (up to the center of mass coordinates)

(2.22)

the dots standing for the oscillator contributions.

A vielbein for the generalized metric

(2.23)

can be constructed from the vielbein for the metric (2.1) and inverse metric (2.9), as follows

(2.24)

In the basis of left and right movers, that we call “LR”, where the metric takes the diagonal form

(2.25)

the vielbein is

(2.26)

Note that this is not the most general parameterisation for the generalized vielbein. We could have used on the first line a vielbein for the (ordinary) metric , and its inverse , and on the second line, which corresponds to the right sector, a different vielbein , giving rise to the same metric . For simplicity we use in most of the text the same vielbein on the left and on the right, except later in section 4 (see in particular Eq (4.6)), where we need to make use of this freedom.

Then the momenta in (2.8) are

(2.27)

2.3 Gauge symmetry enhancement

At special points in moduli space there is an enhancement of gauge symmetry due to the fact that there are extra massless states with non-zero momentum or winding on the torus. From (2.12a) and (2.12b), the massless states satisfy

(2.28a)
(2.28b)

and therefore, . This means there are, from the point of view of the non-compact -dimensional space-time, massless 2-tensors (given by the usual states with no momentum or winding and )444 () denote the oscillation numbers along the non-compact (compact) directions., massless vectors (with or ), and massless scalars (with ).

Let us concentrate on the vectors first, and analyze the case . There are two types of massless states of this form, those with and no momentum or winding, and states with and winding or momentum such that , . The former are the Kaluza-Klein (KK) vectors generating in the left sector, which are massless at any point in moduli space. The fields and their vertex operators are

(2.29)

where , and is the -dimensional momentum in space-time. The massless states with no oscillation but momentum or winding number along internal directions have vertex operators

(2.30)

where label the compact left-moving momenta, related to (2.8a) by

(2.31)

and is a cocycle introduced to ensure the right properties of the OPE between vertex operators (see Appendix B for more details). The reason for introducing new notation for the left-moving momentum will become clear in a moment. The OPE of the currents with those in the KK sector (2.29) is

(2.32)

and allows to identify as the Cartan currents and as the currents corresponding to a root of the holomorphic part of the enhanced algebra. The OPE between two reads

(2.33)

where denotes the inner product. The Killing form is given in (A.6), and is just a in the Cartan-Weyl basis. Singular terms appear in the OPE only if is equal to or . If , then and one gets from (2.33)

(2.34)

If , then is a root and (2.33) reads

(2.35)

The sign is determined by the product defined in Eq. (B.10) and can be reproduced by the cocycles. It makes the OPE invariant under and .

Altogether Eqs (2.32), (2.34) and (2.35) show that the currents satisy the OPE algebra of the holomorphic part of the enhanced symmetry group. This group has rank (the dimension of the torus). Furthermore, the requirement implies that the enhanced symmetry must correspond necessarily to a simply laced algebra. We denote the dimension of the left-moving part of the algebra by . The number of roots is equal to . We will see below some examples of enhanced gauge groups. In Appendix A we collect the necessary definitions, notation and conventions regarding Lie algebras and Lie groups.

The right part of the enhanced gauge algebra is constructed in an analogous way from states with , and either and no momentum or winding, or and momentum and winding such that . The vertex operators for the former are

(2.36)

and for the latter

(2.37)

with

(2.38)

a root of the Lie group corresponding to the right part of the enhanced gauge symmetry. This group has rank as well, but might not be the same as the one on the left. For simplicity, we will consider from now on the case of equal groups on the left and on the right, which is what happens at points of maximal enhancement (to be discussed later) that will be our primary focus. Almost all of the formulas have though a straighforward generalisation to a gauge symmetry enhancement group of rank and dimension .

Regarding the extra massless scalars, one gets a total of . of them are the usual KK scalars with and no momentum or winding, corresponding to the internal metric and -field. The fields and their vertex operators are

(2.39)

They are massless at any point in moduli space, and their vev’s determine the type of symmetry enhancement. At the points in moduli space where the symmetry is enhanced to a group of dimension , there are scalars with , and vertex operators

(2.40)

as well as scalars with , and vertex operators

(2.41)

As we will see, in the effective theory around a point in moduli space where there is symmetry enhancement, these are the Goldstone bosons in the symmetry breaking process. Finally, there are scalars with no oscillation number and . Their vertex operators are

(2.42)

Let us now see explicitly how to find the extra massless vectors and scalars with momentum or winding number. Their existence depends on the location in moduli space, it depends on and . The massless vectors in the left moving sector should have , , and therefore satisfy

(2.43)

The simplest case to analyse is that of a torus with diagonal metric and all the radii at the self-dual point, together with vanishing -field , .) For each torus direction there are two extra massless vectors with . These combine with the KK vectors (2.29) to enhance the symmetry from to . Combining with the right moving sector, one has . Other examples of symmetry enhancement groups are found at points in moduli space which are fixed points of a subgroup of the T-duality group.

Maximal enhancement555“Maximal” stands here for an enhanced semi-simple and simply-laced symmetry group of rank corresponding to a level 1 affine Lie algebra occurs when the background is a fixed point of the symmetry (2.21) up to an identification by a theta shift (2.17) and an transformation (2.18), namely when

(2.44)

The case just discussed is the simplest one (with ), but there are more general examples in which the background is [17]

(2.45)

where sgn denotes the sign function and is the Cartan matrix associated to the corresponding algebra. Note that the matrices and its transpose at the enhancement point acquire a triangular form. Non-maximal enhanced symmetries can be found at fixed points of factorized dualities instead of the full inversion transformation (2.21).

We will analyse in the next section the case of compactifications in detail, where the gauge groups of maximal enhancement are and .

2.4 Compactifications on

Here we discuss in detail the case. The moduli can be joined conveniently into two complex fields as follows. The complex structure is given by

(2.46)

while the Kähler structure is introduced as

(2.47)

where is the determinant of the metric on the torus. The inverse relations read

(2.48)

Later we will need a vielbein for the metric and its inverse, which can be taken to be

(2.49)

The generalized metric, defined in (2.14), reads in terms of and

(2.50)

and the corresponding generalized veilbein (2.24) is

(2.51)

The left and right moving momenta (2.8) in terms of and read666Here we are expressing the vectors and on the tangent space of as complex variables.

(2.52a)
(2.52b)

The moduli space is isomorphic to , where and sweep each factor, respectively. The duality group is generated by the usual and modular transformations, together with the factorized duality exchanging the complex and the Kähler structures777In terms of the transformations given in the previous section, and are the transformations in (2.18) given respectively by and , while is a T-duality transformation .

(2.53)

Worldsheet parity acts by

(2.54)

The fundamental domain is given by two copies of the domain shown in Figure 1.

Figure 1: Fundamental domain for the modulus .

The possible groups of symmetry enhancement for the left or right sector have rank 2, and are thus (no enhancement), or the maximal or . All of these occur on the plane (up to identifications under the discrete symmetries (2.53) and (2.54)). As discussed in the previous subsection, enhancement to occurs in a compactification with metric given by the identity ( all radii equal to the string length) and no -field. This satisfies (2.45), where the Cartan matrix for is given in (A.24), and corresponds to . The Cartan matrix of is given in (A.28), and thus according to (2.45), the enhancement point is reached for , , . This corresponds to , which is at the other corner of the fundamental domain in Figure 1. We will discuss the physics around these two points in detail in the next subsections. enhancement occurs at the borders of the fundamental region, namely at , and . At the interior of the region, the enhancement group is . This asymmetry between the left and right sectors can be understood from the fact that points at the interior are not fixed points of the symmetry (or ). The mirror region, which is to the right of the region displayed in Figure 1 in our conventions, has enhanced gauge symmetry group .

The locations of these groups on the domain are displayed in Figure 2.

Figure 2: Enhancement groups on the domain . The groups and occur at the isolated points on the left and right vertices, respectively, while occurs along the boundaries.

2.4.1 enhancement point

At there is a gauge symmetry enhancement point with gauge group. This corresponds to the two-torus being a product of two circles at the self-dual radii, namely the metric is given by the identity and there is no -field. This satisfies , where is the Cartan Matrix of the algebra given in (A.24). We choose the vectors defined in (2.9) to be

(2.55)

At this point there are massless vectors and massless scalars. The left vectors associated to the Cartan subalgebra are given by (2.29) and those associated to the ladder operators by (2.30), where now the in (2.30) are the roots of the algebra. For the left gauge group, the massless ladder vectors at are those in Table 2.1. There is an identical construction for the right sector, with appropiate and .

root vector
1 0 1 0 (2,0) (0,0)
0 1 0 1 (0,2) (0,0)
1 0 -1 0 (0,0) (2,0)
0 1 0 -1 (0,0) (0,2)
Table 2.1: Massless vectors with momentum and winding at the enhancement point . Only those associated with positive roots are shown.

2.4.2 enhancement point

At there is a symmetry enhancement point. The resulting gauge group is . The metric and -field are given by

(2.56)

where , being the Cartan matrix of the algebra. We choose the vectors defined in (2.9) to be

(2.57)

The generalized metric (2.14) (which is given in (2.50) in terms of and ) is

(2.58)

This satisfies (2.44) with and where is defined in footnote 7.

There are massless vectors and massless scalars at this point in moduli space. The left vectors associated to the Cartan subalgebra are given by (2.29) and those associated to the ladder operators by (2.30), where now are the roots of the algebra. For the left gauge group, the ladder vectors that are massless at the SU(3) point are those in Table 2.2. There is a similar construction for the right sector, with appropiate and . In Table 2.3 we give some of the 64 massless scalars, to be used in section 5.2.

Note that the vector in Table 2.2 is not the same as the one denoted in Table 2.1 corresponding to the case, as the notation refers to the roots of each algebra. Comparing the two tables, we can see that there is an overlap between them and this will be important in section 6.

root vector
1 0 1 0 (2,-1) (0,0)
-1 1 0 1 (-1,2) (0,0)
0 1 1 1 (1,1) (0,0)
1 -1 -1 0 (0,0) (2,-1)
0 1 0 -1 (0,0) (-1,2)
1 0 -1 -1 (0,0) (1,1)
Table 2.2: Massless vectors with momentum and winding at the enhancement point