Cluster Algebras, Steinmann Relations, and the Lie Cobracket in planar N = 4 sym

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1 Steinmann Relations, and in sym (Niels Bohr Institute) Galileo Galilei Institute November 8, 2018 Based on work in collaboration with John Golden , 190x.xxxxx

2 Outline Planar N = 4 supersymmetric Yang-Mills (sym) theory Symmetries and Infrared and Helicity Polylogarithms and Cluster-Algebraic Polylogarithms, the Coaction, and Cluster-Algebraic in N = 4 sym Decomposing the

3 N = 4 sym SUSY Ward identities many relations among amplitudes with different helicity structure Conformal theory no running of the coupling or UV divergences AdS 5 S 5 dual theory multiple ways to calculate quantities of interest Supersymmetric the types of functions that version of QCD show up here also appear in QCD

4 Planar Limit and Dual Conformal Symmetry We work with in the N c limit with fixed g 2 = g 2 YM N c/(16π 2 ) All non-planar graphs are suppressed in this limit, giving rise to a natural ordering of external particles This ordering can be used to define a set of dual coordinates p µ α i σα µ = λ α λ i i α = x α i α x α i+1 α The coordinates x µ i label the cusps of a light-like polygonal Wilson loop in the dual theory, which respects a superconformal symmetry in this dual space [Alday, Maldacena; Drummond, Henn, Korchemsky, Sokatchev] x 2 x 1 p 1 p 5 p 2 p 3 x 5 p 4 x 3 x 4

5 Helicity and Infrared Since we are working with all massless particles, our amplitudes A n must be renormalized in the infrared Infrared divergences are universal and entirely accounted for by the BDS Ansatz [Bern, Dixon, Smirnov] In the dual theory, the BDS Ansatz constitutes a particular solution to an anomalous conformal Ward identity that determines the Wilson loop up to a function of dual conformal invariants [Drummond, Henn, Korchemsky, Sokatchev] finite function of dual conformal invariants { ( }} ){ A n = A BDS n exp(r n) 1 + Pn NMHV + P N2 MHV n + + Pn MHV }{{}}{{} IR structure helicity structure

6 Dual Conformal Invariants We can construct dual conformally invariant cross ratios out of combinations of Mandelstam invariants x 2 ij = (x i x j) 2 = (p i + p i p j 1) 2 that remain invariant under the dual inversion generator I(x α i α α xαi ) = x 2 i I(x 2 ij) = x2 ij x 2 i x2 j These can first be constructed for n = 6 since x 2 i,i+1 = p 2 i = 0 x 3 x 2 u = x2 13x 2 46, v = x2 24x 2 51, w = x2 35x 2 62 x 2 14 x2 36 x 2 25 x2 41 x 2 36 x2 52 x 4 x 1 In general, we can form 3n 15 independent ratios x 5 x 6

7 Loops and Legs in Planar N = 4 Legs Loops MHV NMHV Ω [Bern, Caron-Huot, Dixon, Drummond, Duhr, Foster, Gürdoğan, He, Henn, von Hippel, Golden, Kosower, AJM, Papathanasiou, Pennington, Roiban, Smirnov, Spradlin, Vergu, Volovich,... ] Unexpected and striking structure exists in the the direction of both higher loops and legs Galois Coaction Principle Cluster-Algebraic This talk will focus on using these polylogarithmic amplitudes (especially the two-loop MHV ones) as a data mine

8 Polylogarithms Loop-level contributions to MHV (and NMHV) amplitudes are expected to be multiple polylogarithms of uniform transcendental weight 2L, meaning that the derivatives of these functions satisfy df = i F s i d log s i for some set of symbol letters {s i}, where F s i is a multiple polylogarithm of weight 2L 1 [Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka] The symbol letters {s i} can in general be algebraic functions of dual conformal invariants Examples of such functions (and their special values) include log(z), iπ, Li m(z), and ζ m. The classical polylogarithms Li m(z) involve only the symbol letters {z, 1 z} Li 1(z) = log(1 z), Li m(z) = z 0 Li m 1(t) dt t

9 The Coaction Multiple polylogarithms are endowed with a coaction that maps functions to a tensor space of lower-weight functions [Goncharov; Brown] H w p+q=w H p H dr q If we iterate this map w 1 times we will arrive at a function s symbol, in terms of which all identities reduce to familiar logarithmic identities The location of branch cuts is determined by the 1,w 1 component of the coproduct, up to terms involving transcendental constants The derivatives of a function are encoded in the w 1,1 component of its coproduct 1,...,1Li m(z) = log(1 z) log z log z

10 Symbol Alphabets and Discontinuities The symbol exposes the discontinuity structure of polylogarithms In six-particle kinematics there are only 9 symbol letters: S 6 = {u, v, w, 1 u, 1 v, 1 w, y u, y v, y w} s i...k = (p i + + p k ) 2, u = s12s45 s 123s 345 y u = 1 + u v w (1 u v w) 2 4uvw 1 + u v w + (1 u v w) 2 4uvw Only letters whose vanishing locus coincides with internal propagators going on shell can appear in the first symbol entry In seven-particle kinematics there are 42 analogous symbol letters, 14 of which are parity odd For more than seven particles, symbol alphabets not as well understood algebraic roots appear in symbol letters even at one loop in N 2 MHV amplitudes [Prlina, Spradlin, Stankowicz, Stanojevic, Volovich]

11 The Steinmann Relations 2 1 The Steinmann relations tell us that amplitudes cannot have double discontinuities in partially overlapping channels [Steinmann; Cahill, Stapp] vs Disc s234 (Disc s345 (A n)) = 0 log ( ) ( u vw log w ) ( uv... log u ) ( vw log v ) uw...

12 The Steinmann Relations 2 1 The Steinmann relations tell us that amplitudes cannot have double discontinuities in partially overlapping channels [Steinmann; Cahill, Stapp] vs Disc s234 (Disc s345 (A n)) = 0 log ( ) ( u vw log w ) ( uv... log u ) ( vw log v ) uw......in fact, the Steinmann relations constrain not just double discontinuities, but all iterated discontinuities [Caron-Huot, Dixon, von Hippel, AJM, Papathanasiou] For six and seven particles, this appears to be equivalent to requiring cluster adjacency [Drummond, Foster, Gürdoğan]

13 Infrared Normalization Steinmann functions don t form a ring Disc si 1, i, i + 1 [ A (1) n ] 0 [ [ ( ) ]] 2 Disc s234 Disc s345 A (1) 0 The BDS ansatz exponentiates the one-loop amplitude, leading to products of amplitudes starting at two loops (and obscuring the Steinmann relations) n

14 Infrared Normalization Steinmann functions don t form a ring Disc si 1, i, i + 1 [ A (1) n ] 0 [ [ ( ) ]] 2 Disc s234 Disc s345 A (1) 0 The BDS ansatz exponentiates the one-loop amplitude, leading to products of amplitudes starting at two loops (and obscuring the Steinmann relations) This is fixed by the BDS-like ansatz, which only depends on two-particle invariants n A BDS n exp(r n) A BDS-like n E MHV n The BDS-like ansatz only scrambles Steinmann relations involving two-particle invariants, which are obfuscated in massless kinematics anyways

15 Infrared Normalization However, the BDS-like ansatz only exists for particle multiplicities that are not a multiple of four [Alday, Maldacena, Sever, Vieira; Yang; Dixon, Drummond, Harrington, AJM, Papathanasiou, Spradlin]

16 Infrared Normalization However, the BDS-like ansatz only exists for particle multiplicities that are not a multiple of four [Alday, Maldacena, Sever, Vieira; Yang; Dixon, Drummond, Harrington, AJM, Papathanasiou, Spradlin] To unpack this statement: there exists a unique decomposition of the one-loop MHV amplitude taking the form A (1) MHV,n = Xn(ɛ, {si,i+1}) }{{} IR structure + Y n({s i,...,i+j}) }{{} dual conformal invariant for all particle multiplicities n that are not a multiple of four Exponentiating the function X n rather than the full one-loop amplitude accounts for the full infrared structure of this theory, yet is invisible to the operation of taking discontinuities in threeand higher-particle channels

17 Infrared Normalization This is problematic if we want to test the equivalence of the Steinmann relations and cluster adjacency in eight-particle kinematics However, if this is test is our only objective the last slide makes clear there is a way out: normalize the amplitude by a minimal BDS ansatz only consisting of the infrared-divergent part of the one-loop amplitude It can be explicitly checked that this restores not only all (higher-particle) Steinmann relations, but also all cluster adjacency relations this provides further evidence that the these conditions are equivalent (when cluster adjacency can be unambiguously applied) [Golden, AJM (to appear)]

18 Lie Cobracket Polylogarithms also come equipped with a structure k 1 δ(f ) (ρ i ρ k i )ρ(f ) i=1 ρ(s 1 s k ) = k 1 ) (ρ(s 1 s k 1 ) s k ρ(s 2 s k ) s 1 k

19 Lie Cobracket Polylogarithms also come equipped with a structure ρ(s 1 s k ) = k 1 k k 1 δ(f ) (ρ i ρ k i )ρ(f ) i=1 (ρ(s 1 s k 1 ) s k ρ(s 2 s k ) s 1 ) The cobracket of classical polylogarithms is especially simple: where δ ( Li k ( z) ) = {z} k 1 {z} 1, k > 2 δ ( Li 2( z) ) = {1 + z} 1 {z} 1 {z} 1 = ρ(log(z)), {z} k = ρ( Li k ( z))

20 Lie Cobracket Polylogarithms also come equipped with a structure ρ(s 1 s k ) = k 1 k k 1 δ(f ) (ρ i ρ k i )ρ(f ) i=1 (ρ(s 1 s k 1 ) s k ρ(s 2 s k ) s 1 ) The cobracket of classical polylogarithms is especially simple: where δ ( Li k ( z) ) = {z} k 1 {z} 1, k > 2 δ ( Li 2( z) ) = {1 + z} 1 {z} 1 {z} 1 = ρ(log(z)), {z} k = ρ( Li k ( z)) In fact, any weight four function that has no δ 2,2 component can be written in terms of classical polylogarithms [Dan; Gangl; Goncharov, Rudenko]

21 Different Levels of Polylogarithmic Full Function S Analytic δ Nonclassical

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23 clusters

24 cluster variables

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37 Gr(4, 6) A 3 CLUSTER ALGEBRAS: AN INTRODUCTION Figure 4. The exchange graph of the cluster algebra of type A3, which coincides with the 1-skeleton of the associahedron [Williams]

38 Cluster Coordinates cluster h16i h12i h56i h15i h13i h35i h23i h34i 1 6 cluster h45i 5 A-coordinates Gr(2, 6) Gr(4, 6) cluster cluster the variables Lie Cobracket a -type: h2456i, h2346i, h3456i h1456i h2456i Symmetries h1246ih3456i and h2345i h2346i h1256i h1456ih2346i x -type: h1246i h1256ih2346i Cluster h1236ih2456i Algebras h1234i h1236i Polylogarithms, h1234ih2456i the h1246ih2345i X -coordinates = u(1 v) v(1 u)y uy v = v(1 w) w(1 v)y vy w = w(1 u) u(1 w)y uy w Gr(2, 6) $ Gr(4, 6) R=α, α Zi = (λ α i, x β α i λ iβ ), abcd = ɛ RST U Za R Zb S Zc T Zd U

39 More generally, clusters can be defined to be quiver diagrams that have a cluster coordinate associated with every node f 11 f 21. f 00 f k1 f 12 f 22 f k2 f 13 f 23 f k3 Gr(k, n) f 00 f 00 f 00.. f 00. f 00.. f 00 f 1l f 2l. f 00 f kl 1,..., k l n k f ij = { i + 1,..., k, k + j,..., i + j + k 1, i l j + 1, 1,..., i + j l 1, i + 1,..., k, k + j,..., n, i > l j + 1.

40 We can translate between clusters in A-coordinates and X -coordinates using x i = j a b ji j b ij = (# of arrows i j) (# of arrows j i)

41 A cluster algebra is the closure of a given quiver under cluster mutation a k a k = i b ik >0 a b ik i + i b ik <0 a b ik i b ij, if k {i, j}, b b ij, if b ik b kj 0, ij = b ij + b ik b kj, if b ik, b kj > 0, b ij b ik b kj, if b ik, b kj < 0. x 1 x k i = (, ) i = k, x i 1 + x sgn(b bik ik), i k k

42 Cluster-Algebraic in Planar N = 4 Cluster algebras appear in sym in a number of striking ways δ(f ) [Building Blocks] The cobracket of all two-loop MHV amplitudes can be expressed in terms of Bloch group elements evaluated on cluster X -coordinates, {X } k [Golden, Paulos, Spradlin, Volovich] [Cluster Adjacency] The cobracket of all two-loop MHV can be expressed as a linear combination of terms {X i} 2 {X j} 2 and {X k } 3 {X l } 1 where each pair of X -coordinates appears together in a cluster of Gr(4, n) [Golden, Spradlin] [ ] The nonclassical part of all two-loop MHV amplitudes can be decomposed into functions defined on their A 2 and A 3 subalgebras [Golden, Paulos, Spradlin, Volovich]

43 Cluster-Algebraic in Planar N = 4 Cluster algebras appear in sym in a number of striking ways S(F ) [Building Blocks] The symbol alphabets for n {6, 7} are precisely cluster coordinates on the Grassmannian Gr(4, n), and all symbol letters in the two-loop MHV amplitudes are also cluster coordinates on Gr(4, n) [Golden, Goncharov, Spradlin, Vergu, Volovich; Drummond, Papathanasiou, Spradlin] [Cluster Adjacency] In the symbol of (appropriately normalized) amplitudes in which no algebraic roots arise, each pair of adjacent A-coordinates appears together in a cluster of Gr(4, n) [Drummond, Foster, Gürdoğan]

44 Cluster-Algebraic in Planar N = 4 Cluster algebras appear in sym in a number of striking ways F [Building Blocks] The two-loop MHV amplitudes are expressible as (products of) functions taking only negative cluster X -coordinate coordinates, Li n1,...,n d ( X i,..., X j) [Golden, Spradlin]

45 Cluster-Algebraic in Planar N = 4 Cluster algebras appear in sym in a number of striking ways I [Building Blocks] The integrands in this theory are encoded by plabic graphs, which are dual to cluster algebras [Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka] [Cluster Adjacency] Cluster adjacency translates to the statement that cluster coordinates only appear in adjacent entries of the symbol or cobracket when the boundaries corresponding to their zero-loci are simultaneously accessible

46 Cluster Polylogarithms Using cluster A- or X -coordinates, we can define polylogarithms on any cluster algebra that can be represented as a quiver In particular, we can consider functions that live on the cluster subalgebras of Gr(4, n) The union of all A- or X -coordinates on the clusters in a (sub)algebra provide a symbol alphabet

47 Cluster Polylogarithms Cluster polylogarithms on the subalgebras of Gr(4,n) efficiently capture the nonclassical structure of R n (2) (or equivalently E n (2) ) There exists only a single nonclassical polylogarithm defined on A 2, and only two on A 3, but they have special properties Physically: δ 2,2(R n (2) ) = A 3 Gr(4,n) d i δ 2,2(f (i) A ) = 3 A 2 Gr(4,n) c i δ 2,2(f (i) A ) 2 Mathematically: - f A2 act as a basis for all nonclassical polylogarithms, while - f A3 acts as a basis for all nonclassical cluster - polylogarithms whose cobracket satisfies cluster adjacency [Golden, Paulos, Spradlin, Volovich]

48 Cluster Polylogarithms Cluster polylogarithms on the subalgebras of Gr(4,n) efficiently capture the nonclassical structure of R n (2) (or equivalently E n (2) ) There exists only a single nonclassical polylogarithm defined on A 2, and only two on A 3, but they have special properties Physically: δ 2,2(R n (2) ) = A 3 Gr(4,n) d i δ 2,2(f (i) A ) = 3 A 2 Gr(4,n) c i δ 2,2(f (i) A ) 2 Mathematically: - f A2 act as a basis for all nonclassical polylogarithms, while - f A3 acts as a basis for all nonclassical cluster - polylogarithms whose cobracket satisfies cluster adjacency [Golden, Paulos, Spradlin, Volovich] This basis of f A2 and f A3 functions is massively overcomplete... what about larger subalgebras of Gr(4,7)?

49 Cluster Polytopes Gr(4,7) = E 6 associahedron 833 vertices 1785 squares A 1 x A Apentagons 2 associahedra 476 A 3 associahedra 14 D 5 associahedra

50 Cluster Polytopes Gr(4,7) = E 6 associahedron 833 vertices 1785 squares A 1 x A Apentagons 2 associahedra 476 A 3 associahedra 14 D 5 associahedra

51 Type Nonclassical Cobrackets appendix we tabulate the number of independent + + weight-four + cluster polylogthat Cluster havea 2 apolylogarithms nonzero Lie 1(0) cobracket on 0Gr(4, 1(0) 7) ' E 6 0and its 0 suablagebras. In A 3 6(1) 0 1(0) 0 1(1) A 4 21 (6) 0 3 (0) 0 0 Automorphism Signature Type D 4 Nonclassical 34 Cobrackets (9) A (21) 2 (1) 5 (1) 2 (0) 5 (3) A 2 D 5 1(0) 116 (42) 0 1(0) 0 0 A 3 E 6 6(1) 448 (195) 0 1(0) 0 1(1) A 4 21 (6) 0 3 (0) 0 0 D 4 34 D(9) 4 Automorphism Signatures + AD 45 + D (21) D 4 D4 2 (1) D 4 + D5 4 (1) 2 (0) D5 4 (3) D D (42) S 3 S 3 S 3 S 3 S 3 S 3 S 3 S E (195) S 3 2(0) 0 + S 3 1(0) 0 + S 3 1(0) 0 3 1(0) 0 S3 0 0 S3 1(1) 0 S D 4 + D 4 + D 5 + D 5 + S 3 Z S 2 3 Z 2 5(2) 0 0 (0) 0 D 4 Automorphism Signatures Nonclassical + D D 5 5 Automorphism Polylogarithms Signatures D 4 + D4 D 5 D5 + S 3 Z + S 2 Z S 3 2(0) 9(2) 00 S3 0 0 D 4 + D 4 D 5 + D 5 + Z + 2 S 3 Z 2 S 3 + S 3 01(0) 3 (1) 0 S3 1(1) 0 D 5 D5 D 4 D4 Z Z 2 S 3 S S 3 71(0) (5) 0 S3 0 0

52 D 5 and A 5 are special in E 6, as only a single orbit of each type exists under the E 6 automorphism group It follows that all D 5- and A 5-constructible polylogarithms in E 6 necessarily take the form: D 5 E 6 f D5 (x i...) = 6 1 i=0 j=0 A 5 E 6 f A5 (x i...) = ) (±1) i (±1) j Z j 2,E 6 σe i 6 (f D5 (x i...) 6 i=0 ) (±1) i σe i 6 (f A5 (x i...)

53 D 5 and A 5 are special in E 6, as only a single orbit of each type exists under the E 6 automorphism group It follows that all D 5- and A 5-constructible polylogarithms in E 6 necessarily take the form: D 5 E 6 f D5 (x i...) = 6 1 i=0 j=0 A 5 E 6 f A5 (x i...) = ) (±1) i (±1) j Z j 2,E 6 σe i 6 (f D5 (x i...) 6 i=0 ) (±1) i σe i 6 (f A5 (x i...) Surprisingly, a D 5 and A 5 decomposition of δ 2,2(R (2) 7 δ 2,2(R (2) n ) = D 5 Gr(4,7) δ 2,2(f D 5 ) = A 5 Gr(4,7) ) both exist δ 2,2(f A 5 )

54 ...moreover, we can play the same game with the new f D5 and functions f A5 there exists only a single orbit of A 4 subalgebras in each D 5 and A 5

55 ...moreover, we can play the same game with the new f D5 and functions f A5 there exists only a single orbit of A 4 subalgebras in each D 5 and A 5 Both f D5 and f A5 turn out to be decomposable into the same A 4 function: R (2) 7 = = D 5 Gr(4,7) A 5 Gr(4,7) f + A 4 (x 1 x 2 x 3 x 4) +... A 4 D 5 f + A 4 (x 1 x 2 x 3 x 4) +... A 4 A 5 [Golden, AJM]

56 In fact, many nested decompositions are possible (although, none involving D 4), each making different properties of δ 2,2(R (2) 7 ) manifest i jf R (2) j i 7 i fjf D j i 5 i f j AF j i 5 i jf f Aj i 3 i f jf + A j i 4 i f j F + A j i 3 i fjf A j i 2

57 The same game can be played in eight-particle kinematics, particularly using the new functions f D 5, f A 5, and found in seven-particle kinematics [Golden, AJM (to appear)] f + A 4 It is completely systematic, starting from such a representation of their nonclassical component, to generate the full analytic expression for R (2) 8 or E (2) 8 [Duhr, Gangl, Rhodes; Golden, Spradlin] s of Gr(4,n) can also be associated with R-invariants, perhaps allowing a similar story to be developed in the NMHV sector [Drummond, Foster, Gürdoğan]

58 A great deal of surprising structure remains to be explained in In particular, the role of cluster algebras in this theory deserves to be better understood The meaning of these nonclassical decompositions remains obscure The big looming question is whether any similar types of structure can be found that extend beyond the polylogarithmic parts of this theory (or even to amplitudes involving algebraic roots) [Paulos, Spradlin, Volovich; Caron-Huot, Larsen; Bourjaily, Herrmann, Trnka]

59 ( S A MHV n Thanks! A MHV (2) n ) ( (2) δ cluster adjacency negative X -coordinate arguments A MHV n subalgebra constructibility (2) ) Cluster algebras