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Deposited in DRO: 30 January 2017 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Bhattacharya, Jyotirmoy and Lipstein, Arthur E. (2016) '6d dual conformal symmetry and minimal volumes in AdS.', Journal of high energy physics., 2016 (12). p. 105. Further information on publisher's website: https://doi.org/10.1007/JHEP12(2016)105

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Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk JHEP12(2016)105 Springer December 8, 2016 December 12, 2016 December 20, 2016 = 4 super Yang- November 16, 2016 : : : : N Revised 10.1007/JHEP12(2016)105 Received Accepted Published doi: , Published for SISSA by dimensions, the result is very similar to the corresponding  2 − . 3 1611.02179 = 4 super Yang-Mills theory. We confirm this result holographically by The Authors. , and we comment on its possible interpretation. We also study 2-loop 4- c N AdS-CFT Correspondence, M-Theory, p-branes, Scattering Amplitudes

/ The S-matrix of a theory often exhibits symmetries which are not manifest , [email protected] E-mail: [email protected] Centre for Particle TheorySouth & Road, Department Durham of DH1 Mathematical 3LE, Sciences, Durham United Kingdom University, Open Access Article funded by SCOAP ArXiv ePrint: diverges as 1 point integrands with 6dall-loop dual formula conformal for symmetry the and 4-point amplitude. speculate on theKeywords: existence of an its structure suggests athe Lagrangian loop with momentum more inamplitude 6 than of two derivatives.generalizing On the integrating Alday-Maldacena solution out for ato minimal a area minimal string in volume Anti-deseams M2-brane Sitter correspond ending space on to a a pillow-shaped null-polygon. surface This in involves the careful boundary treatment whose of a prefactor which amplitudes are imposed by theMills dual theory conformal and symmetry the ofof planar ABJM dual 4d theory. conformal symmetry Motivatedworldvolume in by theory six this, of dimensions, we M5-branes whichconformal investigate may (if symmetry the provide it uniquely useful consequences enjoys fixes insight such into the a the integrand symmetry). of the We one-loop find 4-point that amplitude, 6d and dual Abstract: from the viewpoint of its Lagrangian. For instance, powerful constraints on scattering Jyotirmoy Bhattacharya and Arthur E. Lipstein 6d dual conformal symmetryin and AdS minimal volumes JHEP12(2016)105 ]). 7 ], M-theory 2 [ ]. In such an 6 5 S = 4 super Yang- × 5 N 5 16 – 1 – 4 18 13 ], respectively. 4 9 [ 8 4 ] establishes a correspondence between string or M-theory is large, the theory is entirely dominated by planar diagrams. 10 6 S 1 11 × N 7 ], which admits an expansion in terms of the ’t Hooft parameter, 10 5 8 1 14 ], and AdS 3 [ 7 S × 4 The second and third examples of the AdS/CFT conjecture are more challenging be- In the first example, the worldvolume theory on D3-branes is 4d 3.1 General setup 3.2 Single cusp3.3 solution Four cusp3.4 solution Regulated on-shell action 2.1 Review of2.2 dual conformal symmetry One-loop four-point 2.3 Bubbles and2.4 triangles Higher loops , and in the limit when N In the planar limit, the theory is believed to because solvable M-theory (for arises a as review,formulate see the the for strong worldvolume example theories coupling [ of limit M-branes. of This string was however, theory, recently making accomplished it difficult to M5-branes, which are conjectured to beon dual AdS to IIB string theory on AdS Mills theory (SYM) [ which in turn canexpansion, also the be topologies classified with by higher the genus topologies are suppressed of by Riemann the surfaces rank [ of the gauge group The AdS/CFT conjecturein [ the nearenergy horizon world-volume geometry theory of of theframework a of branes. gauge-gravity stack duality Although with of this diverse applications,examples idea D-branes the of has most this or prominent conjecture now and involve M-branes grown maximally early into supersymmetric to a field theories the wide on effective D3, M2 low- and 1 Introduction 4 Discussion A Details of 2-loop 4-point amplitude B Details of regularized action 3 Minimal volumes in AdS Contents 1 Introduction 2 6d dual conformal amplitudes JHEP12(2016)105 ]. = 4 ]. It 45 , 14 N 44 – ], which 12 46 = 4 SYM can be N = 4 SYM and ABJM N ] for some attempts in this = 4 SYM, the dual confor- ]. ]. Hence, the origin of dual 24 11 N 41 – – ], it is unclear how to generalize – 9 17 39 orbifold of the space transverse to 16 0) tensor multiplet, which consists , , k 15 Z 2, but allows one to define a tunable ]. In the case of ABJM theory, the am- k > 38 , ]. Hence, dual conformal symmetry imposes ] exhibit a remarkable property known as dual 37 – 2 – When the orbifold parameter is large, it is dual 43 30 1 , – 42 28 . 3 CP × = 4 SYM are nevertheless fixed to all orders by an anomalous 4 4 of the susy when / N = 4 SYM, we could have deduced these theories simply by looking N ]. The key insight was to perform a 8 ]. Moreover, the amplitude/Wilson loop duality of 4d ] and the ABJM theory [ 36 – 27 – 31 25 At loop-level, dual conformal symmetry is broken by IR divergences but the four and Note that dual conformal symmetry is not equivalent to ordinary superconformal sym- Using modern methods for computing on-shell scattering amplitudes, it is possible The third example is the least understood because the worldvolume theory describing Prior to the ABJM theory, there were other proposals which have maximal superconformal symmetry, = 6 susy known as the ABJM theory. 1 for dual conformal invariant S-matrices in 3d and 4d,five respectively. points amplitudes of although the interpretation of these theories is not fully understood [ very powerful constraints on theone planar S-matrix. can For use example, dual in which conformal the symmetry 4-point to tree-level uniquelyThe amplitudes rest fix can of the be the 1-loop tree-level deducedin S-matrix 4-point using principle can integrand, can unitarity then be from methods be usedformulate deduced [ to using ABJM deduce BCFW or the recursion Lagrangian. [ Hence, apriori if we had no idea how to fermionic T-duality in theconformal gravity symmetry dual in is the also ABJM unclear theory appears [ to bemetry rather and mysterious. whensymmetry the for two scattering are amplitudes combined, [ they give rise to infinite dimensional Yangian loop whose contour is obtainedto by tail arranging [ the externalderived momenta from of the the self-duality amplitudeand of head fermionic IIB T-duality string transformations theoryplitude/Wilson under [ loop a duality certain combination does of not bosonic generalize beyond 4-points and the status of the grangians. Moreover,from the the S-matrix point often ofSYM view exhibits [ of symmetries the that Lagrangian.conformal are symmetry For in example, totally the the obscure mal planar planar symmetry limit. amplitudes can of In be the understood context as of the ordinary conformal symmetry of a dual Wilson it to more thantheory one. would In even the admitdirection). absence a of Lagrangian description a (see tunable [ coupling, it is notto clear learn whether a such great a deal about the S-matrix of many theories without reference to their La- multiple M5-branes remains elusive.standing AdS/CFT The and M5-brane M-theory, it theory alsoduality is provides of the not many geometric supersymmetric origin only theories of crucialis electric-magnetic that believed for arise to from under- dimensional be reductionof a a [ 6d self-dual CFT two-form gauge whoseume field, field theory five content for scalars, is a and single a eight M5-brane fermions. (2 is Although well-understood the [ worldvol- the branes which breakscoupling 1 for the theory. TheN Lagrangian is a 3d superconformalto Chern-Simons IIA theory string with theory on AdS for M2-branes [ JHEP12(2016)105 ], 62 = 4 SYM, its integral is N ], as well as at strong coupling = 4 SYM was first confirmed 0) theory is strongly coupled, 0) tensor multiplets assuming , , 51 N – ]. In particular, the 2-loop 4-pt 49 60 – = 4 SYM might extend beyond 1-loop. 54 N ]. = 4 SYM, their Mellin-Barnes representation 64 which exhibit dual conformal symmetry in six – 3 – = 4 SYM, some of the propagators are squared N 2 N ] and 3d [ ]. These all-loop formulae, first conjectured by Bern, 63 47 ]. There is also evidence that a similar all-loop formula exists 53 , ]. On the other hand, since the (2 ], boil down to exponentiating the one-loop amplitude and encoding 52 61 = 4 SYM at leading order. , 48 31 N = 4 SYM, the integrand can be associated with a scalar box diagram 1) SYM amplitudes have dual conformal symmetry in the four-dimensional sense [ , = 4 SYM, and the strong coupling calculation in the dual string theory is N N 3 The all-loop formula for the four-point amplitude of In this note, we find that 6d dual conformal symmetry fixes the 1-loop 4-point in- Given the important role played by dual conformal symmetry in the first two examples Since the M5-brane theory is self-dual, it is unclear how to define a topologicalNote expansion that in terms 6d of (1 an 2 3 in AdS whose boundary corresponds to the null’t polygon Hooft parameter. obtained A bythe similar difficulty space arranging occurs transverse for the to the the ex- M2-brane branes, theory, but which can introduces beas a overcome do tunable by maximal coupling. orbifolding SYM amplitudes in 10d [ conformal symmetry and unitarity.different spacetime Remarkably, structure although than that theseis of very integrands similar, have suggesting a that very the similarity to at strong coupling by Alday and Maldacena who computed the minimal area of a string obtain has a veryessentially different the spacetime same (up structure toture than scheme-dependent was that terms). found of for Recalling thethat that 2-loop a 4-point 4-point very amplitudes amplitude similar with of struc- also the carry dual ABJM out conformal theory, a it symmetry preliminary therefore have analysis appears a of 2-loop universal 4-point structure. integrands consistent We with 6d dual tegrand. Like with massless external legs,which but unlike suggests an underlyingsider triangle Lagrangian and with bubble more diagramsnot with than important dual two conformal at derivative. integrands 4 but show We points. that also they Remarkably, con- are although the 4-point 1-loop integrand that we of the theory. Ourto only a assumptions planar will limit be whereand that a that the semi-classical it theory supergravity has description admitsdimensions. is rational something admissible loop analogous in integrands the bulk, several subtleties withthat this is point not possible of tolocality view. construct and a unitarity tree-level [ For S-matrixit instance, for is (2 it unclear has whatassumptions previously the about been asymptotic the asymptotic states shown states, should locality, be. or even In the this superconformal note, symmetry we will not make any amplitude of the ABJMamplitude of theory has anidentical to almost that identical of structure to the 1-loopof the 4-point AdS/CFT duality described above,for in this the note theory we proceed of to explore M5-branes, its implications assuming it enjoys such a symmetry. There are of course, Dixon, and Smirnov [ the coupling dependence throughbeen the confirmed cusp up to and four collinear loopsusing anomalous in string perturbation dimensions. theory theory [ [ This has for the 4-point amplitude of the ABJM theory [ dual conformal Ward identity [ JHEP12(2016)105 2 / ) d in the gives − t A and = (6 s  dimensions using  = 4 SYM. Finally, 2 − N = 6 d provides more details about B ]. In fact, this was the first hint of the 31 ], and can be associated with the conformal ] is embedded within this more general surface, it is 66 31 , – 4 – we explore the consequences of dual conformal 2 65 5 ]. 70 – 67 = 4 SYM and the ABJM theory, it is possible to encode the 0. This additional divergence is consistent with previous holo- N →  as ]). In / 73 ] for related discussion of Wilson surface operators in the M5-brane theory. We evaluate the on-shell action for the 2-brane solution and find a structure 72 , 4 we present our conclusions and describe future directions. Appendix 71 4 , suggesting the existence of an amplitude/Wilson surface duality and an all-loop , we find a solution describing a minimal volume 2-brane in AdS whose boundary is 3 2 This note is organized as follows. In Note that since the original null polygon of [ Also see [ 4 5 the resulting function does notby depend dual on asymptotic conformal states symmetry and in is the determined to planar all limit. orders Moreover,parameterized it can by be theamplitude computed same picture. at two strong parameters corresponding to the Mandelstam variables that in the planar limit itsymmetric is (the possible definition of to color-ordering definesee in color-ordered gauge amplitudes for theories which example are can cyclically be [ four-point found amplitudes in many with reviews, all possiblesuperamplitude. asymptotic states When this into quantity a is single divided quantity by known the as tree-level four-point a superamplitude, 2 6d dual conformal amplitudes Let us considerwhose a loop hypothetical integrands 6d aresymmetry theory rational. for for such a We which theory, shall a primarily study focusing planar the on limit consequences the 4-point can of case. dual be We conformal defined shall also and assume formula for the 4-point amplitudein analogous section to the BDSmore formula details of about 2-loopthe 4-point integrands calculation and of appendix the on-shell action of the 2-brane in AdS. the Mellin-Barnes technique. Wesection also initiate the studya 2d of surface 2-loop encoding 4-point aand null amplitudes. find polygon. that In We it then has compute asection the very on-shell similar structure action to for the this 1-loop solution 4-point amplitude we computed in graphic calculations of Wilson surfacesanomaly [ of Wilson surfaces [ symmetry in six dimensions. Inintegrand particular, of the we 1-loop find 4-point that amplitude, dual which conformal we integrate symmetry in fixes the null polygon. very similar to the 6dity 1-loop and amplitude, the which existence suggests of anon-shell an amplitude/Wilson action surface all-loop dual- of BDS-like formula the forhas Alday-Maldacena the a string 4-point prefactor solution, amplitude. the that Unlikeand the on-shell depends diverges action like on 1 of the the dimensional 2-brane regularization parameter amplitude/Wilson loop duality. Forcalculation the to M5-brane involve theory, the we minimal wouldwhich volume somehow expect of encodes the an a analogous null M2-brane polygon.Maldacena whose solution, boundary Remarkably, by we is relaxing obtain a a a constraint 2dand solution of surface the the which boundary Alday- minimizes of the volume this of solution a is 2-brane a in pillow-shaped AdS surface whose seams correspond to a ternal momenta of the amplitude head to tail [ JHEP12(2016)105 . (2) 4 = 4 (2.2) (2.1) M N . ) l ( n µ i M p l . λ 2 i µ i = x x =1 ∞ l X +1 , and analyze the possible structure of µ i → x – 5 – (1) 4 µ i − x M µ i = 1 + x n th leg. These coordinates automatically incorporate i M . This quantity can then be expanded in powers of the ’t tree n A / n A . Note that this relies on the ability to cyclically order the external 1 = : n λ M is the momentum of the . Arranging the external momenta of a scattering head-to-tail gives a null polygon whose µ i p Note that dual conformal symmetry is not equivalent to ordinary conformal symmetry and imposes very powerfulSYM, constraints although on this scattering symmetry amplitudes. is broken For by example, IR in divergences, it nevertheless determines the momentum conservation. Dual superconformalsymmetry in symmetry the then dual corresponds space.translations to Note since conformal that the they amplitudes dependpoints are in manifestly on invariant the external under dual dual momentathey space. transform which covariantly correspond Dual under conformal inversions to in symmetry differences the implies of dual the space: nontrivial property that particles and is therefore only well-definedof in the the planar dual limit. space In are equations, defined the by coordinates where 2.1 Review ofDual dual conformal conformal symmetry symmetry can beinto seen by a arranging polygon the external andas momenta shown of writing an in the amplitude figure amplitude as a function of the vertices of this polygon Hooft coupling In the next subsectionsdeduce we the will 1-loop review 4-point the contribution concept of dual conformal symmetry, use it to function in six dimensions, which onlyand relies on does the not assumption of require dualthen knowing conformal symmetry in the asymptotic principle states.superamplitude, be whose The obtained structure four-point is by superamplitude anthe multiplying can important quantity open this question. quantity We by shall the therefore define tree-level four-point Figure 1 vertices are coordinates of the dual space. coupling using holographic methods. Our goal in the present paper is to deduce such a JHEP12(2016)105 (2.3) , where 2 .  = 6. An unusual feature 2 04 D x 2 24 x 2 02 x . 2 j + x 2 ij 2 i x 2 03 x x 2 13 x 2 01 → x 2 ij  2 40 , x 2 24 x x D – 6 – . One loop box. D 0 2 30
2 13 2 0 x x x dx 0 2 20 is a convention tied to our definition of the coupling x x 1 8 D 2 10 → = 6, it is invariant under dual inversions, under which d x Figure 2 D 0 D Z dx ) is manifestly invariant under translations in the dual space. 1 8 2.3 . = 4 (1) 4 M and the factor of j x ], so it would interesting to study the scattering amplitudes of these theories. − i x 75 , = 74 ). This can be represented using a 1-loop box diagram depicted in figure ij x 2.1 in ( if there is an underlyingworth Lagrangian, noting it that may four-derivative correspond termstheories to [ appear a in higher derivative certain theory.Another non-unitary It possibility 6d is is superconformal thatto the these underlying points theory in is section unitary but non-local. We will return Since dual translations and inversionsthis are establishes dual sufficient conformal to symmetry generate ofof the the the integrand integrand dual when is conformal that two group, of the propagators in each term are squared, which suggests that and woudl therefore only modifyNote scheme-dependent that terms the in integrand the in ( 4-pointIt 1-loop is amplitude. also easythe to measure see and that propagators when transform as follows: where λ each edge of theto box define corresponds dual tosubsection, conformal the a triangle former propagator and do in not bubble contribute the integrals, at integrand. but 4-points and as It the we latter is describe evaluate also to in a possible the constant next In 6d, dual conformal symmetryto and have cyclic the invariance fix following the form: one-loop four point integrand a function of dualWard identity, so conformal the cross amplitudes are ratios givenfunction. by and the BDS still formula times satisfy a the nontrivial remainder anomalous dual2.2 conformal One-loop four-point finite part of thean four anomalous and dual five-point conformal amplitudes Ward to identy. all Beyond orders five points in it the is t-Hooft possible coupling to via introduce JHEP12(2016)105 , ]. ,  ) 2 76 z ) t. (2.4) (2.5) ) the − z −  ) with ( 3 ↔ a O 2.3 s 2 + = 6 − 2 + a D/ D 2 ) − − z 1 1 8 , we will obtain a a − 3 + −  2 2 )Γ( 3 which arise from the − π ). Hence, to z  1  D/ − 2 + ). − − t s Γ( )Γ( 2.4 2 Γ( z Γ( z 2 /  ) ln − t s z 1 4 4  a + − πi dz 3  2  a so that the arguments of the gamma )Γ(1 +  terms and would modify the constant ∞ − z is the renormalization scale. i  ∞ 2 1 i t
+ 2 µ a + c  1 − − µ c  : −   1 Z is a complex number chosen such that the 2 , the integral over the new contour does not O 2 + − c −  3 D/ 1 D/ – 7 – <  − − 2 ) )Γ(3 + a s c t µ z )Γ( ( = , and ) z − i i 2 < z a a +  Γ( D/ ) due to the prefactor of 1 3 < 2 z  =1 Γ( a  1 4 i ( iπ  1 2 . The contour for the integral in ( a t =1 s . If we shift the contour to the right while picking up the O − 4 i P 2 − )Γ(  1) ) z = 4 SYM, except for the constant term. Had we multiplied z = )Π = − πi ( + a dz − a 2 1 N (1) 4 ) is IR divergent, so we will regulate it by taking  a − , ∞ Figure 3 as depicted in figure i 2 24 ∞ − M D 2.3 Γ( i +  x 1 c , this would give rise to 2) − c × Γ( − − − 2) = Z E − 1 γ ) = t Γ( E (   and is actually −  γ 2 , i 2 − ( e )  a  1 z 2 13
− = e i − x 0 2 0 , we obtain z 0. Although this breaks dual conformal symmetry, it does so in a controlled and 0, one must choose st  is the Euler-Mascheroni constant and x x 8Γ( 2 = D E d than s − γ =  <  < =1 4 i γ Note that our result for the 1-loop 4-point amplitude six dimensions is the same as The integral in ( This integral can then be evaluated using Mellin-Barnes techniques, as described in [ Π (1) 4 e = 6 Z M where the 1-loop amplitude of by term, which reflects that these terms are scheme-dependent. In section have poles in amplitude is given by the residue at Since functions have positive realterms part. Γ(1 + The integralresidue generates at poles in where arguments of the gammaD functions have positive real part. For the integral in ( In general, an integral of this form has the following Mellin-Barnes representation: where well-understood way as we will see shortly. JHEP12(2016)105 ), but ). From 2.5 3 . = 4, they ν ) i n ν ) + i but has mass 2 ν − ν 2 ij Γ( − x ]. Hence, a bubble . Dual conformal 1 Γ (3 76 5 , ν =1 3 3 i ν 4 legs (for Π 3 + 3 and ν γ − 2
2
4 ν ν n > 2 03 31 2 + . + x γ x 3
1 . Bubble diagrams. 2 ν β ! 2 31 ν
j −
x 2 0 , = 4 SYM. 23 2 3 2 02 x 2 ij 3 i x ν x x 2 0 − N 1 x − α 1 ν Figure 5 ν
2

+
ν β 0 12 2
2 01 x x + 6 x 2 23 1 d x – 8 – ν 0 ( x Z 3 1 2 6 − d = 3 ν Z + ) = α
= bubble 2 12 x M α, β, γ n triangle − 1 i = 4 SYM, the 4-point 2-loop integrand has a double box topology, M 2 . To simplify the analysis we will focus on double box diagrams, . Triangle diagrams. n/ N 6 π = 6 and ( = 3 ν + Figure 4 2 ν triangle + M 1 ν ], one finds that they evaluate to rational functions of the kinematic invariants: For any number of legs, one can define the following dual conformal 1-loop bubble 77 2.4 Higher loops In this section wein will six consider dimensions. 2-loop In 4-pointas amplitudes depicted with in dual figure conformal symmetry Note that this integraldimension zero. must Dimensional be analysis a thereforewhich implies can function that be it of explicitly must the evaluate verifiedcontribution to using kinematic would a the modify constant, invariant formulae the in constantas term appendix we of explained A in the of the 1-loop [ previous 4-point section, amplitude the in constant term ( is scheme-dependent in any case. integrals: this equation, we seecollapse that to bubble such integrals contributions whichof we only will [ exist describe shortly). for Furthermore, using the results symmetry restricts triangle integrals to have the form where of the 4-point amplitude to all orders2.3 as it does in Bubbles andIn triangles the previous section,consider we triangle focused or on bubble 1-loop diagrams, box as diagrams. depicted In in principle, figures one can also the same structure atAdS, strong which coupling suggests by that an computing anomalous the dual minimal conformal volume Ward of identity fixes a the 2-brane finite in part JHEP12(2016)105 = 6, (2.6) D = 4 SYM, even t. ). N ↔ t 2.3 s ↔ + s 2 05 + x 2 2
(which amounts to taking this
2 54 2 05 01 x x x become on-shell and we can re-label 2 53 2 54 ) after cutting the propagators with x x 4 04 2 2 2 2.3

x

2 24 2 52 2 24 2 53 x x x x = 4 super-Yang-Mills and string theory on . These two integrands are given by and 2 2 7 2 13
2 52
02 – 9 – N x . Two loop box. x x 2 04 2 13 x 2 04 x x 2 2 02
x 2 02 ) are remarkably similar to that of 2 x Figure 6
] to verify the validity of the BDS form of the four point 2.6 2 01 2 01 x 31 x 5 5 dimensions and see if they arise from exponentiating the 1-loop x x  D D 2 d d 0 0 , as depicted in figure . Cutting a propagator of the 2-loop box gives a 1-loop box. − x x is no longer integrated over. Discarding on-shell momenta, the two 53 D D , we list all ten integrands consistent with dual conformal symmetry x 0 d d x = 6 A or Z Z D 01 Figure 7 x was exploited in [ since = 4 SYM, as we found at 1-loop. It would be very interesting to integrate these 5 15 S N x × In appendix 5 as 05 amplitude, which would suggest a BDS-like formula. 3 Minimal volumes inAt AdS strong coupling, theAdS duality between two distinguished integrals inthough ( their spacetime representation2-loop looks 4-point completely amplitude different. ofthat a This of 6d suggests dual that conformalexpressions the theory in should be very closely related to momentum on-shell), then the momenta x integrands above reduce to terms appearing in the 1-loopas integrand well ( as their Mellin-Barnes representations. The Mellin-Barnes representation of the For example, if we cut the propagator with momentum case, there are ten possibletwo structures of consistent which with reduce dualmomentum conformal to symmetry the in 1-loop integrand in ( although contributions with triangle or bubble subdiagrams deserve further study. In this JHEP12(2016)105 (3.2) (3.3) (3.4) (3.5) (3.6) (3.1) . µ y , which ends 5 µ y r + by the relation . We will take the 2 4 , r µν , 2  G R = ν b 4 . Y 1 ∂ξ ∂X − − µ . a 1 2 i = 5. Note that 11d supergravity . − y 2 ∂ξ 4 2 ∂X 4 ≥ Y Y µν d ,Y + Σ − + r 1 G 2 2 2 1 2 r  3 r ,... Y = Y 5 + 4 , = 2-brane in the near horizon geometry of , with 2 0 + Y Det d ) are the worldvolume coordinates of the 1 can be embedded in 2 3 2 2 M dy + − s = 4 5 Y ξ Y , ξ 1 − i 3 2 − + d − ∀ – 10 – = , ξ 2 , 2 1 1 0 ν Z ξ Y ,Y 2 3 = 0 + , dX M ,Y i 2 2 µ 5). In the Poincare patch the metric is T y 0 , Y 1 ). This AdS ≥ = = 0 , with the 3-form potential having non-zero legs only along dX − 3 d L 4 3.2 µν 1 Y ξ = 0 2 S − 3 G d Y µ × 7 − (with Z d = slice of ( where S 5 , ]. µ r y are the coordinates of the target space with metric 79 , µ = is the M2-brane tension, ( 78 X µ 2 Y M T In this section we will find a simple generalization of the Alday-Maldacena solution = 0) removed, which can be motivated by noting that a 2-brane is one dimension higher 2 Y In terms offollowing embedding constraints: coordinates, the single cusp solution forThis a is essentially 2-brane the same satisfies as the( single-cusp the Alday-Maldacena solution with one constraint These coordinates are related to the Poincare coordinates by 3.2 Single cusp solution and focus on a AdS target space to be AdS For the rest of the discussion we will set where brane, and a stack ofneglected) M5-branes [ (since the solution is3.1 localized on the General sphere, setup The the action 3-form for an can M2 be brane is given by the induced volume on the brane evaluating the on-shell action for the string world-sheet andthat exponentiating. minimizes the actionadmits for a a solution 2-brane of in AdS the AdS sphere, which describessolution a can stack then of be M5-branes interpreted in as the a supergravity single limit. Hence, our on a four-sided null-polygonas at the the contour boundary. ofΣ This a boundary then Wilson-loop polygon constitutes dual can theloop. to be world-sheet the thought The of four of extremality the pointmotion of amplitude. probe-string following Σ from ending The implies Nambu-Goto on extremal action. that the surface the The boundary four string point Wilson- world-sheet amplitude satisfies is the then equations obtained by of amplitude. This was performed by finding an extremal surface Σ in AdS JHEP12(2016)105 ] ) 0 2 y 31 − 3.6 p (3.9) (3.7) (3.8) ), we 2 , y 1 , y 0 y ), they have a very ) = ( 3 = 0). In fact, it can 3.5 3 , ξ 2 Y . ) gives a 4-cusp solution 2 i 2 , ξ , y 1 2 ], the constraints in ( 3 , y ξ y 1 +1 =3 31 i p X − . , y 0
− 2 y 2
= 0 y . 2 2 2 3 y − Y ) = ( 1 1 − 3 Y , y 1 2 2
, ξ y 2 1 = 2
y 1 2 1 − , ξ y − 1 −
ξ ). Note that this solution describes a minimal- Y 2 1 ) actually describe a 2-brane solution. Indeed, 1 − 0 y 1 – 11 – 3.1 3.6 − q ,Y -brane solutions in AdS: 2 0 v u u t p = y ) translates to = = 0 2 4 3.8 , r Y q 2 , r y ). Choosing ( 4) transformations following [ 2 = 1 , y y r 1 3.5 y = 0 = y 0 0 limit of this solution is a two dimensional surface on the boundary y . t → = r s ) is indeed a solution to the Nambu-Goto equations of motion for a 2-brane , we have plotted this surface by projecting it on the spaces orthogonal to 3.7 8 , respectively. From this figure, we see that the Wilson surface can be visualized 3 ], which we shall describe in the next section. y Note that the 31 In figure and as a pillow with fouridentify the seams edges corresponding with to thevariables the satisfy momenta edges of of a a scattering null amplitude, polygon. the Moreover, resulting if Mandelstam we on-shell action of the minimaldirections string solution of times the a contribution world-volume,the from although the classical this remaining solutions factorization themselves. does not occur atof the AdS level which of may be taken as the contour of a Wilson surface in the 6d dual field theory. These solutions have the remarkable property that their on-shell actions factorize into the where we have made useto of the ( equations of motion followingvolume from ( 2-brane in AdS andby can writing be it in obtained embeddingbe from coordinates the generalized minimal-area to and an string relaxing infinite solution a family in constraint of [ ( In terms of Poincare coordinates ( 3.3 Four cusp solution On performing abecome set of SO(2 If we use thefind worldvolume that diffeomorphsim ( symmetryin to AdS. set After ( realizing howit to is lift straightforward the to single generatein cusp a solution [ four-cusp for solution a for string a to 2-brane that following of the a procedure 2-brane, rather nontrivial that the constraintswhen in the ( constraints are transformeddifferent to structure Poincare coordinates than using the ( Alday-Maldacena solution: than a string and therefore requires one less condition to specify it. Nevertheless, it is JHEP12(2016)105 , , 2 2 ) and (3.10) (3.11) (3.12) of the ξ ξ , . 3.1 2 2 ξ ξ sinh sinh a, b 1 1 2 ξ ξ sinh sinh ) followed by a ξ 1] and we make 1 1 , 2 ξ ξ 3.8 sinh sinh ξ 1 , the surface reduces cosh 3 3 − b b y 1 sinh sinh ξ + + a ξ cosh . b b 2 2 1 ξ ξ 2 a ) ξ + + sinh b 2 2 ξ ξ a cosh cosh 2 sinh . Also note that if we replace a t (1 + 1 1 8 2 a ξ ξ cosh cosh ) = π ), it remains a valid solution of the 1 1 s ξ ξ (2 cosh cosh 3.10 = = = t cosh cosh 3 1 , must take values in [ − = = } 3 1 3 , 1] in ( , 2 , y , , y ), obtained by projecting it on the space orthogonal , ξ – 12 – ) 1 2 2 2 2 b , y , y ξ ξ ξ − we can simply apply a boost to ( [ 2 2 , ξ 3.12 ). When projected orthogonal to − ξ ξ 1 t 2 t ξ ∈ 2 a { (1 sinh sinh sinh ) solves the equations of motion arising from ( ξ 6= = 8 ) 2 2 3 ) s s 1 1 1 sinh sinh ξ ξ 2 ξ ξ ξ π ( ξ 3.10 1 1 sinh f 2 (2 ξ ξ ξ 2 3 1 = 0 ( sinh ξ sinh sinh sinh ξ = b sinh 1 − s b b b sinh sinh ξ 1 sinh = 0, which corresponds to 1 ξ b b − + + + sinh b 1 p 2 2 2 ξ ] to obtain + + 2 sinh ξ ξ ξ b a 2 2 31 cosh 2 ξ ξ b cosh a cosh cosh cosh 1 + 0 gives the following equations for the surface in boundary: ) when a √ 1 1 1 cosh cosh 1 + → ξ ξ ξ a respectively, for √ 1 1 3.9 r ], they are now in Minkowski space rather than AdS, which makes the two ξ ξ a 3 y 31 cosh cosh cosh . Plots of the boundary surface ( cosh cosh = = = , and 0 2 r Taking To generalize this solution to = = 0 y y by an arbitrary function of y 2 0 y y 3 Although these equations aresolution identical [ to thesurfaces 4-cusp distinct. string solution of Alday-Maldacena equations of motion. This is simply related to world-sheet diffeomorphism invariance. It is not difficult toreduces check to that ( ( ξ where the world sheetthe coordinates following identification ofsolution: the Mandelstam variables with the parameters to to a null polygon with its interior filled. rescaling following [ Figure 8 JHEP12(2016)105 2 ), ), , ξ 1 3.14 3.10 (3.14) (3.15) (3.16) (3.17) (3.13) ξ on a circle 7 . ), we find that 2 , , 1 ) for simplicity. , 3.14 3.14 = 0
) = 0 0 2 i  ], where the Lagrangian , y ) and restoring the AdS 1 31 for O 3.1 , y , 0 + y =0 2 (  3 ) M i  T y 3 ( , y R 2 2 , → y . This is not surprising, since , − 2 i 2 ξ − ˜  ) 2 =0 S =  2 r 3 
, y M ) near the cusps. This can be accomplished ξ L 2 T 1  2 ). T y integrals and we obtain =0 3 − and  = 2+ Z ) 2 − R ) into the regulated action ( 3.15
1 3 3.10 (1 3 ). Now we wish to modify the solution in ( 2 2 ξ 3 0 – 13 – = y , ξ reg ξ y ( 1 dξ S = ξ 3.16 i −  reg 2 ) 3.14 L  S 1 ) back into Lagrangian ( 1 + Z (2 + r 3.10 2 q M → T 3 3 ) generalizes to ) = R 2 , y 3.7 = , y 1 =0   ], this regularization was motivated by analytically continuing the T r , y decouples from the 0 31  2 3 y ( ξ r ] and define the regularized action as 1 + 31 = 0 solution refers to that in ( r 0. In [  → , we get r  < R Upon substituting the solution in ( Although it is difficult to obtain a 4-cusp solution for the regularized action in ( On plugging the solution ( where where the the integral over as a solution of the regularizedso that action it ( reduces to theby regulated the solution following ( redefinitions: it suffices to knowsingle-cusp the solution single-cusp in ( solution for the regularized action. In particular, the where near-horizon metric of blackM2-brane D-banes. in the Since bulk, we it areand would computing be then interesting the apply to minimal the first volumereduced dimensionally above of solution. reduce procedure AdS an In to this compute paper, the we on-shell will action stick with of regularization the in dimensionally ( To implement the analogue of dimensionaloutlined regularization in in [ the bulk, we follow the method simply evaluated to unitydescribe on how substituting to the evaluate theregulated. full on-shell solution. action, which In is the divergent and next therefore section, needs3.4 we to will be Regulated on-shell action radius which is independent ofare the the corresponding coordinates coordinates of the string world-sheet in [ JHEP12(2016)105 ] apart (3.18) ) which 31 M5-branes 3.18 ] and circular N 65 2. For more details, / . )  , (  O ), the expectation value C 0) theory [ + 2 + ,  3.18 are scheme-dependent since /  3 2 2 instead of 3 t s  / C 2  + s µ 2   ln 2 . b 3 2 1 4 ; 0. A similar divergence was found in   − + . 2 ln 2 2 dimensions. We shall discuss this point − → 2 ,t reg  1 2  + S 2 , div i 2  2 e S − 8 ln / − ) is given by =  , − + 2 – 14 – 2 4 1 2 s 2 µ ,s 3 π  3.12 M  1 div 2 = F S 1 and the constant term  2 2 C  .  − µ 2 ), and div 2
,t = π
S N  2  4 ,s and that in the near-horizon limit of a stack of div 2 − ], this constant was reported to be 1 − − 1 ], and can be associated with the conformal anomaly of Wilson S 3 p div 31 l 2 Γ = 66 S ) Γ 1 π π reg (2 − terms in S i =
= .  1 ˜ 2 S 4 M T ]. Furthermore, it may be possible to interpret the prefactor in ( O is identical to the regulated on-shell action for the string solution obtained in [ ˜ . as an t’Hooft coupling in the 6 70 S , we finally obtain – B 67 is the renormalization scale, πN N/ ). This suggests that the 6d 4-point amplitude is dual to a Wilson surface and µ = ] is an overall prefactor which diverges as 3 In terms of the on-shell action for the minimal 2-brane in ( 2.5 6 Note that 31  6 p R l mensions. This study isM2-brane motivated worldvolume by theories the have fact dual that conformal the symmetry scattering evenfrom amplitudes though an of it additive D3 is constant.see and hidden In appendix [ goes as further in section 4 Discussion In this note, we have explored dual conformal symmetry for loop amplitudes in six di- the on-shell action forin the [ 2-brane compared toprevious the holographic on-shell calculations of action Wilson of surfacesWilson the of loops the string of 6d 5d obtained (2 SYMsurfaces [ [ Note that the on-shell actionin has ( a very similarcan structure be to described the 1-loop to 4-point all-loop amplitude orders in terms of a BDS-like formula. A novel feature of Note that the they can be altered by rescaling of the Wilson surface with contour in ( (with a similar formula for where Recalling that  and JHEP12(2016)105 . φ 4 , and φ∂ A ) are squared, which 2.3 = 4 SYM. A new feature of the = 4 SYM. = 4 SYM can be computed at N N N , whose Mellin-Barnes representations are very – 15 – ], it was proven that it is not possible to construct dimensions, we obtain a result that is remarkably = 4 SYM and the ABJM theory. We also analyze 2.4  61 2 N − ]. This is certainly a possibility for the M5-brane world- 80 = 6 d 0. This additional divergence was observed in previous holographic → ]. Another possibility is that the 6d theory in question is not unitary.  ], so studying the scattering amplitudes of these theories is an important 82 , 81 75 , , computing the 2-loop Mellin-Barnes integrals we obtained in appendix
74 2  volume theory given that it hastheory been [ argued to beIndeed, a non-gravitational four-derivative self-dual terms string appear inries certain [ non-unitary 6d superconformaldirection theo- for future research. In [ similar to that of the 2-loop 4-pointNote amplitude that of two of thesuggests propagators that in if there the is 1-loop anSuch integrand underlying theories in Lagrangian, are it ( not contains generally termstheories unitary of unless like the string they form theory arise [ from expansions of non-local for the 4-point amplitude.O This woulddetermining require if extending any our linear 1-looping combination of calculation the these 1-loop to integrals amplitude. canintegrands In we arise the describe from end, exponentiat- in it section may only be necessary to consider the two Perhaps the most immediateculations task to would be higher to loops extend and our check perturbative if 4-point they cal- are consistent with an all-loop formula Although our calculations are highly suggestive, the existence of a 6d theory with these Given that the planar scattering amplitudes in If we consider a hypothetical 6d theory which has a planar limit and scattering ampli- • • properties and its relation toa M5 number branes of is questions still that rather deserve speculative. further study: In particular, there are that the 6d 4-point amplitudeorders is using dual a formula to analogous aon-shell to Wilson action the surface for BDS and the formula can 2-brane of which compared be diverges to described as that to of all-loop thecalculations string of in Wilson AdS is surfaces. an overall prefactor also be computed holographically bycorresponds minimizing to the a volume Wilson of surfacesimple a which generalization 2-brane somehow of encodes whose the boundary aof 4-cusp null a Alday-Maldacena polygon. 2-brane solution In in which fact, AdSthe minimizes and we the 1-loop verify find that action a 4-point its amplitude on-shell action we has deduced essentially using the dual same structure conformal as symmetry. This suggests similar to the 4-pointtwo-loop amplitude 4-point of amplitudes in sixdual dimensions and conformal identify symmetry two and integrands compatible unitarity. with strong from thenull-polygonal area Wilson of loop, a one string may worldsheet expect in that AdS the whose amplitudes boundary of corresponds the to 6d a theory can play a role in the scattering amplitudes of M5-branetudes worldvolume with theories. rational (field theoretic)symmetry loop fixes integrands, the we integrand find forgral that the over imposing 4-point loop dual 1-loop momentum conformal amplitude. in After performing the inte- from the point of view of their Lagrangian formulations. This suggests that it may also JHEP12(2016)105 t. (A.1) ↔ s + 7 ν
2 05 x ], where the radius 4 . For the M5-brane ν
23 , 2 54 N/k x 22 ]. It would therefore be 5 83 ν
2 2 53 α x
6 2 24 ν x
1 2 52 α x
3 2 13 orbifold of the space transverse to the M2- ν x
k 2 04 Z x – 16 – 6. This is reminiscent of the conjecture that 5d 1 ν
→ 0) compactified on a circle [ . If we interpret this as the ’t Hooft coupling, we see 2 02 , d x N/ 2 ν
2 01 is sent to infinity. For the M2-brane worldvolume theory, this x ]. 5 86 N x  2 − ], it has been argued that they may be removed by non-perturbative 6 d 84 0 x  ]. 2 − ) is proportional to 85 6 d = 4 SYM, dual superconformal symmetry was shown to arise from self-duality 3.18 Z N = duality in M-theory, perhapsstring by compactifying theory, the performing theory varioustransformations, on combinations and a of then circle bosonic decompactifying towere and back obtain considered fermionic to in IIA [ T-duality M-theory. Some of these ideas divergences [ effects [ In of the dual stringtransformations. theory under It a would combination therefore of be bosonic interesting and to fermionic look T-duality for an analogue of T- that the coupling divergesSYM as is equivalent to theof 6d compactification (2 is proportionalinteresting to to explore the how our 5dWilson results coupling loops relate [ of to the 5d planar SYM. scattering amplitudes Although and perturbative amplitudes of 5d SYM have UV can be accomplished by performingbranes, a which ultimately gives riseworldvolume theory, to our the results ’t suggestthe Hooft that the worldvolume coupling coupling itself. is In tiedin to particular, ( the note dimension that of the prefactor of the on-shell action It would be interesting towhich revisit suggest this that question some of in these light of assumptionsA our may crucial need loop-level to assumption calculations, be indefine relaxed. our such analysis a is limit, theof there the existence must gauge of be group a a planar coupling which limit. can In be order sent to to zero as the rank tree-level amplitudes of self-dual tensor multiplets assuming unitarity and locality. (2) 4 • • A In general, the 2-loop integrand with a double box topology has the following form: ful comments onST/L000407/1. the AL manuscript. is supportedsearch JB Fellowship by holder. the is Royal supported Society by as a the Royal STFC SocietyA University Consolidated Re- Grant Details of 2-loop 4-point amplitude Ultimately, we hope thisquestion line of of how investigation to will formulate yield the new worldvolume insight theory into of the M5-branes. Acknowledgments longstanding We thank Paul Heslop, Simon Ross, Douglas Smith, and Tadashi Takayanagi for help- JHEP12(2016)105 . d . 4 I (A.2) (A.3) + 3) + 2) )   1 ), and we = 4 SYM z ) + + 4 − 2.6 4 z 4 N z z , ) ) − )Γ( 2 − 3 − 1 3 z z . Note that first z 1 z 1 k z z + − ν + − 1 2 12 2 + z ν z ν 2 j 2 3 3 4 4 3 3 2 4 2 − − ν Γ( α − + 2)Γ( + 1)Γ( ) are very similar to that 1 2 Γ(  z +  ) i 1 457  2.6 4 3 2 2 1 1 1 1 1 1 2 + + ν ν D/ α t s z ) 4 4 , respectively. For comparison, 4 z  z − = − z d )Γ( 7 j 2 6 1 (2) 1 1 1 1 1 2 2 3 1 2 4 ν πi z I − z ijk dz 2 1 ν D/ + − z 6 + 1)Γ( 4 + 2)Γ( 1 1 2 3 1 2 1 1 2 1 7 =1 1 ν and Y j ν + z 4 )Γ( 4 2 z z d correspond to the integrals ( z Z 6 (1) 5 , and )Γ( 123 3 2 2 1 1 1 1 1 1 2 + + ) consistent with dual conformal symmetry. I j 1 ν ) 2 + 4 ν − l 3 ν 3 z ν 1 z z − A.1 z D/ + 4 1 2 1 1 3 1 2 1 2 1 Γ( ν i – 17 – 2 + D − − 7 ν , 6 , , 3 5 D/ ) )Γ( = 567 + 1)Γ( 1 1 2 3 1 2 1 1 2 1 , + 2)Γ( ν 1 4 D 4 4567 ν , ) 4 4 z z − ν ij π 1 z z ν − z =2 − 2 + l +1 3 2 2 1 1 1 1 1 1 2 + , + 2 ν 123 )Γ( − 2 D 4 ) ν 2 Q i 2 z 0) and we will denote the Mellin-Barnes integrand as ) has a Mellin-Barnes representation 2 z z z ν Γ( ) , Γ(1 1) D/ t 1 1 2 + 1 2 1 1 3 1 2 1 2 1 )Γ( + Γ( , α ν − Γ( , A.1 π =1 3 4 2 1 ( 2 7 i 1 3 4567 )Γ( z ν s , π z 3 − ν , α 1 P z + 7 , + − = = )Γ( 3 1 (9) (8) (7) (6) (5) (4) (3) (2) (1) 4 2 = No. (10) , ν , z ) ) ) z z D 6 4 4 1 4 ν 23 + , ) + + , ν Γ( , z , z )Γ( 4 , 1 ν , z 5 3 3  4 3 , 2 3 1 z z , z 2 z 1 α − , ν , z , z 1 t 3 , z , 4 2 2 2 ν ν 2 + − + , z (1 − 2 5 , z , z , ν 2 , z Γ( I z D 1 3 ν D/ 1 1 . Table listing sets of exponents in ( z , z z + = 6 . The integral in ( ( ( Γ( 1 , ν 1 Γ( Γ( − d 2 z d α 1 1 4 ( 6 (2) s D × × × z d I , ν I = 4 SYM: ( 4 1 = I d ν 6 (1) Table 1 ( N I I shall denote their Mellinthe Barnes integrands Mellin-Barnes as representation ofis the given planar by 4-pointRemarkably, 2-loop the amplitude Mellin-Barnes in integrands correspondingof to ( in table where and the second sets of exponents listed in table There are ten possibilities consistent with dual conformal symmetry, which we summarize JHEP12(2016)105 , (B.3) (B.4) (B.1) (B.2)  2 b ;  2
3  − 2 b 2 1 O ;  2 +  ) −  ) 2 ,  ξ b b ) + 1 + 2 1 ( .
2 2 1 4 ξ −  O 
)) 1 1 + 1  1 sinh 2 + + F ξ , term, we find an extra additive 1  . 2  2 2 + 1 ξ
2
 I 2 term, in the amplitude. We think 2 0 ) 2 ln  2 2 2 O 2 

b 3 0 (
) sinh(2 )) 2  2 π  2 ) 4  1 2  − + −  O sinh 2 ξ ξ 2 ( − ) cosh( − O 2 − − 2 b 1 1 1 − I O  1 ξ − ] as well. 1 2 − + 2 + Γ ) 7  2 Γ (4) 31 −  2  2 2 π −  2 ) Γ ξ ξ sinh(2 b ) cosh( π + Γ = = = b 1 ξ 2 − 1 √    2 log ) ) ) 2
cosh I − 2 2 2 ) + cosh( 0 (1  ). 1 M cosh(2 ξ ξ ξ  2 ) ξ 1 2 T ξ 2
+ 3 ξ 1 1 + O – 18 – + 3.18 R sinh sinh sinh −  ) + cosh( + ) 1 1 1 = 2 2 (cosh  b ξ ξ ξ b ) sinh( ξ 2 2 ) takes the form  1 4 2 1 − M ξ dξ 4 log(4) ) cosh(2  T + 2+ 1 1 sinh sinh sinh 3 (1
3.17 ξ ) + 2 3 − 2 b b b 2 3 dξ 1 1 R ), which permits any use, distribution and reproduction in 1 2 I ) sinh( 1 ξ ξ 2 I sinh( 2 dξ 1 4 + + +  ∞  − ξ b − 2 2 2 + − ξ ξ ξ −∞ = = = 1 cosh(2 2 16( Z  I 2
  2 cosh(2 T sinh(
2
T  Z 2
b b cosh cosh cosh  1 ; + 1 2 − 2 1 1 1 − − 8(  CC-BY 4.0 1 2 ξ ξ ξ 2 M − b = T Γ 2 − 3 This article is distributed under the terms of the Creative Commons Γ b π iS − 1 2 R ]. This eventually contributes to the constant (cosh (cosh (cosh ; 31 2 2 2  = = = 2  1 2 dξ dξ dξ − I I T 1 1 1 , 1 2 dξ dξ dξ  ∞ ∞ ∞ 1 + + + F −∞ −∞ −∞ 2 Note that on evaluating the onshell Nambu-Goto action, in the Z Z Z 7 , compared to [ 1 4 Attribution License ( any medium, provided the original author(s) and source are credited. this should be present for the case of the string worldsheet in [ and putting all the terms together gives ( Open Access. Using the following expansions The integrals evaluate to where The regulated on-shell action in ( B Details of regularized action JHEP12(2016)105 B 0) , , JHEP ] Int. J. , (1982) (2 , 99 , B 97 (1974) 461 Supergravity ]. (2009) 66 B 109 (2008) 091 Nucl. Phys. B 72 , = 11 ]. 10 d (2008) 105 SPIRE B 811 IN Phys. 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