# Dyons, Superstrings, and Wormholes: Exact Solutions of the Non-Abelian Dirac-Born-Infeld Action

Authors: Edward A. Olszewski
Journal: Advances in High Energy Physics
 (2015)

---

## Abstract

We construct dyon solutions on coincidentD4-branes, obtained by applying T-duality transformations to type I SO(32) superstring theory in 10 dimensions. These solutions, which are exact, are obtained from an action comprising the non-Abelian Dirac-Born-Infeld action and a Wess-Zumino-like action. When one spatial dimension of the D4-branes is taken to be vanishingly small, the dyons are analogous to the ’t Hooft/Polyakov monopole residing in a 3+1-dimensional spacetime, where the component of the Yang-Mills potential transforming as a Lorentz scalar is reinterpreted as a Higgs boson transforming in the adjoint representation of the gauge group. Applying a T-duality transformation to the vanishingly small spatial dimension, we obtain a collection of D3-branes, not all of which are coincident. Two of the D3-branes, distinct from the others, acquire intrinsic, finite curvature and are connected by a wormhole. The dyons possess electric and magnetic charges whose values on each D3-brane are the negative of one another. The gravitational effects, which arise after the T-duality transformation, occur despite the fact that the action of the system does not explicitly include the gravitational interaction. These solutions provide a simple example of the subtle relationship between the Yang-Mills and gravitational interactions, that is, gauge/gravity duality.


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## Body

## 1. Introduction

Theoretically appealing but experimentally elusive, the magnetic monopole has captured the interest of the physics community for more than eight decades. The magnetic monopole (an isolated north or south magnetic pole) is conspicuously absent from the Maxwell theory of electromagnetism. In 1931, Dirac showed that the magnetic monopole can be consistently incorporated into the Maxwell theory with virtually no modification to the theory [1]. In addition, Dirac demonstrated that the existence of a single magnetic monopole necessitates not only that electric charge be quantized but also that the electric and magnetic couplings be inversely proportional to each other, the first suggestion of the so-called weak/strong duality. Subsequently, ’t Hooft [2] and Alexander Polyakov showed that, within the context of the spontaneously broken Yang-Mills gauge theory SO(3), topological magnetic monopole solutions of finite mass must necessarily exist. Furthermore, these solutions possess an internal structure and also exhibit the same weak/strong duality discovered by Dirac. Consequently, Montonen and Olive conjectured that there exists an exact weak/strong electromagnetic duality for the spontaneously broken SO(3) gauge theory [3]. More recently, this conjecture has become credible within the broader context of N=2 or N=4 Super-Yang-Mills theories. Despite the lack of experimental evidence for the existence of magnetic monopoles, physicists still remain optimistic of their existence. Indeed, Guth proposed the inflationary model of the universe, in part, to explain why magnetic monopoles have escaped discovery [4].The focus of our investigation is electrically charged magnetic monopole (dyon) solutions within the context of superstring theory. In Section2, we construct dyon solutions which are exact and closed to first order in the string theory length scale. We, first, begin with a type I SO(32) string theory in ten dimensions, six of the spatial dimensions being compact but arbitrarily large. We, then, apply the group of T-duality transformations to five of the compact spatial dimensions to obtain 16 D4-branes, some of which are coincident. The five T-dualized dimensions of each D4-brane constitute the internal dimensions of a 4+1-dimensional spacetime. Making an appropriate ansatz, we obtain dyon solutions residing on the D4-branes. The solutions are based on an action which includes coupling of the D4-branes to NS-NS closed strings, the non-Abelian Dirac-Born-Infeld action, and coupling to R-R closed strings, a Wess-Zumino-like action. We next apply a T-duality transformation to the D4-branes, resulting in a collection of D3-branes, some of which are coincident and two of which are connected by a wormhole. Finally, we interpret the dyon solutions in the context of gauge/gravity duality.Because of the differences in the literature among the systems of units, sign conventions, and so forth, we present in AppendixA the conventions chosen by us so that direct comparisons can be made between our results and those of other authors.


## 2. Dyons and Dimensional Reduction of Type ISO(32) Theory

In this section, we construct dyon solutions based on superstring theory. We begin with type ISO(32) superstring theory in ten dimensions [5], six of the spatial dimensions of which are compact. Next, we apply the group of T-duality transformations to five of the compact dimensions letting the size, R, of the dimensions become vanishingly small; that is, R→0. These five dimensions are the internal dimensions of spacetime. Strictly speaking, spacetime consists of 16 D4-branes, bounded by 25 orientifold hyperplanes. Each of the D4-branes comprises four spatial dimensions, three unbounded and one compact. In what follows, we assume that none of the D-branes are close to the orientifold hypersurfaces. Thus, the theory describing the closed strings in the vicinity of any of the D4-branes is type II oriented, rather than type II unoriented. In this particular case, since we have applied the T-duality transformation to an odd number of dimensions, the closed string theory is the type IIa oriented theory. Furthermore, each end of an open string must be attached to a D4-brane, which may be the same D4-brane or two different D4-branes. If we assume that the number of coincident D4-branes is n (2≤n≤16), then a U(n) gauge group is associated with the open strings attached to the coincident D4-branes. Given these prerequisite conditions, we now construct dyon solutions which reside on these coincident D4-branes. These solutions are derived from the D-brane action comprising two parts, the Dirac-Born-Infeld action, SDBI, which couples NS-NS closed strings to the D4-brane and the Wess-Zumino-like action, SWZ, which couples R-R closed strings to the D4-brane.


### 2.1. Dyon Solutions onD4-Branes

The dyon solutions are obtained from the equations of motion derived from the action,S, which describes the coupling of closed string fields to a general Dp-brane (which in our case is p=4). The action is [6] (1)S=SDBI+SWZ,where(2)
S
DBI
=
-
τ
p
∫
M
p
+
1
S
T
r
e
-
Φ
-
1
·
det
⁡
G
A
B
+
B
A
B
+
2
π
α
′
F
A
B
1
/
2
,
(3)
S
WZ
=
μ
p
∫
M
p
+
1
∑
p
′
C
p
′
+
1
∧
Tr
⁡
e
2
π
α
′
F
+
B
.Here, τp is the physical tension of the Dp-brane, and μp is its R-R charge (see Appendix B for a discussion of the relationships among the various string parameters).The dyon solutions are based on the following ansatz. The dilaton background,Φ, is constant: (4)Φ=Φ0.And background field B vanishes: (5)BAB=0A,B=0⋯4.The metric G is given by (6)GAB=G~ABIn,where, for our purposes, G~AB is restricted so that G~00=-1 and G~44=1.Forp=4, we can reexpress the determinant in (2) as (7)
det
⁡
G
A
B
+
2
π
α
′
F
A
B
=
det
⁡
G
~
A
B
I
n
+
2
π
α
′
2
!
F
A
B
F
A
B
-
2
π
α
′
2
5
-
4
!
3
2
·
1
2
F
∧
F
∗
E
F
∧
F
∗
E
,where (8)F∧F∗E=detG~AB4!FABFCDϵABCDE.See Appendix C, (C.21), for further details.The termIn is the n-dimensional identity matrix. The value of n is the dimension of the group U(n) associated with the gauge fields residing on the D4-branes. All R-R potentials vanish, except for the one-form potential C(1), which is a constant background field, (9)C1=C4dx4,for some constant value C4. The gauge field, F, is obtained from the gauge potential A(A=AEdxE), where(10)Aμ=Aμxi,μ=0⋯3;i=1⋯3,A4=A4xi.Note that the gauge potentials are static; that is, they do not depend on time, x0, and also do not depend on the spatial coordinate x4. The gauge field F(F=FABdxA∧dxB), a Lie algebra-valued two-form, is given by (11)F=dA-iA∧A.(See Appendix A.) The components of the potentials A0 and A4 are constrained in accordance with the condition (12)A0∧A4=0so that F04=0. To facilitate its interpretation, we express FAB as a five-dimensional matrix which is explicitly partitioned into electric and magnetic fields which reside in four-dimensional spacetime and an additional component of the magnetic field which resides in the additional space dimension; that is,(13)FAB=0E1E2E30-E10B3-B2-D1A4-E2-B30B1-D2A4-E3B2-B10-D3A40D1A4D2A4D3A40.We are seeking dyon solutions. Therefore, with foresight, we make the following assumptions:(14)Ei≡F0iaTa=F0iiTi,Bi≡12ϵijkFjkaTa=12ϵijkFjkiTi,DiA4≡∂iA4-iAi∧A4aTa=F4iaTa=F4iiTi.The parenthetical index (i) indicates that there is no summation of that index; however, if an expression contains two indices i without parentheses, then summation of these two indices is implied. Furthermore, each matrix element in (13) includes a generator of U(n); for example, Ei=Ei(i)T(i). Because we are seeking dyon solutions, we may assume without loss of generality that each T(i) is a generator in the fundamental representation of a local U(1)×SU(2) subgroup of SU(n) (see (44a), (44b), (44c), and (44d)).The action, (2), can be more straightforwardly interpreted from the perspective of four-dimensional spacetime. Since the action does not depend on the coordinate x4, we can trivially eliminate x4 from the action by integrating the x4 coordinate. As a result of the integration, the tension of the D4-brane, τ4, and the Yang-Mills coupling constant, gD4, are replaced by those of the D3-brane, τ3 and gD3 (see (B.3) and (B.5)). Let the size, R4, of the x4-dimension become vanishingly small; that is, R4→0. Then, the field A4 becomes a Lorentz scalar transforming as the adjoint representation of the gauge group, and (14) gives the covariant derivative of A4. From the perspective of four spacetime dimensions, A4 assumes the role of a Higgs boson transforming as the adjoint representation of the gauge group.Substituting (4)–(6) and (13) into (1) and then integrating the x4 coordinate, we obtain (15)SDBI=∫d3+1ξLDBI,where (16)LDBI=-12πα′2gD32STr⁡det⁡GAB+2πα′FAB1/2=-det⁡G~ij2πα′2gD32STr⁡L′.The function L′ is defined as (17)
L
′
=
I
n
-
2
π
α
′
2
E
·
E
-
B
·
B
-
D
A
4
·
D
A
4
+
2
π
α
′
4
B
·
D
A
4
2
-
2
π
α
′
4
E
·
B
2
-
2
π
α
′
4
E
×
D
A
4
·
E
×
D
A
4
1
/
2
.We have used the fact that |det(G~ij)|=|det(G~AB)|. In (16), the ordering of the generators of the algebra, Ta, corresponds with the order of the fields as they appear in the equation; for example,(18)B·DA42=B·DA4B·DA4=G~ijBiiTiDjA4jTj·G~ijBiiTiDjA4jTj.Note that “STr” indicates that the trace is calculated symmetrically; that is, the trace is symmetrized with respect to all gauge indices [6, 7]. The implication is that the evaluation of the trace requires that after the expansion of (16) in powers of the field strengths, all orderings of the field strengths are included with equal weight; that is, products of Ta are replaced by their symmetrized sum, before the trace is evaluated. This is discussed in detail in [6, 7].In (16), the dot product and cross product of two 3-vectors, for example, E and DA4, are defined as E·DA4=G~ijEiDjA4 and (E×DA4)i=εijkEjDkA4.In obtaining (16), we have reexpressed the dilaton Φ, on a D4-brane, in terms of the dilaton Φ′, on a D3-brane, both of which are related by a T-duality transformation in the x4-dimension. Specifically, Φ and Φ′ are related by eΦ′=α′1/2eΦ/R4. The constant dilaton background ϕ0 has been incorporated into the physical tension τp (see Appendix B).Substituting (5), (9), (6), and (13) into (3), we obtain (19)SWZ=μ42!∫M5C1∧Tr⁡2πα′F∧2πα′F.Integrating the x4 coordinate in (19), we obtain (20)SWZ=∫d3+1ξLWZ,where(21)LWZ=θ4π2Tr⁡F∧F=det⁡G~ijθ4π2Tr⁡E·B=det⁡G~ijθ4π2EiiBiiSTr⁡TiTi=det⁡G~ijθ8π2EiiBii,where Ei(i)=F0i(i)T(i). Here, (22)θ≡C42!2πR4α′1/2.In obtaining (21), we have explicitly evaluated μ4 using (B.4). Equation (21) is associated with the Witten effect. Witten has demonstrated that adding term (21) to the Lagrangian of Yang-Mills theory does not alter the classical equations of motion but does alter the electric charge quantization condition in the magnetic monopole sector of the theory [5, 8, 9]. In summary, the action, S, for the D4-brane is given by(23)S=∫d3+1ξL,where(24)L=LDBI+LWZ.The equations of motion which are obtained from (23) are (25)DμPμν=0,where (26)Pμνa=∂L∂Fμνa.In addition, the fields Fμνa satisfy the Bianchi identity(27)DαFβγa=0.To facilitate the ensuing analysis, we transform the Lagrangian density,L, to the Hamiltonian density, H, using the Legendre transformation(28)H=STr⁡P0·E-L,where (29)P0ii≡∂L∂Eii=-det⁡G~ij2πα′2gD32STr⁡XiiL′+det⁡G~ijθ4π2BiiSTr⁡TiTi,where (30)Xii=EiTi+E·BTiBi+DA4×E×DA4iTi.After performing detailed calculations, we obtain (31)H=det⁡G~ij2πα′2gD32STr⁡H′,where (32)
H
′
=
I
n
+
2
π
α
′
2
P
0
·
P
0
+
B
·
B
+
D
A
4
·
D
A
4
+
2
π
α
′
4
B
·
D
A
4
2
+
P
0
·
D
A
4
2
+
P
0
×
B
2
+
D
A
4
×
P
0
×
B
2
2
π
α
′
-
2
I
n
+
D
A
4
·
D
A
4
1
/
2
.The electric fieldEi(i) can be expressed as a function of P0i(i): (33)Eii=∂H∂P0ii=det⁡G~ij2πα′2gD32STr⁡YiiH′.The term Yi(i) is given by(34)
Y
i
i
=
2
π
α
′
2
P
0
i
T
i
+
2
π
α
′
4
·
D
i
A
4
T
i
P
0
·
D
A
4
+
B
×
P
0
×
B
i
T
i
-
B
×
D
A
4
i
T
i
P
0
·
B
×
D
A
4
2
π
α
′
-
2
I
n
+
D
A
4
·
D
A
4
.We seek dyon solutions which are BPS states, that is, whose energyE (E=∫‍d3ξH) is a local minimum. First, we reexpress H(35)
H
=
det
⁡
G
~
i
j
2
π
α
′
2
g
D
3
2
STr
⁡
I
n
+
2
π
α
′
2
cos
⁡
ϕ
P
0
·
D
A
4
+
sin
⁡
ϕ
B
·
D
A
4
2
+
2
π
α
′
2
sin
⁡
ϕ
P
0
·
D
A
4
-
cos
⁡
ϕ
B
·
D
A
4
2
+
P
0
-
cos
⁡
ϕ
D
A
4
2
+
B
-
sin
⁡
ϕ
D
A
4
2
+
2
π
α
′
4
P
0
×
B
2
+
D
A
4
×
P
0
×
B
2
2
π
α
′
-
2
I
n
+
D
A
4
·
D
A
4
1
/
2
.The mixing angle,ψ, between the electric and magnetic fields of the dyon is defined as (36)tan⁡ψ=gmge.The quantities gm and ge are the electric and magnetic charges, respectively, of the dyon. The energy, E, is minimized by constraining the dyon solutions to satisfy(37a)P0=cos⁡ψDA4,(37b)B=sin⁡ψDA4.In (35), the second and third squared terms are zero as a consequence of the constraint. Since P0∝B, the fourth squared term is also zero by virtue of (38)P0×Bk=P0iiB0jj-P0jjB0ii2!TiTjεijk-ifijkTk.Thus, H simplifies so that the energy is (39)
E
=
1
2
π
α
′
2
g
D
3
2
∫
d
3
ξ
det
⁡
G
~
i
j
Tr
⁡
I
n
+
2
π
α
′
2
cos
⁡
ϕ
P
0
·
D
A
4
+
sin
⁡
ϕ
B
·
D
A
4
.Substituting (37a) and (37b) into (32) through (34) and using (39), we find(40)E=P0.In (39), there are two terms which contribute to the mass of the system. The first term within the trace, that is, In, corresponds to the volume of each coincident D4-brane (or D3-brane), which is infinite because the D-branes are not compact. The second term, by virtue of the equations of motion, (26), and the Bianchi identity, (27), can be expressed as a divergence and is therefore a topological invariant. The second term corresponds to the mass of the dyon and is proportional to ge2+gm2 as discussed below.The solutions to (25) and (27) can be straightforwardly obtained from the dyon solutions derived in [10]. Adapting the notation of [10] to the notation used here, we express the vector potential A, (10), in the form (in accordance with our conventions, the Yang-Mills coupling constant appears explicitly in the Lagrangian (A.1). In [8, 10], the coupling constant has been incorporated into the Yang-Mills fields. Thus, to compare results here with those in the references, the fields A and related fields should be divided by gD3)(41)A=Aμdxμ+A4dx4=cos⁡ψSrgD3vα1Trdt+WrTθsin⁡θndϕ-Tϕdθ+gD3vα2T⊥+Qrα1Trdx4,where v is an arbitrary constant. For the Lie group SU(n)(42a)α1=n2n-1,(42b)α2=-n-22n-1.Here,Ti(i=r,θ,ϕ) constitute a representation of the SU(2) subalgebra and T⊥ commutes with each Ti. The quantities r,θ,ϕ are the spherical polar coordinates in three dimensions. The elements Tr,Tθ,Tϕ are related to Tx,Ty,Tz:(43a)Tr=Txsin⁡θcos⁡nmϕ+Tysin⁡θsin⁡nmϕ+Tzcos⁡θ,(43b)Tθ=Txcos⁡θcos⁡nmϕ+Tycos⁡θsin⁡nmϕ-Tzsin⁡θ,(43c)Tϕ=-Txsin⁡nmϕ+Tycos⁡nmϕ.For SU(n), the n-dimensional matrices Tx,Ty,Tz, and T⊥ are given by(44a)Tx=120⋯000⋮⋱⋮⋮⋮0⋯0000⋯0010⋯010,(44b)Ty=120⋯000⋮⋱⋮⋮⋮0⋯0000⋯00-i0⋯0i0,(44c)Tz=120⋯000⋮⋱⋮⋮⋮0⋯0000⋯0100⋯00-1,(44d)T⊥=12nn-2-20⋯000⋱⋮⋮⋮⋮⋯-2000⋯0n-200⋯00n-2.Tx, Ty, Tz, and T⊥ are suitable linear combinations of specific elements of the Cartan subalgebra of SU(n) (see [10] for details). The value of the integer nm in (43a), (43b), and (43c) is the integer multiple of the fundamental unit of dyon’s magnetic charge.These results differ from those of [10]. For a direct comparison, first replace the azimuthal angle, ϕ, in [10] with ϕ′ and extend the domain from [0,2π] to [0,2πnm]; that is, ϕ′∈[0,2πnm]. Now, perform the change of variables ϕ′=nmϕ to the dyon solutions of [10] to obtain those given in (41). In addition, apply the same change of variables to the metric in [10] to obtain the metric G~ij: (45)G~ij=1000r2000r2nm2sin2⁡θ.Here, r∈[0,∞], θ∈[0,π], and ϕ∈[0,2π]. This generalizes the results of [10] which only applies to dyons with one unit of magnetic charge; that is, gm=1/gD3.The solutionsW(r), Q(r), and S(r) are obtained as in [10](46a)Wr=wx=1-xsinh⁡x,(46b)Qr=qx=coth⁡x-1x,(46c)Sr=sx=qx=coth⁡x-1x,where the dimensionless variable x is related to the radial coordinate r: (47)x=sin⁡ψgD3vα1r.The field tensorFAB of a dyon with electric charge ge and magnetic charge gm, (48)gm=nmgD3,can now be obtained from (41). Specifically, (49)Ftr=gegS′rgD3vα1Tr,Ftθ=1-WrgegSrgD3vα1Tθ,Ftϕ=1-WrgegSrgD3vα1nmsin⁡θTϕ,Frθ=-W′rTϕ,Fϕr=-W′rnmsin⁡θTθ,Fθϕ=-Wr2-Wrnmsin⁡θTr,(50)DrA4=Q′rgD3vα1Tr,DθA4=1-WrQrgD3vα1Tθ,DϕA4=1-WrQrgD3vα1nmsin⁡θTϕ.We now show that gauge invariance of action, (23), implies SL(2,Z) invariance. Consider U(1) gauge transformations which are constant at infinity and are also rotations about the axis A^4=A4/|A4|, specifically the gauge transformations [8] (51)δAμa=1gD3vα1DμA4a.Action (23) is invariant under these gauge transformations. According to the Noether method, the generator of these gauge transformations, N, is given by (52)N=∂L∂∂0AμaδAμa.Substituting the Lagrangian density (24) into (52), we obtain (53)N=GegD3+gD32θGm8π2,where(54a)Gm=1vα1∫d3ξSTr⁡DA4·B,(54b)Ge=1gD32vα1∫d3ξTr⁡DA4·P0are the magnetic and electric charge operators. Since rotations of 2π about the axis A^4 must yield the identity for physical states, that is, (55)e2πiN=1,applying the U(1) transformation on the left side of (55) to states in the adjoint representation of SU(n), we find that the eigenstates of N are quantized with eigenvalue (56)N=α1η,where η is an arbitrary integer. Substituting (56) into (53), we obtain (57)ge=α1ηgD3-θ′2πnmgD3,where we have defined θ′ by (58)θ≡α1θ′,and used the fact that (59)gmgD3=nm4π.Taking θ′=0 in (57), we obtain the quantization condition for the electric charge (60)ge=ηα1gD3.The electromagnetic contribution to the mass (rest energy) of the dyon, mem, can be obtained by substituting (54a) and (54b) into (39) and integrating the second term within the trace to obtain (61)mem=vα1ge2+gm2.We can now makeSL(2,Z) symmetry explicit. We first define (62)τ=θ′2π+4πgD32i.If θ′=0, then the weak/strong duality condition gD3→gm=(4π)/gD3 is equivalent to (63)τ⟶-1τ.In (57), the transformation θ′→θ′+2π results in identical physical systems with only states being relabeled. The transformation is equivalent to (64)τ⟶τ+1.Transformations (63) and (64) generate the group SL(2,Z). See [8, 9] for further details.Note that in (54a) and (54b) Ge is, strictly speaking, not the electric charge operator because P0 is not the electric field but rather is its conjugate; however, according to (33) and (34), if DA4 and B become vanishingly small for asymptotically large values of the radial coordinate, then P0 approaches E. Thus, in the asymptotic limit Ge is the electric charge operator. This distinguishing feature is a direct consequence of the fact that our analysis is based on the Born-Infeld action rather than the Yang-Mills-Higgs action. In our case, this point is inconsequential since P0=E, exactly.


### 2.2. Dyon Solutions onD3-Branes

As emphasized previously, the dyon solutions derived in Section2.1, when interpreted from 3+1 spacetime dimensions, that is, the compactified theory in which R4→0, are the ’t Hooft/Polyakov magnetic monopole or dyon, with the potential A4 being a Higgs boson transforming in the adjoint representation of the gauge group U(n). Here, our purpose is to reinterpret these dyon solutions in which R4→0 from the equivalent T-dual theory. In the T-dual theory, the radius R4 is replaced by R4′ (R4′=α′/R4) so that the radius of the x4-dimension R4′→∞. In addition, the potential A4 is reinterpreted as the x4-coordinates of the n  D3-branes embedded in 4 + 1 dimensional spacetime. These coordinates can be directly obtained by diagonalizing A4, (41), using a local gauge transformation which rotates Tr into Tz. The n  x4-coordinates are the diagonal elements of the matrix; that is (we are assuming that after the T-duality transformation the D3-branes are far from any orientifold hyperplanes. This can always be accomplished by adding the to A4 component of the gauge potential a constant U(1) gauge transformation θ0T0, θ0 being a suitable constant (see Appendix A)),(65)A4⟶2πα′gD3vα2T⊥+Qrα1Tz=2πα′gD3vu10⋯000⋱⋮⋮⋮⋮⋯u1000⋯0u2+u~r00⋯00u2-u~r,where(66a)u1=-α2nn-2,(66b)u2=α22n-2n,(66c)u~r=α12Qr.Of the n  D3-branes n-2 of the D3-branes, denoted by D3n-2, are coincident. The x4-coordinate of each is the constant value 2πα′gD3vu1. For the remaining two D3-branes, denoted by D31 and D32, the x4-coordinate of each is a function of the radial coordinate r. Specifically, x4=2πα′gD3v(u2-u~(r)) for D31 and x4=2πα′gD3v(u2+u~(r)) for D32, and as a consequence these two D3-branes have nonvanishing intrinsic curvature. This occurs despite the fact that before the application of the T-duality transformation no gravitational interaction is explicitly present. We now introduce the length scale LD3 which is the separation between D31 and D32, in the asymptotic limit as the radial coordinate r→∞. It is related to previously defined parameters by (67)LD3=2πα′gD3vα1.Another relevant length scale is the size of the dyon, that is, the region of space where all components of the Yang-Mills field, FAB, are nonvanishing. According to (49), (46a), (46b), (46c), and (47), only the radial components of the electric and magnetic fields are long range, with the remaining components of the fields vanishing exponentially for x≫1. Thus, additional structure of the dyon becomes apparent whenever x≲1 or equivalently whenever r≲1/(sin⁡ψgD3vα1). We can therefore define the size of dyon Ld, as measured from asymptotically flat space, that is, r→∞, to be (68)Ld=1sin⁡ψgD3vα1=2πα′sin⁡ψLD3.In Figure 1, we show, for the gauge group SU(5), embedding plots of the 5 D3-branes as a function of the dimensionless radial coordinate, x (x=r/Ld). As x→∞, the x4-coordinate of D32 approaches that of the (5–2) coincident D3-branes, D35–2, in effect, joining them by a wormhole in an asymptotically flat region of space. At x=0, D32 is joined to D31 by another wormhole. (See [11] for a recent discussion of thin shell wormholes exhibiting cylinder symmetry.) As we will show, in general, the intrinsic curvature of the two surfaces in the neighborhood of x=0 is relatively large but, nonetheless, finite. Although these features described in Figure 1 apply to the particular gauge group SU(5), they apply to all SU(n), n≥2. (For n=2 there are no coincident D3-branes.)Figure 1
Embedding Functions of the5  D3-branes for the gauge group SU(5). The scaled coordinate x4/LD3 is plotted as a function of the scaled radial coordinate, r/Ld, for the 5  D3-branes. The radial coordinate, r, is scaled by the size of the dyon, Ld, and the x4 coordinate is scaled by the separation between the two D3-branes D31 and D31 in the asymptotic limit of large r.We now consider in detail the twoD3-branes, D1 and D2, whose x4-coordinates are radial dependent. The T-duality transformation on the x4-dimension pulls back the metric onto D1 and D2, inducing the metric, Gij′, (69)Gij′=G~ijIn+DiA4iTiDjA4jTjδijG~44In.Expanding the right-hand side of (69), we obtain(70)Gij′=G~ij0⋯000⋱⋮⋮⋮⋮⋯G~ij000⋯0G~ij+A~ij00⋯00G~ij+A~ij,where (71)A~ij=gD3Lα122×Q′r2000W~2r000W~2rnm2sin2⁡θ.Here, (72)W~r=1-WrQr,and G~ij is given by (45). In obtaining (70), we have used (50) and the fact that the matrices Tr, Tθ, and Tϕ, (43a), (43b), and (43c), satisfy the relationship (73)Tr2=Tθ2=Tϕ2=1220⋯000⋮⋱⋮⋮⋮0⋯0000⋯0100⋯001.Each of the diagonal entries in the matrix Gij′ corresponds to the metric on one of the n  D3-branes obtained from the T-duality transformation. In the case of the first n-2 entries, corresponding to the D3-branes D3n-2, the metric is flat. In the case of the last two entries corresponding to the D3-branes, D31 and D32, their geometries are identical and intrinsically curved. The only feature which distinguishes these two D3-branes is that the function Q(r) defining the x4-coordinate for D3-brane D2 is replaced by -Q(r) for D31, as evidenced in (65) and (66a) and (66b) and (66c) and also in Figure 1. As a consequence, the electric and magnetic charges of the dyon on D2 are minus the values on D1. The electric and magnetic field lines enter the wormhole from one D3-brane and exit from the other. Figure 2 is an embedding diagram showing the D3-branes D1 and D2 in the neighborhood of the radial coordinate r=0. As there is no event horizon surrounding r=0, the two D3-branes are joined by a wormhole at r=0.Figure 2
Wormhole. Shown is the embedding diagram for the twoD3-branes, D31 and D32, with azimuthal angle suppressed. The domain of the radial coordinate, r, is 0≤r≤15Ld, and the range of the embedding coordinate x4 is -.45LD3≤x4≤+.45LD3.Of particular interest is the intrinsic scalar curvature ofD31 and D32, in the neighborhood of r=0. The scalar curvature can be calculated from the metric Gij′, (70), and its value R(0) at r=0 is (74)R0=216sin⁡ψLD36sin3⁡ψLD34sin2⁡ψ+12πα′22.For a given value of sin⁡ψ, R(0) assumes its maximum value R~(0): (75)R~0=2738sin⁡ψπα′,when LD3=L~D3, where (76)L~D3=123πα′sin⁡ψ1/2.For either LD3→0 or LD3→∞, the scalar curvature R(0)→0; that is, the geometry of D1 and D2 becomes flat, everywhere. The expression for the scalar curvature R(r) is a complicated function of r and not amenable to straightforward interpretation and, therefore, will not be given. In Figure 3, we show a plot of the scalar curvature as a function of the radial coordinate. In this example, LD3=α′, and the dyon has only one unit of magnetic charge so that sin⁡ψ=1. Near r=0, the scalar curvature is positive and finite. As r increases, the scalar curvature becomes slightly negative and asymptotically approaches zero as r→∞. These features of the scalar curvature described for this specific example also apply in general.Figure 3
Scalar curvatureRr. For the case LD3=α′ and sin⁡ψ=1, the dimensionless scalar curvature, R×α′, is plotted as a function of the scaled radial coordinate, r/Ld.Consider dyon solutions for whichLD3≈α′ or less. The F-strings connecting D31 and D32 would be in their ground state, a BPS state. In addition, assume that gD3≪1; then as r→0 from an asymptotically flat region of space, within either D31 or D32, the string length scale will be reached before the gravitational interaction becomes dominant at the length scale of O(gD31/4α′) [5]. Thus, action, (2), which does not include the gravitational interaction, should apply, and consequently the dyon solutions derived should be accurate. On the other hand, let gD3→1/gD3 so LD3≫1; then the D-string, also a BPS state, becomes lighter than the F-string. As a consequence of weak/strong duality, the dyon solutions should still be applicable with the F-strings being replaced by D-strings and the dyon electric and magnetic charges being interchanged.After applying aT-duality transformation to the dyon solutions obtained in Section 2.1, we have obtained dyon solutions residing on D3-branes where the effect of the gravitational interaction is apparent. This occurs despite the fact that action, (2), does not explicitly include the gravitational interaction. The presence of gravitational effects in this case is an example of how, in string theory, one-loop open string interactions, that is, Yang-Mills interactions, are related to tree level, closed string interactions, that is, gravitational interactions. In Figure 4, we depict, for illustrative purposes, two parallel Dp-branes in close proximity. The two Dp-branes can interact through open strings which connect the two Dp-branes. In the figure, we show the one-loop vacuum graph for such an interaction which can be interpreted as an open string moving in a loop. Alternatively, the interaction can be interpreted as a closed string being exchanged between two Dp-branes. In a certain sense, spin-2 gravitons, that is, the closed strings in their massless state, comprise a bound state of spin-one Yang-Mills bosons, that is, open strings in their massless state.Figure 4
One-Loop String Diagram. Shown is the diagram for an open string, whose end points are fixed on two differentDp-branes, propagating in a loop, τ being the time variable on the world sheet. Alternatively, interchanging τ with σ, the same diagram can be interpreted as a closed string being exchanged between the two Dp-branes.Prior to the application of theT-duality transformation, the open strings can propagate anywhere in the D4-brane. After the T-duality transformation, the open strings are constrained to propagate only within D3-branes, whereas the closed strings can still propagate in the bulk region between the D3-branes; that is, gravitons can propagate in spacetime dimensions not allowed for the Yang-Mills bosons. Based on certain general assumptions, Weinberg and Witten have shown the impossibility of constructing a spin-2 graviton as a bound state of spin-1 gauge fields [12]. One of the assumptions on which the proof is based is that the spin-1 gauge bosons and spin-2 gravitons propagate in the same spacetime dimensions. In the example presented here, this assumption is violated so that the conclusion of their theorem is avoided. These dyon solutions, thus, provide a simple example of gauge/gravity duality, which is discussed in detail in the work of Polchinski [13].


## 2.1. Dyon Solutions onD4-Branes

The dyon solutions are obtained from the equations of motion derived from the action,S, which describes the coupling of closed string fields to a general Dp-brane (which in our case is p=4). The action is [6] (1)S=SDBI+SWZ,where(2)
S
DBI
=
-
τ
p
∫
M
p
+
1
S
T
r
e
-
Φ
-
1
·
det
⁡
G
A
B
+
B
A
B
+
2
π
α
′
F
A
B
1
/
2
,
(3)
S
WZ
=
μ
p
∫
M
p
+
1
∑
p
′
C
p
′
+
1
∧
Tr
⁡
e
2
π
α
′
F
+
B
.Here, τp is the physical tension of the Dp-brane, and μp is its R-R charge (see Appendix B for a discussion of the relationships among the various string parameters).The dyon solutions are based on the following ansatz. The dilaton background,Φ, is constant: (4)Φ=Φ0.And background field B vanishes: (5)BAB=0A,B=0⋯4.The metric G is given by (6)GAB=G~ABIn,where, for our purposes, G~AB is restricted so that G~00=-1 and G~44=1.Forp=4, we can reexpress the determinant in (2) as (7)
det
⁡
G
A
B
+
2
π
α
′
F
A
B
=
det
⁡
G
~
A
B
I
n
+
2
π
α
′
2
!
F
A
B
F
A
B
-
2
π
α
′
2
5
-
4
!
3
2
·
1
2
F
∧
F
∗
E
F
∧
F
∗
E
,where (8)F∧F∗E=detG~AB4!FABFCDϵABCDE.See Appendix C, (C.21), for further details.The termIn is the n-dimensional identity matrix. The value of n is the dimension of the group U(n) associated with the gauge fields residing on the D4-branes. All R-R potentials vanish, except for the one-form potential C(1), which is a constant background field, (9)C1=C4dx4,for some constant value C4. The gauge field, F, is obtained from the gauge potential A(A=AEdxE), where(10)Aμ=Aμxi,μ=0⋯3;i=1⋯3,A4=A4xi.Note that the gauge potentials are static; that is, they do not depend on time, x0, and also do not depend on the spatial coordinate x4. The gauge field F(F=FABdxA∧dxB), a Lie algebra-valued two-form, is given by (11)F=dA-iA∧A.(See Appendix A.) The components of the potentials A0 and A4 are constrained in accordance with the condition (12)A0∧A4=0so that F04=0. To facilitate its interpretation, we express FAB as a five-dimensional matrix which is explicitly partitioned into electric and magnetic fields which reside in four-dimensional spacetime and an additional component of the magnetic field which resides in the additional space dimension; that is,(13)FAB=0E1E2E30-E10B3-B2-D1A4-E2-B30B1-D2A4-E3B2-B10-D3A40D1A4D2A4D3A40.We are seeking dyon solutions. Therefore, with foresight, we make the following assumptions:(14)Ei≡F0iaTa=F0iiTi,Bi≡12ϵijkFjkaTa=12ϵijkFjkiTi,DiA4≡∂iA4-iAi∧A4aTa=F4iaTa=F4iiTi.The parenthetical index (i) indicates that there is no summation of that index; however, if an expression contains two indices i without parentheses, then summation of these two indices is implied. Furthermore, each matrix element in (13) includes a generator of U(n); for example, Ei=Ei(i)T(i). Because we are seeking dyon solutions, we may assume without loss of generality that each T(i) is a generator in the fundamental representation of a local U(1)×SU(2) subgroup of SU(n) (see (44a), (44b), (44c), and (44d)).The action, (2), can be more straightforwardly interpreted from the perspective of four-dimensional spacetime. Since the action does not depend on the coordinate x4, we can trivially eliminate x4 from the action by integrating the x4 coordinate. As a result of the integration, the tension of the D4-brane, τ4, and the Yang-Mills coupling constant, gD4, are replaced by those of the D3-brane, τ3 and gD3 (see (B.3) and (B.5)). Let the size, R4, of the x4-dimension become vanishingly small; that is, R4→0. Then, the field A4 becomes a Lorentz scalar transforming as the adjoint representation of the gauge group, and (14) gives the covariant derivative of A4. From the perspective of four spacetime dimensions, A4 assumes the role of a Higgs boson transforming as the adjoint representation of the gauge group.Substituting (4)–(6) and (13) into (1) and then integrating the x4 coordinate, we obtain (15)SDBI=∫d3+1ξLDBI,where (16)LDBI=-12πα′2gD32STr⁡det⁡GAB+2πα′FAB1/2=-det⁡G~ij2πα′2gD32STr⁡L′.The function L′ is defined as (17)
L
′
=
I
n
-
2
π
α
′
2
E
·
E
-
B
·
B
-
D
A
4
·
D
A
4
+
2
π
α
′
4
B
·
D
A
4
2
-
2
π
α
′
4
E
·
B
2
-
2
π
α
′
4
E
×
D
A
4
·
E
×
D
A
4
1
/
2
.We have used the fact that |det(G~ij)|=|det(G~AB)|. In (16), the ordering of the generators of the algebra, Ta, corresponds with the order of the fields as they appear in the equation; for example,(18)B·DA42=B·DA4B·DA4=G~ijBiiTiDjA4jTj·G~ijBiiTiDjA4jTj.Note that “STr” indicates that the trace is calculated symmetrically; that is, the trace is symmetrized with respect to all gauge indices [6, 7]. The implication is that the evaluation of the trace requires that after the expansion of (16) in powers of the field strengths, all orderings of the field strengths are included with equal weight; that is, products of Ta are replaced by their symmetrized sum, before the trace is evaluated. This is discussed in detail in [6, 7].In (16), the dot product and cross product of two 3-vectors, for example, E and DA4, are defined as E·DA4=G~ijEiDjA4 and (E×DA4)i=εijkEjDkA4.In obtaining (16), we have reexpressed the dilaton Φ, on a D4-brane, in terms of the dilaton Φ′, on a D3-brane, both of which are related by a T-duality transformation in the x4-dimension. Specifically, Φ and Φ′ are related by eΦ′=α′1/2eΦ/R4. The constant dilaton background ϕ0 has been incorporated into the physical tension τp (see Appendix B).Substituting (5), (9), (6), and (13) into (3), we obtain (19)SWZ=μ42!∫M5C1∧Tr⁡2πα′F∧2πα′F.Integrating the x4 coordinate in (19), we obtain (20)SWZ=∫d3+1ξLWZ,where(21)LWZ=θ4π2Tr⁡F∧F=det⁡G~ijθ4π2Tr⁡E·B=det⁡G~ijθ4π2EiiBiiSTr⁡TiTi=det⁡G~ijθ8π2EiiBii,where Ei(i)=F0i(i)T(i). Here, (22)θ≡C42!2πR4α′1/2.In obtaining (21), we have explicitly evaluated μ4 using (B.4). Equation (21) is associated with the Witten effect. Witten has demonstrated that adding term (21) to the Lagrangian of Yang-Mills theory does not alter the classical equations of motion but does alter the electric charge quantization condition in the magnetic monopole sector of the theory [5, 8, 9]. In summary, the action, S, for the D4-brane is given by(23)S=∫d3+1ξL,where(24)L=LDBI+LWZ.The equations of motion which are obtained from (23) are (25)DμPμν=0,where (26)Pμνa=∂L∂Fμνa.In addition, the fields Fμνa satisfy the Bianchi identity(27)DαFβγa=0.To facilitate the ensuing analysis, we transform the Lagrangian density,L, to the Hamiltonian density, H, using the Legendre transformation(28)H=STr⁡P0·E-L,where (29)P0ii≡∂L∂Eii=-det⁡G~ij2πα′2gD32STr⁡XiiL′+det⁡G~ijθ4π2BiiSTr⁡TiTi,where (30)Xii=EiTi+E·BTiBi+DA4×E×DA4iTi.After performing detailed calculations, we obtain (31)H=det⁡G~ij2πα′2gD32STr⁡H′,where (32)
H
′
=
I
n
+
2
π
α
′
2
P
0
·
P
0
+
B
·
B
+
D
A
4
·
D
A
4
+
2
π
α
′
4
B
·
D
A
4
2
+
P
0
·
D
A
4
2
+
P
0
×
B
2
+
D
A
4
×
P
0
×
B
2
2
π
α
′
-
2
I
n
+
D
A
4
·
D
A
4
1
/
2
.The electric fieldEi(i) can be expressed as a function of P0i(i): (33)Eii=∂H∂P0ii=det⁡G~ij2πα′2gD32STr⁡YiiH′.The term Yi(i) is given by(34)
Y
i
i
=
2
π
α
′
2
P
0
i
T
i
+
2
π
α
′
4
·
D
i
A
4
T
i
P
0
·
D
A
4
+
B
×
P
0
×
B
i
T
i
-
B
×
D
A
4
i
T
i
P
0
·
B
×
D
A
4
2
π
α
′
-
2
I
n
+
D
A
4
·
D
A
4
.We seek dyon solutions which are BPS states, that is, whose energyE (E=∫‍d3ξH) is a local minimum. First, we reexpress H(35)
H
=
det
⁡
G
~
i
j
2
π
α
′
2
g
D
3
2
STr
⁡
I
n
+
2
π
α
′
2
cos
⁡
ϕ
P
0
·
D
A
4
+
sin
⁡
ϕ
B
·
D
A
4
2
+
2
π
α
′
2
sin
⁡
ϕ
P
0
·
D
A
4
-
cos
⁡
ϕ
B
·
D
A
4
2
+
P
0
-
cos
⁡
ϕ
D
A
4
2
+
B
-
sin
⁡
ϕ
D
A
4
2
+
2
π
α
′
4
P
0
×
B
2
+
D
A
4
×
P
0
×
B
2
2
π
α
′
-
2
I
n
+
D
A
4
·
D
A
4
1
/
2
.The mixing angle,ψ, between the electric and magnetic fields of the dyon is defined as (36)tan⁡ψ=gmge.The quantities gm and ge are the electric and magnetic charges, respectively, of the dyon. The energy, E, is minimized by constraining the dyon solutions to satisfy(37a)P0=cos⁡ψDA4,(37b)B=sin⁡ψDA4.In (35), the second and third squared terms are zero as a consequence of the constraint. Since P0∝B, the fourth squared term is also zero by virtue of (38)P0×Bk=P0iiB0jj-P0jjB0ii2!TiTjεijk-ifijkTk.Thus, H simplifies so that the energy is (39)
E
=
1
2
π
α
′
2
g
D
3
2
∫
d
3
ξ
det
⁡
G
~
i
j
Tr
⁡
I
n
+
2
π
α
′
2
cos
⁡
ϕ
P
0
·
D
A
4
+
sin
⁡
ϕ
B
·
D
A
4
.Substituting (37a) and (37b) into (32) through (34) and using (39), we find(40)E=P0.In (39), there are two terms which contribute to the mass of the system. The first term within the trace, that is, In, corresponds to the volume of each coincident D4-brane (or D3-brane), which is infinite because the D-branes are not compact. The second term, by virtue of the equations of motion, (26), and the Bianchi identity, (27), can be expressed as a divergence and is therefore a topological invariant. The second term corresponds to the mass of the dyon and is proportional to ge2+gm2 as discussed below.The solutions to (25) and (27) can be straightforwardly obtained from the dyon solutions derived in [10]. Adapting the notation of [10] to the notation used here, we express the vector potential A, (10), in the form (in accordance with our conventions, the Yang-Mills coupling constant appears explicitly in the Lagrangian (A.1). In [8, 10], the coupling constant has been incorporated into the Yang-Mills fields. Thus, to compare results here with those in the references, the fields A and related fields should be divided by gD3)(41)A=Aμdxμ+A4dx4=cos⁡ψSrgD3vα1Trdt+WrTθsin⁡θndϕ-Tϕdθ+gD3vα2T⊥+Qrα1Trdx4,where v is an arbitrary constant. For the Lie group SU(n)(42a)α1=n2n-1,(42b)α2=-n-22n-1.Here,Ti(i=r,θ,ϕ) constitute a representation of the SU(2) subalgebra and T⊥ commutes with each Ti. The quantities r,θ,ϕ are the spherical polar coordinates in three dimensions. The elements Tr,Tθ,Tϕ are related to Tx,Ty,Tz:(43a)Tr=Txsin⁡θcos⁡nmϕ+Tysin⁡θsin⁡nmϕ+Tzcos⁡θ,(43b)Tθ=Txcos⁡θcos⁡nmϕ+Tycos⁡θsin⁡nmϕ-Tzsin⁡θ,(43c)Tϕ=-Txsin⁡nmϕ+Tycos⁡nmϕ.For SU(n), the n-dimensional matrices Tx,Ty,Tz, and T⊥ are given by(44a)Tx=120⋯000⋮⋱⋮⋮⋮0⋯0000⋯0010⋯010,(44b)Ty=120⋯000⋮⋱⋮⋮⋮0⋯0000⋯00-i0⋯0i0,(44c)Tz=120⋯000⋮⋱⋮⋮⋮0⋯0000⋯0100⋯00-1,(44d)T⊥=12nn-2-20⋯000⋱⋮⋮⋮⋮⋯-2000⋯0n-200⋯00n-2.Tx, Ty, Tz, and T⊥ are suitable linear combinations of specific elements of the Cartan subalgebra of SU(n) (see [10] for details). The value of the integer nm in (43a), (43b), and (43c) is the integer multiple of the fundamental unit of dyon’s magnetic charge.These results differ from those of [10]. For a direct comparison, first replace the azimuthal angle, ϕ, in [10] with ϕ′ and extend the domain from [0,2π] to [0,2πnm]; that is, ϕ′∈[0,2πnm]. Now, perform the change of variables ϕ′=nmϕ to the dyon solutions of [10] to obtain those given in (41). In addition, apply the same change of variables to the metric in [10] to obtain the metric G~ij: (45)G~ij=1000r2000r2nm2sin2⁡θ.Here, r∈[0,∞], θ∈[0,π], and ϕ∈[0,2π]. This generalizes the results of [10] which only applies to dyons with one unit of magnetic charge; that is, gm=1/gD3.The solutionsW(r), Q(r), and S(r) are obtained as in [10](46a)Wr=wx=1-xsinh⁡x,(46b)Qr=qx=coth⁡x-1x,(46c)Sr=sx=qx=coth⁡x-1x,where the dimensionless variable x is related to the radial coordinate r: (47)x=sin⁡ψgD3vα1r.The field tensorFAB of a dyon with electric charge ge and magnetic charge gm, (48)gm=nmgD3,can now be obtained from (41). Specifically, (49)Ftr=gegS′rgD3vα1Tr,Ftθ=1-WrgegSrgD3vα1Tθ,Ftϕ=1-WrgegSrgD3vα1nmsin⁡θTϕ,Frθ=-W′rTϕ,Fϕr=-W′rnmsin⁡θTθ,Fθϕ=-Wr2-Wrnmsin⁡θTr,(50)DrA4=Q′rgD3vα1Tr,DθA4=1-WrQrgD3vα1Tθ,DϕA4=1-WrQrgD3vα1nmsin⁡θTϕ.We now show that gauge invariance of action, (23), implies SL(2,Z) invariance. Consider U(1) gauge transformations which are constant at infinity and are also rotations about the axis A^4=A4/|A4|, specifically the gauge transformations [8] (51)δAμa=1gD3vα1DμA4a.Action (23) is invariant under these gauge transformations. According to the Noether method, the generator of these gauge transformations, N, is given by (52)N=∂L∂∂0AμaδAμa.Substituting the Lagrangian density (24) into (52), we obtain (53)N=GegD3+gD32θGm8π2,where(54a)Gm=1vα1∫d3ξSTr⁡DA4·B,(54b)Ge=1gD32vα1∫d3ξTr⁡DA4·P0are the magnetic and electric charge operators. Since rotations of 2π about the axis A^4 must yield the identity for physical states, that is, (55)e2πiN=1,applying the U(1) transformation on the left side of (55) to states in the adjoint representation of SU(n), we find that the eigenstates of N are quantized with eigenvalue (56)N=α1η,where η is an arbitrary integer. Substituting (56) into (53), we obtain (57)ge=α1ηgD3-θ′2πnmgD3,where we have defined θ′ by (58)θ≡α1θ′,and used the fact that (59)gmgD3=nm4π.Taking θ′=0 in (57), we obtain the quantization condition for the electric charge (60)ge=ηα1gD3.The electromagnetic contribution to the mass (rest energy) of the dyon, mem, can be obtained by substituting (54a) and (54b) into (39) and integrating the second term within the trace to obtain (61)mem=vα1ge2+gm2.We can now makeSL(2,Z) symmetry explicit. We first define (62)τ=θ′2π+4πgD32i.If θ′=0, then the weak/strong duality condition gD3→gm=(4π)/gD3 is equivalent to (63)τ⟶-1τ.In (57), the transformation θ′→θ′+2π results in identical physical systems with only states being relabeled. The transformation is equivalent to (64)τ⟶τ+1.Transformations (63) and (64) generate the group SL(2,Z). See [8, 9] for further details.Note that in (54a) and (54b) Ge is, strictly speaking, not the electric charge operator because P0 is not the electric field but rather is its conjugate; however, according to (33) and (34), if DA4 and B become vanishingly small for asymptotically large values of the radial coordinate, then P0 approaches E. Thus, in the asymptotic limit Ge is the electric charge operator. This distinguishing feature is a direct consequence of the fact that our analysis is based on the Born-Infeld action rather than the Yang-Mills-Higgs action. In our case, this point is inconsequential since P0=E, exactly.


## 2.2. Dyon Solutions onD3-Branes

As emphasized previously, the dyon solutions derived in Section2.1, when interpreted from 3+1 spacetime dimensions, that is, the compactified theory in which R4→0, are the ’t Hooft/Polyakov magnetic monopole or dyon, with the potential A4 being a Higgs boson transforming in the adjoint representation of the gauge group U(n). Here, our purpose is to reinterpret these dyon solutions in which R4→0 from the equivalent T-dual theory. In the T-dual theory, the radius R4 is replaced by R4′ (R4′=α′/R4) so that the radius of the x4-dimension R4′→∞. In addition, the potential A4 is reinterpreted as the x4-coordinates of the n  D3-branes embedded in 4 + 1 dimensional spacetime. These coordinates can be directly obtained by diagonalizing A4, (41), using a local gauge transformation which rotates Tr into Tz. The n  x4-coordinates are the diagonal elements of the matrix; that is (we are assuming that after the T-duality transformation the D3-branes are far from any orientifold hyperplanes. This can always be accomplished by adding the to A4 component of the gauge potential a constant U(1) gauge transformation θ0T0, θ0 being a suitable constant (see Appendix A)),(65)A4⟶2πα′gD3vα2T⊥+Qrα1Tz=2πα′gD3vu10⋯000⋱⋮⋮⋮⋮⋯u1000⋯0u2+u~r00⋯00u2-u~r,where(66a)u1=-α2nn-2,(66b)u2=α22n-2n,(66c)u~r=α12Qr.Of the n  D3-branes n-2 of the D3-branes, denoted by D3n-2, are coincident. The x4-coordinate of each is the constant value 2πα′gD3vu1. For the remaining two D3-branes, denoted by D31 and D32, the x4-coordinate of each is a function of the radial coordinate r. Specifically, x4=2πα′gD3v(u2-u~(r)) for D31 and x4=2πα′gD3v(u2+u~(r)) for D32, and as a consequence these two D3-branes have nonvanishing intrinsic curvature. This occurs despite the fact that before the application of the T-duality transformation no gravitational interaction is explicitly present. We now introduce the length scale LD3 which is the separation between D31 and D32, in the asymptotic limit as the radial coordinate r→∞. It is related to previously defined parameters by (67)LD3=2πα′gD3vα1.Another relevant length scale is the size of the dyon, that is, the region of space where all components of the Yang-Mills field, FAB, are nonvanishing. According to (49), (46a), (46b), (46c), and (47), only the radial components of the electric and magnetic fields are long range, with the remaining components of the fields vanishing exponentially for x≫1. Thus, additional structure of the dyon becomes apparent whenever x≲1 or equivalently whenever r≲1/(sin⁡ψgD3vα1). We can therefore define the size of dyon Ld, as measured from asymptotically flat space, that is, r→∞, to be (68)Ld=1sin⁡ψgD3vα1=2πα′sin⁡ψLD3.In Figure 1, we show, for the gauge group SU(5), embedding plots of the 5 D3-branes as a function of the dimensionless radial coordinate, x (x=r/Ld). As x→∞, the x4-coordinate of D32 approaches that of the (5–2) coincident D3-branes, D35–2, in effect, joining them by a wormhole in an asymptotically flat region of space. At x=0, D32 is joined to D31 by another wormhole. (See [11] for a recent discussion of thin shell wormholes exhibiting cylinder symmetry.) As we will show, in general, the intrinsic curvature of the two surfaces in the neighborhood of x=0 is relatively large but, nonetheless, finite. Although these features described in Figure 1 apply to the particular gauge group SU(5), they apply to all SU(n), n≥2. (For n=2 there are no coincident D3-branes.)Figure 1
Embedding Functions of the5  D3-branes for the gauge group SU(5). The scaled coordinate x4/LD3 is plotted as a function of the scaled radial coordinate, r/Ld, for the 5  D3-branes. The radial coordinate, r, is scaled by the size of the dyon, Ld, and the x4 coordinate is scaled by the separation between the two D3-branes D31 and D31 in the asymptotic limit of large r.We now consider in detail the twoD3-branes, D1 and D2, whose x4-coordinates are radial dependent. The T-duality transformation on the x4-dimension pulls back the metric onto D1 and D2, inducing the metric, Gij′, (69)Gij′=G~ijIn+DiA4iTiDjA4jTjδijG~44In.Expanding the right-hand side of (69), we obtain(70)Gij′=G~ij0⋯000⋱⋮⋮⋮⋮⋯G~ij000⋯0G~ij+A~ij00⋯00G~ij+A~ij,where (71)A~ij=gD3Lα122×Q′r2000W~2r000W~2rnm2sin2⁡θ.Here, (72)W~r=1-WrQr,and G~ij is given by (45). In obtaining (70), we have used (50) and the fact that the matrices Tr, Tθ, and Tϕ, (43a), (43b), and (43c), satisfy the relationship (73)Tr2=Tθ2=Tϕ2=1220⋯000⋮⋱⋮⋮⋮0⋯0000⋯0100⋯001.Each of the diagonal entries in the matrix Gij′ corresponds to the metric on one of the n  D3-branes obtained from the T-duality transformation. In the case of the first n-2 entries, corresponding to the D3-branes D3n-2, the metric is flat. In the case of the last two entries corresponding to the D3-branes, D31 and D32, their geometries are identical and intrinsically curved. The only feature which distinguishes these two D3-branes is that the function Q(r) defining the x4-coordinate for D3-brane D2 is replaced by -Q(r) for D31, as evidenced in (65) and (66a) and (66b) and (66c) and also in Figure 1. As a consequence, the electric and magnetic charges of the dyon on D2 are minus the values on D1. The electric and magnetic field lines enter the wormhole from one D3-brane and exit from the other. Figure 2 is an embedding diagram showing the D3-branes D1 and D2 in the neighborhood of the radial coordinate r=0. As there is no event horizon surrounding r=0, the two D3-branes are joined by a wormhole at r=0.Figure 2
Wormhole. Shown is the embedding diagram for the twoD3-branes, D31 and D32, with azimuthal angle suppressed. The domain of the radial coordinate, r, is 0≤r≤15Ld, and the range of the embedding coordinate x4 is -.45LD3≤x4≤+.45LD3.Of particular interest is the intrinsic scalar curvature ofD31 and D32, in the neighborhood of r=0. The scalar curvature can be calculated from the metric Gij′, (70), and its value R(0) at r=0 is (74)R0=216sin⁡ψLD36sin3⁡ψLD34sin2⁡ψ+12πα′22.For a given value of sin⁡ψ, R(0) assumes its maximum value R~(0): (75)R~0=2738sin⁡ψπα′,when LD3=L~D3, where (76)L~D3=123πα′sin⁡ψ1/2.For either LD3→0 or LD3→∞, the scalar curvature R(0)→0; that is, the geometry of D1 and D2 becomes flat, everywhere. The expression for the scalar curvature R(r) is a complicated function of r and not amenable to straightforward interpretation and, therefore, will not be given. In Figure 3, we show a plot of the scalar curvature as a function of the radial coordinate. In this example, LD3=α′, and the dyon has only one unit of magnetic charge so that sin⁡ψ=1. Near r=0, the scalar curvature is positive and finite. As r increases, the scalar curvature becomes slightly negative and asymptotically approaches zero as r→∞. These features of the scalar curvature described for this specific example also apply in general.Figure 3
Scalar curvatureRr. For the case LD3=α′ and sin⁡ψ=1, the dimensionless scalar curvature, R×α′, is plotted as a function of the scaled radial coordinate, r/Ld.Consider dyon solutions for whichLD3≈α′ or less. The F-strings connecting D31 and D32 would be in their ground state, a BPS state. In addition, assume that gD3≪1; then as r→0 from an asymptotically flat region of space, within either D31 or D32, the string length scale will be reached before the gravitational interaction becomes dominant at the length scale of O(gD31/4α′) [5]. Thus, action, (2), which does not include the gravitational interaction, should apply, and consequently the dyon solutions derived should be accurate. On the other hand, let gD3→1/gD3 so LD3≫1; then the D-string, also a BPS state, becomes lighter than the F-string. As a consequence of weak/strong duality, the dyon solutions should still be applicable with the F-strings being replaced by D-strings and the dyon electric and magnetic charges being interchanged.After applying aT-duality transformation to the dyon solutions obtained in Section 2.1, we have obtained dyon solutions residing on D3-branes where the effect of the gravitational interaction is apparent. This occurs despite the fact that action, (2), does not explicitly include the gravitational interaction. The presence of gravitational effects in this case is an example of how, in string theory, one-loop open string interactions, that is, Yang-Mills interactions, are related to tree level, closed string interactions, that is, gravitational interactions. In Figure 4, we depict, for illustrative purposes, two parallel Dp-branes in close proximity. The two Dp-branes can interact through open strings which connect the two Dp-branes. In the figure, we show the one-loop vacuum graph for such an interaction which can be interpreted as an open string moving in a loop. Alternatively, the interaction can be interpreted as a closed string being exchanged between two Dp-branes. In a certain sense, spin-2 gravitons, that is, the closed strings in their massless state, comprise a bound state of spin-one Yang-Mills bosons, that is, open strings in their massless state.Figure 4
One-Loop String Diagram. Shown is the diagram for an open string, whose end points are fixed on two differentDp-branes, propagating in a loop, τ being the time variable on the world sheet. Alternatively, interchanging τ with σ, the same diagram can be interpreted as a closed string being exchanged between the two Dp-branes.Prior to the application of theT-duality transformation, the open strings can propagate anywhere in the D4-brane. After the T-duality transformation, the open strings are constrained to propagate only within D3-branes, whereas the closed strings can still propagate in the bulk region between the D3-branes; that is, gravitons can propagate in spacetime dimensions not allowed for the Yang-Mills bosons. Based on certain general assumptions, Weinberg and Witten have shown the impossibility of constructing a spin-2 graviton as a bound state of spin-1 gauge fields [12]. One of the assumptions on which the proof is based is that the spin-1 gauge bosons and spin-2 gravitons propagate in the same spacetime dimensions. In the example presented here, this assumption is violated so that the conclusion of their theorem is avoided. These dyon solutions, thus, provide a simple example of gauge/gravity duality, which is discussed in detail in the work of Polchinski [13].


## 3. Conclusions

We have investigated dyon solutions within the context of superstring theory. Beginning with type ISO(32) superstring theory in ten dimensions, six of the spatial dimensions of which are compact, we have applied the group of T-duality transformations to five of the compact dimensions. The result is 16 D4-branes, a number n (2≤n≤16) of which are coincident. The five T-dualized dimensions, whose size is taken to be vanishingly small, become the five internal spacetime dimensions while the remaining five dimensions correspond to the external 4+1-dimensional spacetime. Making a suitable ansatz for the gauge fields residing on the n coincident D4-branes, we have obtained dyon solutions from an action consisting of two terms: the 4+1-dimensional, non-Abelian Dirac-Born-Infeld action and a Wess-Zumino-like action. The former action gives the low energy effective coupling of D4-branes to NS-NS closed strings and the latter of D4-branes to R-R closed strings. The method of solution involves transforming the 4+1-dimensional action from the Lagrangian formalism to the Hamiltonian formalism and then seeking solutions which minimize the energy. The resulting dyon solutions, which are BPS states, reside on the n  D4-branes and are therefore associated with a supersymmetric U(n) gauge theory in 4+1 spacetime dimensions. These dyon solutions can be alternatively understood in the limit when the size of the remaining compact spacetime dimension, x4, approaches zero. In this situation, the 4+1-dimensional spacetime is reduced to a 3+1-dimensional spacetime. As a consequence, the A4 component of the vector potential becomes a Lorentz scalar with respect to 3+1-dimensional spacetime and can be interpreted as a Higgs boson transforming as the adjoint representation of the U(n) gauge group, analogous to the Higgs boson associated with the ’t Hooft/Polyakov magnetic monopole. Finally, we perform a T-duality transformation in the x4-direction. As a result, n-2 of the D4-branes are transformed into n-2 coincident D3-branes, whose intrinsic geometry is flat. The remaining two D4-branes are transformed into two separate D3-branes whose intrinsic geometry is curved. As depicted in Figure 3, the two D3-branes are joined by wormhole at r=0. The scalar curvature of each D3-brane reaches a maximum, finite value, at r=0 and approaches zero as r→∞. The dyon resides on these two D3-branes. Furthermore, the values of electric and magnetic charges of the dyon on one D3-brane are minus the values on the other D3-brane, and as a consequence the electric and magnetic field lines enter the wormhole from one D3-brane and exit from the other D3-brane.TheT-duality transformation in the x4-direction causes two of the D3-branes to acquire intrinsic curvature. This occurs despite the fact that the Lagrangian density from which the dyon solutions have been obtained does not explicitly include the gravitational interaction. This can be understood heuristically from the open string, one-loop vacuum graph given in Figure 4. From one perspective, the graph describes an open string, whose ends are fixed on two different D3-branes, moving in a loop, or, alternatively, the exchange of a closed string between two D3-branes. Thus, the gravitational interaction, that is, the closed string interaction, and the Yang-Mills interaction, that is, the open string interaction, appear as alternative descriptions of the same interaction. This simple example is suggestive of the subtle, but profound, connection between the Yang-Mills and gravitational interactions, specifically gauge/gravity duality.



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