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August, 2016


Non-geometric Five-branes in Heterotic Supergravity


Shin Sasaki111shin-s(at)kitasato-u.ac.jp and Masaya Yata222phymasa(at)nus.edu.sg


Department of Physics

Kitasato University

Sagamihara 252-0373, Japan

Department of Physics,

National University of Singapore

2, Science Drive 3, Singapore 117542, Singapore


We study T-duality chains of five-branes in heterotic supergravity where the first order -corrections are present. By performing the -corrected T-duality transformations of the heterotic NS5-brane solutions, we obtain the KK5-brane and the exotic -brane solutions associated with the symmetric, the neutral and the gauge NS5-branes. We find that the Yang-Mills gauge field in these solutions satisfies the self-duality condition in the three- and two-dimensional transverse spaces to the brane world-volumes. The monodromy structures of the -brane solutions are investigated by the -corrected generalized metric. Our analysis shows that the symmetric -brane solution, which satisfies the standard embedding condition, is a T-fold and it exhibits the non-geometric nature. We also find that the neutral -brane solution is a T-fold at least at . On the other hand, the gauge -brane solution is not a T-fold but show unusual structures of space-time.

1 Introduction

Extended objects such as branes play an important role in superstring theories. The U-duality [1], which makes non-trivial connections among consistent superstring theories, relates various branes in each theory. It was shown that M-theory compactified on has the U-duality symmetry group in lower dimensions [2, 3, 4]. BPS branes in superstring theories form lower-dimensional multiplets under the U-duality group. For example, when we consider M-theory compactified on , we have the BPS point particle multiplet. The higher dimensional origin of parts of these point particles is the ordinary branes wrapped/unwrapped on cycles in . Here, the ordinary branes are waves, F-strings, D-branes, NS5-branes, Kaluza-Klein (KK) branes. However, there are states whose higher dimensional origin does not trace back to the ordinary branes. These states are called exotic states and their higher dimensional origin is known as exotic branes [5, 6].

Among other things, an exotic brane in type II string theories known as the -brane, has been studied intensively [7, 8, 9, 10]. The most tractable duality in string theory is the T-duality. Type II string theories compactified on has T-duality symmetry group . The exotic -brane is obtained by performing the T-duality transformations along the transverse directions to the NS5-brane world-volume. As its notation suggests, the -brane has two isometries in the transverse directions to the brane world-volume [3]111In this paper, we use the notation in the sense that they are obtained by the T-duality transformations twice on the heterotic NS5-branes.. Its tension is proportional to where is the string coupling constant. Therefore the -brane is a solitonic object of co-dimension two. These co-dimension two objects in string theory, sometimes called defect branes [11], exhibit specific properties [12, 13]. When one goes around the center of the co-dimension two branes in the transverse two directions and come back to the original point, the background geometry of the branes changes according to the non-trivial monodromy. Therefore the metric and other background fields are generically governed by multi-valued functions. In this sense, they are called non-geometric [14]. However when the monodromy is given by the symmetry group of the theory, which is generically the U-duality group in string theory, then the non-geometry becomes healthy candidate of solutions to string theory. This kind of solution is called U-fold. An important nature of the exotic branes is that they are non-geometric objects [15]. Therefore they are called non-geometric branes or Q-branes [16]. In particular, the -brane in type II string theories is a T-fold, whose monodromy is given by the T-duality group .

The purpose of this paper is to study exotic branes in heterotic string theories. Compared with type II string theories, exotic branes in type I and heterotic string theories have been poorly understood. This is due to the non-Abelian gauge field living in the space-time in these theories. Notably, the Yang-Mills gauge field enters into the space-time action as the first order -corrections. Due to the -corrections, the Buscher rule [17] of the T-duality transformation for heterotic supergravity is modified [18, 19, 20]. The most famous extended objects in heterotic supergravity theories are the heterotic NS5-brane solutions. There are three distinct NS5-brane solutions in heterotic supergravity [21, 22, 23]. We will perform the T-duality transformations to these heterotic NS5-branes by the -corrected Buscher rule and obtain new five-brane solutions. We will then study the monodromy structures of these solutions.

The organization of this paper is as follows. In the next section, we introduce the heterotic NS5-brane solutions known as the symmetric, neutral and gauge types [21, 22]. In section 3, we introduce the isometries along the transverse directions to the NS5-brane world-volumes and perform the T-duality transformation by the -corrected Buscher rule. We obtain the heterotic Kaluza-Klein five-brane (KK5-brane) solutions associated with the three types of solutions. We then perform the second T-duality transformations on the KK5-branes and write down the exotic -brane solutions. In section 4, we examine the monodromy structure of the heterotic -brane solutions by the -corrected generalized metric. We show that the monodromy is given by the T-duality group for the symmetric and the neutral solutions while the gauge solution remains geometric. Section 5 is devoted to conclusion and discussions. The explicit form of the smeared gauge KK5-brane solution is found in appendix.

2 Heterotic NS5-brane solutions

In this section, we introduce the NS5-brane solutions in ten-dimensional heterotic supergravity which is the low-energy effective theory of heterotic superstring theory. Heterotic supergravity consists of the ten-dimensional gravity multiplet coupled with the vector multiplet. The relevant bosonic fields in heterotic supergravity are the vielbein , the dilaton , the NS-NS -field and the Yang-Mills gauge field . Here are the curved space indices while are the local Lorentz indices. The Yang-Mills gauge field is in the adjoint representation of the gauge group which is or . We employ the convention such that the gauge field is represented by anti-hermitian matrices and the trace is taken over the matrices of the fundamental representation.

A remarkable property of heterotic supergravity is that the Yang-Mills gauge field contributes to the action as the first order -correction. The famous anomaly cancellation mechanism and the supersymmetry completion result in the Riemann curvature square term which involves higher derivative corrections in the same order in [24]222The authors would like to thank Tetsuji Kimura for his introducing [24].. The ten-dimensional heterotic supergravity action for the bosonic fields at the first order in is given by

(1)

Here we employ the convention such that where and are the gravitational and the gauge coupling constants in ten dimensions. The ten-dimensional metric in the string frame is defined through the vielbein as where the metric in the local Lorentz frame is . The Ricci scalar is constructed from the spin connection :

(2)

The spin connection is expressed by the vielbein and its inverse:

(3)

The modified spin connection which enters into the action in the terms is defined as

(4)

where and the modified -flux is defined by

(5)

Here the field strength of the -field is defined by

(6)

The Yang-Mills and the Lorentz Chern-Simons terms which appear in (5) are defined as follows:

(7)

Here the symbol stands for the anti-symmetrization of indices with weight . Note that the modified -flux is iteratively defined through the relations (4) and (5) order by order in . The modified -flux obeys the following Bianchi identity:

(8)

where is the -valued curvature 2-form. The component of the Yang-Mills gauge field strength 2-form is

(9)

Here are the gauge indices and is the structure constant for the Lie algebra associated with . Note that the terms and in the action (1) are higher derivative corrections to the ordinary second order derivative terms.

The heterotic NS5-brane solutions satisfy the equation of motion derived from the action (1) at . There are three distinct NS5-brane solutions known as the symmetric, the neutral and the gauge types in heterotic supergravity [21, 22]. They are 1/2 BPS configurations and preserve a half of the sixteen supercharges. They satisfy the following ansatz:

(10)

where the indices , represent the world-volume and the transverse directions to the NS5-branes and is the Levi-Civita symbol. The 1/2 BPS condition leads to the self-duality condition for the Yang-Mills gauge field:

(11)

and other components vanish. Here the Hodge dual field strength in the transverse four-dimensions is defined by . The harmonic function is determined by the Yang-Mills gauge field configurations of the solutions. These heterotic NS5-branes are characterized by two charges. One is the topological charge associated with the gauge instantons:

(12)

where the integral is defined in the transverse four-space. The other is the charge associated with the modified -flux:

(13)

Here is the asymptotic three-sphere surrounding the NS5-brane. In the following we briefly introduce the three distinct NS5-brane solutions.

Symmetric solution

The most tractable NS5-brane solution in heterotic theory may be the so-called symmetric solution. The Yang-Mills gauge field which satisfies the self-duality condition (11) is given by an instanton solution in four-dimensions. The gauge field takes value in the subgroup of the gauge group and the explicit solution is given by [22],

(14)

where is the self-dual part of the Lorentz generator. This is written in the following form,

(15)

where is the ’t Hooft symbol which satisfies the self-duality condition in terms of the indices :

(16)

The anti-hermitian matrices are defined as

(17)

They satisfy the algebra

(18)

The solution (14) is nothing but the BPST one-instanton in the non-singular gauge [25]. The constant is the asymptotic value of the dilaton. Compared with the general BPST instanton solution, the symmetric solution has a fixed finite instanton size . This solution has charges . A remarkable fact about the symmetric solution is that the dilaton configuration, hence the harmonic function , is obtained through the standard embedding ansatz:

(19)

Here the indices of the gauge field are identified with the indices of the subgroup of the local Lorentz group in the transverse directions. If the relation (19) holds, the Chern-Simons terms in the modified -flux (5) cancel out. Therefore the -flux only comes from the -field and the Bianchi identity (8) becomes . From the Bianchi identity and the relation between and in (10), the -field is determined through the following condition:

(20)

Although the symmetric NS5-brane solution (10) with (14) is a solution to the equation of motion derived from the action (1) at , the string sigma-model analysis indicates that the symmetric solution is an exact solution valid at all orders in [22]. Indeed, the symmetric solution has the sigma model description and it is protected against -corrections.

Neutral solution

The neutral solution is the one with charges [22]. For this solution, the Yang-Mills gauge field becomes trivial and the harmonic function is given by

(21)

Since the gauge field does not appear in the neutral solution, this is a solution to type II supergravities. Indeed, this is the type II NS5-brane solution. It is remarkable that the neutral solution (10) with (21) for heterotic supergravity is valid at the first order in . For this solution, we find the curvature is order . Therefore the Lorentz Chern-Simons term in the modified -flux (5) becomes a higher order correction in and negligible. Then the Bianchi identity again becomes . With the solution (21) at hand, the -field is given by that of the type II NS5-brane and its components, which are determined by (20), are the same as the ones of the symmetric solution in the linear order in . We note that the neutral solution has the worldsheet sigma model description and receives higher order -corrections in heterotic theories [22].

Gauge solution

The last is the so called gauge solution which has been originally found in [21]. The Yang-Mills gauge field is again given by the BPST instanton. As in the case of the neutral solution, the curvature term in the modified -flux is neglected as it is a higher order in . However, the Yang-Mills gauge field still contributes to the -flux in the gauge solution and the Bianchi identity becomes . From the Bianchi identity, the harmonic function is determined to be

(22)

Compared with the symmetric solution (14), the size modulus is not fixed for the gauge solution and it has charges . Similar to the neutral solution, the gauge solution is obtained by a perturbative series of and the functional forms (10), (22) are valid at the first order in . Indeed, the gauge solution has the worldsheet sigma model description and is expected to receive higher order -corrections [22]. The position of the instanton corresponds to that of the NS5-brane. Therefore the instanton with finite size resides in the core of the NS5-brane. It has been discussed in [26], when the instanton shrink to zero size , a gauge multiplet on the brane world-volume becomes massless and the gauge symmetry is enhanced. The gauge NS5-brane in the heterotic theory in the zero size limit is related to the D5-brane in type I theory by the S-duality.

We make a comment on the -field for the gauge NS5-brane solution. A careful analysis reveals that only the Yang-Mills Chern-Simons term contributes to the modified -flux and we find at least at . Therefore the -field is not excited in the gauge NS5-brane solution. Then, the components of the -field take a constant value:

(23)

3 T-duality chains of five-branes and -corrected Buscher rule

In this section, we derive new five-brane solutions in heterotic supergravity in the family of the T-duality chains. Since the gauge field enters into the action (1) as the first order -correction, the T-duality transformation in the heterotic supergravity should be modified from the standard Buscher rule [17]. The first order -corrections to the Buscher rule in heterotic supergravity have been written down in [19]. This is given by

(24)

where we have decomposed the indices . The indices and specify an isometry and non-isometry directions respectively. In (24), we have defined the following quantity,

(25)

where we have defined . We call (24) with (25) the heterotic Buscher rule. Under the transformation (24), the actions of and heterotic supergravities dimensionally reduced on coincide. Note that when the Yang-Mills gauge field and the higher derivative corrections coming from are turned off, the relation (24) reduces to the ordinary Buscher rule for the NS-NS backgrounds in type II supergravities.

3.1 Heterotic KK5-branes

Before going to the exotic -branes, we first write down the heterotic KK5-brane solutions. In order to perform the T-duality transformation for the heterotic NS5-brane solutions, we introduce a isometry along the transverse direction to the brane world-volumes 333If we perform the T-duality transformations along the world-volume direction of the NS5-branes in the heterotic theory, we obtain the identical solutions in the theory and vice versa.. To this end, we first compactify the -direction with the radius and consider the periodic array of the NS5-branes. Then by taking the small radius limit , we introduce the isometry to the heterotic NS5-brane solutions. The self-duality condition (11) reduces to the monopole equation where . Therefore the resulting solutions are called the heterotic monopoles or smeared NS5-branes. The explicit forms of the smeared NS5-brane solutions which originate from the periodic arrays of the symmetric, the neutral and the gauge solutions have been written down in [27, 29]. By performing the T-duality transformation of the solutions along the isometry direction, we obtain the KK5-brane solutions associated with the three types of the NS5-branes. In the following, we calculate the T-duality transformations of the NS5-brane solutions and derive the KK5-brane solutions.

Symmetric KK5-brane

For the symmetric solution where the standard embedding condition is satisfied, the ansatz and the Bianchi identity leads to the condition . The periodic array of the symmetric NS5-brane solution is governed by the following harmonic function on [29]:

(26)

Taking the compactification radius small , the sum in (26) is approximated by the integral over . We call this procedure as smearing. The result is444 The constant would diverge in the limit but this is an artifact of the smearing procedure. We can find the harmonic function (27) which finite by solving the Laplace equation in three dimensions.

(27)

The corresponding solution is the smeared symmetric NS5-brane of co-dimension three discussed in [27]. We note that the smeared symmetric NS5-brane is an H-monopole whose quantized charge is well-defined. On the other hand, the Yang-Mills monopole charge for the smeared solution is not defined anymore [29]. We now perform the T-duality transformation on the smeared symmetric NS5-brane solution. For a solution where the standard embedding (19) is satisfied, the heterotic Buscher rule (24) is quite simplified. This is because the relation Tr( holds and the last term in (25) vanishes.

We perform the T-duality transformation by utilizing the heterotic Buscher rule (24). Then we obtain the symmetric KK5-brane solution:

(28)

where the harmonic function is given in (27) and is determined by the following relation:

(29)

The dilaton and the NS-NS -field vanish but the Yang-Mills gauge field remains non-trivial. The symmetric KK5-brane solution is not a purely geometric solution but the geometry is dressed up with the Yang-Mills gauge fields. The gauge field is obtained as

(30)

We find that the Yang-Mills gauge field configuration (30) satisfies the anti-monopole (Bogomol’nyi) equation

(31)

This equation is nothing but the anti-self-duality condition in disguise555 The flip of the sign in the Hodge dualized field strength comes from the choice of the freedom for the overall sign in the heterotic Buscher rule (24). If we choose another sign in front of in the right hand side of (24), the gauge field satisfies the self-duality condition instead of the anti-self-duality condition. . Since the field configuration (30) satisfies the equation , this belongs to the general class of spherically symmetric solutions discussed in [28].

Meanwhile, by using the relation (29), the modified spin connection associated with the geometry (28) is calculated to be

(32)

From the results (30) and (32), we find that the standard embedding condition (19) still holds for the symmetric KK5-brane solution.

Neutral KK5-brane

We next study the neutral KK5-brane solution. For the neutral NS5-brane solution, we have the trivial Yang-Mills gauge field . Again, the smearing procedure is applicable to the neutral solution. The smeared neutral NS5-brane solution is governed by the harmonic function (27). As we have claimed in the previous section, the modified spin connection is in the for the neutral solution. Therefore it is negligible in in the heterotic Buscher rule (25). Applying the heterotic Buscher rule, we find that the metric, the -field and the dilaton for the neutral KK5-brane solution are given by (28) and the gauge field remains vanishing . Therefore the neutral KK5-brane is a purely geometric Taub-NUT solution at least at the leading order in . Indeed, this is nothing but the KK5-brane solution in type II theories.

Gauge KK5-brane

For the gauge NS5-brane solution, the periodic harmonic function (27) does not work as a solution since is not satisfied for the gauge solution. The ansatz (10) and the Bianchi identity indicates . In order to find a solution of co-dimension three associated with the gauge solution, we perform the singular gauge transformation of the solution (22). Then the gauge field becomes

(33)

where is the antisymmetric and anti-self-dual matrix and is written as

(34)

This is the BPST instanton in the singular gauge. The solution is obtained by the famous ’t Hooft ansatz:

(35)

The function satisfies and given by for the -instanton solution. Here are the positions of the instantons. Note that the solution (33) corresponds to . Using this fact, we can perform the smearing procedure for along the line of obtaining (27). This periodic array of the instantons is just the calorons of the Harrington-Shepard type [30]. The harmonic function for the smeared gauge NS5-brane is determined by the Laplace equation whose source is given by the limit of the calorons, namely, the smeared instantons. After the smearing, we find where and is a rescaled size modulus. Then the smeared gauge NS5-brane solution is found to be

(36)

As in the case of the gauge NS5-brane solution, we find that the -fields in the smeared gauge NS5-brane solution becomes trivial. This property is also found in the heterotic monopole solution [29]. We also note that the smeared gauge NS5-brane does not exhibit the H-monopole property. Indeed, we find the modified -flux behaves like and the charge vanishes. This is in contrast to the symmetric and the neutral solutions. We also comment that the solution (36) based on the smeared instanton does not have a finite monopole charge. This is again in contrast to the monopole solution in [29].

Since the -field is trivial in the gauge solution, we can set these components as

(37)

where are constants. The other components do not appear in the solution and they can be set to zero. Now we perform the T-duality transformation of the smeared gauge NS5-brane solution along the -direction. After calculations, we find the following gauge KK5-brane solution:

(38)

The solution seems complicated but one finds that the Yang-Mills gauge field satisfies the anti-monopole equation (31). Notably, the function becomes negative at a finite value of . The solution is ill-defined near the center of the brane. This property is similar to the symmetric and neutral NS5-brane solutions with whose physical interpretation is unclear [29].

3.2 Heterotic -branes

Now we are in a position where the second T-duality transformation is performed on the KK5-brane solutions. We introduce another isometry along the transverse direction to the KK5-brane world-volumes. By performing the second T-duality transformation along the -direction, we obtain the exotic -brane solutions associated with the symmetric, the neutral and the gauge solutions.

Symmetric -brane

The harmonic function in is obtained by the periodic array of the KK5-brane. This is given by

(39)

Taking the compactification radius small , the sum in (39) is approximated by the integral over with the cutoff scale . The result is [15]

(40)

where and is a constant which specifies the region where the solution is valid [31]. The constant diverges in the large cutoff scale limit . Since the harmonic function only depends on , the NS-NS -field does so. The only non-zero component of the -field is determined by the following relations:

(41)

The explicit form of is found to be

(42)

If we employ the coordinate system where is the angular coordinate in the -plane, then is proportional to . This result leads to the fact that the space-time metric, the dilaton and the NS-NS -field in the solution are not single-valued anymore. The logarithmic behaviour of the harmonic function is characteristic to co-dimension two objects. We call the solution (28) on which the harmonic function is replaced by (40) the smeared symmetric KK5-brane solution. Indeed, the heterotic vortex and the domain wall are discussed in [32, 33] where the harmonic function behaves as logarithmic and linear functions.

Since the heterotic KK5-brane solution satisfies the standard embedding condition, this again simplify the heterotic Buscher rule (24). By performing the second T-duality transformation along the -direction, we obtain the following symmetric -brane solution:

(43)

The metric, the dilaton and the NS-NS -field in (43) are the same with the ones in type II theory. However, in heterotic theory, the Yang-Mills gauge field remains non-trivial. The gauge field is calculated as

(44)

We find that (44) satisfies the vortex-like equation in two dimensions:

(45)

where is a complexified adjoint scalar field. The vortex-like equation (45) is obtained by dimensionally reducing the self-duality condition (11) to two dimensions. We note that since the Yang-Mills gauge field contains the explicit angular coordinate in , is not a single-valued function also in the symmetric -brane solution.

The modified spin connection associated with the solution (43) is calculated as