JHEP07(2020)091 Springer July 14, 2020 June 15, 2020 April 28, 2020 : : : , Published Received Accepted Published for SISSA by https://doi.org/10.1007/JHEP07(2020)091 [email protected] , . 3 2004.07181 The Authors. c Scattering Amplitudes, Wilson, ’t Hooft and Polyakov loops

The Kerr-Schild double copy relates exact solutions of gauge and gravity theo- , [email protected] Queen Mary University of London,E-mail: 327 Mile End Road, [email protected] E1 4NS, U.K. Centre for Research in String Theory, School of Physics and Astronomy, Open Access Article funded by SCOAP between the classical double copy and the BCJ doubleKeywords: copy for scattering amplitudes. ArXiv ePrint: magnetic monopoles associated withsame arbitrary solution gauge (the groups pure cantopological NUT information be on metric) double both in copied sidesthe gravity. to of nature the the We of double further copy theline correspondence, describe gauge independently operators, how group. of and to we This match argue up information through is specific neatly examples expressed that in they terms provide of a useful Wilson bridge theory object, which linearises thethe Yang-Mills double equations. copy This for appearstheory scattering plays to a amplitudes, be crucial in at role. odds whichof Furthermore, it with the classical is non-abelian fields not yet nature —gravity clear of such whether theories. or the as not gauge non-trivial global In properties topology this — paper, can we be clarify matched these between issues gauge by and explicitly demonstrating how Abstract: ries. In all previous examples, the gravity solution is associated with an abelian-like gauge Luigi Alfonsi, Chris D. White and Sam Wikeley Topology and Wilson lines:double global copy aspects of the JHEP07(2020)091 ]. 85 – 82 5 23 20 8 12 bundles N 13 Z ], the double copy has remained an intensively- 3 )/ – 1 – – 10 18 1 N ], some of which are exact solutions of their respective 3 81 – 44 8 21 5 ], either with or without supersymmetry. The double copy has also been 1 43 22 – 4 , 2 3.2.3 Topology of the non-abelian monopole 3.2.1 Gauge3.2.2 field of a non-abelian Allowed magnetic monopole charges 5.1 The all-order5.2 structure of infrared The singularities Regge limit 3.1 Abelian case 3.2 Non-abelian case perturbative solutions in gauge andcorrespondence gravity between theories, different this quantum clearlytraditional field indicates theories, approach a profound that to new may thinkingthis even about is suggest field that to our theory findcorrespondence, needs explicit such as revisiting. non-perturbative the solutions biadjoint One in scalar way theory theories to considered that in probe enter refs. the [ double copy extended to classical solutions [ field equations. There areclassical ongoing hopes General that Relativity, this including maywave for provide detection. astrophysical new applications calculational In such tools parallelunderpinnings as for to of gravitational this the effort, doubleany) however, of copy it its correspondence, is validity. and worth If, to examining for try the example, conceptual to the double ascertain copy the can limits be (if extended to arbitrary (non)- Since its inception just over a decadestudied ago relationship [ between gauge and gravitying theories. amplitudes, Its and first there application was isloop to by orders scatter- now [ a large amount of evidence for its validity at arbitrary 1 Introduction 6 Conclusion A Topological classification of SU( 4 Topology of the Taub-NUT solution 5 Wilson lines and the double copy 3 (Non)-abelian magnetic monopoles and their topology Contents 1 Introduction 2 The Kerr-Schild double copy JHEP07(2020)091 ] ]. 85 44 ] which considered the 86 gravity solution, as shown Gravity data — such as topological same global Double copy Double copy Double – 2 – gravity solution, regardless of the gauge group. has a precedent in the study of infrared singularities of same Abelian 1 Non−abelian Gauge ] involves products of fields at the same spacetime point, and thus 44 ] for yet more examples). However, this would appear to forbid the ability Transformation ], and was also considered for solutions of SU(2) gauge theory in ref. [ 36 27 , Yang-Mills equations, and thus looks like an abelian gauge theory solution [ 35 . This follows from the fact that colour information is removed when performing . Conceptual structure of the double copy for exact solutions, in which (non-)abelian 1 In this paper, we will see that it is indeed possible to relate exact solutions with non- For scattering amplitudes and (non-linear) perturbative classical solutions, the non- There is, however, another approach to extending the conceptual framework of the trivial topology between gauge and gravity theories in a meaningful way. First, we will (see refs. [ to map topological information between gauge(e.g. and characteristic gravity classes) solutions: of topological agroup invariants gauge one field considers, usually and rely crucially this on information which is particular irrelevant gauge after taking the double copy. However, by analogy with amplitudes we mustclassical be solutions able with to find different a gauge constructionin groups such figure that map exact to the the double copy, sostarted that with. a given Indeed,amplitudes gravity figure [ solution does not care which gauge group one abelian nature of thecopy gauge procedure. theory As plays is— a by time-independent now crucial Kerr-Schild well-known, role there metricslinearised in exists — the a whose formulation special single of classIn copy of the what gravity gauge double sense solutions solution the obeys gauge the theory solution is genuinely non-abelian is then rather obscure. information — between them. Thissingle was copy examined of recently the in Eguchi-Hanson ref. instanton,that [ alas topological without providing information any can conclusive evidence further be investigation, double-copied. and Nevertheless, is the thus idea the clearly main deserves aim of this paper. or quantum context havehave involved locality local built in, quantities.initiated even if Scattering in this amplitudes, ref. is foris not [ example, manifest. again Furthermore, manifestly theand local. classical gravity double theories, copy If it it must be is possible true to that match there up is a deep connection between gauge Figure 1 objects must double copy to the double copy. This is to note that nearly all previous incarnations in either a classical JHEP07(2020)091 . , , ], 1 5 µν 89 η (2.2) (2.1) ]. There is an established . ]. For general solutions, this 88 . , 3 , we briefly review the Kerr- ] for a modern review), which ν = 0 – k 87 1 2 µ 90 µ , we study the topology of Taub- . 4 ∂k φk · 6 = µν multiplying an outer product of a vector , k φ = 0 , h ν µν k – 3 – µ κh k + µν g µν = η ν , we examine (non-)abelian monopoles, and describe the = k ) metric signature throughout. 3 is the Newton constant. The Kerr-Schild form corresponds µ − µν k , g N − µν , G ] to be the Taub-NUT solution [ η 1 − , 45 , where ]. Furthermore, we will see that all of our patching conditions — in either N 85 πG ]. 32 29 is the graviton field representing the deviation from the Minkowski metric , √ . The relevance and implications of Wilson lines will be outlined in section 27 3 µν = h κ The structure of this paper is as follows. In section Note that we use the (+ with itself, where the latter must satisfy null and geodesic properties: 1 µ to the graviton decomposing intok a scalar field can be defined as follows: Here and solution with adouble gravitational copy counterpart, for in scattering amplitudes amust where be way appropriate carried that [ out overlaps order-by-orderspecial in family with of the the exact relevant gravity coupling knownsingle solutions constants. BCJ copies. is However, known, These a which are all certain the have well-defined Kerr-Schild gauge metrics theory (see e.g. [ Finally, we discuss our results and conclude in section 2 The Kerr-Schild double copy The aim of the classical double copy is to associate a given non-abelian gauge theory Schild double copy. Incharacterisation section of their non-trivial topology.NUT In spacetime, section and relatesection this to the single copy gauge theory solutions considered in recently in ref. [ gauge or gravity theoriesThis — will allow can us be toand make expressed previous contact in between results the concerning terms study the oflimits of structure [ magnetic certain of monopoles scattering Wilson in amplitudes line this in paper, operators. special kinematic copy is already known [ patching condition for the Taub-NUT metric thatand characterises we its non-trivial will topology see [ independently that of this the precisely natureof corresponds of an to exact the the solution gauge similarand whose group. which condition fully local in generalises Thus, and gauge the global we preliminary theory, properties observations provide regarding obey an SU(2) the explicit monopoles construction made example of figure monopole. For general gaugethe groups, latter this may is or may knownalways not to be be have characterised classified a by by non-trivialspatial a regions. a topology, certain known Furthermore, where the patching topological non-abelian condition invariant.which monopole between However, can it it always gauge takes be can the fields written form in in a of gauge different a in trivially dressed abelian-like monopole solution, whose double take a particular solution of pure gauge theory, namely a (singular) point-like magnetic JHEP07(2020)091 , µ k (2.7) (2.5) (2.6) (2.3) (2.4) ) is not and φ in gravity 2.3 ]. This has M , which gives 45 N ]. Clearly eq. ( ) as 91 NUT charge 2.3 . ), in which the colour structure is an arbitrary constant colour µ , a 2.4 will suffice. We may furthermore ν c ψl l ) different vectors that obey certain µ a a , . c , , µ ) and write T l µ µ D a c abel µ , Nψl A 2.4 φk µ ) ) A k + a ) a + (˜ a ν T T µ k a T a c µ a c φk c – 4 – ) , in which both abelian and non-abelian classical a = (˜ = ( 1 = ( T Mφk µ µ a ) respectively, such that for the case of a pure NUT µ c A A = A 2.6 ]. = ( µν h µ 78 , A 60 , 57 Particularly relevant for this paper is the Taub-NUT solution in grav- 2 are distinct scalar fields, and is a generator of the gauge group. The extension to symmetries other ]. a ψ ), and double copies the gauge field as if it were abelian. Note that this is the solution of a purely abelian gauge theory. If the latter has a known ] analysed this solution in an abelian gauge theory, and concluded that it was 44 T a . ] proved that for every time-independent Kerr-Schild graviton, one may then 45 T ], which was examined from a double copy point of view in ref. [ a 44 and abel µ c ]: 88 φ A , 92 There is a growing list of specific cases of the Kerr-Schild double copy which have 87 Going the other way, the double copy of a point charge may be more general than a pure gravity 2 the first and secondcharge terms in in the eq. gravity ( theory, one may follow eq. ( solution, as discussed in refs. [ create a gauge field Reference [ a dyon, possessing both electric and magnetic monopole charge. These are represented by form [ where mutual orthogonality conditions. Suchin an general, ansatz but will happens not to linearise do the so Einstein for Taub-NUT. equations One may then single copy the solution to tric) charge [ ity [ a Schwarzschild-like mass term,rise but to in a rotational addition charactercoordinates, a in the so-called the gravitational relevant field graviton at infinity. field In can particular be (Plebanski) written in a so-called double Kerr-Schild charge ( provides an explicit realisation ofsolutions figure are taken to map to the same gravitybeen solution. examined in detail.giving Arguably rise the to simplest is the that Schwarzschild of metric, a for point-like mass which the single copy is a point-like (elec- where Kerr-Schild double copy, any solution ofcompletely the factorises, form of can eq. also ( be easily double copied. One simply strips off the colour which obeys the linearisedvector, Yang-Mills equations. and Here than time translational invariance hasunique: been considered for in a ref.pick particular [ any gauge gauge group, group. any Indeed, choice it of is instructive to write eq. ( more easily found. Eachand such ref. solution [ is characterisedconstruct by a the single explicit copy gauge forms field of This ansatz turns out to linearise the Einstein equations, such that exact solutions can be JHEP07(2020)091 . , cor- π = θ ) represents Dirac string 2.7 ) is the appropriate a T a c = 0 (i.e. the location of (˜ g r the gauge group. One refers dφ. φ G A + 1) (2.8) ) has a singularity at dθ θ − 2.8 A θ + cos dr , r 0 . The spatial part of this manifold can be A , } 0 = 0 , – 5 – , where there is one fibre attached to each point dz (0 ] did not explicitly check that eq. ( z ˜ g , namely manifolds that look locally like a product A 45 − fibre + M − { = -axis, which is usually referred to as the dy z y D µ A A as the + G dx x . Reference [ A g denotes Minkowski spacetime, and ) is straightforward, and always results in a pure NUT charge and 2.7 M ), and denoted the magnetic charge parameter that would arise in a principal fibre bundles ]. To briefly summarise for the uninitiated: gauge fields can be defined 93 t, r, θ, ϕ , where base space G as the M × 3 The Dirac monopole is singular at the origin and thus the base space is Minkowski This idea was first discussed by Wu and Yang, who gave an elegant fibre bundle Here, and for the rest of the paper, spherical coordinates are defined such that M 3 spacetime with the origin removed, to in the base space.differ from Whilst this such globally. a Thus,bundle bundle for with a may the given look same gaugefields local locally group, is structure. like there then a may The entirely be classification product equivalent more of to space, than topologically the it one classification non-trivial might fibre of gauge fibre bundles. fields defined in each can then be glued together. interpretation [ as connections on space Whilst it is notthe possible singular point-like to monopole), remove theby the orientation performing physical of a singularity the gauge string at transformation.singular singularity away can To from construct be the a modified origin, gaugethe one field gauge must that field employ is at is everywhere least non-singular non- two in coordinate each. patches, Where such these that coordinate patches overlap the gauge gauge theories. 3.1 Abelian case The gauge potential forresponding the to Dirac the monopole entire of negative eq. ( 3 (Non)-abelian magnetic monopolesIn and the previous their section, we topology havegauge seen that groups (non)-abelian all magnetic map monopoles to withcharacterise the arbitrary a non-trivial pure topology of NUT monopole charge solutions, in in both gravity. abelian In and non-abelian this section, we discuss how to a genuine magnetic monopolesee solution in in the the following section case thatnon-abelian of this generalisation a is indeed of non-abelian the the gauge case,double such theory. magnetic that copy charge. We ˜ will of Inin line eq. gravity. with ( the above comments, the is the well-known gaugegauge potential theory for of a electromagnetism.coordinates Dirac ( We magnetic have monopole herepurely in expressed abelian the theory the by U(1) result ˜ (abelian) in spherical polar where JHEP07(2020)091 ) is (3.1) (3.2) (3.3) 2.8 in each patch, as S µ C A N S and A A (b) N µ A , . ) . ϕ . The monopole field of eq. ( ( C 1 ) has a string-like singularity on the + 1) − S θ × R 3.1 µ 2 ˜ gϕ ) has a string-like singularity that cuts the ∂ S ) cos ig 2 , ϕ 2.8 e 0 ( , S 0 i g , – 6 – ) = (0 ϕ − ( ˜ g N µ S − A ). This completes the construction of a single-valued, . This is straightforward to achieve in practice. Firstly, = = C 3.2 S µ S µ A A . 2 (a) by the gauge transformation N µ ) is already non-singular in the northern hemisphere, and so can be taken A without it being ill-defined at one point. This itself explains the presence , provided that they are related by a gauge transformation in the region 2 2.8 2 S axis, and is thus non-singular throughout the southern hemisphere as required. . (a) The Dirac monopole field of eq. ( is the coupling constant, and z g . One may then define the gauge field N µ We indeed then need at least two coordinate patches in order to define the gauge field A is an element of thepositive gauge group. The fieldFurthermore, the of two eq. fields can ( bethe matched gauge at the transformation equator, in wherenon-singular eq. they gauge ( are equivalent field up away to from the origin. where as This is related to throughout all space,respectively. which we We can are takeshown free in to to figure be define theof separate overlap, northern namely gauge on and the fields equator southernthe field hemispheres of eq. ( the base space is nowtangential topologically to equivalent the to surface ofvector the field sphere, on and it isof well the known that Dirac it string, ispoint, which impossible may as to be shown define in taken a figure to puncture the sphere at infinity at the singular sphere at infinity atmay the take south non-singular pole; gaugepatched (b) fields together Wu-Yang in construction by the a of northern gauge a transformation and non-singular on southern gauge the hemispheres, field: equator such one thatcontinuously they deformed are into the surface of a sphere at infinity. Adding the time direction, Figure 2 JHEP07(2020)091 4 N µ A (3.7) (3.4) (3.5) (3.6) Dirac ). For this θ ( represents the C ) represents the g ) as should correspond varies in the range 3.7 increases from 0 to 3.4 θ A θ (a), culminating in an (b). The curve obtained 3 and 3 )) implies the famous B . The latter, up to a constant ) on the 2-sphere at infinity 3.3 ˜ θ g, . ( π # ) also corresponds to the identity C µ π ] uses a unit radius. ( A . However, eq. ( , = 4 . The set of topologically non-trivial µ 93 U . One may associate each such curve ,
B µν G θ S µ Z dx corresponding to the 2-sphere at infin- Σ ) A d ∈ θ (0) in eq. ( S ( − µν S C F I N µ S , n A ig ) = – 7 – 2 n first Chern number " π ZZ µ (2 = = dx S ˜ g B g C ). This justifies the statement that ˜ ). I Φ 2. Upon switching to the southern gauge field, there is ) = exp 3.3 θ = 3.4 π/ ( B U ), ( = Φ θ 3.2 increases from zero proceeds clockwise from the lower point as at . We may depict this as shown in figure θ S µ A (0) is an infinitesimally small loop around the north pole, and thus A C 2, the definition of the gauge field jumps from the northern field π/ = group elements (indicated by the dotted line). Thus, as θ is the area element on the surface ) be single-valued (i.e. that µν Σ 3.2 d = 0, the curve There is another way to derive the quantisation condition, which makes the relevant The patching together of two distinct gauge fields constitutes a non-trivial topology, ], a closed curve is swept out in the gauge group In fact, any sphere surrounding the origin will do e.g. ref. [ θ equivalent 4 (0) corresponds to the identity element of the gauge group. As , π , the curves gradually sweep over the sphere as shown in figure phase that a chargedto particle be would single-valued experience on uponto the traversing equator, the it curve follows[0 that the points element. At to the southern field in the gauge groupshown, reaching as point a discontinuity and the curve continues from point At U π infinitesimally small loop at the south pole, such that in the following section.characterised by Consider constant the values of familywith the of a polar group circles angle element as follows: bundle can be seen fromeach the of fact which is that classified therefactor, by are the is discretely so-called given different precisely U(1) principle by bundles, eq. ( topological aspects clearer, and which will also pave the way for the non-abelian discussion amount of magnetic charge.of Furthermore, eq. the ( requirement that thequantisation gauge condition transformation which relates the electric and magnetic charge. The relation to the topology of the fibre southern hemispheres, one may apply Stokes’ theorem to rewrite eq. ( where we have used eqs. ( which turns out to beflux related is to given the by magnetic charge. To see this, note that thewhere magnetic ity. Separating the surface integral into two separate contributions for the northern and JHEP07(2020)091 π B in = B (3.9) (3.8) θ (3.10) and ) — the and associated A G 3.7 A ), where for G = 0 and ( 1 θ π B , (b) A Z , where θ ∈ = 0. The remaining compo- ] closely. Our ultimate aim a 0 94 A , πn, n ) First, let us address the form of a 2 − 5 + 2 r first homotopy group ( S µ Z O A µ + ). dx ) ) θ 3.6 ( – 8 – r θ, φ C ( (U(1)) = ), ( I a i . 1 a a g π 3.5 T = = a µ a i A N µ A A = µ µ dx A ) θ ( C I g (a) , namely the possible ways in which one can join the points ]. G 95 , 94 (b) correspond to equivalent group elements imposes — via eq. ( . (a) A family of curves for fixed values of the polar angle 3 (b). These closed loops are classified by the 3 Most of the literature on non-abelian monopoles focuses on spontaneously broken theories with addi- 5 see e.g. refs. [ nents of the gauge field can then be expanded in inverse powers of the radial coordinate: tional scalar fields, due to the non-singular nature of the solutions. For useful reviews of the singular case, is to identify thepret relevant this quantity from that the specifiesnon-abelian point their monopole of non-trivial field view topology, and of to the inter- double3.2.1 copy. Gauge fieldAssuming of a a static non-abelian field, monopole we may choose a gauge such that which is precisely equivalent to eqs. ( 3.2 Non-abelian case In this section,monopoles, we where we review will the follow properties the presentation of of singular ref. point-like [ non-abelian magnetic an integer numberin of figure times. Putcondition another way, the fact that the two points figure the present case one has i.e. the set ofgroup integers manifold under of addition. U(1) is This a circle, simple such result that follows any from closed the loop fact may that wind around the this circle correspond to the north and southwith poles the respectively; (b) family closed of loop curves. in the gauge group gauge fields is then inclosed one-to-one loops correspondence in with the set of topologically inequivalent Figure 3 JHEP07(2020)091 ) is φ 3.12 A (3.19) (3.15) (3.16) (3.17) (3.11) (3.12) (3.13) (3.14) (3.18) , where 0 , resulting in r > r φ dependence, and 2 and − r r ,  ) . ] . , , . ν , θ, φ 0 A r , = 0 ( = 0 µ r , 1 ] = 0 ] = 0 − A A θφ , 0 [ µν φθ r F U F F = 0 a ig . jk r η d terms in the field strength of eq. ( η η ∂ φ r F 0 = 0. This can be done by finding a gauge  i √ − r √ √ . This gives rise to a magnetic field 0 θ φ U A r , , µ Z ijk ∂ θ G F µ φ  A A i ∂ g + A θ 1 2 as the only non-zero component of the gauge ν A A  1 ∂ [ [ = ∂ – 9 – − − θ φ ig ig 1 − U θφ = A exp r − − sin ν is defined by F a i P a  A φθ B µν µ θ UA T F F ∂ ∂ ) = η η a µν → = √ √ F r φ µ ∂ by integrating along lines of constant µν ∂ = A r, θ, φ F ( θ 1 µν A − F U = 0. This leaves θ ) terms in the potential can be omitted. To avoid any issues arising 2 A − belong to the Lie algebra of . Hence, the only non-vanishing component of the field strength is r a i ( r ) results in a O 3.17 and is the absolute value of the determinant of the Minkowski metric. The field denotes path ordering. This describes an integration along radial lines, with the a i η P A is the small but non-vanishing radius of a sphere centred on the monopole. To simplify 0 Equation ( where equations give rise to two non-trivial equations of motion: independent of The Yang-Mills field equations in spherical coordinates are can be implemented for a gauge in which field, the form of whichconsidered may here be are determined time-independent, fromvanish. so the equations the Furthermore, of the motion. assumptions made The monopoles thus far imply that at large distances where lower bound protecting the integral from the singularity at the origin. A similar procedure the analysis, we first choosetransformation a such gauge that in which A solution to this expression is For a point-like monopoletherefore solution we the require afrom magnetic the field unavoidable with singularity atr the origin, we consider the field only for where the field strength where JHEP07(2020)091 has -axis. π (3.27) (3.24) (3.26) (3.21) (3.23) (3.25) (3.20) (3.22) z 4 / ), yields 3.18 = 0 to avoid another θ , ) ϕ , and the factor of 1 ( 1 G − S θ , µ by analogy with the abelian case, vanishes at . ∂ , . )  φ . N µ cos 1) ϕ ϕ ) ( A ) , A , φ + 1) − S φ M ) π ( ( , π i g θ π θ 2 π I Q 4 M 4 M (2 − ig = ) = 0 Q Q S )  (cos φ (cos − ( ϕ M ( M M π π Q M 1 – 10 – ) + 4 4 ig Q Q − φ ) = Q e ( (0) = S φ φ S = = ( S µ ∂ ) = exp M φ S φ A ϕ M ) ( = A A S ϕ φ ( S A -axis, we require that ) in the second equation of motion, eq. ( z = and we will once again end up with a Dirac string. Choosing ) is defined with a Dirac string aligned along the negative N µ 3.21 θ A ) are matrices in the Lie algebra of , we may impose the single-valuedness condition, 3.23 φ ( M 3.1 ) and ( Q is a constant matrix. The general solution for the gauge field is therefore 3.20 M ) and Q φ ( ). Thus, faced with a point-like magnetic monopole in an arbitrary gauge group, M 2.7 which leads to a generalised form of the Dirac quantisation condition: where Similarly to section for use in thegauge southern transformation hemisphere. These two gauge fields are related by a non-abelian 3.2.2 Allowed magneticThe charges gauge field of eq. ( We may define this toand define be a a second “northern” potential gauge whose field only non-zero component is The importance of this fromeq. a ( double copy perspectiveone is can that always it choose is a preciselyto gauge of a such the pure that form NUT the of charge double in copy gravity. is straightforward, and will lead Utilising eqs. ( and hence this to lie along thesingularity negative at the north pole. This necessitates that where been included by convention.to As be in well-defined the for all abelian case, it is not possible for this potential which admits a general solution JHEP07(2020)091 . ]. ∗ G 96 . To (3.32) (3.28) (3.29) (3.31) (3.30) G of the . After 0 and the i k H G , while the ) is satisfied ]. w hence reduces 97 , are fixed by the 3.29 G 94 . We can therefore w ∗ G that is dual to ∗ gauge group can be found in ref. [ ), results in G and weights ∗ 6 G 3.27 α , 2 represent the possible magnetic electric | ) ) i i ( . k 0 ( 0 , ) are gauge-equivalent, where α k Z k α k | · Z · ( i , 0 ∈ 0 as an element of the root lattice of has roots n k ∈ k k 0 H k w i · G . The electric weights X 2k k k , n π = − and g . To analyse this condition further, one may must always satisfy N,N ∗ n k ). In general more solutions will exist. Further 2 ) k i – 11 – = 4 G ( = w = α M ) = 3.31 i are the roots of the dual group α ) is to take k n w 2 · Q 2 ( ], we refer to as the and their magnetic duals | · 0 α i k w α 96 k is the universal covering of the algebra then all possible G | and weights X 3.29 2 w / in the hyperplane perpendicular to a third weight ∗ G = α α k = and weights ]: ∗ . The electric group α α ∗ 96 [ G ) is known as a magnetic weight vector. Taking this form of the r G has roots ∗ Weyl reflection G , . . . ., k 1 k gauge group is a weight vector of the representation in which the Cartan generators are ex- = ( are integers and denotes the identity element in i I k w n aıey one wouldNa¨ıvely, expect an infinite number of possible magnetic charges, correspond- For useful reviews of group theory relevant for the present context, see e.g. refs. [ 6 corresponds to a factoring out Weyl reflections, the allowed magnetic weights may still live in a number of ing to arbitrary weightsto in physically the distinct dual or weightmagnetic allowable lattice. charges magnetic associated monopoles. However, with not For weights example, all one of may these show correspond that magnetic group fields present in thecharges. theory, If and both the groups magneticby share a weights the rescaling. same Lie Furthermore,magnetic algebra, if then weights are their specified root vectors byexamples will eq. of differ electric ( only gauge groups where consider two systems offollowing roots the and terminology weights of classifyingmagnetic ref. two [ separate gauge groups, which, Hence, a possible solution toRecall eq. that ( the rootwhich lattice contains the of root afor vectors Lie themselves. group Thus, is the the constraint sublattice in of eq. the ( group’s weight lattice monopoles. The magnetic weightssee are this, the weights recall of that a roots Lie group where pressed. Classifying the possibleto monopoles determining which in magnetic an weight arbitrary vectors gauge correspond group to physically distinct and stable write the magneticCartan charge subalgebra matrix of as a linear combination ofwhere the generators charge matrix in the generalised quantisation condition, eq. ( where JHEP07(2020)091 . N and Z / (3.33) (3.34) ) N . This . Thus 2 Z 1 stable N ˜ G / S ) of the Lie − , where the ˜ N G N G ) within each = SU( 2 M ∗ Q G = SU( 2, the gauge field along the contour. , µ G π/ A  S µ = ) allow for is the centre of A θ N µ K dx , C # 2) is the equator of the sphere I µ A π/ ig , where ( µ  , one may write the general patching C dx H (b). Similarly, we may also construct ) exp ˜ ≡ θ G/K ) are therefore dynamically unstable and 2 ( by P C N C I  B H ig " U ), the magnetic group will be – 12 – and = SU( , and N = A exp G H  P N µ lie in the same sublattice as the weight at the origin of A = SU( µ N ) = θ Z , each of which can be associated with an element of the G ( dx / ) 3 C U must correspond to equivalent group elements. Denoting the I N B ig sublattices of the weight lattice of SU(  , this traces out a curve in the electric gauge group π N and ]. We are again considering singular monopoles, so that the origin exp . Weights which lie within the same sublattice can be connected by A 93 P both correspond to the identity element. At , π ]. As the origin of the weight lattice corresponds to the zero magnetic charge , we reviewed the Wu-Yang fibre bundle interpretation of the Dirac monopole, ), as the constitutes an element of 98 (a). This is the general condition that encodes the non-trivial topology of the N 3.1 = 0 sublattices denotes path ordering of the (non-commuting) gauge fields H 3 (b), where θ U P 3 varies from 0 to It is important to note that this instability of monopoles with non-minimal magnetic = SU( θ ∗ condition between the northern and southern gauge fields as where in figure As points switches from its northern tofigure southern form, leading oncegroup of again such to transformations the between situation shown in the family of curvesgauge of group, figure where non-trivial topology. In thisnon-abelian section, case we [ describeof the spacetime generalisation is of this removed,in and picture turn to spatial necessitates the slices at arethe then northern least topologically and two equivalent southern coordinate to hemispheres patches, of which figure we may again choose to be to the abelian case, as we now3.2.3 describe. Topology ofIn the section non-abelian monopole in which two gauge fields are patched together, thus generating a gauge configuration with However, all weights of SU( the lattice. All monopolewill solutions reduce for to the vacuumG solution. This, however, ismonopole not the solutions. case This for has an elegant topological interpretation that is directly related sublattice as the origin will decay to the vacuumweights state. within a givenIf sublattice either necessitates the a electric careful oralgebra, specification magnetic then of gauge the group the other isif will gauge the we be groups. universal consider the covering Yang-Mills adjoint group with group an integral sum ofthat roots, monopoles with while higher weights valuesthey in of always magnetic different decay charge sublattices to are configurations dynamically cannot.sublattice possessing unstable, [ such the It that minimum can valueconfiguration, be of this shown tr( implies that all monopoles defined by magnetic weights within the same different JHEP07(2020)091 A of ), the (3.35) ] for a N 100 = U(1). A ∗ G ), whose form is , the centre of the 3.34 K Woodward classes ), which is proportional 3.4 ), which follows directly from G ( 1 π , Z ∈ K. by any element of . The identity element corresponds to N = A Z  , n ) for an abelian electric group πin 2 ) = – 13 – ˜ 3.9 e G/K G ] is a widely studied exact solution of GR, whose  ( . The relevance for the double copy is as follows. = 1 1 π , where it is the latter that gives rise to the rotational π 88 H A , N U 87 can be related to , we find B ] formalised this by pointing out that the classical double copy N 1 stable monopoles, consistent with the discussion of the previous 45 ) reduces to eq. ( Z / NUT charge − in then given by ) , the general Taub-NUT solution has a Schwarzschild-like mass term . The relevant first homotopy group is then ] have suggested that given characteristic classes in gauge theories may 3.34 N 2 N ˜ ) follows. This can in turn be related to eq. ( H ) characterises the different ways in which one may join the points G 86 G (b) to form a closed curve. As an example, we may take an electric group 3.9 ( 1 3 π = SU( , in which case G ], as we discuss in appendix in figure ˜ G/K 99 B Note that eq. ( = , and an additional discussed in section M aspect of the field.and Analogies the between Taub-NUT magnetic solution monopolesreview). have in been However, non-abelian ref. made gauge [ many theories times before (see e.g. ref. [ a gravitational counterpart. 4 Topology of theThe Taub-NUT solution Taub-NUT solution ofgravitational refs. field [ has a rotational character that does not die off at spatial infinity. As mation is irrelevant for the gravityno side problem, of as the it double copy. isacteristic Here simply classes. we not see The true that appropriate that thereof quantity the is the that in double gauge classifies fact copy theory the shouldindependent solution “double apply is of to copiable” instead the individual topology the gauge char- patching group. condition We of will eq. see ( in the following section that this indeed has classify the topology ofref. solutions [ using what wePrevious may studies refer [ to asbe the related to similarpuzzle, quantities in in that gravity theories, characteristic under classes the depend double upon copy. the This gauge creates group, a whereas colour infor- from which eq. ( to the first Chern numberFor that other classifies gauge the groups, non-trivial topology however,topological the of invariant. U(1) patching For principal condition example, bundles. will iffirst not one Chern be replaces number relatable the vanishes to gauge (due the group to same U(1) tracelessness with of SU( the generators), but one can instead section. general element of Thus for monopole configurations that arestable. topologically This equivalent leaves to the vacuum, and hence un- by the first homotopy groupthe of fact the electric that gauge group and G universal covering singular non-abelian monopole. As in the abelian case, this non-trivial topology is classified JHEP07(2020)091 ) ). , 2), 2.5  2.5 (4.7) (4.4) (4.5) (4.6) (4.1) (4.2) (4.3) 2 π/ Ω , ≤ d  θ 2 + Ω 2 ) d , 2 + 2 0, in which case Ω κN 2 ( d ) dr ) → 2 − . 2 2 2 N κN r ( M dr N  + − ) 2 + 2 2 r ) ], and is a direct analogue 2 ( r  − M ) κN 103 [ 2 2 ) p as a “northern” time coordinate dr + ( t )  2 2 κN r r ( ( 1 M + ( − θdϕ − , = f 2 2 0 2 . In the northern hemisphere ( r ]  ( κNϕ, − 2 ]. 2 dϕ − ] πκN + sin = 0, and thus that the Misner string has been + 4 2 Misner string 1) , r ] 2 102 dϕ θ must be periodic with period S ) , t – 14 – ∓ = 8 − dϕ 1) dθ S r θ t 0 = 101 1) t − = 2 , − N θ 2 − for later convenience. The metric has a coordinate 79 t N r (cos and Ω θ )( + d + N (cos 2 κN t r κN r (cos N − + 2 → r ( κN + 2 N N,S dt The double copy is manifest in a particular coordinate system in + 2 dt ) = )[ [ r 7 r ) is well-defined, and one regards dt ( 2 2 ( . This is the so-called [ ) (which applies on the equator on which the two coordinate patches ) ) f f π 2 2 4.4 ], such that the metric assumes the double Kerr-Schild form of eq. ( ) ) 4.5 = κN κN = are the Schwarzschild-like mass and NUT charge parameters of eq. ( 92 ( 2 κN κN θ ( N − + ( ds 2 2 , eq. ( − + ( r r π 2 2 2 r r and = ) reduces to . In the southern hemisphere, one may transform to a “southern” time coordinate = M N 2 4.1 ϕ < N,S 2 t The Taub-NUT solution has also been used to examine the double copy of the electromagnetic duality 7 ds ds ≤ ≡ 0 overlap) then implies that both symmetry that operates in gauge theories [ where the upper (lower) sign correspondssees to that the the northern southern (southern) metric casemoved is respectively. into singular the One at northern hemisphere. Given that the azimuthal coordinate lies in the range t such that the northern and southern line elements are given by singularity at of the Dirac stringone for a may magnetic use monopole. twonorthern different The and coordinate known southern solution hemisphere, patches, tothe as obtained this metric in by problem of figure splitting is eq. spatial that ( slices into a eq. ( where we have rescaled quantities are given by Here respectively. The case of a pure NUT charge is given by the limit where is the squared element of solid angle in four spacetime dimensions, and the additional the radial coordinate. (2,2) signature [ However, once the doubletransformation copy to between more solutions conventionalwith is coordinates. the known, Taub-NUT line we To element this may in make end, spherical a polar it coordinates, coordinate is convenient to work implies an exact relationship between dyons and Taub-NUT, which is true for all values of JHEP07(2020)091 S t (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) . The ϕ time holonomy increases, a path is fixed, or vice versa. θ 0 8 t . . 2 is reached. At this 0 2 t π/ 1) ∓ = (0) is infinitely small, such , θ θ are independent of C ϕ N,S 0 } (a), consisting of circles on the (cos 2 ϕ ]. The full NUT charge appearing 3 N,S 0 πκh h , , . { κN , i i Z 105 = 2 , µν N,S ) take the form dx ∈ dx = 4 i dϕ 0 i κh 104 0 4.6 00 h = 0, the loop g g ϕ ) of figure + N,S ϕϕ C N,S 0 ) yields θ θ , n ) as forming a map from the set of curves in C I h ( 0 µν I κ C C η – 15 – 4.11 , h I = nN 4.13 = = | κ 1) | t = t = ∓ as a free parameter, in terms of which ∆ ∆ | N,S µν | | 0 N θ g ) can be written purely in terms of the graviton field, as ) N θ ( 4.9 (cos corresponds to the fact that the transformation between N,S N t n ∆ | = 2 ) form a subgroup of the general group of diffeomorphisms that ϕ ), which from eqs. ( N,S 0 h 4.13 , the difference in time coordinate upon returning to the starting point is a spatial index, and the lack of manifest Lorentz covariance on the 4.12 C } 3 , ), one may either take 2 , ] 4.7 1 ) is then given by is a basic unit of NUT charge. Indeed, it is known that this property is required for 106 = 0, which constitutes the identity element of the group. As ∈ { 0 is defined only up to an integer multiple of the time period (a) to the diffeomorphism group. At 4.6 t , such that the metrics arising from eq. ( i N 3 N t From the double copy perspective, it is desirable to obtain the above periodicity condi- From eq. ( 8 → ∞ that ∆ is traced out in the diffeomorphism group untilThis the is analogous equator to thetheory. quantisation condition between the electric and magnetic charges in abelian gauge where we have usedtime shifts the of fact eq.act that in ( the gravity. components Onefigure may thus view eq. ( Let us now consider2-sphere the at family spatial of infinity.holonomy curves in One eq. may ( associate each such curve with a value of the time Then the time holonomy of eq. ( r where the only non-zero components of the northern and southern gravitons are given by satisfies [ where right-hand side follows from thehand explicit side. choice of the This physical takes time coordinate a on suggestive the form left- if one considers asymptotically large distances tion from a procedure thatcarry makes this its relationship out with as theupon follows. gauge theory trying First, manifest. to a We standard synchronisegiven may closed result clocks loop in around GR a is closed that there contour. is a That is, upon traversing a the spacetime to be fullyin spherically eq. symmetric ( [ where the arbitrary integer and where JHEP07(2020)091 , , ] ]. m π π S µν N,S h 108 t 107 → ) into (4.17) (4.18) (4.16) (4.19) (4.14) (4.15) θ and 4.11 N µν h that traverses a m is quantised according m . ). First, note that the phase Z ∈ ) is quantised. Reference [ 3.34 , πκN.  4.15 i 0 = 8 h ) as required. i | πn, n , 2) , 0 . 4.7 dx 4 n 0 π = 2 C 1 π/ t 2 κN ( I  i = 4 N S 0 = t h = ∆ – 16 – = 0) that it is given in the weak field limit by iκm m −  m i ∆ mκN 00 (b), that we have previously used for elements of a N 0 ∆ h | − | h ). For this phase to be well-defined on the equator ) yields eq. ( 3 , this implies that the mass  ) yields 2) 0 i t π/ 4.12 dx 4.18 ( 4.11 Φ = exp S C t I entering the phase of eq. ( ∆ | ). κm m 4.7 as in eq. ( ) implies ] considered a gravitational analogue of the magnetic monopole, by consid- ) becomes infinitely small again, and we are back at the identity element. in space, and finding (if 0 θ t ( 108 C 4.17 where the two coordinate patches overlap, one must have that the difference C ) yields 3 4.13 The description of the quantisation condition in terms of the gravitational phase expe- There is another way to reach the same conclusion, that makes contact with Dirac’s so that combining this with eq. ( rienced by a test particle allowsis one directly to analogous write to down the aaround gauge patching a condition theory loop for condition Taub-NUT defines of that a eq. map ( from the base spacetime to the group U(1), so that the curves to Equation ( which says that the mass was unsure about whatgiven to the take discussion for aboverefers this regarding to the mass. the periodicity energy However, of ofcoordinate its a the static is interpretation time wavefunction compact becomes in coordinate. with the clear period presence The of mass the NUT charge. If the time Substituting the results of eqs. ( which is directly relatedof to figure eq. ( in phases evaluated withsuch that the northern and southern graviton fields is a multiple of 2 original argument for the quantisationReference of [ electric charge inering an the abelian phase gauge experienced theorygiven by loop [ a non-relativistic test particle of mass eq. ( The only way that thewith time period holonomies can coincide is then to impose periodicity of the curve This is precisely the(non-)abelian picture of gauge figure group. Incorrespond order to for the the gravitons samemust to physical be be solution, physically patched equivalent. the on However, the time substituting equator the holonomies i.e. explicit arising results to from of eqs. ( point, the graviton field jumps from its northern to its southern form. Finally, as JHEP07(2020)091 B ) are (4.20) (4.21) and 4.20 A . κ 9 ]. The Chern 89 ) tells us that it is ), before applying , 4.5 3.34  µ S 0 h µ ). Indeed, interpreting the is proportional to the first ), and this description makes dx 4.8 N C , I j 3.35 dx ). A similar conclusion was reached is an element of the group of trans- iκm ∧ ). The full NUT charge is given by 3.8  i can be written as H in the above construction. Furthermore, U dx π 4.11 j exp (b), where the latter is now interpreted as n 0 h H 3 i U ∂ ] due to our explicit inclusion of factors of S = – 17 – 89 , according to eq. (  ZZ 0 = 0, and µ N 0 N π κ h 8 00 ) is a precise counterpart of eq. ( µ h ) — that is the precise analogue of its counterpart in = dx 4.20 C N 4.20 I iκm  ) as a field strength, one finds that exp 4.21 (b). A general such element is written in eq. ( (a) lead to the description in figure ], which formally defined the NUT charge according to the integral is the 2-sphere at spatial infinity. This can be verified by separating the field into 3 3 89 S Above, we have seen that one may describe the pure NUT charge in gravity using one Our definition differs slightly to that of ref. [ 9 indeed possible to mapcopy, global such information that from the gaugeshould theory latter be to becomes no gravity more surprise under than thatform, the the merely given double that patching a colour condition local structure on is statement. the stripped gravity off Furthermore, side upon it takes taking an the abelian-like double copy. The patching a patching condition —(abelian) eq. gauge ( theory, andfield. which The characterises ingredients the wenot non-trivial have new. topology used However, to of they formulatedouble the have copy. the graviton not The patching previously fact condition that been of eq. analysed eq. ( from ( the point of view of the the time variable. of two descriptions, bothhemispheres of and which are define equivalent. therelated graviton That field by is, differently a one in may diffeomorphism. each, divide such In space that into the the two weak definitions field are and non-relativistic limit, one may write number is integer-valued, and thusthe amounts U(1) to fibres intime this coordinate, case rather correspond thanpath. the to group However, the the of group pictures phases of area experienced ultimately time by non-relativistic equivalent translations particles given particle moving that with is along the a governed phase a periodic directly experienced by by the Hamiltonian, which is conjugate to where northern and southern partsStokes’ related by theorem the and time usingan translations integer of the eq. multiple results ( of ofintegrand the eqs. of basic ( eq. unit ( Chern number, that specifies the non-trivial topology of a U(1) bundle [ to U(1), namely by theby first ref. homotopy group [ of eq. ( where we have usedformations the that fact leaves that thein phase figure invariant, and whichclear thus that connects the the topology of points theory. the That Taub-NUT solution is, is the similar non-trivial to topology the case is of classified an by abelian maps gauge from the equator at infinity a path in theand group southern manifold fields of are U(1). related by The a requirement multiple that of the 2 phases for the northern of figure JHEP07(2020)091 ) ], 109 3.34 ] and ] for (in 109 116 , applied to ) is defined operator, or – 1 4.20 112 ]. Instead of relating 93 Wilson line ]: the double copy for the Taub-NUT solution ], not just at asymptotic infinity. Furthermore, – 18 – 109 45 ) then form an explicit realisation of figure 4.20 ], which is particularly convenient from a double copy point of ], which is defined only at asymptotic infinity, and thus not for- 118 , ) and ( 111 ] deserves further study. It is also worth emphasising that the results , 117 3.34 , 109 110 30 – in this case due to the closed nature of the contour. Such operators appear 28 ] are useful for the study of Taub-NUT metrics, independently of the double copy. 109 Motivated by the gauge theory case, we define a gravitational Wilson line in terms of It is possible to relate the gauge theory and gravity patching conditions to previous Wilson loop view. In this section, wein point terms out of how Wilson the lines,literature. results of and the how previous this section relates can to be other expressed examples ofthe the phase double experienced copy by in a the scalar test particle in a gravitational field. In the full covariant information must be compared atknown different is points that in Wilson spacetime. linessome have What also cases is been perhaps very) defined less early inhave well- works gravity: been on see used this e.g. refs. to subject.and [ try gravity In [ to recent set times, up gravitational common Wilson languages lines between non-abelian gauge theories In the previousmagnetic sections, monopole we can have be seen— written that that in involves the terms the of non-trivialequator gauge-covariant of a topology phase the certain experienced 2-sphere of patching at by aa condition infinity. a This — (non)-abelian corresponds particle eq. to moving ( ain around so-called many the places in the study of quantum field theory, arising whenever gauge-dependent of ref. [ 5 Wilson lines and the double copy the fact that gauge transformationsbe on associated the gauge with theory diffeomorphisms sideliterature on of on the the both correspondence gravity exact should sidethe classical is traditional solutions well-known and formulation in scattering ofotherwise) the amplitudes. Taub-NUT of double in We ref. copy this have [ thus paper, used but stress that the relevance (or the analysis carried outgraviton here, most patching, notably and the alsoin use the the of fact weak a field Wu-Yang-like that formulationence limit, our for between and the patching our thus ethos condition at andapplies of asymptotic that to infinity. eq. of the However, ( ref. there entire [ is classical a solutions crucial [ differ- gether graviton fields analogouslythe to northern and the southern Wu-Yang fields constructionsupertranslation by [ [ a diffeomorphism, the authorsmally instead equivalent use to a a BMS diffeomorphism.no dual They longer then needed, argue which thatrise avoids a the to periodic well-known pathological problem time closed coordinate that time-like is the curves. Taub-NUT metric There gives are similarities between ref. [ global properties of exact solutions. work in the contextBefore of moving scattering on, amplitudes, however, it whichwhich is we provides worth shall drawing an do attention alternative to in the way the recent to following study section. describe of ref. the [ Taub-NUT solution by patching to- conditions of eqs. ( JHEP07(2020)091 (and (5.3) (5.4) (5.5) (5.6) (5.1) (5.2) to be µν C h , so that we define , ), one must replace C ) by taking  ]. Furthermore, one
3 5.2 ij # , is the conventional time 4.15 h 2 j t  / ˙ ) x , 1 i ) is the Wilson line operator x   x ( ) ν 5.2 x µν + ˙ . For the Taub-NUT solution ( ds ) to eq. ( i h t dx a µ 0 , where ν µ A h 5.4 i a the mass of the test particle. This ds ds mt x dx dx , T m µ a parameter along the curve with the µ = ) can be generalised for massless particles. In µν µν + 2 ˙ g ds ds . s s dx dx 5.2  ν κh ], where it was used to analyse properties 00 . Equation ( h 2 ds ds κ s ds + dx ds C C 117 C , Z → µν dt Z ) with the action for a massless point particle, involving Z → η , and replace this with a second kinematic factor – 19 – )), and g C ig a 2 a 5.1 iκ I  = im = 0. Then, in the non-relativistic (small velocity) T 115  T , " 3.34 µν 2 00 30 exp g ). To go from eq. ( h iκm – ) as required.  P 28 5.2 ) = exp to obtain 4.15 ) = exp ) = C κ C ( C . Here we may make contact with eq. ( ( . ) = exp Φ( 10 grav ) in C grav ]). ( Φ . Φ 5.1 into the length parameter 119 grav m Φ ) reduces to eq. ( ) is a parametrisation of the curve, and 5.3 is a generator of the gauge group in the appropriate representation (n.b. this is s ( a µ ) is a double copy of its gauge theory counterpart x T 5.2 There is a very well-defined sense in which the gravitational Wilson line operator of Although we considered massive particles above, eq. ( 10 that case, one must replacean the einbein exponent (see in e.g. eq. ( ref. [ which is precisely themust also BCJ strip prescription off the forrepresenting colour scattering the generator amplitudes tangent vector [ to the Wilson line contour: where absorbed into the gaugesame field mass in dimension eq. as ( the in coupling eq. constant ( with its gravitational counterpart: considered in the previouslimit, section, eq. ( eq. ( the equator of the 2-spherecoordinate: at infinity, and taking where the dot represents differentiation with respect to where we have ignored anthat will overall cancel normalisation in any constant normalised thatabsorbed expectation the is value mass of independent Wilson line of operators).that was We have considered also in e.g.of refs. scattering [ amplitudes. one may expand eq. ( where expression simplifies in the weak field limit. Writing theory, this must be proportional to the proper length of a given path JHEP07(2020)091 ] for that (5.8) (5.7) ]. ]. 123 122 – 121 , 120 , 117 117 soft function . a  a µ S dsA ∞ 0 ] would have been drastically Z ]), and that of gravity is known µ i 27 p a 124 T ig  , µ i exp ]): , which has a simple physical interpretation: sp A H·S These singularities have a universal form that 124 = (see e.g. ref. [ ≡ P = – 20 – µ i i 11 12 x Φ A that is completely finite, and , associated with the exchanged gauge bosons becoming + 0 hard particles. One may thus write (see e.g. ref. [

i n Φ i Y

0 hard function * . = 1 -point scattering amplitude with virtual gluon or graviton radiation, one ] used the above properties to present all-loop order evidence for the S n 27 infrared (IR) divergences is a so-called H Here, we wish to point out that the analysis of ref. [ Reference [ Interestingly, the soft function of QED is known exactly if there are no propagating fermions. In (non)-abelian gauge theories, one also encounters singularities when radiation is collinear with the 12 11 of the outgoing hard particles (see e.g. ref. [ external particles in the interaction, although such singularities are absent in gravity [ simpler using Wilson lines. It iscan known be in (non)-abelian expressed gauge as theories thatcorrespond a the to soft vacuum function expectation the physical value of trajectories Wilson line operators, whose contours kinematics could be satisifiedcould at be arbitrary neglected. loopQED orders, or Furthermore, if QCD the terms both authorsrealisation outside mapped of noted of to figure that the the the same soft IR gravitational limit results, singluarities thus of providing either an explicit BCJ double copy.any That given is, order the in authorsobtain perturbation the showed theory known that in IR either singularities oneFeynman-diagrammatic of QED arguments, may gravity. or making The isolate clear QCD, somewhat IR how and lengthy the analysis singularities double BCJ used at copy duality intricate between them colour to and calculating additional terms in theinfrared singularities logarithm to of all the orders soft inin function perturbation QCD and theory. amounts QED to The is structure summing onlyexactly. of up partially the known That soft function is, itin has gravity now terminates been at first well-established order that in the the logarithm gravitational of coupling the constant soft [ function interaction that produced the a pedagogical review) where collects all the soft singularities. The latter is known to have an exponential form, where If one dresses an encounters “soft” (i.e. having vanishing 4-momenta). factors off from the fullsoft scattering amplitude radiation has an infinite Compton wavelength, and thus cannot resolve the underlying plitude double copy. Indeed,can be there entirely are expressed existing inthe terms cases double of of copy Wilson is the lines,realised precisely amplitude such in the that double the replacements the copy existing made underlying literature, that mechanism above. and of so This5.1 it has is not worthwhile necessarily to The been briefly all-order review these structure examples. of infrared singularities This mirrors the replacement of colour information by kinematics in the scattering am- JHEP07(2020)091 ] s 64 (5.9) (5.10) .  µν dsh ∞ 0 , 2 scattering in the Regge Z 2 ν i ) 3 p → p µ i p − 2 iκ 1 ]. Going beyond the leading soft p  ), namely that the Wilson line ex- 27 . ), we might write a similar definition = ( 2 exp 5.6 p (a), and the Mandelstam invariants 5.2 ≡ ' ), ( 4 , t 2 scattering, which can be expressed (for .,i 4 2 ) 5.5 2 → p grav , p Φ + 1 – 21 – p 1 of 2 p , ' + 3 = ( 0 p ), so that it does not matter if one starts with a QED

.,i 5.8 ), the particles suffer a very small deflection owing to the t, s grav Φ Regge limit 5.10 i  − in eq. ( Y

s a 0 T * = . ] that the double copy of pure gauge theory is not pure gravity, but a ]: grav 125 S 117 constitute the squared centre of mass energy and momentum transfer respectively. t There is potentially a rather subtle flaw in the above argument, namely that for VEVs incoming ones i.e. The lack of recoilarguments means to that the the previous particles section,limit can we arrive can only at be change the by described idea a that (in 2 phase position and, space) using by similar a vacuum expectation value of Wilson lines where we have labelledand 4-momenta as inIn the figure Regge limitsmall of momentum eq. transfer. ( line Thus, trajectories, in such that position the space, outgoing particles they become follow approximately approximately collinear with straight- the Another kinematic limit of scatteringable amplitudes is in the which all-order high-energymassless information or particles) is as obtain- IR singularities at eachapproximation, order one in may perturbation indeed theory befor [ sensitive an to interesting the discussion additional of matter this content point). (see e.g.5.2 [ The Regge limit must be written in ato form which the manifestly decouples fact lefttheory [ and containing right-indices. a This dilaton is and related do axion not (two-form) pose field. a Inand problem: any by case, standard both the power-counting are axion arguments scalar and will degrees dilaton not of contribute freedom to the in structure four of spacetime leading dimensions, and non-abelian results boththe reproduce colour the generator sameor gravity QCD result Wilson is line. obvious; one strips off of Wilson lines to double copy in arbitrary circumstances, the propagator for the graviton The all-order double copy offollows IR simply singularities obtained from using the diagrammaticponents observations arguments themselves of then double eqs. copy ( in a precise way. Furthermore, the fact that both abelian thus can onlyproperties change to by form a partwith phase. the of gravitational a Wilson For scattering linein operator this amplitude, gravity of [ it phase eq. can ( to only have be the a right Wilson gauge line. covariance Armed The physics of this result is that hard particles emitting soft radiation cannot recoil, and JHEP07(2020)091 ], 34 – (5.11) impact 31 .  ) z + ], which showed properties of so- 3 4 p p sp ( . µ 127 z , global dsA 126 z ∞ −∞ (b) Z µ p s ). The Wilson line language thus ig  1 2 5.2 p p exp P ) = z – 22 – , where the magnitude of the latter is the p, z 2 scattering; (b) In the Regge limit, one may model the Φ( → 3 4 p , p i 0 | ) z , 2 p (a) 0)Φ( , 1 p 2 1 Φ( p p | ) are directly related to the BCJ double copy for scattering amplitudes. The 0 ] demonstrated that the same picture works for gravity if the QCD Wilson 5.6 29 , or distance of closest approach: → h . (a) Momentum labels for 2 | ), ( t | 5.5 s To clarify these issues, we first reviewed how to express a general singular monopole Above, we have seen two examples in which the Wilson line double copy replacements of A solution for arbitrary gauge groups. By a suitable gauge choice, this can always be expressed non-trivial global structure forinvariants monopole that solutions depend on is the usuallydifferent field characterised strength. gauge by groups, However, topological these which invariantswith can makes a be it unique different gravity difficult for counterpart. to see how to associate this information 6 Conclusion In this paper, wenormally have expressed examined as the afields. double local We copy statement have relating for insteadlutions the exact considered between graviton classical theories, whether field solutions, using it to the is which products magnetic possible is monopole of to in gauge map gauge theory as a test case. A fact that Wilson linesthe also magnetic underly monopole our and identificationdifferent NUT of manifestations charge global suggests of topological that the information theybe double for form done copy. a with This useful gravitational bridge, also Wilson relating lines, begs an the issue question which of certainly what merits else further study. can in each theory turnsreproduces out previous to studies be of thewhere very Regge the different. limit results Furthermore, based overlap. the on diagrammatic Wilson arguments line [ doubleeqs. copy ( This framework was firstthat developed known in properties the ofReference classic the [ Regge work limit of in refs.lines QCD [ are emerge replaced from with themakes equivalent Wilson the ones line calculations based approach. in on QCD eq. and ( gravity look essentially identical, although the physics separated by a transverse displacement parameter Figure 4 scattering by two Wilson lines separated by a transverse impact parameter JHEP07(2020)091 ], )). 93 3.34 ], but the ]) has not 29 27 ]. This provides an explicit 45 bundles N Z ], and the resulting patching condition can )/ N 108 – 23 – , the appearance of Wilson lines makes the double copy par- 5 ], and which result in a periodic time coordinate. This treats 103 for exact classical solutions, and extends the recent SU(2) results of 1 , we reviewed how the non-trivial topology of a Dirac monopole can be 3.2 ]), using the known time translations that remove the Misner string singularity in ] to arbitrary gauge groups. 85 109 We hope that the results of this study provide useful food for thought in widening the As we argued in section We then reviewed the Wu-Yang fibre bundle construction for magnetic monopoles [ described by the first Chern class. In the case of other gauge groups, it is not necessarily and innovation programme764850 “SAGEX”. under the Marielodowska-Curie Sk A grant agreement Topological No. classification ofIn SU( section We thank Prarit Agarwal,Papageorgakis George for Barnes, useful Davidence discussions. Berman, and Technology Adrian Facilities Council This Padellaro (STFC)theory, Consolidated work and Grant gauge ST/P000754/1 Costis has “String theory been and supported duality”, by and the by U.K. the Sci- European Union Horizon 2020 research tentially have a key roledouble copy to literature play on in scattering establishing amplitudes a and bridge, that where relatingremit appropriate, to of between classical the the solutions. double copy yet further, including non-perturbativeAcknowledgments and/or global aspects. saw that previous double copyin properties of terms scattering of amplitudes Wilson hadother lines. neat (the interpretations One double of copy thesebeen structure (the previously of understood Regge IR in limit) singularities thisare was in way already other fixed-angle in known scattering the uses [ literature. [ of This gravitational strongly Wilson suggests lines that there in a double copy context, and that they po- to the Aharonov-Bohm effect inthen magnetism be [ expressed in terms of gravitational Wilson lines. ticularly natural, given that thesimple relevant gauge replacements theory of and coupling gravity constants, operators and are colour related information by by kinematics. We also ref. [ each coordinate patch [ the NUT solution as atime U(1) translations bundle, (modulo where the onerelativistic time may period), particle take the being or fibres the taken to group around correspond of a either phases loop. to experienced The by a latter non- description maps most cleanly in which one defines separate gaugehemispheres. fields associated These with are the patched northern togetherof and on southern string-like the spatial equator, singularities, to but createpatching topologically a condition, global non-trivial. field which that The can isthe topology free always is phase be classified experienced expressed by inNext, by the we terms a showed of that particle Wilson a upon lines similar being construction representing can taken be around applied the in gravity equator (see (eq. also the ( recent copy to gravity is straightforward:monopole one simply with removes a the colour purerealisation matrix, of NUT and figure charge, replaces the asref. [ follows from ref. [ as a constant colour matrix multiplying an abelian-like monopole field, so that the double JHEP07(2020)091 ) ) N
base A.2 ) (A.4) (A.5) (A.1) (A.2) (A.3) 2 ) each N S N ]. PU( → · · · 99 0 π )) )-bundle, after N N → .
) ) bundles on N SU( N , Z N 2 / ), we see that there are ) S ( SU( N 2 A.3 N 0 of the PU( H π Z SU( . → ). More formally, we can write → ∼ = N ) ) = ) N ) Z N N N N Z Z Z , ( , ) = 2 0 2 . Each topologically distinct monopole N π S 2 PU( S ( ( Z S 2 2 ( . Such manifolds arise in our context due to → 0 H H 3 π
) Woodward class → ∼ = ∼ = N – 24 –
quotient

) ) ) N PU( ) N N 1 N π relating the various groups appearing above: PU( PU( PU( , , SU( ), which permits any use, distribution and reproduction in → 2 1 2 , we consider magnetic solutions associated with an electric ) as the 2nd S
π , otherwise known as PU( S ) N 1 N 1 N Z 3.2 Z H , H / 2 ) center , which is homotopic to → SU( S } ( N
0 2 1 ) CC-BY 4.0 π N H N − { Z This article is distributed under the terms of the Creative Commons → = 2 we have PU(2)=SO(3) and the 2nd Woodward class reduces to the ∈ 3 = SU(
SU( ] R N N , short exact sequence ω G 2 Z S 2, one must use the Woodward classes. 1 ) groups there is indeed a known description, which we thought worth pointing 1 π N H ]. For N > 99 As discussed in section topologically different monopole solutions associated with electric group PU( · · · → · · · → class as the relevant characteristicwith class for monopoles in SU(2) gaugeOpen theory. For Access. SU( Attribution License ( any medium, provided the original author(s) and source are credited. We refer to [ ref. [ 2nd Stiefel-Whitney class. In other words, the Stiefel-Whitney class replaces the first Chern implies the exact sequence and hence the isomorphism of cohomology groups the fact that all ofare the defined magnetic monopoles on we considersolution are corresponds singular at to the a origin,N distinct and fibre thus bundle, andcorresponding from to eq. a ( differentclassify element the of solutions the using first the homotopy group. dual description However, of we can cohomology also classes, where eq. ( The first homotopy group on themanifolds, left-hand as side explained is previously what using classifies figure PU( from which we obtain the isomorphism: Associated with thishomotopy sequence groups: is a corresponding exact sequence relating the following the following patching together gauge fields infor the northern SU( and southern spatialout hemispheres. in However, this appendix. These results were firstgauge obtained by group Woodward in ref. 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