# Supersymmetric Dyons, Superstrings, and Rotating Wormholes

Authors: E. Olszewski
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

We construct supersymmetric dyon solutions based on the ‘t Hooft/Polyakov monopole. We show that these solutions satisfyκ-symmetry constraints and can therefore be generalized to supersymmetric solutions of type I SO32 string theory. After applying a T-duality transformation to these solutions, we obtain two D3-branes connected by a wormhole, embedded in an M5-brane. We analyze the geometries of each D3-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 1/ρ2, ρ 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.


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## Body

## 1. Introduction

In a previous study, we have investigated spin 0 dyons within the context of type ISO32 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 T-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 [2–6]) which connect two D3-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 combinedκ-symmetry and supersymmetry so that they are also solutions of superstring theory. After applying a suitable coordinate/gauge transformation, followed by a T-duality transformation, we obtain rotating wormhole solutions which are both magnetically and electrically charged.We now outline the steps in our analysis. In Section2, we review dimensional reduction of D=10, N=1 supersymmetry to D=6, N=2 and then to D=4, N=4 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 D=6, N=2 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 κ-symmetry and supersymmetry and are therefore solutions of type IIB superstring theory, which we then recast as solutions of type I SO32 superstring theory, residing on an M5-brane. In Section 5, we apply a T-duality transformation to the superstring solutions obtained in Section 4, reducing the theory from D=1+4 to D=1+3. The result is two rotating dyons of equal but opposite charge, each residing on a curved D3-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 D dimensions, the Levi-Cività symbol is ε012⋯D=1. 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 x,y, and z, where x≡x1, y≡x2, and z≡x3) 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, ηMN, is mostly positive. The gamma matricies satisfy the following relations: ΓM†=ΓM. Also, we employ the Lorentz-Heaviside units of electromagnetism so that c=ℏ=ε0=μ0=1. As a consequence, the Dirac quantization condition is gegm=4πnm/2, ge (gm) being the electric (magnetic) charge and nm being an integer.


## 2. Dimensional Reduction ofD=10, N=1 Supersymmetry

In this section, we describe the dimensional reduction of theD=10, N=1 supersymmetric Yang Mills theory, first to the D=6, N=2 theory, then to the D=4, N=4 theory. This reduction is performed, specifically, with the purpose of demonstrating how dyons in D=4 can be naturally described as evolving from this dimensional reduction process.We begin with theD=10, N=1 supersymmetric Lagrangian density [9]
(1)L=−14FMNaFaMN−i12λa¯ΓMDMλa,where
(2)Fμνa=∂μAνa−∂νAμa−igD9fabcAμb,Aνc.The quantitygD9 is the Yang-Mills coupling constant in ten dimensions (note that gD92=gD322π6α′3, where gD3 is the Yang-Mills coupling constant in four dimensions and α′ is the string coupling constant. See Appendix B of reference [1]), and fabc are the structure constants of the gauge group. Here, the gaugino field, λ, is the supersymmetric partner of the gauge field. The action is invariant under the supersymmetric transformations
(3a)δAMa=−iζ¯ΓMλa,(3b)δλa=12FMNaΓMNζ,where ΓMN=ΓMΓN−ΓNΓM. The gaugino field λa and supersymmetric parameter ζ are the 32 component Majorana spinors with positive chirality, i.e., λ¯=λaTC, where C is the charge conjugation matrix and Γ10λa=+1λa, where the chirality matrix Γ10=i−4ε01⋯9Γ0Γ1⋯Γ9 (the chirality matrix in D dimensions is ΓD≡Kε01⋯D−1Γ0Γ1⋯ΓD−1, where D=2k+2 and K=i−k for the Minkowski signature and K=i−k+1 for the Euclidean signature).Using Noether’s theorem, we obtain the supercurrent by varying the Lagrangian density with respect to the fieldsXX=AMaorλa[10],
(4)ζJM+ζ†J†M≡∑XδLδ∂MX−KM,where KM is a function whose divergence is the variation of the Lagrangian density under supersymmetry transformations, i.e., ∂MKM=δL. The supercharges, Qα, are obtained from the supercurrents
(5a)Qα=∫d9xJα0,(5b)Qα†=∫d9xJα†0.The supercharges, which are the generators of supersymmetry transformations,(6)ζ†Q†+ζQ,X=δX,can be obtained from equation (4). Alternatively, we can compare equation (6) directly to equations (3a) and (3b) and obtain
(7)Qα=−12∫dx9FMNaλa†Γ0ΓMNα.In deriving equation (7), we have used the equal-time, canonical anticommutation and commutation relations
(8a)λαax⟶,t,λβ†by⟶,t=δabδαβδ9x⟶−y⟶,(8b)AMax⟶,t,ENby⟶,t=−ηMNδabδ9x⟶−y⟶.The fieldENb is the canonical momentum conjugate to ANb (ENb=F0Nb) and iλ¯αaΓ0 is the canonical momentum conjugate to λαa.We now calculate the anticommutatorQα,Qβ†. 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 D=4, N=2 supersymmetry, whereas this calculation is based on the sequential, dimensional reduction from D=10, N=1 supersymmetry to D=6, N=2 supersymmetry and finally to D=4, N=4 supersymmetry. Our reason for presenting the calculation is to demonstrate the relationship between dyons in D=4, N=4 supersymmetry and superstrings in the type I SO32 theory. The anticommutator is evaluated as
(9)zeta2ζ1†Q,Q†=ζ2Q,ζ1†Q†==ζ2ζ1†−12222∫dx9FMNaFKLaΓ0ΓMΓNΓK†ΓL†.In evaluating equation (9), it is helpful to organize the terms as follows: in the first group, all terms where M,N and K,L assume different values; in the second group, bothM,N are contracted withK,L, resulting in terms with noΓ matrices; and in the third group, one ofM,N is contracted with one ofK,L, resulting in terms with two Γ matrices. In the first group, terms which contain Γ0 or Γ0† vanish because Γ0†=−Γ0. 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 P0, the energy, i.e.,
(10)P0=−∫dx9F0aMF0Ma−14η00FMNaFaMN.In obtaining this result, we have used the fact thatΓ0†=−Γ0 and assumed that all surface integrals vanish. The third group comprises of terms which contain the product ΓMΓN. If both M,N≠0, 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 Γ0ΓNorΓ0†ΓN, N≠0. Each of these terms evaluates to
(11)PN=∫dx9F0aMFNMa.Thus,(12)Q,Q†=P0+Γ0ΓNPN.In preparation for constructing dyon solutions in four dimensions, we constrain the Majorana spinorsζ and λa in D=10, also, to be states of positive chirality in D=6, i.e., Γ6χ=+1χ,χ=ζ,λa (the chirality matrix in D dimensions is ΓD≡Kε01⋯D−1Γ0Γ1⋯ΓD−1, where D=2k+2 and K=i−k for the Minkowski signature and K=i−k+1 for the Euclidean signature). We next reexpress the spinors χ in terms of projections, i.e.,
(13a)χ=χ++χ−,(13b)χ+=χ+,+1+χ+,−1,(13c)χ−=χ−,+1+χ−,−1,where
(14)χ+,±1=1+Γ1021+Γ621+Γ′621±Γ0Γ42χ,χ−,±1=1+Γ1021+Γ621−Γ′621±Γ0Γ42χ.Here,Γ′6 is the chirality matrix for dimensions 0, 1, 4, 5, 6, and 7:
(15)Γ′6=1i2ϵ014567Γ0Γ1Γ4Γ5Γ6Γ7.We note, in particular, that theζ±,±1 in the s-basis [9] are
(16)ζ+,±1=a+,±1∗1010101010±0110011010+a+,±11001010101±0101100101,ζ−,±1=a−,±1∗1010100101±0110010101+a−,±11001011010±0101101010,where a+,±1 and a−,±1 are the arbitrary complex constants.With foresight, we make the following assumptions:(1)
All potential functionsAMa=AMaxi, i.e., depend only on the three space coordinates and are time independent(2)
A6a=A7a=A8a=A9a=0(3)
A4 and A5 may or may not commute(4)
A5a asymptotically approaches a nonvanishing vacuum state, while A4a may vanish asymptotically, i.e,
(17a)limr⟶∞A4aA4a=v2cos2ψ,(17b)limr⟶∞A5aA5a=v2sin2ψ,for 0<ψ≤π/2 and v nonvanishingThe reduction from ten to six dimensions is trivial. SinceA6a through A9a vanish, only the gamma matrices Γ0 through Γ5 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., ΓN×I4, where N=0,⋯,5. The gamma matrices act on the first three component spinors of χ, while the four dimensional identity matrix acts on the remaining two. The only significant consequence of the dimensional reduction is that the spinor χ is replaced by two spinors:
(18a)χ+=χ+,+1+χ+,−1,(18b)χ−=χ−,+1+χ−,−1,and correspondingly the supercharge Q to two supercharges:
(19a)Q+=Q+,+1+Q+,−1,(19b)Q−=Q−,+1+Q−,−1.Thus, dimensional reduction results in a transitioning fromD=10, N=1 supersymmetry to D=6, N=2 supersymmetry with both supercharges being eigenstates of positive chirality in six dimensions, i.e., Γ6Q±=+Q±. 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
(20)FijaFkNaΓiΓjΓk†ΓN†,where N=4,5. Boundary terms derived from FijaF45a involve a curl integrated over a surface at infinity. Such terms, which can be expressed as a line integral, vanish asymptotically if F4ia and F5ia approach zero faster than 1/r as r⟶∞. This is the case for monopole or dyon solutions which asymptotically approach zero as 1/r2. The remaining terms can be expressed as a divergence which becomes a surface integral at the boundary. If Fija approach zero as 1/r2 as r⟶∞ as is the case for monopole and dyon solutions, the surface integral is nonvanishing. Specifically, the contribution from the first group of terms is
(21)Γ1Γ2Γ3Γ4gmvcosψ+Γ1Γ2Γ3Γ5gmvsinψ,where the magnetic charge gm is obtained from the relationship
(22)gmvsinψ=∫S∞BaiA5adSi.We have used the fact that the asymptotic behavior ofA4a is given by equation (17a) and that the magnetic field is given by
(23)Bak=ϵkij2!Fija.In obtaining equation (22), we have used
(24)ΓiΓjΓkΓ4∂kFijaA4a=ΓiΓjΓkΓ5FijaFk4a+ϵkij2!DkFijaA4a.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 thex4 and x5 directions. The relevant terms from equation (9)
(25)∫dx6∫dx3F0KaFKNaΓ0Γ0ΓNΓ0†,where N=4,5. 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
(26)Γ0ΓN∂kF0KaANa=Γ0ΓNF0KaFKNa+DkF0aKANainto 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:
(27)Γ0Γ4gevcosψ+Γ0Γ5gevsinψ.Here, we have used the fact that the electric charge is obtained:(28)gevcosψ=∫S∞EaiA4adSi,where Eai=Eia=Fi0a. Substituting equation (21) and equation (28) into equation (12), we obtain
(29)Qa,Qb†=δabP0+Γ0ΓiPi+Γ0Γ4gevcosψ+Γ0Γ5gevsinψ+Γ1Γ2Γ3Γ4gmvcosψ+Γ1Γ2Γ3Γ5gmvsinψ,for a,b=+,− (we use Fraktur font to denote “+” or “-”). Simplifying the terms involving central charges, we obtain
(30)Qa,Qb†=δabP0+Γ0ΓiPi+gvexpiΓ4ψ′Γ0Γ4cosψ+iΓ4sinψΓ0Γ4Γ6Γ6Γ0Γ4.The chargeg and the angle ψ′ are defined by (because of our choice of metric, i.e., η00=−1, electromagnetic duality implies ∗Ba⟶Ea and ∗Ea⟶−Ba)
(31)g=gm2+ge2,tanψ′=gmge.SinceΓ6χ=1χ, we can simplify equation (30).
(32)Qa,Qb†=δabP0+Γ0ΓiPi+gvexpiΓ4ψ′+ψΓ0Γ4Γ0Γ4.Before reducing from six to four dimensions, we note that(33)Qa,±1=1±Γ0Γ42Qa.In the rest frame of the system, i.e.,Pi=0 and P0=M, M being the rest energy of the system, we can show
(34)Qa,+1,Qb,+1†=δabM+gvexpiΓ4ψ′+ψ,Qa,−1,Qb,−1†=δabM−gvexp−iΓ4ψ′+ψ,Qa,+1,Qb,−1†=0.Alternatively, we define(35)Qa1=1+Γ42Qa,+1,Qa2=1−Γ42Qa,−1.We can show by direct substitution of equation (35) into equation (34) that
(36)Qai,Qbj†=δabδijM+Γ4gvexp−iΓ4ψ′+ψ,for i,j=1,2. 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.
(37)Γμ=γμ×I2×I4,Γ4=γ5×I2×I4,Γ4=γ5×σ1×I4,Γ5=γ5×σ2×I4.Here,γμ and γ5 are the four dimensional gamma matrices, σ1 and σ2 are Pauli matrices, and I2 and I4 are the identity matrices in two and four dimensions, respectively. Finally, the reduction from six dimensions to four dimensions requires that Γ4 and Γ4 from equation (37) be substituted into equation (36). In reducing from D=6, N=2 to D=4, N=4 supersymmetry, each supercharge Qa is replaced by two supercharges Qa,±1.The supersymmetry algebra, equation (36), obtained from dimensional reduction of D=10, N=1 supersymmetry, differs from that of Witten and Olive [11] which is based on D=4, N=2 supersymmetry. The most obvious distinction is that there are two sets of supercharges, i.e., a=+,−. 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 A4a and A5a 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 ψ=0 in our analysis. In our subsequent analysis of dyons with spin, Section 3, we do not eliminate one of A4a and A5a by a coordinate rotation, analogous to the chiral rotation. The reason is that our analysis is complicated because A4a and A5a, in general, do not commute. Instead, we are able to set ψ′=ψ, 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 theD=6 supersymmetric extension of the Yang-Mills-Higgs action. This methodology shows, implicitly, the relationship between dyons with spin in D=4 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 SO3 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
(38)L=−14FμνaFμνa+12DμΦaDμΦa−VΦaΦa,where
(39)Fμνa=∂μAνa−∂νAμa−igD3fabcAμb,Aνc.The Higgs fieldΦa is scalar transforming according to the adjoint representation of the gauge group, and consequently, its covariant derivative is
(40)DμΦa=∂μΦa−igD3fabcAμb,Φc.The quantitygD3 is the Yang-Mills coupling constant in four dimensions, and fabc are the structure constants of the gauge group. For our purposes, we assume that the gauge group is SU2 (or a group which contains SU2 as a subgroup). In addition, we require that the potential VΦaΦa 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 Φa.
(41a)Eia=cosψDiΦa,(41b)Bia=sinψDiΦa,where
(42a)Eia=Fi0a,(42b)Bia=εijkFjka.The electric and magnetic fields are obtained fromEia and Bia:
(43a)Ei=EiaΦav,(43b)Bi=BiaΦav.Here,(44)v2=limr⟶∞ΦaΦa.See equation (50) below.In equations (41a) and (41b), the electric qe and magnetic qm charges are
(45a)qe=qcosψ,(45b)qm=qsinψ,where q=qe2+qm2. For these solutions ψ′=ψ, see equation (31).From the perspective of six dimensions, the functionΦa can be reinterpreted as gauge fields
(46a)A4a=Φacosψ,(46b)A5a=Φasinψ.This follows because the Higgs fieldΦa does not depend on the coordinates of dimensions four and five so that under gauge transformations, the components A4a and A5a transform in the same manner as Φa. In six dimensions, the dyon is described in terms of the potential function
(47)A=Aμdxμ+A4dx4+A5dx5=cosψvQrTrdt+WrgD3Tθsinθnmdϕ−Tϕdθ+cosψvQrTrdx4+sinψvQrTrdx5,where v is vacuum expectation value of Φa in the asymptotic limit of large r (see equation (17a)). The magnetic charge of the dyon is gm=4πnm/gD3, for nm an integer, which is the Higgs field winding number. Tr,Tθ, and Tϕ constitute a representation of the SU2 algebra. The quantities r,θ, and ϕ are the spherical polar coordinates in three dimensions (in the transformation to spherical polar coordinates, we have chosen the x-axis, rather that the z-axis, to be the azimuthal axis. The motivation for this choice is to provide consistency with our choice of Γ matrices. Specifically, spin states are chosen to be eigenvalues of the spin operator Sx. See equation (69)). The elements Tr,Tθ, and Tϕ are related to Ta,a=1,2,3:
(48a)Tr≡T·er=Taera=Tysinθcosnmϕ+Tzsinθsinnmϕ+Txcosθ,(48b)Tθ≡T·eθ=Taeθa=Tycosθcosnmϕ+Tzcosθsinnmϕ−Txsinθ,(48c)Tϕ≡T·eϕ=Taeϕa=−Tysinnmϕ+Tzcosnmϕ,where the Ta are generators of an SU2 subalgebra of SO32 (the gauge group SO32 is relevant for our discussion of superstrings in Section 4).
(49a)er=erae^xa=cosθe^x1+sinθcosnmϕe^x2+sinθsinnmϕe^x3,(49b)eθ=eθae^xa=−sinθe^x1+cosθcosnmϕe^x2+cosθsinnmϕe^x3,(49c)eϕ=eϕae^xa=−sinnmϕe^x2+cosnmϕe^x3.Here, thee^xa,a=1,2,3 are unit vectors in the x, y, and z directions, respectively.The Higgs field is(50)ΦaTa=vQreraTa=vQrTr.Using equations (48a), (48b), and (48c), we can express the DΦaTa in spherical polar coordinates
(51)DrΦaTa=vQr′Tr,DθΦaTa=v1−WrQrTθ,DϕΦaTa=v1−WrQrnmsinθTϕ.The solutionsWr and Qr are obtained as in reference [14].
(52a)Wr=wu=1−usinhu,(52b)Qr=qu=cothu−1u,where the dimensionless variable u is related to the radial coordinate r:
(53)u=rLdyon.The quantityLdyon characterizes the size of the dyon, i.e., the region of space in which it exhibits internal structure:
(54)Ldyon=1sinψMgluon,where the mass of the gluon, resulting from spontaneous symmetry breaking, is
(55)Mgluon=gD3v.In addition, the mass of the dyon is related to the mass of a gluon(56)Mdyon=gv=ggD3Mgluon.For our purposes, we also require that solutions be invariant underSL2,Z transformations, weak/strong duality, so that we include in the Lagrangian density Witten’s θ term [15].
(57)Lθ=−θgD3232π2Fμνa∗Faμν.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, qe, is given as
(58)qe=negD3−nmgD3θ2π,wherene 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 D=10 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 theD=6, N=2 supersymmetric Yang-Mills theory, obtained from the dimensional reduction of the D=10, N=1 theory, presented in Section 2. The D=6, N=2 theory comprises two supercharges of positive chirality in six dimensions, Qa,a=+,−. The theory is invariant under supersymmetry transformations generated by supercharges Qa (the gamma matrices in D=10 are represented as ΓN×I4, where ΓN are six dimensional gamma matrices. See Section 2).
(59a)δA0a=∑aδAa0a=−iζ¯aΓ0λaa=−iζa†λaa,(59b)δAia=∑aδAaia=−iζ¯aΓiλaa=−iζa†Γ0Γiλaa,(59c)δA4a=∑aδAa4a=−iζ¯aΓ4λaa=−iζa†Γ0Γ4λaa,(59d)δA5a=∑aδAa5a=−iζ¯aΓ5λaa=−ζa†Γ6Γ0Γ4Γ4λaa,(59e)δλa=∑aδλaa=∑a12FMNaΓMNζa=∑aEaΓ0−ΓiFi5aΓ0Γ0Γ5−iBaΓ0Γ4−iΓiFi4aΓ0Γ4Γ6Γ0Γ5ζa.Supersymmetry is broken by a part ofζa which is an eigenstate of Γ0Γ4 with eigenvalue -1, i.e., ζa,−1. Substituting equations (41a) and (41b) and equations (42a) and (42b) into equation (59a) and equation (59b), we obtain
(60a)δAaMa=0,(60b)δλa,−1a=2EaΓ0−iBaΓ0Γ4ζa,−1,(60c)δλa,+1a=0.As a characteristic of BPS states, half of the supersymmetries are broken, i.e., forζa,−1, and half are unbroken, i.e., for ζa,+1. The dimensional reduction to D=4 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 D=4, N=2 supersymmetric Yang-Mills-Higgs theory. In contrast, we begin with the D=10, N=1 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, AMa, unchanged. In reference [8], Kastor and Na have shown that, because of the nonlinearity inherent in the supersymmetry transformations, there are nonvanishing contributions to AMa 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 ψ=π/2. 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(61)Ψ=expδΨ¯=Ψ¯+δΨ¯+12!δ2Ψ¯+13!δ3Ψ¯+14!δ4Ψ¯,where Ψ represents both bosonic and fermionic fields after the transformation and Ψ¯ 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 ζa,−1. Substituting equation (60a) and equation (60b) in equation (61), we obtain
(62a)δ2A0a=∑aδ2Aa0a=2ζa,−1†Γ0Γ4Γjζa,−1Bja,(62b)δ2Aia=∑aδ2Aaia=−i2ζa,−1†ΓiΓjζa,−1Eja,(62c)δ2A4a=∑aδ2Aa4a=2ζa,−1†Γ0Γ4Γjζa,−1Bja,(62d)δ2A5a=∑aδ2Aa5a=2ζa,−1†Γ0Γ4Γjζa,−1Eja,(62e)δλa=∑aδλa,−1a=∑a2EaΓ0−iBaΓ0Γ4ζa,−1.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, a−,±1∗ in ζ−,±1, equation (16), by the operators aa,−1 and aa,−1† and then integrating the anticommutator of the fermionic zero modes, equation (62e),
(63)δab∫dx3dy3δλa,−1a,δλb,−1b†.Using equations (8a), (8b), and (16), we obtain
(64a)aa,−1,ab,−1†=14Mδab,(64b)aa,−1†,ab,−1†=0,(64c)aa,−1,ab,−1=0,where we have used the fact that the mass of the dyon is
(65)M=∫dx3E⟶a·E⟶a+B⟶a·B⟶a=gv.Applying equation (64a), (64b), and (64c) in the evaluation of equation (62a), (62b), (62c), (62d), and (62e), we obtain
(66a)δ2A0a=−2μ⟶m·1gDΦ⟶a,(66b)δ2A⟵a=2μ⟶e×1gDΦ⟶a,(66c)δ2A4a=2μ⟶m·1gDΦ⟶a,(66d)δ2A5a=2μ⟶e·1gDΦ⟶a,(66e)δλ−1a=∑a2DΦΓ0cosψ+sinψ−iΓ4ζa,−1.Here, the electric dipole moment, due to the spinning magnetic charge, is(67)μ⟶m≡qm2Mdyonζa,−1†S⟶ζa,−1,and the magnetic dipole moment, due spinning electric charge, is
(68)μ⟶e≡qe2Mdyonζa,−1†S⟶ζa,−1.The spin operator is defined in terms of the Lorentz generators of the rotation group, i.e.,(69)Sl≡ϵjkl−i4Γj,Γk.Because the supersymmetric spinors,ζa,−1, are eigenstates of Sx (with eigenvalue 1/2), then
(70)ζa,−1†S⟶ζa,−1=ζa,−1†Sxζa,−1e^x=12e^x.The complex constants,a−,±1 in ζ−,±1, equation (16), are arbitrary, and consequently, different sets of dyon solutions are obtained when quantizing the fermionic modes. Specifically, choosing both a−,+1=0 and a−,−1=0 results in spin 0 dyon solutions. Choosing either a−,+1=0 or a−,−1=0 yields two sets of spin 1/2 dyons with Sx=+1/2. Alternatively, interchanging a−,±1∗ with a−,±1 yields dyon solutions with Sx=−1/2. Setting both constants not equal to zero, simultaneously, we obtain spin 1 dyon solutions where Sx=±1,0. Considering all of these dyon solutions in total, we can evaluate μ⟶m and μ⟶e, explicitly, where for the spin 0 dyon, Sx=0, for the two spin 1/2 dyons Sx=±1/2, and for the spin 1 dyon Sx=±1,0.The potential functionsδ2A0a and δ2A⟵a are amenable to straightforward interpretation. Given that
(71)limr⟶∞1gDΦ⟶a=1r2eranm,then δ2Aa0a and δ2Aaia in the limit of large r 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 N=2 super Yang-Mills theory [8]).It is apparent that the electric dipole field derived from the potentialδ2A0a is equal but opposite to the field derived from the potential δ2A4a. Not as obvious is the fact that the magnetic dipole field derived from the potential δ2A⟵a is also equal but opposite to that derived from the potential δ2A5a. This relationship follows directly from the fact that DiDiΦa=0. 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 ISO32 Superstring Theory

The purpose of this section is to generalize the results of Section3 to the superstring theory. As we show, the solutions obtained in Section 3 correspond, in superstring theory, to D3-branes, which are embedded in an M5-brane compactified on a type IIB torus [16].First, the arena for discussing the dyon solutions of Section3 is the M5-brane. The M5-brane is a 5+1 hypersurface propagating in D=1+10dimensions [17]. The underlying theory is based on a single copy of the D=11 Majorana fermions which in D=10 superstring theory reduces to the two Majorana-Weyl fermions. The defining characteristic of these fermions, ε, is that they satisfy a constraint equation, i.e., κ-symmetry:
(72)Γ6ϵ=ϵ.This is precisely the constraint placed on the spinors, equation (14), defining the dyon solutions in Section 3. Consequently, from the perspective of D=11, the dyon solutions obtained previously live, in fact, on an M5-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 BAB=0, where the only nonvanishing R-R potential is C1, which is a constant background).
(73)S=SDBI+SWZ,where
(74)SDBI=−τp∫Mp+1STre−Φ−detg+2πα′F,(75)SWZ=μp∫Mp+1PC1∧STre2πα′F.Here,τp is the physical tension of the Dp-brane, μp is its R-R charge, and gαβ=PGMN is the pullback of the background metric GMN. 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 FAB, computation of the symmetric trace yields ambiguous results in terms of order F6 [18, 19].The fermionic action based on the Lagrangian density (equation (1)) is replaced by the fermionic Dp-bane action:
(76)SF=τp2∫Mp+1e−Φ−detg+2πα′Fθ¯1−ΓDpM~−1αβΓβDα−Δθ,whereΔ vanishes since spacetime background is flat for the cases we are considering. Here,
(77)Mαβ=gαβ+Fαβ,and Γα=PΓM. For type IIB D(2n+1)-branes,
(78)ΓD2n+1=∑q+r=n+1A2n+1q,rBq,r,and for type IIA D(2n)-branes,
(79)ΓD2n=∑q+r=n+1A2nq,rBq,r,where the ΓD2n for the type IIA theory and the ΓD2n+1 for the IIB theory differ by a factor of -1 in references [17, 20]. The reason is derived from the fact that Γ10, denoted Γ10 in [20], is defined with indices raised, whereas in [17], Γ10 is defined with indices lowered. We adopt the same convention for Γ10, as [20].
(80)A2n+1q,r=−1r+1iσ2σ3rϵα1⋯α2qβ1⋯β2rFα1α2⋯Fα2q−1α2qΓβ1⋯β2r,(81)A2nq,r=−1r+1Γ10ϵα1⋯α2qβ1⋯β2rFα1α2⋯Fα2q−1α2qΓβ1⋯β2r,(82)Bq,r=q!2r!2q−detg+2πα′F.Since our interest is the type ISO32, our focus will be the type IIB theory to which the type I SO32 is related. For the type IIB theory, θ is a 64 component double spinor:
(83)θ=θ1θ2.Eachθi,i=1,2 is the 32 component Majorana-Weyl spinor of positive chirality, i.e., Γ10θi=+1θi. In equation (80), the Pauli matrices act on the spinorial index i in θi. For the Abelian gauge theory, the fermionic action is invariant under κ-symmetry which acts on fermions:
(84)δθ¯=κ¯1+ΓD2n+1,The actionSF, 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 F2[18]. At first, we ignore these problems and assume that the action SF applies to the non-Abelian theory, in which case D corresponds to the gauge covariant derivative of the applicable non-Abelian gauge theory.We now show that the BPS solutions given in Section3 are exact solutions of type I SO32 superstring theory. Since the type I SO32 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 ε:
(85)ϵ=ϵ1ϵ2,which preserve the bosonic nature of these solutions, i.e., θ remains zero, and furthermore, the bosonic solutions remain unchanged to first order. To satisfy the condition that θ=0, the combined κ- and supersymmetry transformations must vanish, i.e.,
(86)sθ=δκθ+ϵ=0.Here, theκ-symmetry transformation is
(87)δκθ=1+ΓD2n+1κ,Simón has shown that this condition is satisfied when(88)ΓD2n+1ϵ=ϵ.Simón has solved equation (88) for a supersymmetric D3-brane configuration, i.e., n=1, 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.Forn=1, equation (78) becomes
(89)ΓD3=STr14!−detg+2πα′Fϵα0⋯α3Γα0⋯α3iσ2+6Fα0α1aTaΓα2α3σ1+3Fα0α1aTaFα2α3bTbiσ2.Substituting equation (89) into equation (88) and rearranging terms, we obtain
(90)STr−detg+FI2ϵ=STr1+ΓiΓ0DiΦaTacosψσ3+sinψσ1−ΓiΓ0σ3EiaTa+ΓiΓjEiaTaDjΦbTbcosψσ3+sinψσ1−ΓiΓ0BiaTaσ1+BaiTaDiΦbTbcosψσ3+sinψσ1BiaTaEibTbiσ2ϵ,where I2 is the identity matrix in two dimensions. In transitioning from equation (89) to equation (90), we have imposed the projection constraints
(91)−Γ4σ2ϵ=ϵ,(92)Γ0ΓΦε=ε.The matrixΓΦ is defined as
(93)ΓΦ≡Γ4cosψ+Γ5sinψ.Substituting equations (41a), (41b), and equation (51) into the square root term in equation (90), we obtain (see the appendix for details)
(94)Tr1+sin2ψZr2Tr2+Zθ2Tθ2+Zϕ2Tϕ22I2ϵ,where
(95)Zr=DrΦr,Zθ=DθΦθr,Zϕ=DϕΦϕrsinθ.Making the same substitutions into the terms to the right of the equal sign in equation (90), we obtain
(96)Tr1+sin2ψZr2Tr2+Zθ2Tθ2+Zϕ2Tϕ2σ12ϵ.The reduction of the right-hand side of equation (90) proceeds, for the most part, as in [17] without requiring that the symmetric trace condition, with one exception. The second term in the second line of equation (90) requires invoking the symmetric trace condition to vanish. In obtaining both equation (94) and equation (96), we have used the fact
(97)BiaTa=BiaTa=sinψZiTaδia,EiaTa=EiaTa=cosψZiTaδia.UsingZi, defined in equation (95), rather than DΦiaδai in equation (94), is equivalent, geometrically, to transforming from the orthogonal basis vectors of spherical polar coordinates ∂r,∂θ,∂ϕ to the orthonormal basis vectors, ∂r,θ^,ϕ^, i.e., ds2=dr2+r2dθ2+r2sin2θdϕ2=dr2+θ∧2+ϕ∧2. The reason for this replacement is to facilitate a comparison of results, presented here, with those presented in [17], where the metric tensor is given in an orthonormal basis.We, now, solve the constraint equations, equation (91) and equation (92) for ε, obtaining (because we have two independent supercharges, (a=+,−), there are two solutions for ε. We have omitted labeling ε with an additional subscript a so that the notation is less cluttered)
(98)ϵ=ζa,+1−iΓ4ζa,+1.Up to a phase, which we take to be zero, we find thatζa,−1 (equation (16)) is given by
(99)ζa,−1=ϵ2=−iΓ4ϵ1=−iΓ4ζa,+1.The relationship between the type IIB theory, (N=2, D=10) supersymmetry, discussed here, and relevant type I theory SO32, (N=1, D=10) supersymmetry, can be gleaned from equation (66e) and equation (99) (it is worth noting that there is a supersymmetric version of the non-Abelian Dirac-Born-Infeld action (equation (73)). Its construction is based on a generalization of the principles used here to extend the supersymmetric results of Section 3 to superstring theory. See, for example, the work of Bergshoeff et al. [21]). We define
(100)ϵTypeI=cosψϵ2−sinψϵ1,so that
(101)δλa,−1a=2DΦΓ0ϵTypeI.In summary, we have shown that the dyon solutions obtained in Section3 satisfy the κ-symmetry constraint, equation (88), and are therefore, also, solutions of type IIB (type I SO32) superstring theory.


## 5.T-Duality, Gauge/Gravity Duality, and Wormholes

In this section, we applyT-duality transformations to the superstring solutions derived in Section 4 and study the duality between the supersymmetric string theoretic solutions obtained therein and their gravitational analogue. Specifically, we apply the T-duality transformations to spatial dimensions x4 and x5 of the M5-brane, transforming the gauge potential functions, A4aTa and A5aTa, into embedding coordinates, 2πα′A4aTa and 2πα′A5aTa. In order that such transformation be interpreted, straightforwardly, the potentials should not depend on the coordinates x4 or x5 and furthermore should also commute. The metric obtained on the two resulting D3-branes is derived by pulling back the metric induced by the embedding coordinates [22], i.e., the metric, gμν, is given by
(102)gμν=ημν+∑M,N=45ηMNSTrDμAMaTaDνANbTb.After including the back reaction in theT-duality transformations, we find that the potential functions A4aTa and A5aTa, equations (46a) and (46b), in general, do not commute complicating their interpretation as embedding coordinates. Since the noncommutativity is present only for solutions with nonvanishing spin, we organize this section into two subsections: the first dealing with the case of vanishing spin and second dealing with the more complicated case of nonvanishing spin, which includes both spin 1/2 and spin 1 solutions.


### 5.1. Case 1: Spin 0 Solutions

Before applyingT-duality transformations, we perform a coordinate transformation in the x4,x5 plane which induces a gauge transformation on A4a and A5a, thereby eliminating A4a. Since, for the spin 0 case (see equation (50)):
(103a)A4aTa=ΦaTacosψ,(103b)A5aTa=ΦaTasinψ.By rotating thex4,x5 coordinate axes through an angle π/2−ψ, we transform A4a and A5a:
(104)A4aTa⟶sinψA4aTa−cosψA5aTa=0,A5aTa⟶cosψA4aTa+sinψA5aTa=ΦaTa=vQrTr.We note that this transformation leaves unchanged the components of the Minkowski metricηMN,M,N=4,5. After diagonalizing the matrix Tr, we apply a T-duality transformation to the x5-coordinate axis. As a consequence, we obtain two D3-branes embedded in a subspace of the M5-brane, where the embedding coordinates of the two D3-branes are x5=±LD3QrLD3=2πα′v=2πα′Mgluon/gD3 In addition, the value of the electric or magnetic charge associated with one D3-brane is opposite in sign of the corresponding charge on the other D3-brane (see Figure 1).Figure 1
Wormhole. Shown is the embedding diagram of the twoD3-branes with azimuthal angle suppressed. The radial coordinate, r, has been replaced with the dimensionless coordinate r/Ldyon, and the embedding coordinate x5 has been replaced with the dimensionless coordinate x5/LD3.The geometrical interpretation ofLD3 is straightforward. It is one half of the separation between the D3-branes in the asymptotic region of space, i.e., r⟶∞ (alternatively, we can transform A4aTa⟶A0aTa−A4aTa=0, which, also, results in the transformation of the Minkowski metric, coincidentally, identical in form to equation (102). Furthermore, the quantity LD3 now depends only on the magnetic charge, qm, and not on the electric charge, qe). This is a consequence of the fact that limr⟶∞Qr=1. As noted previously, Ldyon is the characteristic size of the dyon (see equation (54)).Using equations (102) and (104), we can calculate the metric tensor
(105)gμνdxμdxν=−dtdt+4π2v~2α′2gD32sin2ψd/duqu2+α′sin2ψgD32v~2dudu+4π2v~2α′gD32qu2w~2usin2ψ+u2α′sin2ψgD32v~2dθdθ+4π2v~2α′gD32qu2w~2usin2ψ+u2α′sin2ψgD32v~2sin2θdϕdϕ,where w~u=−1+wu and v~=vα′, and the functions qu and wu are defined in equations (52a) and (52b). It is straightforward to calculate the scalar curvature. Details of performing this calculation can be found elsewhere [1]. We omit presenting the scalar curvature here, since it comprises a large number of terms and is not amenable to obvious interpretation. Nonetheless, we can show that the scalar curvature
(106)limr⟶∞R⟶−24π2sinψ4gD32v~6Ldyon6α′r6,ifψ≠0,0,ifψ=0,so that the geometry of eachD3-brane is asymptotically flat. Furthermore, we can also show that the scalar curvature, R, is finite everywhere. In particular, for small values of r, the scalar curvature is given by
(107)limr⟶0R⟶216π2sin4ψgD32v~64π2sin2ψv~4+92α′+rLdyon28v~6gD32π2sinψ4100π2sinψ2v~4−108954π2sinψ2v~4+93α′.We are constraining the Yang-Mills coupling constant on theD3-branes such that 0≤gD32≤4π. Solutions when gD32>4π are obtained using weak/strong duality; i.e., the dual theory is obtained by interchanging electric and magnetic charge and letting gD32⟶4π/gD32 .In Figure2, we compare plots of the scalar curvature of a magnetic monopole, with one unit of magnetic charge, to a dyon with one unit of magnetic and one unit of electric charge. The mass of the gluon Mgluon=MP/2 (the Planck mass, MP≈1/α′), and the Yang-Mills coupling constant gD3=1. Note that the maximum scalar curvature of the dyon is less that of the monopole. In addition, it can be shown that when ψ⟶0 i.e., qe⟶∞, the two D3 -branes merge into a single D3 -brane whose scalar curvature R⟶0. Furthermore, it can also be shown, independent of the value of ψ, that as r⟶∞, the metric gμν⟶ημν and that the geometry of each D3-brane is asymptotically flat.Figure 2
Scalar curvature: depicted in (a) and (b) are the scalar curvatures of a magnetic monopole and dyon, each without spin. In each figure, the scalar curvature,R, in units of 1/α′, is plotted as a function of the dimensionless, spherical coordinates r/Ldyon, and θ. The mass of the gluon Mgluon=MP/2. The Yang-Mills coupling constant gD3=1.
(a)
Magnetic monopole: the magnetic charge of the monopole isqm=4π, the electric charge qe=0, i.e., ne=0, and the θ-term=0. Its mass is 4πMgluon(b)
Dyon: the magnetic charge of the dyon isqm=4π, the electric charge qe=1/2, i.e., ne=0, and the θ-term=−π. Its mass is 1/4+16π2Mgluon


### 5.2. Case 2: Spin 1/2 and Spin 1 Solutions

For the case of nonvanishing spin, the potential functionsA4aTa and A5aTa do not commute, except in the spin 1 case when the x component of the spin, Sx, vanishes. We consider, first, the case when Sx≠0.For the case of nonvanishing spin,(108a)A4aTa=ΦaTacosψ+2μ⟶m·1gDΦ⟶aTa,(108b)A5aTa=ΦaTasinψ+2μ⟶e·1gDΦ⟶aTa.See equation (50), equation (66c), and equation (66d). The noncommutativity of A4aTa and A5aTa is derived from the fact that ΦaTa and e^x·DΦ⟶aTa do not commute. We resolve the problem of noncommutativity by performing the following coordinate transformation:
(109)x4⟶x4−cos2ψx0−sin2ψx5.This induces the following gauge transformation:(110)A4aTa⟶A4aTa−cos2ψA0aTa−sin2ψA5aTa=0,thereby eliminating the noncommutativity of A4aTa and A5aTa. In addition, the metric tensor ηMN is transformed to
(111)ηMN⟶−sin2ψ2cos2ψsin2ψcos2ψcos2ψ1sin2ψsin2ψcos2ψsin2ψ−cos2ψ2+2,for components M,N=0,4,5. The remaining ηMN are unchanged.WhenSx≠0, we limit our consideration to solutions where the electric charge, qe=0, i.e., magnetic monopole solutions. Our reason for this limitation is that some calculations, including the curvature tensor, are calculatingly challenging and furthermore comprise such a large number of terms that they are not straightforward to interpret. For the magnetic monopole solutions, ψ=π/2 so that the metric simplifies to
(112)ηMN⟶0−10−110001.Using equation (102), equation (112), and equation (108b), we can calculate the metric tensor, gμν. For these solutions,
(113)g00=AB.Here,(114a)A=4−12π2v~2α′gD32w~u2qu2+u2×Sx2qu4α′2π2w~u2gD36sin2θ,(114b)B=u24π2v~2α′gD32w~u2qu2+u2.The spatial components of the metric,gij, are the same as those of the spin 0 case, equation (105), with an important difference in the ϕ component of the metric. Because of the nonvanishing component of the spin Sx, reference frames retrogress about the x-axis so that the ϕ coordinate is transformed:
(115)ϕ⟶ϕ−Ωt.The angular speed,Ω, is
(116)Ω=−8π2α′qu3w~2ugD35v~2Sx4π2α′qu2w~2ugD32v~2+u2.Relative to spatial infinity, inertial frames are dragged with speed(117)vΩ=LdyonusinθΩ.We note that these solutions are, strictly speaking, only accurate toOα′ or, equivalently, accurate for values of the gluon mass, Mgluon, less than the Planck mass, MP. In fact, we can show that whenever, Mgluon¡MP, the speed with which inertial frames are dragged is less than the speed of light, thus avoiding the possibility of closed time-like curves. In Figure 3, we show a plot of the spatial dependence of vΩ for Mgluon=1/2MP.Figure 3
Frame dragging. Shown in the plot is the speed, with which reference frames are dragged around the wormhole relative to a stationary reference frame in a region of space far from the wormhole, i.e.,r⟶∞. The speed, vΩ, is given as a multiple of the speed of light.The scalar curvature in the case of nonvanishing spin is markedly different from that of vanishing spin. Regarding its general features, we can show that the scalar curvature,R∝1/ρ2, is the cylindrical polar coordinate, ρ⟶0. Here, ρ≡y2+z2. In particular, as r⟶0, i.e., ρ,x⟶0,
(118)limρ,x⟶0R⟶−Ldyonρ218gD32v~24π2v~4+9α′,where v~=vα′. Furthermore, we can also show that
(119)limx⟶∞R⟶−2ρ2+Ldyon2gD32v~2ρ2α′.In Figure4(a), we show a plot of the scalar curvature when Mgluon=1/2MP and gD3=1. In Figure 4(b), we show a plot of ρ2R, evaluated at ρ=0. The purpose of scaling by ρ2 is to remove the divergent part of the scalar curvature.Figure 4
Rotating magnetic monopole: in (a), the scalar curvature,R, in units of 1/α′, is plotted as a function of the dimensionless, cylindrical cooordinates ρ/Ldyon and x/Ldyon. In (b), the scalar curvature, which has been rescaled to remove its divergent behavior, is plotted as a function of x/Ldyon at ρ=0. The charge of the monopole is qm=4π, i.e., the electric charge qe=0 and the θ-term=0. Its mass is 4πMgluon. The Yang-Mills coupling constant gD3=1.
(a)(b)Surprisingly, the scalar curvature is independent ofSx. The reason is that the R00 component of the Ricci tensor is the only component which depends on Sx. Specifically, R00∝Sx2. In addition, the g00 component of the metric tensor is the only component which depends on Sx, i.e., g00∝Sx2. Thus, after contracting the metric tensor with the Ricci tensor, the scalar curvature is independent of Sx. As a result, after contracting the metric tensor with the Ricci tensor, the scalar curvature is independent of Sx. This is consequential for the spin one monopole solutions when Sx=0. For spin one, when Sx=0, equations (108a) and (108b) reduce to equations (103a) and (103b), which would seem to indicate that the scalar curvature is the same as for the spin 0 case. Alternatively, as well as preferably, we can obtain the case Sx=0 as the limit Sx⟶0 for the case of nonvanishing Sx. Taking the limit, Sx⟶0, we find that g00⟶0,R00⟶0, while all other components of the metric tensor, the Ricci tensor, and scalar curvature remain unchanged. This analysis demonstrates that the case of a spin one, magnetic monopole with Sx=0, is inherently different from the that of a magnetic monopole with vanishing spin.It is interesting to contrast the geometries of vanishing spin with nonvanishing spin solutions in the asymptotic region of space. For the spin zero monopole, asr⟶∞, the metric approaches the Minkowski metric, and the scalar curvature R⟶0. For the nonvanishing spin monopole, the metric also approaches the Minkowski metric, transformed using “light cone” coordinates; however in contrast, as ρ⟶∞, the scalar curvature
(120)R⟶−2gD32v~2α′.


## 5.1. Case 1: Spin 0 Solutions

Before applyingT-duality transformations, we perform a coordinate transformation in the x4,x5 plane which induces a gauge transformation on A4a and A5a, thereby eliminating A4a. Since, for the spin 0 case (see equation (50)):
(103a)A4aTa=ΦaTacosψ,(103b)A5aTa=ΦaTasinψ.By rotating thex4,x5 coordinate axes through an angle π/2−ψ, we transform A4a and A5a:
(104)A4aTa⟶sinψA4aTa−cosψA5aTa=0,A5aTa⟶cosψA4aTa+sinψA5aTa=ΦaTa=vQrTr.We note that this transformation leaves unchanged the components of the Minkowski metricηMN,M,N=4,5. After diagonalizing the matrix Tr, we apply a T-duality transformation to the x5-coordinate axis. As a consequence, we obtain two D3-branes embedded in a subspace of the M5-brane, where the embedding coordinates of the two D3-branes are x5=±LD3QrLD3=2πα′v=2πα′Mgluon/gD3 In addition, the value of the electric or magnetic charge associated with one D3-brane is opposite in sign of the corresponding charge on the other D3-brane (see Figure 1).Figure 1
Wormhole. Shown is the embedding diagram of the twoD3-branes with azimuthal angle suppressed. The radial coordinate, r, has been replaced with the dimensionless coordinate r/Ldyon, and the embedding coordinate x5 has been replaced with the dimensionless coordinate x5/LD3.The geometrical interpretation ofLD3 is straightforward. It is one half of the separation between the D3-branes in the asymptotic region of space, i.e., r⟶∞ (alternatively, we can transform A4aTa⟶A0aTa−A4aTa=0, which, also, results in the transformation of the Minkowski metric, coincidentally, identical in form to equation (102). Furthermore, the quantity LD3 now depends only on the magnetic charge, qm, and not on the electric charge, qe). This is a consequence of the fact that limr⟶∞Qr=1. As noted previously, Ldyon is the characteristic size of the dyon (see equation (54)).Using equations (102) and (104), we can calculate the metric tensor
(105)gμνdxμdxν=−dtdt+4π2v~2α′2gD32sin2ψd/duqu2+α′sin2ψgD32v~2dudu+4π2v~2α′gD32qu2w~2usin2ψ+u2α′sin2ψgD32v~2dθdθ+4π2v~2α′gD32qu2w~2usin2ψ+u2α′sin2ψgD32v~2sin2θdϕdϕ,where w~u=−1+wu and v~=vα′, and the functions qu and wu are defined in equations (52a) and (52b). It is straightforward to calculate the scalar curvature. Details of performing this calculation can be found elsewhere [1]. We omit presenting the scalar curvature here, since it comprises a large number of terms and is not amenable to obvious interpretation. Nonetheless, we can show that the scalar curvature
(106)limr⟶∞R⟶−24π2sinψ4gD32v~6Ldyon6α′r6,ifψ≠0,0,ifψ=0,so that the geometry of eachD3-brane is asymptotically flat. Furthermore, we can also show that the scalar curvature, R, is finite everywhere. In particular, for small values of r, the scalar curvature is given by
(107)limr⟶0R⟶216π2sin4ψgD32v~64π2sin2ψv~4+92α′+rLdyon28v~6gD32π2sinψ4100π2sinψ2v~4−108954π2sinψ2v~4+93α′.We are constraining the Yang-Mills coupling constant on theD3-branes such that 0≤gD32≤4π. Solutions when gD32>4π are obtained using weak/strong duality; i.e., the dual theory is obtained by interchanging electric and magnetic charge and letting gD32⟶4π/gD32 .In Figure2, we compare plots of the scalar curvature of a magnetic monopole, with one unit of magnetic charge, to a dyon with one unit of magnetic and one unit of electric charge. The mass of the gluon Mgluon=MP/2 (the Planck mass, MP≈1/α′), and the Yang-Mills coupling constant gD3=1. Note that the maximum scalar curvature of the dyon is less that of the monopole. In addition, it can be shown that when ψ⟶0 i.e., qe⟶∞, the two D3 -branes merge into a single D3 -brane whose scalar curvature R⟶0. Furthermore, it can also be shown, independent of the value of ψ, that as r⟶∞, the metric gμν⟶ημν and that the geometry of each D3-brane is asymptotically flat.Figure 2
Scalar curvature: depicted in (a) and (b) are the scalar curvatures of a magnetic monopole and dyon, each without spin. In each figure, the scalar curvature,R, in units of 1/α′, is plotted as a function of the dimensionless, spherical coordinates r/Ldyon, and θ. The mass of the gluon Mgluon=MP/2. The Yang-Mills coupling constant gD3=1.
(a)
Magnetic monopole: the magnetic charge of the monopole isqm=4π, the electric charge qe=0, i.e., ne=0, and the θ-term=0. Its mass is 4πMgluon(b)
Dyon: the magnetic charge of the dyon isqm=4π, the electric charge qe=1/2, i.e., ne=0, and the θ-term=−π. Its mass is 1/4+16π2Mgluon


## 5.2. Case 2: Spin 1/2 and Spin 1 Solutions

For the case of nonvanishing spin, the potential functionsA4aTa and A5aTa do not commute, except in the spin 1 case when the x component of the spin, Sx, vanishes. We consider, first, the case when Sx≠0.For the case of nonvanishing spin,(108a)A4aTa=ΦaTacosψ+2μ⟶m·1gDΦ⟶aTa,(108b)A5aTa=ΦaTasinψ+2μ⟶e·1gDΦ⟶aTa.See equation (50), equation (66c), and equation (66d). The noncommutativity of A4aTa and A5aTa is derived from the fact that ΦaTa and e^x·DΦ⟶aTa do not commute. We resolve the problem of noncommutativity by performing the following coordinate transformation:
(109)x4⟶x4−cos2ψx0−sin2ψx5.This induces the following gauge transformation:(110)A4aTa⟶A4aTa−cos2ψA0aTa−sin2ψA5aTa=0,thereby eliminating the noncommutativity of A4aTa and A5aTa. In addition, the metric tensor ηMN is transformed to
(111)ηMN⟶−sin2ψ2cos2ψsin2ψcos2ψcos2ψ1sin2ψsin2ψcos2ψsin2ψ−cos2ψ2+2,for components M,N=0,4,5. The remaining ηMN are unchanged.WhenSx≠0, we limit our consideration to solutions where the electric charge, qe=0, i.e., magnetic monopole solutions. Our reason for this limitation is that some calculations, including the curvature tensor, are calculatingly challenging and furthermore comprise such a large number of terms that they are not straightforward to interpret. For the magnetic monopole solutions, ψ=π/2 so that the metric simplifies to
(112)ηMN⟶0−10−110001.Using equation (102), equation (112), and equation (108b), we can calculate the metric tensor, gμν. For these solutions,
(113)g00=AB.Here,(114a)A=4−12π2v~2α′gD32w~u2qu2+u2×Sx2qu4α′2π2w~u2gD36sin2θ,(114b)B=u24π2v~2α′gD32w~u2qu2+u2.The spatial components of the metric,gij, are the same as those of the spin 0 case, equation (105), with an important difference in the ϕ component of the metric. Because of the nonvanishing component of the spin Sx, reference frames retrogress about the x-axis so that the ϕ coordinate is transformed:
(115)ϕ⟶ϕ−Ωt.The angular speed,Ω, is
(116)Ω=−8π2α′qu3w~2ugD35v~2Sx4π2α′qu2w~2ugD32v~2+u2.Relative to spatial infinity, inertial frames are dragged with speed(117)vΩ=LdyonusinθΩ.We note that these solutions are, strictly speaking, only accurate toOα′ or, equivalently, accurate for values of the gluon mass, Mgluon, less than the Planck mass, MP. In fact, we can show that whenever, Mgluon¡MP, the speed with which inertial frames are dragged is less than the speed of light, thus avoiding the possibility of closed time-like curves. In Figure 3, we show a plot of the spatial dependence of vΩ for Mgluon=1/2MP.Figure 3
Frame dragging. Shown in the plot is the speed, with which reference frames are dragged around the wormhole relative to a stationary reference frame in a region of space far from the wormhole, i.e.,r⟶∞. The speed, vΩ, is given as a multiple of the speed of light.The scalar curvature in the case of nonvanishing spin is markedly different from that of vanishing spin. Regarding its general features, we can show that the scalar curvature,R∝1/ρ2, is the cylindrical polar coordinate, ρ⟶0. Here, ρ≡y2+z2. In particular, as r⟶0, i.e., ρ,x⟶0,
(118)limρ,x⟶0R⟶−Ldyonρ218gD32v~24π2v~4+9α′,where v~=vα′. Furthermore, we can also show that
(119)limx⟶∞R⟶−2ρ2+Ldyon2gD32v~2ρ2α′.In Figure4(a), we show a plot of the scalar curvature when Mgluon=1/2MP and gD3=1. In Figure 4(b), we show a plot of ρ2R, evaluated at ρ=0. The purpose of scaling by ρ2 is to remove the divergent part of the scalar curvature.Figure 4
Rotating magnetic monopole: in (a), the scalar curvature,R, in units of 1/α′, is plotted as a function of the dimensionless, cylindrical cooordinates ρ/Ldyon and x/Ldyon. In (b), the scalar curvature, which has been rescaled to remove its divergent behavior, is plotted as a function of x/Ldyon at ρ=0. The charge of the monopole is qm=4π, i.e., the electric charge qe=0 and the θ-term=0. Its mass is 4πMgluon. The Yang-Mills coupling constant gD3=1.
(a)(b)Surprisingly, the scalar curvature is independent ofSx. The reason is that the R00 component of the Ricci tensor is the only component which depends on Sx. Specifically, R00∝Sx2. In addition, the g00 component of the metric tensor is the only component which depends on Sx, i.e., g00∝Sx2. Thus, after contracting the metric tensor with the Ricci tensor, the scalar curvature is independent of Sx. As a result, after contracting the metric tensor with the Ricci tensor, the scalar curvature is independent of Sx. This is consequential for the spin one monopole solutions when Sx=0. For spin one, when Sx=0, equations (108a) and (108b) reduce to equations (103a) and (103b), which would seem to indicate that the scalar curvature is the same as for the spin 0 case. Alternatively, as well as preferably, we can obtain the case Sx=0 as the limit Sx⟶0 for the case of nonvanishing Sx. Taking the limit, Sx⟶0, we find that g00⟶0,R00⟶0, while all other components of the metric tensor, the Ricci tensor, and scalar curvature remain unchanged. This analysis demonstrates that the case of a spin one, magnetic monopole with Sx=0, is inherently different from the that of a magnetic monopole with vanishing spin.It is interesting to contrast the geometries of vanishing spin with nonvanishing spin solutions in the asymptotic region of space. For the spin zero monopole, asr⟶∞, the metric approaches the Minkowski metric, and the scalar curvature R⟶0. For the nonvanishing spin monopole, the metric also approaches the Minkowski metric, transformed using “light cone” coordinates; however in contrast, as ρ⟶∞, the scalar curvature
(120)R⟶−2gD32v~2α′.


## 6. Conclusions

In this study, we have investigated the superstring analogue of the ‘t Hooft/Polyakov monopole. We have conducted this study in several steps. First, because superstring theory naturally resides in ten dimensions, we have reviewed the dimensional reduction ofD=10, N=1 supersymmetry in a way that specifically applies to this study. The theory underlying the ‘t Hooft/Polyakov monopole is based on a real-valued scalar boson, which undergoes spontaneous symmetry breaking. In this study, we assume this boson to be complex-valued so that the monopole (dyon) possesses both magnetic and electric charge. We, next, recast the scalar dyon theory as a supersymmetric gauge theory in six dimensions. The complex scalar field is replaced by two real fields which correspond to components of the gauge field in the two extra dimensions. Applying supersymmetry transformations to the gauge fields, we obtain a theory comprising of dyons with spin zero, one half, or one. In addition to possessing both magnetic and electric charge, the dyon possesses both electric and magnetic dipole moments. We show that both the gyromagnetic and gyroelectric ratios of each are exactly two, as would be expected [23] (the gyroelectric ratio has been reported, previously, by Kastor and Na). Next, we reinterpret the supersymmetric dyon solutions as solutions in N=2 type IIB superstring theory. For N=2 type IIB superstring theory, the number of on-shell bosonic and fermionic degrees of freedom, in general, is unequal, the number of fermionic degrees is sixteen, and the number of bosonic degrees is eight. Supersymmetry requires that the degrees of freedom of each be equal. To prove that the supersymmetric dyon solutions are also supersymmetric solutions in N=2 type IIB string theory, we show that the solutions satisfy κ-symmetry constraint equations, which, when satisfied, remove half of the fermionic degrees of freedom. We then recast the solutions in the type IIB theory as solutions in the type I SO32 theory. We, next, perform a T-duality transformation on the two components of the gauge field in the two extra dimensions. The T-duality transformation is complicated by the fact that the two gauge fields do not commute, a complication we resolve by eliminating one of the two components of the gauge field by a judiciously chosen coordinate/gauge transformation. The transformed solution is a rotating wormhole joining two D3-branes. The electric or magnetic charge of the dyon associated with each D3-brane is opposite in sign to that of the other D3-brane. Finally, we analyze the geometry of the D3-branes for two cases: one corresponding to a dyon with vanishing spin and the other corresponding to a magnetic monopole with nonvanishing spin. For the case of vanishing spin, we calculate the metric tensor and scalar curvature. We find that the scalar curvature is finite, everywhere (we have obtained comparable results in a previous study [1]). In particular, the scalar curvature vanishes, asymptotically far from the throat of the wormhole. For the case of nonvanishing spin, we similarly calculate the metric tensor and find that the spin of the magnetic monopole causes frame dragging, equation (117). We then calculate the scalar curvature. Unlike the case of vanishing spin, the scalar curvature diverges along the spin quantization axis. Specifically, it diverges as 1/ρ2, ρ being the radial, cylindrical coordinate. Also, in contrast to the case of vanishing spin, we find that as ρ⟶∞, the scalar curvature, on the boundary, approaches a constant negative value, equation (120).In summary, we note that the wormhole solutions, obtained in this study, provide an example of a gauge, gravity duality. Furthermore, because they correspond to BPS states and are based on supersymmetry where quantum corrections are expected to be well controlled, we expect such quantum corrections not to modify these solutions in a significant way. Consequently, the underlying theoretical principles may provide some insight in formulating a theory of quantum gravity.



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