Variations on the Dirac String

PHYSICAL REVIEW D 98, 105013 (2018)

Variations on the Dirac string

Brad Cownden* Department of Physics & Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada † Andrew R. Frey Department of Physics, University of Winnipeg, 515 Portage Avenue, Winnipeg, Manitoba R3B 2E9, Canada and Department of Physics & Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada

(Received 1 August 2018; published 27 November 2018)

Dirac’s original solution of the nontrivial Bianchi identity for magnetic monopoles [1], which redefines the field strength along the Dirac string, diagonalizes the gauge and monopole degrees of freedom. We provide a variant of the Dirac string, which we motivate through a formal expansion of the Bianchi identity. We show how to use our variant prescription to study monopole electrodynamics without reference to a dual potential and provide some applications.

DOI: 10.1103/PhysRevD.98.105013

I. INTRODUCTION dual potential must be superposed in a “democratic” formalism (which makes it ill-suited to the quantum Magnetic monopoles, while experimentally elusive [2], mechanics of charges and monopoles). This type of hold a special place in particle physics; they complete approach has been advocated at least since [5]. electric-magnetic duality, provide a reason for charge (iii) As pointed out originally in [6,7], the vector poten- quantization, and arise in the spontaneous breaking of tial is not a globally defined function but a section of many grand unification models. At the same time, there is a a fiber bundle. When the field strength has a non- long tradition of reformulating the description of the trivial Bianchi identity, the potential is defined in at monopole’s interaction with the electromagnetic field since ’ least two coordinate patches; the potentials in the Dirac s original work [1,3]: two patches are related by a gauge transformation in (i) Monopoles arise as solitons of broken gauge theo- the overlap of the two patches. While mathemati- ries; the microscopic description as a semiclassical ’ cally rigorous, treating the potential in this manner field configuration describes the monopole s inter- mixes the gauge and monopole d.o.f. because the actions but requires tracking the full degrees of overlap region moves with the monopole. This freedom (d.o.f.) of that gauge theory. While of formalism also obscures the coupling between the course necessary for processes such as monopole potential and monopole (e.g., [7] switches to a dual creation and annihilation (see e.g., [4]), it may be potential formalism to derive the monopole’s computationally excessive when an effective de- classical equation of motion, while [8] following scription is valid. [9] combined Dirac’s formalism below with the (ii) In the absence of electric charges, the field strength ⋆ ≡ ˜ rigorous gauge patching procedure). can be defined in terms of a dual potential F dA, (iv) The potential in a single coordinate patch extends to which couples to monopoles in the same way the all of spacetime except for a half-line singularity vector potential couples to charges. Of course, if the extending from the monopole (Dirac’s famous string fields of both charges and monopoles are of interest, singularity). In [1], Dirac noted that the electromag- the field strengths from both the potential and the netic field strength can be written in terms of a globally defined potential and an extra term sup- *[email protected] ported on the string singularity. This approach † [email protected] separates the electromagnetic and monopole d.o.f. and elucidates the monopole-field strength coupling. Published by the American Physical Society under the terms of However, it suffers some conceptual difficulties, the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to including a singular field strength along the (arbi- the author(s) and the published article’s title, journal citation, trarily chosen) string and a constraint that electric and DOI. Funded by SCOAP3. charges cannot intersect the Dirac string world

2470-0010=2018=98(10)=105013(7) 105013-1 Published by the American Physical Society BRAD COWNDEN and ANDREW R. FREY PHYS. REV. D 98, 105013 (2018) Z sheet. Pragmatically, the semi-infinite string can be μ ¯ ν ¯ μ 4 ð⋆ d ⋆ GÞ ¼ dY ∧ dY ∂νδ ðx − YÞ awkward. NZ In this paper, we present a new hybridization of Dirac’s ¯ 4 ¯ μ string formalism with the rigorous gauge patching pre- ¼ − d½δ ðx − YÞdY scription by allowing the Dirac string to end on an N 1 ˜μ − 1 ˜μ unphysical reference monopole, and we review the deri- ¼ð =gÞ| ð =gÞ| ð4Þ vation of Dirac’s string prescription from the dual potential M M 1 formalism with an emphasis on the origin of the string. We in terms of the currents of monopoles along and . also advocate for a particular configuration for the string As a result, the field strength defined as F ¼ dA − g ⋆ G and give a simple physical picture for its origin. The fixed automatically solves the Bianchi identity for the dynamical M prescription for the string means that the string world sheet monopole along as long as A includes the potential for a M embedding coordinates depend on the monopole position monopole with reference world line (with appropriate along the entire world sheet, unlike Dirac’s original gauge patching). formalism; our approach presents computational advan- It is worth pausing now to discuss the configuration of ’ tages in several circumstances. The extension of our results the string. In Dirac s formalism, the string is a dynamical to higher-dimensional monopolelike branes appears in the object with no kinetic term (in modern parlance, a tension- less string), so the world sheet N is completely arbitrary companion paper [10] by one of us. Throughout this paper, 2 we work in four-dimensional Minkowski spacetime for except for the specification of its boundary. Dirac fur- M simplicity. We list our conventions, particularly signs, in thermore locates the worldline at spatial infinity, so the the Appendix. string becomes semi-infinite. However, we are free to specify the configuration of the string; a simple choice in a given reference frame is to choose the reference II. DIRAC’S STRING VARIABLES M ⃗ monopole to be stationary ( at constant position X) ⃗ The Maxwell equations and Bianchi identities with and the string to be the straight line from X to the magnetic sources are monopole position X⃗ðtÞ at each time. We can build an arbitrary world sheet in this way by then extending a Dirac M M d ⋆ F ¼ −⋆j; dF ¼ −⋆|;˜ ð1Þ string from to another reference worldline , and so on, and then letting the reference worldlines approach each μ ˜μ other. On the other hand, with a single line segment, the where j is the electric current and | is the magnetic μ current. For a single monopole, the current is world sheet embedding coordinates Y depend on the monopole position at all world sheet points τ, σ unlike Z in Dirac’s formalism. We motivate this prescription for the μ μ 4 μ 3 ⃗ |˜ ¼ g dτ∂τX ðτÞδ ðx − XÞ¼gu ðtÞδ ðx⃗− XÞ; ð2Þ Dirac string configuration in Sec. IV. M It is also instructive to consider the Dirac quantization of charge for different formalisms. With distinct gauge M ’ where g is the monopole charge, is the monopole s patches, quantization enforces the requirement that a τ μ worldline, is the worldline time, and X is the monopole single-value wave function in one coordinate patch remains position. The second equality follows in a fixed reference single valued after the gauge transformation in passing to τ μ frame with a worldline static gauge ¼ t; u is the another patch. In Dirac’s original formalism, there is only a ’ monopole s 4-velocity. single gauge, but the wave function picks up a phase under ’ Dirac s key observation stems from the fact that any motion of the string equal to g times the electric flux conserved current (in Minkowski spacetime) can be written through the surface swept out by the world sheet—the ˜ as the divergence of a two-form, or ⋆| ¼ d ⋆ H for some surface is noncontractible since the charge cannot intersect H. In fact, the field of an isolated monopole provides such the world sheet, as we see below. Since the surface swept an H subject to the additional constraint of the sourceless out by the string is closed (including the point at infinity), Maxwell equations. Instead, following Dirac [1], we define charge quantization is given by the condition that the wave Z function remain single valued if we sweep the string around μν μ ν 4 the charge. When the reference monopole is at a finite G ðxÞ ≡ dY¯ ∧ dY¯ δ ðx − YÞ; ð3Þ N 1An additional sign enters at the last equality from the world where N is the world sheet of a string with boundary sheet orientation as described in the Appendix. ∂N M − M M 2This arbitrariness reflects how a gauge transformation modi- ¼ ( is a so-far arbitrary worldline), Yμ τ; σ is the string position at world sheet coordinates fies where the potential of a monopole becomes singular (or more ð Þ precisely the patching required for the potential), so the string can τ σ σ M M ¯ τ∂ σ∂ , with increasing from to , and d¼d τ þd σ never develop a tension, even quantum mechanically, for an is the world sheet exterior derivative. This has divergence unbroken gauge symmetry.

105013-2 VARIATIONS ON THE DIRAC STRING PHYS. REV. D 98, 105013 (2018) position, both mechanisms for charge quantization are Meanwhile, the electric charge couples to the redefined possible. In particular, if the string world sheet is arbitrary potential A, so its equation of motion turns out as usual M but is fixed, we can sweep the string in a closed surface except for contact terms with the Dirac string, which are around a charge, as per Dirac. With a set prescription for the forbidden due to Dirac’s condition. world sheet as discussed above, however, the world sheet It is important to note that our derivation of the monopole ⃗ equation of motion treated the potential A as an indepen- moves only if we move the reference position X, so the string sweeps out open surfaces, removing the quantization dent d.o.f. in contrast to the case without a Dirac string and condition. On the other hand, the potential must be defined with gauge patches. In that case, there is no coupling between the monopole and gauge field, only a hidden in patches around the reference point, so single-valuedness μ of the wave function still leads to charge quantization. dependence of A on X which should be treated as an The Dirac string formalism also provides a direct means explicit dependence. The Dirac string removes this explicit of finding the monopole equation of motion, which is dependence from the vector potential, but A does still have otherwise carried out indirectly through the dual potential a “hidden” explicit dependence on the arbitrary reference formalism. Including the string coupling G in the field position. The equation of motion for the reference position μνδ δ λ 0 strength, the variation of the Maxwell action with respect to is F Fμν= X ¼ , which enforces the condition that the the monopole and world sheet positions is field strength is independent of the reference position. This condition determines the explicit dependence of the poten- Z μ X δ g ϵ 4 μν δ λρ tial on . SMax ¼ 4 μνλρ d xF ðxÞ G ðxÞ Z Z g 4 μν ¯ λ ¯ ρ α 4 III. THE STRING FROM THE ¼ ϵμνλρ d x F ðxÞfdY ∧ dY δY ∂Yα δ ðx − YÞ 4 N DUAL POTENTIAL þ½d¯δYλ ∧ dY¯ ρ þ dY¯ λ ∧ d¯δYρδ4ðx − YÞg: ð5Þ The dual potential formalism is useful in many applications since it translates the electrodynamics of monopoles We can now restrict to a world sheet integral, first into the more familiar electrodynamics of charges (and can converting the derivative on the delta function to one with even allow for the interaction of charges and monopoles respect to xα and integrating by parts. Then, with all partial since electromagnetism is linear). Here we review a derivatives now with respect to Y, derivation of the Dirac string from the dual potential, Z simplified from versions presented in [11,12]. In particular, n we start with the dual field strength and potential and a δ g ϵ ¯ μν δ λ ¯ ρ SMax ¼ μνλρ d½F ðYÞ Y dY 2 N magnetic current in the absence of electric charges (which 1 lead to a Dirac string for the dual field strength) in order to ∂ μν ¯ α ∧ ¯ λδ ρ emphasize that the Poincar´e duality itself leads to the Dirac þ 2 αF ðYÞðdY dY Y o string for the field strength in the presence of monopoles. þ dY¯ λ ∧ dY¯ ρδYα þ dY¯ ρ ∧ dY¯ αδYλÞ : ð6Þ In form notation, the action for the dual electromagnetism with a single monopole is Z The latter lines of (6) reorganize to 1 − ˜ ∧⋆˜ ˜ ∧⋆˜ Z S ¼ 2 F F þ A | ; ð9Þ g ⋆ ¯ ν ∧ ¯ λδ ρ 2 ðd FÞνλρdY dY Y ; ð7Þ N where A˜, F˜ are the dual potential and field strength and |˜ is the monopole current (2). To dualize back to the “usual” which would yield an interaction between the (unphysical) potential in the absence of monopoles, we treat F˜ as the string—and therefore the monopole—with the electric independent variable and add a Lagrange multiplier term current when the gauge fields are on shell. To avoid AdF˜ to enforce the Bianchi identity for F˜ since A˜ does not this unphysical result, Dirac imposed the additional ˜ condition that charges not intersect the string. The first appear. Solving the equation of motion for F and sub- term, on the other hand, gives an integral over M stituting gives the usual Maxwell action. M 3 With the monopole current, we must first find a way to (assuming fixed), so it combines with the variation of the monopole’s kinetic term to give the magnetic Lorentz eliminate the dual potential from (9). We proceed by force equation recalling that any conserved current can be written as the divergence of a two-form (in other words, any coclosed ν form in Minkowski spacetime is coexact). In fact, the ∂τpμ ¼ −gð⋆FÞ ∂τX : ð8Þ μν Maxwell equation d ⋆ F˜ ¼ ⋆|˜ shows that |˜ can be written in terms of the monopole’s field strength (which also 3With sign determined per footnote 1. satisfies dF˜ ¼ 0). We have also seen above that the

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∇⃗ ⃗ Dirac string coupling G is another such two-form up to the where is the gradient with respect to X and subtraction of a current along the reference worldline M , δ ⃗ ⃗− ⃗ X ¼ X X. Converting one of the derivatives to one so we can write with respect to x⃗, the terms in the sum can be written as a Z divergence, so defining 1 S ¼ − F˜ ∧⋆ F˜ þ A˜ ∧ d ⋆ ðgG þ F˜ Þ 2 ⃗ ∇⃗ ⃗ Z B¼ ×A 1 ∞ ˜ ˜ ˜ ˜ 0 ˜ X 1 ¼ − F ∧⋆ F þ F ∧⋆ ðgG þ F Þ − A ∧ dF ; ð10Þ ⃗ ⃗ ⃗ 3 ⃗ 2 −gδX ðδXÞn;i1in ð∇ Þn δ ðx⃗−X Þ; ð14Þ n 1 ! i1in n¼0 ð þ Þ where F˜ is the dual field strength of the reference monopole ⃗ ⃗ and we have added the Lagrange multiplier term. This last where A includes the potential of a fixed monopole at X term can be rewritten as FdA˜ 0 up to a total derivative, so the and solves the scalar equation of (12). Subtracting this ⋆ ˜ ⋆ ˜ − 0 contribution from the vector part of the Bianchi identity equation of motion is F ¼ ðgG þ FÞ dA . The action is therefore classically equivalent to similarly extracts a curl from the expansion of the delta Z function. We can therefore define 1 − 0− ⋆ ˜ − ⋆ ∧⋆ 0− ⋆ ˜ − ⋆ S ¼ ðdA F g GÞ ðdA F g GÞ: ⃗ ⃗ ⃗ ⃗ ⃗ 2 E ¼ −∇Φ − ∂tA þ gð∂tδX × δXÞ X∞ 1 ð11Þ n þ ⃗ ⃗ 3 ⃗ × ðδXÞn;i1in ð∇ Þn δ ðx⃗− X Þð15Þ ðn þ 2Þ! i1in −⋆ ˜ ’ n¼0 We recognize F ¼ F as the reference monopole s 0 field strength, so we define dA þ F ¼ dA in terms of a to solve the Bianchi identity. potential A with the appropriate gauge patches for the To confirm that this approach separates the gauge and reference monopole. We are leftR with F ¼ dA − g ⋆ G monopole d.o.f., we can vary the Maxwell action with and the usual Maxwell action − F ⋆ F=2 including the respect to X⃗ and, with some manipulation, find the Dirac string. magnetic Lorentz force equation by treating the potentials as independent variables. As with the Dirac string, the IV. AN INTERPRETATION OF THE STRING variation of the action also includes terms proportional to the sourceless Maxwell equations and derivatives of We can gain new insight into the Dirac string through the δ3 ⃗− ⃗ ðx XÞ. Bianchi identity. For clarity, we pick a reference frame and As an alternate approach, we can consider the Dirac work in static gauge for the monopole (i.e., worldline time string with M a static worldline at X⃗ . Then, as we τ 0 ¼ Y ). In nonrelativistic notation, the Bianchi identity is suggested in the section above, take N at any fixed time to be the line segment from X⃗ to X⃗. Then we can choose a ∇⃗ ⃗ δ3 ⃗− ⃗ ∇⃗ ⃗ ∂ ⃗ − ∂ ⃗δ3 ⃗− ⃗ · B ¼ g ðx XÞ; × E þ tB ¼ g tX ðx XÞ: 5 0 τ σ τ ⃗τ σ ⃗ σδ⃗ static gauge t ¼ Y ð ; Þ¼ and Yð ; Þ¼X þ XðtÞ ð12Þ with 0 ≤ σ ≤ 1. Then the Dirac string coupling becomes Z 1 Our goal is to diagonalize the d.o.f., removing the explicit 0i σδ iδ3 ⃗− ⃗ − σδ⃗ G ¼ d X ðx X XÞ dependence of Aμ on the monopole position. We carry out a 0 Z formal expansion of the monopole current around a static X∞ 1 1 3 ⃗ δ i δ ⃗ n;i1in ∇⃗ n δ ⃗− ⃗ σσn reference monopole at fixed position X ; therefore, we end ¼ X ð XÞ ð Þi1i ðx XÞ d n! n 0 ⃗ n¼0 up with a static gauge patching procedure around X and will see that the remaining terms can be organized as a ð16Þ contribution to the field strength.4 We begin by writing and similarly

∞ 3 3 X δ ⃗− ⃗ δ ⃗− ⃗ 1 ðx XÞ¼ ðx XÞ ij ∂ iδ j − ∂ jδ i δ ⃗ n;i1in ∇⃗ n G ¼ð tX X tX X Þ ð XÞ ð Þi1i ∞ n! n X 1 n¼0 3 Z δ i1 δ in ∇ ∇ δ ⃗− ⃗ þ X X i1 i ðx XÞ; 1 n! n δ3 ⃗− ⃗ σσnþ1 n¼1 × ðx XÞ d ð17Þ 0 ð13Þ

5The argument goes through with little change for any function 4This is the approach taken in [13] for D3-branes. of τ.

105013-4 VARIATIONS ON THE DIRAC STRING PHYS. REV. D 98, 105013 (2018) after expanding the delta function. Carrying out the integral We must take care to convert the t derivative to a τ and using the usual relation between the field strength Fμν derivative and integrate by parts where the delta function and fields E⃗, B⃗, we find (14) and (15). in time is differentiated. We are left with Z So the particular solution of the Bianchi identity that we 1 found by expanding the delta function around X⃗ is a ⃗ ∂ ∂ ⃗ δ ⃗ σσδ3 ⃗− ⃗ − σδ⃗ |eff ¼ g t ð tX × XÞ d ðx X XÞ 0 particular realization of the Dirac string. As a result, we see Z that the expansion of the delta function separates the 1 − δ ⃗ ∇⃗ σδ3 ⃗− ⃗ − σδ⃗ g X × d ðx X XÞ : ð21Þ explicit dependence of the field strength on the monopole 0 position from the potential. This also tells us that the Dirac string is a way of treating a monopole’s motion as a This is worth two comments. First, the current for ⃗ fluctuation (even a large one) around a fixed position. The arbitrary linear motion of the monopole, with X chosen key difference with Dirac’s arbitrary string is that this to lie on the line, is interpretation suggests treating the embedding coordinates Z 1 of the string as dependent on the monopole position along ⃗ − δ ⃗ ∇⃗ σδ3 ⃗− ⃗ − σδ⃗ |eff ¼ g X × d ðx X XÞ : ð22Þ the entire world sheet. In fact, we have done this explicitly 0 in deriving (16) and (17). For future reference, it is useful to give the exact Taking δX⃗ to lie along the z axis, Eq. (22) is the expressions for the fields with the linear string configuration: surface current in the ϕ direction of an infinitely tightly Z ⃗ 1 wound, infinitesimally thin solenoid stretching from X ⃗ −∇⃗Φ − ∂ ⃗ ∂ ⃗ δ ⃗ σσδ3 ⃗− ⃗ − σδ ⃗ ⃗ E ¼ tA þ gð tX × XÞ d ðx X XÞ; to X. The direction and magnitude of the current are Z 0 ⃗ 1 precisely such that the flux into the solenoid at X (which ⃗ ∇⃗ ⃗− δ ⃗ σδ3 ⃗− ⃗ − σδ ⃗ B ¼ × A g X d ðx X XÞ: ð18Þ is spherically symmetric since the solenoid is thin) pre- 0 cisely cancels the magnetic flux due to the reference ⃗ ⃗ ⃗ The displacement D and field H are determined as usual from monopole at X. Similarly, the flux out of the solenoid ⃗ E⃗, B⃗in linear media, including the Dirac string contribution. at X precisely reproduces the flux of the dynamical monopole. Second, the current is conserved, as it must be, if we allow for singular distributions. The solenoidal V. THE STRING AS A SOURCE current (22) is clearly conserved, and the other terms take ρ −∇⃗ ⃗⃗ ∂ ⃗ The Dirac string in the field strength automatically the form eff ¼ · k; |eff ¼ tk, which is manifestly solves the Bianchi identity, so the Bianchi identity no conserved. longer determines the field strength associated with the In media, the effective charge and current follow from ⃗ ⃗ dynamic monopole. Instead, as Dirac realized, the string ∇ · D⃗ and ∇ × H⃗, so derivatives of the permittivity and coupling G leads to an effective electric current as a source permeability will appear in general. For piecewise constant for dA, the field strength away from the Dirac string. In dielectric constant and permeability, it is natural to define particular, d ⋆ dA ¼ −gdG, so the effective current is the effective charge and current in a piecewise manner ⋆ jeff ¼ g dG. as well. For practical use, we again choose a line segment Specifying a string configuration that depends explicitly configuration as in Sec. IV. The effective charge is on the monopole position along the world sheet gives a well-defined effective current from the string. In principle, ρ 0 eff ¼ j we can now solve for the electromagnetic fields for arbi- eff Z 1 trary monopole motion as a superposition of the magnetic − ∂ ⃗ δ ⃗ ∇⃗ σσδ3 ⃗− ⃗ − σδ ⃗ ¼ gð tX × XÞ · d ðx X XÞ ; ð19Þ field from the fixed reference monopole and the effective 0 current. The linear string configuration seems particularly where the gradient is with respect to x⃗. Evaluating the well suited to this type of calculation. effective current in the static gauge is slightly subtler. We have VI. APPLICATIONS Z Z As we have noted previously, there are technical diffi- 1 − ϵ τ σ σ∂ jδ k culties in the theory of electrodynamics with monopoles. jeff;i ¼ g i0jk d d ð τX X Þ 0 Here we present several possible applications in which the × ∂ ½δðt − τÞδ3ðx⃗− X⃗ − σδX⃗Þ linear Dirac string configuration yields simplifications. t Z 1 We indicated above that one such application is a direct ϵ σδ k∂ δ3 ⃗− ⃗ − σδ⃗ determination of radiation from moving monopoles, þ g ij0k d X j ðx X XÞ: ð20Þ 0 including energy loss in dielectric materials (such as

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Cherenkov radiation),6 which may be useful for monopole for the derivation of brane equations of motion. It is search experiments. While an infinite, arbitrarily moving particularly helpful for a careful accounting of d.o.f., as Dirac string is unwieldy, the linear Dirac string configu- needed in dimensional reduction. Determining the lower- ration gives a well-defined current, which is simply a dimensional effective action is also an off-shell calcula- growing or shrinking solenoid in many contexts. In the tion, and extra terms from the generalized Dirac string presence of materials, as noted, it is necessary to write the world volume (which vanish on shell) are critical to fields nonrelativistically as E⃗, B⃗and include the permit- account for all the kinetic terms required by supergravity tivity and permeability in defining D⃗, H⃗. [10,13]. We have emphasized that the Dirac string formulation separates the gauge and monopole d.o.f.; as a result, it ACKNOWLEDGMENTS provides a basis for Hamiltonian and therefore quantum treatments. In this form, the introduction of extra unphys- A. R. F. thanks K. Dasgupta, N. Afshordi, and L. Boyle ical d.o.f. for the Dirac string leads to constraints [16].On for interesting discussions. B. C. and A. R. F. are supported the other hand, when the Dirac string takes the linear by the Natural Sciences and Engineering Research Council configuration, the entire string depends on the monopole of Canada Discovery Grant program. Part of this work was position. In the variation of the action, these appear through supported by the Perimeter Institute for Theoretical terms proportional to the sourceless Maxwell equations Physics. Research at Perimeter Institute is supported by [see Eq. (7)] and are trivial on shell. On the other hand, the Government of Canada through the Department of these terms can contribute off shell, for example in the Innovation, Science and Economic Development and by path integral. Understanding how these contribute to the the Province of Ontario through the Ministry of Research quantum mechanics of monopoles is an interesting and Innovation. question. Meanwhile, the analogous terms for D3-branes (see the discussion below) also play an important role in APPENDIX: CONVENTIONS the four-dimensional effective action of type IIB string theory [10,13]. Here we briefly lay out our conventions, including signs. The Dirac string formalism straightforwardly extends to To start, we take the mostly plus metric convention with curved spacetimes and higher dimensions, and the linear ϵ0123 ¼þ1. The Hodge star for a differential p form is given ν1ν by ⋆F 1=p! ϵμ μ p Fν ν ,so⋆⋆F configuration for the string becomes a world sheet with ð Þμ1μ4−p ¼ð Þ 1 4−p 1 p ¼ geodesics as constant-time slices. A second end point to the ð−1Þpð4−pÞþ1. M string on therefore allows us to use the Dirac string for With standard conventions (see [14,17]), the Maxwell monopoles on compact manifolds. While the magnetic equations with magnetic currents included are Gauss law constraint (net magnetic charge on a compact manifold vanishes) means that any Dirac string can end on ⃗ ⃗ ∇ · E⃗ ρ; ∇ × B⃗− ∂ E⃗ |;⃗ oppositely charged physical monopoles, an arbitrary refer- ¼ t ¼ ence end point allows for a cleaner separation of the ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ∇ · B ¼ ρ˜; ∇ × E þ ∂tB ¼ −|;˜ ðA1Þ dynamics of the different monopoles. Furthermore, it allows us to avoid having multiple Dirac strings end on where ρ; |⃗are the electric charge and current and ρ˜; |⃗˜ are one monopole in the case that the monopoles on a compact the magnetic. In relativistic notation, we take Aμ ¼ðΦ; A⃗Þ, manifold carry different numbers of magnetic quanta μ 1 j ¼ðρ; |⃗Þ (and likewise for the magnetic current), (e.g., there are two monopoles of charge þ and one of − ϵ charge −2). Again, in the case of higher-dimensional so F0i ¼ Ei;Fij ¼ ijkBk. The Maxwell equations monopolelike branes, monopole charge can dissolve into become the flux of other fields, so the Dirac string from a monopole μν ν ρ may not even have another monopole on which to end, ∂μF ¼ −j ; ∂μFνλ þ ∂νFλμ þ ∂λFμν ¼ −ϵμνλρ|˜ ; ðA2Þ necessitating the reference end point. Finally, higher-dimensional branes of string theory are or d ⋆ F ¼ −⋆j and dF ¼ −⋆|˜ in terms of forms. The magnetic sources for various rank form fields. As in dual field strength F˜ ≡⋆F therefore satisfies the dual electromagnetism, a mathematically rigorous treatment Maxwell equations dF˜ ¼ −⋆j and d ⋆ F˜ ¼þ⋆|˜.As considers the potentials as sections, while a Dirac-like a result, the dual electric current is usually defined as formalism allows the separation of the gauge and brane −|˜; for simplicity of comparison, we do not introduce d.o.f. (see [13]; one of us will detail this formalism in the this sign. forthcoming [10]). The formalism therefore provides an Finally, we define the Dirac string coupling G as a alternative to the democratic (dual potential) formalism form integral over the string world sheet coordinates τ, σ. We choose the orientation by taking integration measure 6See [14,15] for the energy loss rate in different approximations. d2σ ¼ dτ ∧ dσ ¼ −dσ ∧ dτ.

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