Abstract

We construct supersymmetric dyon solutions based on the ‘t Hooft/Polyakov monopole. We show that these solutions satisfy -symmetry constraints and can therefore be generalized to supersymmetric solutions of type I string theory. After applying a -duality transformation to these solutions, we obtain two -branes connected by a wormhole, embedded in an -brane. We analyze the geometries of each -brane for two cases: one corresponding to a dyon with vanishing spin and the other corresponding to a magnetic monopole with nonvanishing spin. In the case of the vanishing spin, the scalar curvature is finite everywhere. In the case of the nonvanishing spin, we find a frame dragging effect due to the spin. We also find that the scalar curvature diverges along the spin quantization axis as , being the cylindrical, radial coordinate defined with respect to the spin axis. These solutions demonstrate the subtle relationship between the Yang-Mills and gravitational interactions, i.e., gauge/gravity duality.

1. Introduction

In a previous study, we have investigated spin 0 dyons within the context of type I superstring theory in 10 dimensions [1]. Based on the ‘t Hooft/Polyakov monopole, we have constructed dyon solutions which are exact solutions of the non-Abelian Dirac-Born-Infeld action and the Wess-Zumino-like action. After applying a -duality transformation to the solutions, we have obtained solutions corresponding to electrically and magnetically charged wormholes (for additional information about wormholes and their physical properties, please consult the following references [26]) which connect two -branes.

In this study, we extend our previous work to include solutions with nonvanishing spin. Specifically, we have applied supersymmetry transformations to the solutions obtained previously, yielding spin 1/2 and spin 1 dyons. We then show that the solutions also preserve a combined -symmetry and supersymmetry so that they are also solutions of superstring theory. After applying a suitable coordinate/gauge transformation, followed by a -duality transformation, we obtain rotating wormhole solutions which are both magnetically and electrically charged.

We now outline the steps in our analysis. In Section 2, we review dimensional reduction of , supersymmetry to , and then to , supersymmetry. This reduction is carried out with the purpose of showing, explicitly, the connection between dyons in four dimensions and dyons derived from superstrings in ten dimensions. In Section 3, we use the results of Section 2 to reinterpret the spin 0 dyon solutions in four spacetime dimensions [7] as a gauge field dimensionally reduced from ten to six spacetime dimensions. We then apply supersymmetry transformations to the gauge fields, thereby recasting the supersymmetric dyon solutions in four dimensions as a , supersymmetric gauge theory. As a corollary of our analysis, we extend the work of Kastor and Na [8], which applies to supersymmetric magnetic monopoles to include supersymmetric dyons. In Section 4, we show that the solutions obtained in Section 3 preserve combined -symmetry and supersymmetry and are therefore solutions of type IIB superstring theory, which we then recast as solutions of type I superstring theory, residing on an -brane. In Section 5, we apply a -duality transformation to the superstring solutions obtained in Section 4, reducing the theory from to . The result is two rotating dyons of equal but opposite charge, each residing on a curved -brane, connected to one another by a wormhole. Finally, we present numerical and graphical examples, depicting the scalar curvature and frame dragging effect.

Concerning the system of units and sign conventions, we adhere to the same conventions as in our previous work [1]. Specifically, in dimensions, the Levi-Cività symbol is . Greek letters denote spacetime indices, i.e., 0, 1, 2, and 3. Uncapitalized Roman letters denote either the spatial indices 1, 2, 3 (alternatively, 3-space coordinates are denoted as , and , where , , and ) or the indices of the generators of the gauge group. Capitalized Roman indices denote indices of ten spacetime dimensions, i.e., 0, 1, 2, ... 9. The signature of the metric, , is mostly positive. The gamma matricies satisfy the following relations: . Also, we employ the Lorentz-Heaviside units of electromagnetism so that . As a consequence, the Dirac quantization condition is , () being the electric (magnetic) charge and being an integer.

2. Dimensional Reduction of , Supersymmetry

In this section, we describe the dimensional reduction of the , supersymmetric Yang Mills theory, first to the , theory, then to the , theory. This reduction is performed, specifically, with the purpose of demonstrating how dyons in can be naturally described as evolving from this dimensional reduction process.

We begin with the , supersymmetric Lagrangian density [9] where

The quantity is the Yang-Mills coupling constant in ten dimensions (note that , where is the Yang-Mills coupling constant in four dimensions and is the string coupling constant. See Appendix B of reference [1]), and are the structure constants of the gauge group. Here, the gaugino field, , is the supersymmetric partner of the gauge field. The action is invariant under the supersymmetric transformations where . The gaugino field and supersymmetric parameter are the 32 component Majorana spinors with positive chirality, i.e., , where is the charge conjugation matrix and , where the chirality matrix (the chirality matrix in dimensions is , where and for the Minkowski signature and for the Euclidean signature).

Using Noether’s theorem, we obtain the supercurrent by varying the Lagrangian density with respect to the fields [10], where is a function whose divergence is the variation of the Lagrangian density under supersymmetry transformations, i.e., . The supercharges, , are obtained from the supercurrents

The supercharges, which are the generators of supersymmetry transformations, can be obtained from equation (4). Alternatively, we can compare equation (6) directly to equations (3a) and (3b) and obtain

In deriving equation (7), we have used the equal-time, canonical anticommutation and commutation relations

The field is the canonical momentum conjugate to () and is the canonical momentum conjugate to .

We now calculate the anticommutator . This calculation, though similar to that of Witten and Olive [11], differs in that their calculation is based on monopole solutions resulting from the Higgs field embedded in , supersymmetry, whereas this calculation is based on the sequential, dimensional reduction from , supersymmetry to , supersymmetry and finally to , supersymmetry. Our reason for presenting the calculation is to demonstrate the relationship between dyons in , supersymmetry and superstrings in the type I theory. The anticommutator is evaluated as

In evaluating equation (9), it is helpful to organize the terms as follows: in the first group, all terms where and assume different values; in the second group, both are contracted with, resulting in terms with no matrices; and in the third group, one of is contracted with one of, resulting in terms with two matrices. In the first group, terms which contain or vanish because . Each of the remaining terms can be expressed as a divergence. Such terms are typically assumed to vanish sufficiently fast at the boundary so that these terms make no contribution; however, these terms will become relevant when we consider dyon solutions and their associated central charges (see equation (21)). The second group evaluates , the energy, i.e.,

In obtaining this result, we have used the fact that and assumed that all surface integrals vanish. The third group comprises of terms which contain the product . If both , the term vanishes by symmetry arguments and properties of the gamma matricies. The only terms which are nonvanishing from this group are those that contain , . Each of these terms evaluates to

Thus,

In preparation for constructing dyon solutions in four dimensions, we constrain the Majorana spinors and in , also, to be states of positive chirality in , i.e., (the chirality matrix in dimensions is , where and for the Minkowski signature and for the Euclidean signature). We next reexpress the spinors in terms of projections, i.e., where

Here, is the chirality matrix for dimensions 0, 1, 4, 5, 6, and 7:

We note, in particular, that the in the -basis [9] are where and are the arbitrary complex constants.

With foresight, we make the following assumptions: (1)All potential functions , i.e., depend only on the three space coordinates and are time independent(2)(3) and may or may not commute(4) asymptotically approaches a nonvanishing vacuum state, while may vanish asymptotically, i.e, for and nonvanishing

The reduction from ten to six dimensions is trivial. Since through vanish, only the gamma matrices through appear in the supercharges. In reducing from ten to six dimensions, the ten dimensional gamma matrices may be represented as a direct product of six dimensional gamma matrices and a four dimensional identity matrix, i.e., , where . The gamma matrices act on the first three component spinors of , while the four dimensional identity matrix acts on the remaining two. The only significant consequence of the dimensional reduction is that the spinor is replaced by two spinors: and correspondingly the supercharge to two supercharges:

Thus, dimensional reduction results in a transitioning from , supersymmetry to , supersymmetry with both supercharges being eigenstates of positive chirality in six dimensions, i.e., . The central charges are derived from two groups of terms in the anticommutator, the first group and the third group. A typical nonvanishing boundary term from the first group is derived from where . Boundary terms derived from involve a curl integrated over a surface at infinity. Such terms, which can be expressed as a line integral, vanish asymptotically if and approach zero faster than as . This is the case for monopole or dyon solutions which asymptotically approach zero as . The remaining terms can be expressed as a divergence which becomes a surface integral at the boundary. If approach zero as as as is the case for monopole and dyon solutions, the surface integral is nonvanishing. Specifically, the contribution from the first group of terms is where the magnetic charge is obtained from the relationship

We have used the fact that the asymptotic behavior of is given by equation (17a) and that the magnetic field is given by

In obtaining equation (22), we have used

In equation (24), the second term to the right of the equal sign vanishes by virtue of the equations of motion, specifically that the divergence of the magnetic field vanishes.

The contribution to the central charges from the third group of terms corresponds to the momentum in the and directions. The relevant terms from equation (9) where . The portion of the integral over the six dimensional space yields the volume of the six dimensional space which we normalize to one. The remaining part of the integral can be expressed as a divergence which by virtue of equation (17b) yields a nonvanishing surface contribution. Substituting the following expression into equation (25) and using the fact the last term in equation (26) which vanishes by virtue of the equations of motion, i.e., the divergence of the electric field vanishes, we obtain the additional contributions to the central charges:

Here, we have used the fact that the electric charge is obtained: where . Substituting equation (21) and equation (28) into equation (12), we obtain for (we use Fraktur font to denote “+” or “-”). Simplifying the terms involving central charges, we obtain

The charge and the angle are defined by (because of our choice of metric, i.e., , electromagnetic duality implies and )

Since , we can simplify equation (30).

Before reducing from six to four dimensions, we note that

In the rest frame of the system, i.e., and , being the rest energy of the system, we can show

Alternatively, we define

We can show by direct substitution of equation (35) into equation (34) that for . The reduction from six to four dimensions is relatively straightforward. In reducing from ten to six to four dimensions, the requisite ten dimensional gamma matrices are represented.

Here, and are the four dimensional gamma matrices, and are Pauli matrices, and and are the identity matrices in two and four dimensions, respectively. Finally, the reduction from six dimensions to four dimensions requires that and from equation (37) be substituted into equation (36). In reducing from , to , supersymmetry, each supercharge is replaced by two supercharges .

The supersymmetry algebra, equation (36), obtained from dimensional reduction of , supersymmetry, differs from that of Witten and Olive [11] which is based on , supersymmetry. The most obvious distinction is that there are two sets of supercharges, i.e., . In our construction of dyons with spin in Section 3, the second set of supercharges generates spin 1 dyon solutions in addition to spin 1/2 and spin 0 solutions. The other distinction is derived from the fact that the components of the vector potential and in our analysis replace the components of the Higgs field, in Witten and Olive’s analysis. Witten and Olive removes one of these components of the Higgs field by performing a chiral rotation, which would, in a certain sense, be equivalent to setting in our analysis. In our subsequent analysis of dyons with spin, Section 3, we do not eliminate one of and by a coordinate rotation, analogous to the chiral rotation. The reason is that our analysis is complicated because and , in general, do not commute. Instead, we are able to set , which is a direct consequence of the dyon solutions being BPS states.

3. Dyons with Spin

In this section, we review the construction of dyons with spin. One method of incorporating spin is to construct dyon solutions from the supersymmetric extension of the Yang-Mills-Higgs action. This methodology shows, implicitly, the relationship between dyons with spin in and superstrings. We begin the analysis with a discussion of the ‘t Hooft/Polyakov monopole which is derived from the Yang-Mills-Higgs Lagrangian density. ‘t Hooft [12] and Polyakov [13] have shown that within the context of the spontaneously broken, the Yang-Mills gauge theory magnetic monopole solutions of finite mass must necessarily exist and furthermore possess an internal structure. These solutions, which possess zero spin, are derived from the Yang-Mills-Higgs Lagrangian where

The Higgs field is scalar transforming according to the adjoint representation of the gauge group, and consequently, its covariant derivative is

The quantity is the Yang-Mills coupling constant in four dimensions, and are the structure constants of the gauge group. For our purposes, we assume that the gauge group is (or a group which contains as a subgroup). In addition, we require that the potential vanishes so that the magnetic monopole solutions are BPS states, which are solvable in closed form [1, 7, 8, 14]. Straightforwardly, one can also show that these solutions can be modified to be electrically charged as well as magnetically charged. As a consequence of the solutions being BPS states, one can show that the electric and magnetic components of the fields are related to . where

The electric and magnetic fields are obtained from and :

Here,

See equation (50) below.

In equations (41a) and (41b), the electric and magnetic charges are where . For these solutions , see equation (31).

From the perspective of six dimensions, the function can be reinterpreted as gauge fields

This follows because the Higgs field does not depend on the coordinates of dimensions four and five so that under gauge transformations, the components and transform in the same manner as . In six dimensions, the dyon is described in terms of the potential function where is vacuum expectation value of in the asymptotic limit of large (see equation (17a)). The magnetic charge of the dyon is , for an integer, which is the Higgs field winding number. ,, and constitute a representation of the algebra. The quantities ,, and are the spherical polar coordinates in three dimensions (in the transformation to spherical polar coordinates, we have chosen the -axis, rather that the -axis, to be the azimuthal axis. The motivation for this choice is to provide consistency with our choice of matrices. Specifically, spin states are chosen to be eigenvalues of the spin operator . See equation (69)). The elements ,, and are related to : where the are generators of an subalgebra of (the gauge group is relevant for our discussion of superstrings in Section 4).

Here, the are unit vectors in the , , and directions, respectively.

The Higgs field is

Using equations (48a), (48b), and (48c), we can express the in spherical polar coordinates

The solutions and are obtained as in reference [14]. where the dimensionless variable is related to the radial coordinate :

The quantity characterizes the size of the dyon, i.e., the region of space in which it exhibits internal structure: where the mass of the gluon, resulting from spontaneous symmetry breaking, is

In addition, the mass of the dyon is related to the mass of a gluon

For our purposes, we also require that solutions be invariant under transformations, weak/strong duality, so that we include in the Lagrangian density Witten’s term [15].

This term contributes only a surface term to the action and therefore does not affect the classical equations of motion. In the monopole sector of the theory, however, the term does have a nontrivial effect in that it shifts the allowed values of the electric charge [7]. The electric charge, , is given as

where is an integer.

The dyon solutions, equation (47), also satisfy the equations of motion derived from the supersymmetric Lagrangian density, equation (1) with a gaugino field set equal to zero. The solutions, equations (46a), (46b), and (47), satisfy the assumptions placed on the supersymmetric solutions discussed in Section 2, with the additional property that the solutions are also BPS states.

In order to construct dyon solutions with spin, we begin with the , supersymmetric Yang-Mills theory, obtained from the dimensional reduction of the , theory, presented in Section 2. The , theory comprises two supercharges of positive chirality in six dimensions, . The theory is invariant under supersymmetry transformations generated by supercharges (the gamma matrices in are represented as , where are six dimensional gamma matrices. See Section 2).

Supersymmetry is broken by a part of which is an eigenstate of with eigenvalue -1, i.e., . Substituting equations (41a) and (41b) and equations (42a) and (42b) into equation (59a) and equation (59b), we obtain

As a characteristic of BPS states, half of the supersymmetries are broken, i.e., for , and half are unbroken, i.e., for . The dimensional reduction to is trivial. The six dimensional gamma matrices are replaced by those given in equation (37). It is notable that in our analysis, there are two broken supercharges, a result which differs from those of others. See Harvey, for example, [7]. The difference is a consequence of the fact these other analyses begin with the , supersymmetric Yang-Mills-Higgs theory. In contrast, we begin with the , supersymmetric Yang-Mills theory with only gauge fields, and through dimensional reduction, we obtain a second supercharge. For these dyon solutions, the gaugino field has been explicitly set to zero. The broken supersymmetry transformations, which are generated by the two supercharges, each result in a nonvanishing contribution to the fermion (gaugino) field. Furthermore, these transformations which break supersymmetry do not change the energy of the system so that these nonvanishing fermionic “zero” modes can be considered as deformations of the dyon background which keep the energy of the dyon fixed [7]. Since each of these fermionic modes carries spin 1/2, it is possible to construct dyon states, i.e., deformed dyon backgrounds, with either spin 1/2 or spin 1.

To first order the supersymmetry transformation, equation (59a) leaves the potential function, , unchanged. In reference [8], Kastor and Na have shown that, because of the nonlinearity inherent in the supersymmetry transformations, there are nonvanishing contributions to when higher order corrections to the supersymmetry transformations are taken into account. Their methodology utilizes an iterative procedure to calculate higher order corrections to the supersymmetry transformations. They perform their analysis using magnetic monopole solutions, i.e., dyons with vanishing electric charge or . Since the changes resulting from the inclusion of electric charge are not immediately obvious, we review their methodology when electric charge is included in the analysis.

They begin with an iterative expansion of the supersymmetry transformations where represents both bosonic and fermionic fields after the transformation and the bosonic fields before the transformation. This series can be interpreted as follows: the second term to the right of the second equal sign is obtained directly from equations (60a), (60b), and (60c). The third term is obtained by substituting the second term into equations (60a), (60b), and (60c). The series terminates after the fourth term because of the Grassman nature of . Substituting equation (60a) and equation (60b) in equation (61), we obtain

Following Kastor and Na [8], we evaluate the matrix elements in equations (62a), (62b), (62c), (62d), and (62e). We first quantize the fermionic zero modes. This involves replacing the complex constants, in , equation (16), by the operators and and then integrating the anticommutator of the fermionic zero modes, equation (62e),

Using equations (8a), (8b), and (16), we obtain where we have used the fact that the mass of the dyon is

Applying equation (64a), (64b), and (64c) in the evaluation of equation (62a), (62b), (62c), (62d), and (62e), we obtain

Here, the electric dipole moment, due to the spinning magnetic charge, is and the magnetic dipole moment, due spinning electric charge, is

The spin operator is defined in terms of the Lorentz generators of the rotation group, i.e.,

Because the supersymmetric spinors, , are eigenstates of (with eigenvalue 1/2), then

The complex constants, in , equation (16), are arbitrary, and consequently, different sets of dyon solutions are obtained when quantizing the fermionic modes. Specifically, choosing both and results in spin 0 dyon solutions. Choosing either or yields two sets of spin dyons with . Alternatively, interchanging with yields dyon solutions with . Setting both constants not equal to zero, simultaneously, we obtain spin 1 dyon solutions where . Considering all of these dyon solutions in total, we can evaluate and , explicitly, where for the spin 0 dyon, , for the two spin 1/2 dyons , and for the spin 1 dyon .

The potential functions and are amenable to straightforward interpretation. Given that then and in the limit of large approach the classical electric and magnetic dipole potentials. The factor of 2 preceding each dipole moment is the gyromagnetic (“gyroelectric”) ratio (Kastor and Na have previously obtained the gyroelectric ratio in their analysis of magnetic monopoles within the super Yang-Mills theory [8]).

It is apparent that the electric dipole field derived from the potential is equal but opposite to the field derived from the potential . Not as obvious is the fact that the magnetic dipole field derived from the potential is also equal but opposite to that derived from the potential . This relationship follows directly from the fact that . A similar situation occurs in the Maxwell theory in which the magnetic field derived from the vector, dipole potential is, except for a minus sign, identical in form to the electric field derived from the scalar, dipole potential.

4. Dyons, Type IIB, and Type I Superstring Theory

The purpose of this section is to generalize the results of Section 3 to the superstring theory. As we show, the solutions obtained in Section 3 correspond, in superstring theory, to -branes, which are embedded in an -brane compactified on a type IIB torus [16].

First, the arena for discussing the dyon solutions of Section 3 is the -brane. The -brane is a hypersurface propagating in dimensions [17]. The underlying theory is based on a single copy of the Majorana fermions which in superstring theory reduces to the two Majorana-Weyl fermions. The defining characteristic of these fermions, , is that they satisfy a constraint equation, i.e., -symmetry:

This is precisely the constraint placed on the spinors, equation (14), defining the dyon solutions in Section 3. Consequently, from the perspective of , the dyon solutions obtained previously live, in fact, on an -brane.

The application of supersymmetry to string theory is fraught with significant, nontrivial technical issues. First, in the case of superstring theory, the bosonic part of the action based on the Lagrangian density (equation (1)) is replaced by the Dp-brane action which is given by the non-Abelian Dirac-Born-Infeld plus Wess-Zumino-like actions (note: the antisymmetric tensor , where the only nonvanishing R-R potential is , which is a constant background). where

Here, is the physical tension of the Dp-brane, is its R-R charge, and is the pullback of the background metric . STr indicates a symmetric trace for terms involving products of the generators of the gauge group (see reference [1] and references therein). In equation (74), it is known that after expanding the square root as a power series in , computation of the symmetric trace yields ambiguous results in terms of order [18, 19].

The fermionic action based on the Lagrangian density (equation (1)) is replaced by the fermionic Dp-bane action:

where vanishes since spacetime background is flat for the cases we are considering. Here, and . For type IIB D(2n+1)-branes, and for type IIA D(2n)-branes, where the for the type IIA theory and the for the IIB theory differ by a factor of -1 in references [17, 20]. The reason is derived from the fact that , denoted in [20], is defined with indices raised, whereas in [17], is defined with indices lowered. We adopt the same convention for , as [20].

Since our interest is the type I , our focus will be the type IIB theory to which the type I is related. For the type IIB theory, is a 64 component double spinor:

Each is the 32 component Majorana-Weyl spinor of positive chirality, i.e., . In equation (80), the Pauli matrices act on the spinorial index in . For the Abelian gauge theory, the fermionic action is invariant under -symmetry which acts on fermions:

The action , equation (76), corresponding to the fermionic sector of the theory, strictly speaking, only applies to abelian gauge theories. The extension to non-Abelian gauge theories is plagued with problems similar to those occurring in the bosonic action. Specifically, expansion of the square root in terms of the gauge fields yields products of generators of the algebra whose symmetric trace is known to result in inconsistencies at order [18]. At first, we ignore these problems and assume that the action applies to the non-Abelian theory, in which case corresponds to the gauge covariant derivative of the applicable non-Abelian gauge theory.

We now show that the BPS solutions given in Section 3 are exact solutions of type I superstring theory. Since the type I theory is derived from the type IIB theory, we, initially, focus on the type IIB theory. In [1], we have shown that the BPS solutions presented in Section 3 are also solutions of the equations of motion derived from the non-Abelian DBI action, equation (73), and are therefore solutions of the type IIB theory with the fermionic degrees of freedom equal to zero. In general, these bosonic solutions are not supersymmetric. In [17], Simón has shown that whether such a set of bosonic solutions preserves supersymmetry is equivalent to determining if there exist supersymmetry transformations : which preserve the bosonic nature of these solutions, i.e., remains zero, and furthermore, the bosonic solutions remain unchanged to first order. To satisfy the condition that , the combined - and supersymmetry transformations must vanish, i.e.,

Here, the -symmetry transformation is

Simón has shown that this condition is satisfied when

Simón has solved equation (88) for a supersymmetric -brane configuration, i.e., , with the Abelian gauge field residing on the brane. We now show how the solutions obtained by Simón can be straightforwardly extended to the BPS solutions with non-Abelian gauge fields, given in Section 3.

For , equation (78) becomes

Substituting equation (89) into equation (88) and rearranging terms, we obtain where is the identity matrix in two dimensions. In transitioning from equation (89) to equation (90), we have imposed the projection constraints

The matrix is defined as

Substituting equations (41a), (41b), and equation (51) into the square root term in equation (90), we obtain (see the appendix for details) where