JHEP04(2018)016 Springer April 4, 2018 : March 20, 2018 March 31, 2018 : : December 7, 2017 : Revised Published Accepted Received Published for SISSA by https://doi.org/10.1007/JHEP04(2018)016 . 3 1612.07370 The Authors. c 1/N Expansion, Scattering Amplitudes, Supersymmetric Gauge Theory

Describing the geometry of the dual amplituhedron without reference to a , [email protected] Mani L. Bhaumik InstituteUniversity for of Theoretical Physics, California Department atLos of Los Angeles, Physics Angeles, CA and 90095, Astronomy, U.S.A. E-mail: Open Access Article funded by SCOAP ArXiv ePrint: the geometry of theidentities underlying between different polytopes representations in of a the triangulation-independent amplitudes manifest. way, andKeywords: make Abstract: particular triangulation isdetermining an the volume open of the problem. tree-levelcontour NMHV integrals dual of In amplituhedron. logarithms We this serve showas as that well note certain natural as we building the volumes blocks introduce of for general a computing polytopes this in new volume any dimension. way These of building blocks encode Michael Enciso Logarithms and volumes of polytopes JHEP04(2018)016 ]. Schemat- 6 , 5 10 ]. 18 – 1 – 4 17 ]. This region encodes the amplitude via a volume 15 8 = 4 super-Yang-Mills theory (sYM) [ 6 , 3 7 15 , N 5 18 19 12 20 7 4 3 ] and references therein). While many of these developments 13 3 – 13 1 12 1 4.3.2 Four dimensions 4.3.1 Three dimensions One of the major breakthroughs in the study of maximally supersymmetric gauge the- 4.1 Towards the4.2 vertex objects Two-dimensional vertex4.3 objects Higher-dimensional vertex objects 3.1 One dimension 3.2 Two dimensions 3.3 Higher dimensions 2.1 Projective2.2 geometry Volumes of2.3 simplices Volumes of2.4 general polytopes The vertex2.5 formalism Applications to NMHV amplitudes ories is the discovery of theand amplituhedron, loop-level an integrands object in that planar ically, encodes and all tree-level specializing amplitudes toparticular the positive case Grassmannian [ of tree amplitudes, the amplituhedron is a region of a formulations of quantum fieldbeen theory immensely have streamlined been in uncovered, comparison(see and with the many the computations recent standard have reviews Feynmanhave diagram [ applications approach in theoriescomputational with simplicity various of amounts of maximallythem (including supersymmetric ideal no) testing gauge supersymmetry, grounds and the for gravity new theories ideas make [ 1 Introduction Recent years have seen tremendous progress ingauge understanding and scattering gravity theories. amplitudes in New both mathematical structures that are not apparent in textbook 5 Conclusion and outlook 4 Vertex objects from logarithms 3 Volumes and logarithms Contents 1 Introduction 2 Polytopes in projective space JHEP04(2018)016 1 a geometric k > 1 are therefore viewed as ]. For ≥ 10 , k 9 ] by using contours that are closed 9 that is dual to the space in which the ] we showed that we obtain the volume ] the authors computed these volumes by 4 9 12 CP – 2 – ]. ]. In this picture, computing tree amplitudes and 11 , 14 , ]. Additionally, the method we introduce is independent 10 7 13 MHV tree amplitudes with 1 can also be derived using global residue theorems (GRTs) k k > ]. N ], and relations between different representations of the amplitude 9 , 7 5 , our method differs from that in ref. [ ], while the geometry of the underlying space whose volume corresponds 3 15 ] we provided a definition of “combinatorial polytopes” which incorporates a 12 and will now refer to as “vertex objects”. The reason for this naming convention = 1) dual amplituhedron directly in the space in which the polytope lives. The basic n The vertex objects satisfy a simple relation that allows us to easily derive many non- In ref. [ In this note we introduce a new way of computing the volume of the tree-level NMHV For tree-level NMHV amplitudes, the amplitude obtained in this way is naturally k ...i MHV amplitudes with 1 k i on an auxiliary Grassmannianloop [ integrands is equivalentin to the specifying Grassmannian the [ correctfollow from contour the for GRTs. Introducing athe this particular amplitudes auxiliary space [ integrand manifests the Yangian symmetry of objects. These observations motivate usfor to computing view volumes the of vertex objects polytopes. as basic building blocks trivial identities between differentwe representations will of review the in tree-level theN next NMHV section. amplitude, These as identities and their more complex analogues for is that the subscriptsnatural of way these that vertex we objects willof review correspond a shortly. polytope to In by the ref. summingof vertices these [ expressing of vertex the polytopes objects volume over in ofto the a be a vertices known, of polytope and the does the polytope. not volume of This require the way any polytope triangulation is of uniquely the expressed polytope in terms of these vertex such triangulation. general class of polytopes.nor For these even polytopes connectivity are neither necessary. convexityF (and We introduced therefore a positivity) set of new objects that we denote by will see in section (i.e., without boundary) and canonicallycontrast specified to, by for the integrands example,specified themselves. “dlog” by This representations the is integrand of in itselfof amplitudes, [ any where particular the triangulation of contour the is underlying not polytope, and can be used to recover any (or objects in this methodprojective are space containing contour the integrals polytope. withintegrating In a simple, ref. particular [ closed volume contours formplacing in over the the the information complex underlying about polytope the in polytope the in dual the space, contour thus (which has boundaries). As we interpreted as the volumeamplituhedron of lives a [ polytopea in type a of “generalizedunderstanding volume” of of the a dualthat dual amplituhedron such amplituhedron is a [ picture unclear, should though exist there [ are strong indications actor from this formFor what loop integrands remains the (up same to isof true some a but fermionic particular with integrations) the generalizationrestrict amplituhedron is of ourselves corresponding the to to the amplitude. a the positive region tree-level Grassmannian. case. In the rest of this note we form with logarithmic singularities on its boundary, and after stripping off a canonical pref- JHEP04(2018)016 (2.1) MHV in the k Z and each H ∗ 2 we show how CP . 4 . } ∗ n i, j { CP , where the line dual to ⊂ . We will refer to linearly 1 1 ∗ − n ] and the first appendix of n 16 CP CP ], where the vertex objects encode is labeled by }' 12 j 1)-dimensional hyperplane = 0 − and α in the dual i n A give three lines in the dual 2 respectively as lines, planes, and hyper- 1 α 2 − Z n CP – 3 – k > ≡ CP + 1 homogenous coordinates, one for each value in A · n α 3 Z Z | ’s with ∗ k n CP , and CP α 2 Z ∈ , α is defined by α 1 A Z n ’s, and 1, but doing so will likely illuminate the underlying geometry of the 2 ≡ { CP CP is a linearly embedded , and the intersection of lines Z i k > ∈ is a Riemann sphere though we will still refer to it as a line. Z H . Each such point defines an ( ’s, α 1 H 1 Z we show how contour integrals of logarithms naturally arise in computing the CP CP by placing a single linear constraint on the homogenous coordinates of the dual 3 , . . . , n ∗ n ], which provide more complete discussions of these ideas. = 0 CP 17 Intersections of lines, planes, and hyperplanes always exist in projective geometry. For A point The outline of this note is as follows: in the next section we briefly review some is labeled by α α i example, a example, three points pair of lines intersects inZ a unique point. This is shown in figure The subspace embedded planes, even though the underlying topology of these spaces may be rather different. For ref. [ of dual elements. Namely, we have the remaining sections of thisintegrals note in we the show space how containing these the vertex polytope. objects2.1 are given as contour Projective geometry In this brief review of projective geometry we follow ref. [ some key facts about (complex)tion projective of spaces, volumes we of willthen polytopes review briefly in describe the the affine standard formalism generaliza- spacethe introduced to geometry in that ref. of [ of polytopesvertex polytopes objects as in are projective well used space. as to give We manifest their certain volumes. properties of Finally, we the review NMHV tree-amplitude. how these In the vertex objects correspond to a particular combination of these2 integrals. Polytopes in projectiveIn space this section we review the ideas that will be needed in later sections. After discussing key properties of complexpolytopes projective to space projective and spaces. theare standard We expressed generalization will as of also volumes volumesIn briefly of of section describe polytopes how and NMHVareas how tree of the amplitudes quadrilaterals vertex and objects their are higher-dimensional defined analogues. and In used. section naturally given by contour integrals in themanifests dual space the directly, we relations give between athe formalism different that introduction representations both of of an the auxiliaryamplitudes amplitude space. with while This avoiding formalismdual has amplituhedron. not been extended to N to the amplitude gets obscured. By showing that the vertex objects discussed above are JHEP04(2018)016 i ∗ n (2.4) (2.2) (2.3) CP . The define 2 n CP CP intersect in in ∗ 3 in ’s are explicitly n ≤ i α i α 2 ≤ Z 1 CP Z } , with the value of . We can write its α i n implicitly. α 2 n Z } { and j, i α 1 , { ...Z Z 1 = α 1 } [123] Z , and we have that n ), with faces (edges) labeled by . i, j ∗ i ≡ ∗ {    n n ...α , y 0 1 0 i 1 i α x P CP    CP ε 1 = Z in ⊂ 3 α 2 2 i ≡ Z Z ), as shown in figure uniquely define a point in the dual − n 2 3 ih i n H 3 n P , y Z 3 3 CP ,P dual hyperplanes. 2 ...Z CP . We note that Z x – 4 – and 1 1)-dimensional hyperplanes in Z } 2 ' n    1 i i Z 1 Z corresponding to three points 2 1 − h y ), ( Z x i, j Z i Z h ih 2 { n    H H P , y in the dual space that define them. The 2 2 ≡ ∩ 3 Z x 1 1 ≤ iα i Z Z ≤ ), ( h 1 W H 1 } 1 2 labeled by α i , y ∗ 1 ) coordinates in the text. Z = 2 i { x is denoted by distinct points in A , y j CP i n x and i 2)-dimensional hyperplane. Namely, two points − ] 9 n 1)-dimensional hyperplanes . A triangle in affine space defined by vertices ( . Three lines in as [ − ] and review how to express the volume of simplices in a projective way. n A 9 We begin by considering the area of a two-simplex, or a triangle, in real affine space More generally, any two distinct ( taken from context. We have also defined where we have introduced then notation with vertices located atarea ( 2.2 Volumes ofThere simplices is afirst natural understanding generalization this of extensionpolytopes the for follows volume the immediately of case byref. a considering of [ sums polytope a of simplex, to simplices. the projective volume We space. will of therefore more By follow general We therefore see that via the simultaneous intersection of their corresponding to the points defined in terms of the ( a unique ( two ( Figure 2 Figure 1 intersection of lines JHEP04(2018)016 in . 2 (2.9) (2.7) (2.8) (2.5) (2.6) x ) is also [12345] and . 2.3 + 1 points 1 ≡ x i D . P + 1)] γ 3 1 D Z , corresponding to ( 2 W α 2 β Z , and the volume of 3 . 1 ∗ Z ... ’s. The antisymmetry D Z W ! 5 α i 0 1 [12 ’s. Equation ( Z and Z ) is not projectively well- CP

αβγ ih iα ≡ ε α 1 ≡ 2.3 P i W Z 2 ≡ α P ’s — which, according to the Z α 1 3 and therefore the scaling of the α P i 1 between two points − ] ∗ Z Z 9 2 D 5 L ,Z Z . γ 4 CP i 3 and Z ...Z P , 4 W i in terms of their duals and the point at 2 ih i β 2 5 +1 Z ! 2 P α i Z Z D 1 2 1 , as expected, and it expresses the length of ih 1 x W 4 2 Z Z Z P . We note that ( W h

Z 5 x h 1 2 D 3 – 5 – i Z αβγ Z ≡ − Z 4 ε ... h 2 CP 1 +1 Z i and iα Z ≡ 3 x D = P 1 α all have three homogenous coordinates, in line with Z α ’s, corresponding to the two possible orientations of 2 1 Z = L α +1 i h ih α + 1 hyperplanes in the dual W L D Z ...Z ,W P P ,Z 5 D 1 iβ γ Z Z 2 4 h W ...Z (or its dual). We have simply “lifted” the affine coordinates Z W 2 3 αβ β , and 2 Z 1 ε Z iα 2 ih W CP ≡ Z W P there are α . It is projective and antisymmetric in 1 ih , D αβγ α α Z D i ε P 4 P Z ) is projectively well-defined in the ≡ Z CP ...Z 3 α 2.3 1 1 Z since it defines the line at infinity in in Z Z 2 ) indeed reproduces h α Z ! 1 . Translating the above formula gives +1 2.6 P ’s. 1 D α Z 5 D h iα ≤ i = 1 W 4! ≤ 1 V The dimension most relevant for scattering amplitudes is four, so for completeness we This generalizes to volumes of simplices in any dimension. For any It will be instructive to see explicitly how this works for one-dimensional simplices as Equation ( } = ,...,Z α i can be written as α 1 V Z will explicitly write theZ volume of a four-simplex, bounded by the five faces defined by This expression is projectivecorresponds and to the totally two antisymmetric possible in orientations of the the simplex. the two different orientations of the line. { the simplex bounded by these hyperplanes is given by [ Equation ( the line defined byinfinity the defined endpoints by well. A one-simplex isR simply a line, and the distance Here we have defined area — the scalinga we different choose non-zero here number) correspondscompletely in to the antisymmetric the third in choice component the the of of triangle. placing the 1 (as opposed to of the discussion in the previoustheir subsection, determine domain the of faces definitiondefined of to in the extend triangle to — thus allowing We note that the their being elements of into a particular coordinate patch of projective space by placing a 1 in the third component as well as JHEP04(2018)016 , α 1 Z (2.12) (2.10) (2.11) . according to the points 4 i , . These four lines are depicted in ∗ 2 [432] , , CP − [124] [432] viewed as the difference of two triangles. − − 3 – 6 – = [123] = [431] [124] = [431] A A − defined by four lines labeled by ∗ 2 [123] CP we also see that we can write the area of the same region as 3 , we can view any sum of simplices as the volume of a general define four lines in the dual D 2 CP that define them. in 2 α 4 . The quadrilateral shown in figure Z CP in . A quadrilateral in 4 ≤ and are respectively labeled by 1, 2, 3, and 4. , and i α 3 ≤ 3 1 Figure 4 Z By inspection of figure The area of the shaded quadrilateral can be written as } , α i α 2 Z by viewing this areafaces 4, as 3, the and difference 1,we and have between the the triangle area bounded of by the the faces triangle 4, 3, bounded andwhich, by 2. when the one We therefore unravels the see definition that of these 3-brackets, is a non-trivial relation. which is the areatriangle of bounded the by triangle the bounded faces 1, by 2, the and faces 4. 1, This 2, is and depicted 3 in minus figure the area of the Z figure 2.3 Volumes of generalFor a polytopes fixed dimension polytope, expressed through some particular triangulation. For example, four points Figure 3 { JHEP04(2018)016 - i } 4 D , ... 1 h . We { (2.13) } ij F { . . We found that 5 ij , and then walking F } 1 , 3 { -dimensional polytope in } 4 D , 1 } → { real dimensions. A third issue with 1 , 3 , then walking along line 2 to arrive at D } 4 of vertex objects defined as a particular , and our goal will be to give it a precise 2 } { } → { . This is depicted in figure 3 ij 3 } , F 4 2 { , – 7 – 1 { } → { 2 , 4 ] we instead focused solely on the combinatorial structure , which is a space of 2 12 D } → { 4 , CP 1 { ]. In this program one considers convex polytopes, which places posi- 5 , then walking along line 3 to arrive at the vertex ” means to travel along the line whose label is common to the vertex on } ] for details and the higher-dimensional cases. ] we introduced a collection 3 → , 12 12 2 { In ref. [ This set of instructions can be succinctly summarized by the list (1423), which we The amplituhedron makes precise sense of these polytopes as a region in a positive Proving this relation through repeated application of Schouten identities on the -dimensional space as being some full-dimensional region carved out by a finite number D where each “ either side of the arrow. sum of volumes ofvertex of simplices. a two-dimensional These polytope is objects labeled are by two referred lines, to as is as each vertex objects because a along line 3 to arrive back at the vertex define to be shorthand for — as mentioned above, the lines areintersection actually structure Riemann of spheres these — it objects. doesstructure, correctly We therefore depict saying the define that this this polytopeand by is walking its along the intersection line “quadrilateral” 4 tothe defined arrive vertex by at the starting vertex at the vertex refer to ref. [ 2.4 The vertex formalism We consider again thedefinition. quadrilateral While in this figure figure does not correctly depict the topology of the objects involved tivity constraints on the externalgeneral kinematics. kinematics. One In thenof ref. analytically polytopes. continues [ to We consider thennecessarily gave convex a or precise evenideas connected. definition in of two In dimensions, a the as general next well type as subsection introduce of we the polytope will two-dimensional vertex that briefly objects is review not these many triangulations that correspond toapparent the same certain polytope. geometric qualities Some of triangulations may the make underlying polytope whileGrassmannian, masking and others. for the NMHVa case projective under space consideration, [ this Grassmannian is simply underlying space from“inside” a or real “outside” is affine lost.a space Moreover, one to generally thinks a of a of complex hyperplanes. projective However, by space, complexifyingdimensional our any compact polytopes notion space, in we of end uptrying talking to about define a polytope as a sum of volumes of simplices is that there are (infinitely) brackets quickly shows that this geometric proofrelations is more in convenient, especially higher for analogous dimensions.few reasons. However, this Forquadrilateral geometric one, in proof we our is have twois not not different very that been triangulations. precise, our careful for notion to A a of keep second track a and of polytope more the serious itself orientation ambiguity is of rather the tenuous. Namely, once we extend our JHEP04(2018)016 for ij (2.17) (2.15) (2.16) F . 1 ) defines. } 6, defining 1 , ≤ 2.13 5 i ≤ } → { 5 , 1 ) ) , ). Indeed, all possible ) to the left hand side } 3 12 41 α i F F Z 2.12 2.14 { − + . } → { 24 3 , F , and that they satisfy [432] 6 ji + , − F ] (2.14) ] is the volume of the two-simplex + ([231] 12 ] by a factor of 2. − ) in a different order also shows that F 42 } → { ijk ( [124] ijk 12 = 6 . We can then define the disconnected F , − − 2.15 = [ ij 6 5 ), we find + = [431] F ki 14 over the vertices of this quadrilateral. Using 31 F F 2.14 F defined solely through the intersection of its faces. 31 } → { + = = [123] = [123] – 8 – 3 F + 5 , jk + 4 31 23 F F F 23 . + F k + + ij + } → { 23 42 F 4 42 F F , F 2 , and + and equation ( + j + , where we recall that [ , ij 42 14 i k 14 F F F F } → { ) to the left hand side of ( + 2 , 6 14 , and ]. F j 2.14 12 , i } → { 6 , 1 . The quadrilateral depicted in figure ), giving a simple algebraic method for proving many non-trivial identities amongst } → { 1 , This example is a special case of a more general phenomenon — given any set of We consider the sum These vertex objects differ from those introduced in ref. [ 5 2.15 1 { six lines, as shownpolygon on shown the on left the hand right side hand of side figure of this figure by the instructions sums of simplices [ vertex-connecting instructions defining anyeach polygon, vertex yields summing the the areadisconnected of corresponding ones. that polygon. For example, This suppose process we works for have general six polygons, elements even This gives a quick andtriangulations rigorous of proof the of quadrilateral the can non-trivialof be identity ( obtained ( by applying ( which is precisely the volumeApplying of equation the ( quadrilateral that the list of vertices in ( bounded by the three lines the antisymmetry of each these vertex objects are antisymmetric, so that for any choice of Figure 5 JHEP04(2018)016 is ∗ 2 (2.19) (2.20) (2.18) CP elements over and . } } N 2 j, i , { 1 { ]. This form of = . 12 } } 2 . i, j , i that defines them, and { 1 35 of this polygon can be i α i F Z A + 63 } → { 1 F be a collection of , i + } n i α i 56 Z F { + , } → { . It is implied that n +1 45 according to the } k i F , i i k 1 i, j i + { − F n i 24 n =1 – 9 – F k X . The most general polygon in this dual ∗ + → { = 2 62 A ... CP F + is denoted by } → 16 j 3 F , i 2 + i and i 51 ), corresponding to the instructions F n } → { = 2 . . . i lines in the dual A , i 2 1 i i 1 N { i of this polygon is then given by the following sum over the vertices: A . A general polygon (right) defined solely through a set of instructions for traversing the defining ). 2 We note that many different lists give rise to the same polygon. For example, any cyclic The general result can be stated as follows. Let 2.14 CP of the polygon itself — its vertices and howpermutation we of traverse a them. list givesthe the list same (12121234), polygon. sinceover More the again trivially, latter before the corresponds list moving (1234) on. to is staying However, identical on the to final the result vertex in terms of the vertex objects (up and from this expressionthe any area particular is triangulation independent can of be any obtained particular [ triangulation and is inherently tied to the data in given by a list ( The area This can be checked against anytriangulation particular of triangulation this of this polygon polygon.of can ( Additionally, be any obtained from this expression through repeated use Analogously to the case of(51624563). the It quadrilateral, is this then setwritten of the instructions simply case, corresponds as rather to surprisingly, that the the list area Figure 6 intersections of six linesthe (left). intersection of Lines two are lines labeled by JHEP04(2018)016 (2.21) can be ) and the = 4 planar completely } n N NMHV 2.22 ijk M in F { ] (2.24) ) and ( . In four dimensions ] (2.23) , n NMHV ijk 41 mijk ] we show how to extend M F 2.14 F 12 ]. Indeed, + ijklm 9 ] + [ [ ] (2.22) 34 = [ ∗ F ), and their higher-dimensional 4 lmij ijkl + + 1)] (2.25) CP j mijk 2.23 23 ( = [ F ] + [ j F ), ( + lij + F + 1) klmi 12 − i 2.22 lmij F ). ( i F ] of the simplex and encoding where the poles kli ∗ ] + [ + [ + F yields the same result as the list (1234). Indeed, – 10 – 2.24 21 n i,j ij + X F ijklm F jklm klmi [ = + jkl ∂ F F 12 ] + [ + F − ) is reminiscent of the formula of vertex objects that are totally antisymmetric in their n + NMHV ijkl ijk ]. ) does the same, and is also a genuine equality between the jklm } F 21 M F 2.23 12 F . ijkl ] = [ + 2.23 ) are fundamentally different than those on the right of equa- F + -point NMHV tree-level superamplitude { . We continue to use the term “vertex objects” because for a three- n 12 ijkl 2.23 F F ijklm [ ∂ i, j, k, l, m i, j, k, l ) is dependent only on the equivalence class of lists, where equivalence of ] we also defined the corresponding vertex objects in higher dimensions. For 2.20 12 ], describing the boundary ). Similar statements can be made about equations ( ] are. Equation ( 9 , and these planes determine the subscripts of a given α i 2.24 Z We note that equation ( The volume of any polytope is given by the sum over its vertices of these vertex In ref. [ ijklm Quite surprisingly, the sYM can be writtenrepresented as as the volume of a polytope in the left of equationtion ( ( lower-dimensional analogues of equation ( 2.5 Applications to NMHV amplitudes given in [ of [ volume of the simplex and objects that correspond to its vertices. Thus the objects on objects. This expressioncan of be the recovered volume from isanalogues. this unique, Additionally, the expression and expression using any ofobjects triangulation ( the also of volume encodes of the the geometry agives polytope of polytope their all in volumes lower-dimensional terms boundary as of polytopes well the [ and readily vertex subscripts and that satisfy for any choice of for any choice of dimensional polytope a vertexby is a defined by thewe intersection defined of a three planes, collection each defined lists is defined by theirthis determining definition the of same polygon polygon. to arbitrary In higher-dimensional ref. polytopes. [ example, in three dimensionsantisymmetric we in defined their a subscripts and collection satisfying of vertex objects to the list (12121234) is simply which, after using the antisymmetry of the sum in ( to trivial cancellations) is identical. For example, the sum of these objects corresponding JHEP04(2018)016 , i α i ... Z and h α ∗ (2.28) (2.29) (2.26) (2.27) Z . encoding +1) i ( 4 i ∗ 4561 is understood F CP +1) j + ( i, j F ]. 3461 + 5 ) on each simplex in i F [ ) is non-trivial. These ∗ points in + n +1) 2.23 j 2.27 , . ( j 3456 can be obtained. F F . The sum on + + 4 ∗ ) and ( 6 NMHV CP +1) 2361 j M ( 2.26 F ) and in the last equality we used the = 6 we have cyclic polytopes j + n +1) i 2.23 ( F 2356 F + + – 11 – +1) ) are equal directly — namely, they are identical , then for j α ( 1 2345 j Z F 2.27 +1) = = [12345] + [12356] + [13456] = [23456] + [23461] + [24561] + i ]. The geometric interpretation is that they correspond ( i + 1)] α ∗ were initially found by performing two different BCFW , is a reference vector in F ]. Part of the utility of the vertex objects is to show that j Z has many different expressions depending on our choice of 19 ( 1256 α , ∗ ), any triangulation of j +1) + ) and ( F j 14 6 6 Z NMHV NMHV j ( , , then we have j 18 + α 2 , 6 M M NMHV ) is not obvious at the level of Schouten identities on the 2.23 13 + 1) 9 +1) ] for further discussion of this vertex formalism. In the next 2.26 Z n i NMHV +1) i ( M i i ( ( 12 1245 = ∗ i i M 2.12 ∗ F ), we find F F α , [ intersect. ∗ n Z + n n n i,j i,j i,j , we have +1 X X X 2.27 α implicitly inside the five-brackets in the sum are j n Z = = = } 1234 α i F Z ) or ( = { , and polytopes of this form are known as , and n NMHV n α j 2.26 as a subscript. This shows manifestly that the amplitude is independent of M Z ∗ , 6 We refer to ref. [ For general For any given NMHV . For example, if we choose +1 M α α i ∗ that the underlying polytopeZ has vertices only where the four hyperplanes defined by two sections we showof that logarithms. these vertex objects are naturally defined as contour integrals where in the second equalitycyclicity we of used the equation sum ( with and antisymmetry of the vertex objects to cancel in pairs any terms The amplitude is thereforeexpression uniquely and expressed equation in ( terms of the vertex objects. From this auxiliary Grassmannian [ the right hand sideswhen of expressed ( in termeither ( of these objects. By using equation ( brackets, the equivalence of thetwo right representations hand of sides ofshifts ( on the amplitudeto [ two different triangulationstroduction, of their the equality same can also underlying be polytope. understood by As using discussed a in global the residue in- theorem in an while if we choose Just as the relation ( the external kinematics and modulo Z where the JHEP04(2018)016 } 2 ] in , 9 1 (3.4) (3.1) (3.2) (3.3) { along the real axis. 2 x dx. πi dx. ) as a contour integral in 2 , since it is not present in , 3.3 2 2 4 x )  ≤ 1 2 x X dx, x x · ≤ DX 1 to the point x − − P  ( 1 Z x x 1 2 x  x x  πi 1 2 , as expected. X X − − 2 · · log x x = x 1 2 — the discontinuity of the logarithm across   Z Z − dx ) we can rewrite (

1  2 2 1 – 12 – log x x x x Disc 2.7 ≤ − − log = 2 x x x I x ≤ L 1 ≤ I x x πi 1 2 Z ≤ log 1 πi 1 x 2 = = Z 2 L = x πi 1 L 2 − is as as Disc 1 = x 2 πi x L = to L variables to be complex, we can define the complex logarithm function 1 x x with its branch cut connecting the point
2 1 x x ] the vertex objects are defined as a particular sum of simplices. Thus, in some as − − x x Making the same definitions as in ( As mentioned in the introduction, our integrals differ from those discussed in ref. [ ∗ 12 1 of a line from where the contour surroundsaround the the pole cut. at infinity) Evaluating recovers this explicitly (forCP example, by going Unwrapping the contour allows one to drop the “Disc” from the integrand and obtain We can then rewrite 2 its branch cut — giving By allowing the log (i.e., non-spurious) vertices of the polytope play a role. 3.1 One dimension As a warmup, we begin ourL discussion in one dimension. Another way of writing the length is a spurious vertexthe in underlying the polytope triangulationwill but see, depicted shows the in up integrals figure instraightforward we application individual use of terms Cauchy’s have invertex residue closed objects theorem. the contours, triangulation. used so Moreover, they evaluating in them As give the rise corresponds we vertex to to formalism the a discussed above, in which only the genuine building blocks for computing volumes of polytopes. that the latter involvevolumes contours in with this boundaries way leads onto to the the spurious underlying presence poles) of polytope. associated spurious to vertices Evaluating (which a correspond particular physically triangulation. For example, the vertex In [ sense, writing the volume ofchoosing a a polytope in particular terms triangulation.naturally of defined these in However, objects terms of may weindependent contour be will of integrals viewed of simplices. now as logarithms, This simply show thus further giving that motivates them the an these view existence that objects the are vertex objects are basic 3 Volumes and logarithms JHEP04(2018)016 , ≡ α 3 Z α (3.8) (3.9) (3.5) (3.6) (3.7) X (3.10) . This 2.3 and ∗ 1 , 4 ) CP X that goes around · ). If we swap 3 DX , ) P of weight three. The 1 3 ( ) 2.10 ∗ S 2  . This gives X
· X X integral DX CP X X . · · · · P ∗ 1 2 ( 5 6 2 Z Z -dimensional “quadrilaterals”, Z Z  CP D  X X · · , ). log 3 4 , , Z Z  i 3.6  P X X i 2 [342]) · · 2 [1245] + [1246] [124] Z [432] that we obtained in section log : 3 4 − Z ∗ − − ih 1 Z Z − 3  Z P  h 1 X X CP ) and pick up a minus sign from the change in · · – 13 – Z ([341] h log 1 2 [1236] 3.6 − Z Z  = [123] = −  = X X A L . The contour is a three-torus ( · · A δ log 1 2 [123] = [431] Z Z ). In this way, the length of a line is naturally represented dX I is the canonical volume form on −  γ 2 2.6 = [1235] γ ) dX log 1 V πi in equation ( β dX is the canonical volume form (of weight two) on β (2 I α 2 β dX 3 Z ) = dX α dX α 1 X πi A α X and and then go around the branch cut of log (2 X
α 1 αβγδ αβγ αβ = ε ε Z ε X X · · 3 4 V ≡ ≡ ≡ Z Z with DX DX DX α 4 . By explicitly evaluating this integral we find Z We have expressed a two-dimensional area as a closed contour integral whose con- ! 1 x where the branch cut of each logarithm. We find that or hypercubes. 3.3 Higher dimensions Consider the following contour integral in tour specification comes naturallycompute with in the this integrand wayvertex are itself. objects quadrilaterals, are The defined obtained objects byuse from four whose these these lines. integrals area kinds to we of Before compute integrals, describing the we how volume quickly the of discuss three- how and we can orientation of the contour, one readily sees that thus proving the identity [123] identity is now made manifest by the integrand of ( cut of log which is precisely theand area of the quadrilateral given in equation ( where contour is again defined by the integrand in a canonical way: first go around the branch in agreement with equation ( as a contour integral of a logarithm. 3.2 Two dimensions Motivated by the one-dimensional result, we consider the where JHEP04(2018)016 - 7 D (3.11) (3.12) , . As in the defining the 4 D ) by swapping, [5642]) , α 2 − ) is triangulating +1 3.9 D ) 3.10 X ,...,Z [5641] DX · α 1 − Z P = 4 gives the volume of a ( D  , and picking up a minus sign [5632] α 6 X − X · Z · 1 − D 2 D and 2 Z α ([5631] 5 Z Z − ) applied to  = , which shows the superposition of the four log 3.12 V 7 ... faces in “pairs”. With – 14 –  , or with D α 4 X X Z · · 1 2 is the three-dimensional analog of figure Z Z 8 and  [3426]) and α 3 − Z log . A triangulation of the cube using four simplices. I . Figure with D [3425] ) 8 α 2 − 1 πi Z (2 Figure 7 ) leaving the volume of a “cube”. and = [3416] α − 1 V 3.10 Z is the natural generalization of the lower-dimensional volume forms and the ([3415] DX − We note that these (hyper-)cubes are not directly related to the polytopes that are For completeness we write down the contour integral that gives the volume of a As in the two-dimensional case, there is more than one expression for the volume of this faces, we have a generalization of the lower-dimensional cases: = D V where contour goes around the branch cut of each logarithm. relevant for scattering amplitudes: equation ( from the change in orientation of the contour. dimensional “hypercube” bounded by 2 2 which correspond to thecorresponding different to ways figure oftwo-dimensional decomposing case, the these identities cube are manifest analogouslyfor from example, to the integrand figure in ( cube. Namely, just as weby could viewing get it two different asexpressions expressions the for for difference the the area between volume ofwe of two a also different the quadrilateral have cube pairs as of a triangles, superposition we of can four get simplices. three In particular, polytope bounded by 3 pairsa of “cube” faces. with One faces way6 1 to opposite and see each that 2 other equation oppositesimplices is ( each in by other, ( examining 3 figure and 4 opposite each other, and 5 and This corresponds to the volume of a three-dimensional “cube”, where we simply mean a JHEP04(2018)016 (4.4) (4.1) (4.2) (4.3) , , . ] ] ] Q Q Q [23 [31 ) corresponding [12 − − − 2.25 = [231] = [312] = [123] 3 3 3 , defining a reference line in ) ) ) α X X X Q · · · DX DX DX . P P P ( ( (    X X X X X X , showing the three possible ways of forming · · · · · · = 2[123] are all different contours, each being the 1 2 3 4 Q Q . Q Z Z Z 31 31 7 T    γ + – 15 – log log log 23    T , and X X X X X X + 23 · · · · · · γ 3 1 2 3 12 1 2 , T Z Z Z Z Z Z 12    γ log log log 31 23 12 γ γ γ I I I = 8 particles has 8 codimension-1 faces and 20 vertices. However, as 2 2 2 ) ) ) n 1 1 1 πi πi πi (2 (2 (2 ≡ ≡ ≡ 31 23 12 is the same contour that we have described before, only now we are making it T T T . The three-dimensional analogue of figure 12 γ . Cyclicly permuting 1, 2, and 3, we define ∗ 2 It is important tocontour defined note by that the integrandgoes of the around corresponding the branch integral cut —them of namely, up, the each logarithm. contour we that find Performing that these integrations and summing as well as where explicit. We have alsoCP introduced a fixed reference vector 4.1 Towards the vertexWe motivate objects the vertex objects byintegrals first of seeing logarithms. how to We define recover the volume of a simplex from vertex objects and thus topolytopes compute relevant the for volumes scattering of amplitudes. general polytopes, including the4 cyclic Vertex objects from logarithms four-dimensional hypercube, which has 8four-dimensional codimension-1 cyclic faces polytope and whose 16to vertices, volume the whereas is scattering the given of bywe equation will ( show in the next section, these volumes of hypercubes can be used to obtain the Figure 8 a triangulation analogous to that shown in figure JHEP04(2018)016 ) ) 12 X X X γ · · · i j  (4.7) (4.8) (4.5) (4.6) . Z Z k  Z X X , · · 3 3 3  3 1 ) ) ) ) that 3 3 Z Z ) ) X X X . X · · ·  3 X X DX DX DX ) · , as follows: · · P P P 3 } DX DX ( ( ( X P P · Z


( ( +log DX ) branch cut. We i, j

X X X P 3  X ( · · · ) X X · ) X X · · X Q Q Q }\{ · ·  · Q X 1 2 2 3 3 3 DX · , ) Z Z Z Z P 2 , ( log log log , X Q  3 ·
1 ) DX    3 log log P X X ) ( ) and that of either log( · X X X X X X log ( ∈ { ·  
· · · · · · +log X X X DX 3 · · k  · i j P 3 3 1 2 2 1 X X X X X Z  DX ( · Z Z · · · · Z Z Z Z Z Z X , drops out of this sum of integrals X P X X Q 2 3 3 1 ( · ·  · · ij    and = 0 pole as opposed to the log( i Z Z Z Z j ) 1 2 log T X } Z X Z Z Z   X ·  log log log 3 · log X ·  ,  k · , we find 23 31 12 × X X 2 k Q log log Z γ γ γ , · · 31 P Z I I I log log 1 T = 0 pole and the log(  1 2 31 23 γ γ ij   Z Z + + + I I γ – 16 – 2 X γ ∈ { log log (  I ) · I + + 23 2 1   πi T  ) P i, j log . However, this representation depends on an implicit 2 X X X X (2 1 + ) πi α · · · · 12 − γ -independence of this sum results from using the four- 1 i i Q j j (2 πi 12 ’s as I α , Z Z Z Z T ij ), which itself is the result of non-trivial algebra using = (2 − Q  T   , where 2 = = 0 ) ij 2.12 , while present in each to go around the 1 log log T πi α ij ij ], the ij (2 Q γ γ γ dependent I I = − 2 2 ijQ dependent goes around the α [ ) ) − 31 Q 1 1 γ | α πi πi − T ) ] Q | (2 (2 31 + now goes around the branch cut of log( ) T ijk = = ij 31 23 γ + T T ), depending on which term we are considering. If we now consider only the ij = [ + T X 23 + · T 23 ij 12 T T Q + T + 12 By deforming each We begin by rewriting The dependence on 12 -dependent terms in the sum T T ( α ( which is manifestly independent of choice of line at infinity as this defines the branch of, for example, log( where the contour can therefore write the sum of branch cut, and thus pickingunder up the an same overall minus integral sign, and we the can integrand bring vanishes: all of these integrands The contour or log( Q and we are leftresults with twice the volumeterm of identity a in singleSchouten simplex. equation identities. ( At the At levelmore the of manifest, level and the of we integrated the will explore integrands, them however, here these in cancellations some become detail. JHEP04(2018)016 is ’s , )+ 3 ij = 0 1 2 ) T X (4.9) X · · (4.13) (4.10) (4.11) (4.12) X j k X · , Z · Z DX k P ( Q ) . We define ∗ X 2 ( , and )+log( . j ij ] f CP X , X = 0 to · · i i j ’s, but the depen- ij Z Z ijQ γ α i X ) on the dual space. I · Z X 2)[ 2 k ( ) Z − ij 1 πi f = 0. The factor of N (2 ( of the 1 2 X ≡ N · ] and so in particular we have, , 3 − ) j 12 Z   X . ] · ] DX ) = 0 . P ( ijk 31 ijk [ T and its higher-dimensional analogues in = 0 , each defining a line in i,j 2 ) drops out in the above sum, for reasons = [  N . In the following we use integrals similar + 6= α = 0 to X ki α k ki f X Q 23 X as well as all CP · Q   · F T X k + α i, j, k 1 2 · in Q Z + + Q i jk – 17 – 6= N  . f jk Z 12 l ≤ i F T ]) = + ( log ≤ 4.1 1 + α ij i,j } as an integral over a function f ijQ ∂ ij [ α 6= i X ij ∂Q k F Z − F α { ] integrals: , and can also show that for any choice of K  ’s (i.e., for
ji α l ijk 2 f X N X · · ([ Z , that − i j ’s to define the vertex objects, and we will keep the reference } points i,j Z Z . In particular, by again using the fact that log( ij = N 6= α T X  N first goes around all of the branch cuts from k ij ) we wrote Q f 1 2 ij log γ ,...,N = ij 4.10 1 γ ij I F 2 ∈ { and all other ) -independence of this sum of integrals can also be directly checked by differenti- α 1 α πi Q Q ) = 0, it can be easily shown — at the integrand level — that (2 i, j, k 2 1 X X · · i k ≡ In equation ( The Z Z ij F dence on identical to those discussed in section We readily see that for any Each individual vertex object depends on These are (up to a factor of 2) the vertex objects of ref. [ where the contour and then aroundconventional. the Integrating branch this cut gives from 4.2 Two-dimensional vertex objects Suppose that we have the following collection of and therefore that this sumto is those independent of definingboundaries the in these expressions.will be The independent sums of these ofare boundaries, these similar and objects to the those integrand-level that proofs found we of here. will these statements be interested in ating with respect to log( our expressions and note the independencewherever of necessary. our expressions We on also thesewould reference note give boundaries a that vanishing a result,specified naive but by summation the that of branch performing the cuts the of integrands integrations their of along respective the the integrands contours gives as non-vanishing results. circles. In the following we will therefore keep JHEP04(2018)016 . , ∗ 3 +1 D . To CP (4.14) (4.15) (4.16) ≤ α 2 . k ∗ Q . 2 ≤ ] , and for 1 , } D i +1 CP k 4 i ) D Z X { ech cohomology · . . . i is antisymmetric DX dimensions there ˇ C ,...,Z 2 P k i ; 1 ( . Motivated by the i D 1 ∗ ij i gives the exact same Z  3 F j ; X X = [ . CP · ) ki · 1 j l 1 ; F − Z ki D Q + F  i ...i ; 1 i + jk i log ; +1 F D jk ) imply that the collection of func- i + F i,j,k F k X ; 6= D ]. The appearance of + l defining fixed reference planes in 4.13 ij k 1) 3 ;  16 F +1 hyperplanes defined by ’s, although it is independent of ij − planes in the dual X X α i F CP D , and noting that each · · ( Z k + ( N 2 k 1 3! . The precise definition of higher-dimensional Z Q ], as is the precise way of summing the vertex – 18 – ... D =  12 ] ech cohomology class on a subspace of , and + ...i k hyperplanes corresponding to j ˇ 1 ; of the 1 C 1. For now, we simply move on to describing how to i log , ij [ i F defining D ...i  N 4 F 3 i 3 k > X X 3! i · 1 · · F CP i j 2 + Z Z contour going around the branch cuts of the logarithms in the ’s as well as property ( ≡ and all 3 , one obtains its volume by summing the vertex objects over the  ij +1 ) f D 1 D α 1 ijk ech cohomology is a natural setting in which to discuss the Penrose S log Q ...i F ˇ CP 3 k C points in ; i 2 ij i such that for any choice of γ is described in ref. [ N F are fixed reference points in I D is an ( D 3 D α 2 be k ...i ) ; 1) 1 Q i 1 ij N πi CP − F γ ≤ ) by the intersection of i (2 form a representative of a -independence, we note that the sum depends on and ≤ α 2 + ( 2 } 1 ≡ α 1 Q } D ij ijk k α i f MHV amplituhedron with ; Q ...i F k { ij Z 2 i In twistor theory, { F 1 i F Each see this where The contour natural way. Antisymmetrizing over in its first two indices, we then define 4.3.1 Three dimensions Let two-dimensional case, we define polytopes in objects over thevertex vertices. objects arise In as thisfor contour dimensions subsection, integrals three we of and will four. logarithms. see We how will these explicitly show higher-dimensional this only Given any polytope in vertices of the polytope.in In section particular, any vertexthis of vertex the one polytope simply is includes defined (as an reviewed Analogous vertex objects can beexist defined objects in any dimension.one Namely, has in the identity For the remainder of thiscurious note connection we to will cohomology, notthe as explore N it this may issue. be Instead,construct important we the for simply higher-dimensional generalizing note vertex these this objects ideas in to terms of4.3 integrals of logarithms. Higher-dimensional vertex objects tions transform, which takes anormed on-shell cohomology field configuration class on on space-timehere [ (a is subspace of of) a twistor different space nature, to and a the finite- role it is playing in this discussion is still unclear. The antisymmetry of the JHEP04(2018)016 , , , 2 } α i 5 ) Z X · (4.18) (4.17) (4.19) points 3!) the defines DX plays a P . α ( ∗ N ) × ,...,N ’s can be α 2 i 1 ; Z  α j Q ; , it is worth be Q j α 1 X kl X ; ∈ { that appears · · N k F Q ; 1 ≤ α m ∗ il i − Q Z Z F ≤ j 1 ;  i } − ; , these α i i, j, k, l k kl ; log Z j F ; ijk { il F + F ) that appear in the two- k i,j,k,l ; i X + 6= ; l . The boundary , ; jl itself, as in the middle line of m ] j 4.15 ; F -independent representation of ’s drops out in this sum. ijk  ik α defining reference hyperplanes in Q m ijk F ’s that make an appearance here − F ijkl X X F i Z 4 · Q · ; l 2 k − ; ]. Moreover, with an explicit choice = [ Z j Q CP jl ; l 12 ; F  ). In the latter, the boundary lij ik F + . Define F functions are (up to a factor of 2 ∗ log i ; − 4 2.25 l ; +  ijk k jk kli ; CP l X X F ; F F · · – 19 – and all other , a manifestly 3 ij k , and thus the arguments used there to prove the − + α 1 F Z Q l ; depends only the reference boundary Q i 4.1 ;  ) terms in equation ( − jkl 4.1 l contour going around the branch cuts of the logarithms ; F jk X X ijk · · k 4 ; 2 log F k F ) − Z ij Q 1  determines a boundary that is used to define the branch + F ]. It then follows that for any choice of S ( ijk X α 1 X · · 12 F 1 4! i j Q ) log( Z Z = X X · is an ( ·  ] i j l are fixed reference points in l ; ; Z Z k k ; α 3 ; log ij ij l [ Q ; γ k hyperplanes in the dual ; F ij 4! γ . The dependence on N 1 · I α l ). This is in line with the fact that the vertex objects encode triangulation- , and 2 4 Z α 2 ) ] is the volume of the three-simplex bounded by the four faces defined by Q ≡ 1 πi 2.29 , (2 α 1 defining , and ijkl ijkl Q α = k F 4 l can be given, at the cost of making an implicit choice of a plane at infinity that . The contour ; Z It is straightforward to show that the Having established that each ∗ k We note that we are currently working in three dimensions, whereas the BCFW/CSW triangulation ; -independence of this sum directly apply. 4 , CP 2 α α ij j ijk F triangulates a four-dimensional polytope.four However, dimensional the case statements we discussed make in here the directly next carry section. over to the where CP in the natural way. We define 4.3.2 Four dimensions The definition of the four-dimensional vertexin objects is similar. Let one has where [ Z independent data about theof underlying endpoints polytope for [ removed the entirely. branch cuts of the logarithmsvertex that objects define defined in ref. [ determines the branches ofcan the be logarithms. viewed Thus, asinfinity. the generalized These “planes boundaries at therefore infinity”, do— the not such intersection play a of a boundary which directequation gives would role ( a enter in line into defining at the any subscripts triangulation of a particular triangulation of theof underlying the polytope, polytope and on thetriangulation. this independence boundary of In follows the from contrast, volume cuts the general of independence the of logarithmssimilar the that volume role. on appear in AsF the discussed definition in of section dimensional case discussed in section Q comparing the meaning of this boundaryin the to BCFW/CSW that triangulation of of the equation dummy ( boundary cyclic sum of the log( JHEP04(2018)016 , α i Z (4.20) is independent of ’s drops out in this α n ijkl ], which can be used , Z ] F 12 , we have m ’s. α ijklm i Z = [ , and mijk and all other of the F α 1 i, j, k, l N + Q lmij F and all + – 20 – α 1 Q functions are precisely the vertex objects defined in 1, since the Grassmannian picture is already well- klmi = 4 super-Yang-Mills, and we have seen logarithms F ijkl N + F k > jklm F + 4!, the 1 cases, likely by first making a connection to the Grassmannian × ijkl . Again, the dependence on k > F α m Z ech cohomology. It would interesting to further explore this connection. ] is the volume of a four-simplex bounded by the five faces defined by ˇ C , and , though it is dependent on α l α 3 Z Q ijklm , ]. It therefore follows that for any choice of α k MHV tree amplitudes for k 12 Z The vertex objects we defined can be used to obtain identities amongst sums of sim- The vertex objects that we have defined are useful for computing NMHV tree-level Up to a factor of 2 and , 2 α α j understood for these moreobjects complicated naturally encodes cases. the geometry Expressing ofcan volumes the be underlying in polytope. found terms for If of the picture, the analogous the then objects vertex this shouldwithout shed a light need for on any the auxiliary geometry spaces. of the dual amplituhedron directly, plices, and these identitiesintegrals can of logarithms therefore directly now inically be the from viewed space the as containing Grassmannian the beingrelation polytope. between picture obtained these discussed This from two approaches differs in contour willto dramat- the help N extend introduction. the method Understanding introduced in the this note It would therefore be interestingtheories to beyond see tree if level. similar ideas Takingas the can momentum planar be limit (super-)twistors used appears for playplanar to less a be supersymmetric theories. fundamental crucial in role Nonetheless, thiscan and it discussion, be these is extended cease worth to exploring to the non-planar if exist sector and in of to non- the what theory. extent this discussion amplitudes in the planarappear limit naturally. of It wouldlevel. be interesting Additionally, to since seeresults our readily how discussion apply these at has ideas tree been might level to generalize limited Yang-Mills to to theories loop with tree-level less amplitudes, (and no) these supersymmetry. these integrals are canonicallyprinciple specified for by combining the these integrands— integrals themselves, the comes and intersections the directly of organizing from itsWe also the faces found geometry a — surprising of and connection thus betweenobjects the the does polytope and integrands of not the rely two-dimensional vertex on any particular triangulation. those polytopes live. 5 Conclusion and outlook In this paper we showedintegrals that of volumes of logarithms general directly polytopes in can the be space computed using in contour which the polytopes live. The contours of where [ Z sum. This completesas the basic building proof blocks thatreference, for have the computing a vertex volumes natural of definition objects as general of simple polytopes contour ref. as integrals [ described in the in same that space in which Q ref. [ Similarly to the two- and three-dimensional cases, each individual JHEP04(2018)016 , 03 , , ]. JHEP ]. , ]. SPIRE ]. IN [ SPIRE IN Lecture Notes in SPIRE A Note on Polytopes ][ IN SPIRE [ , in IN (2017) 071 (2014) 182 ]. ][ ]. 10 12 (2014) 030 math/0609764 SPIRE 10 , QCD and beyond. Proceedings, SPIRE IN A Duality For The S Matrix JHEP IN JHEP ][ , [ , , in arXiv:0902.2987 JHEP [ , arXiv:1308.1697 ]. ]. ]. , , Cambridge University Press, (2016), arXiv:1012.6030 [ What is the Simplest Quantum Field Theory? Towards the Amplituhedron Volume Positive Amplitudes In The Amplituhedron SPIRE SPIRE SPIRE Yangian symmetry of scattering amplitudes in (2009) 046 IN IN IN ]. – 21 – ][ ][ ][ 05 hep-ph/9601359 arXiv:0909.0250 [ (2012) 081 SPIRE 04 IN The Amplituhedron Into the Amplituhedron JHEP ][ , ), which permits any use, distribution and reproduction in ]. ]. ]. ]. Dual Superconformal Invariance, Momentum Twistors and Scattering Amplitudes Scattering Amplitudes in Gauge Theories (2009) 045 JHEP , , pp. 539–584, 11 SPIRE SPIRE SPIRE SPIRE arXiv:0907.5418 arXiv:1412.8478 arXiv:0808.1446 [ [ [ IN IN IN IN ][ ][ ][ ][ CC-BY 4.0 JHEP , This article is distributed under the terms of the Creative Commons Total positivity, Grassmannians and networks , Springer (2014). Calculating scattering amplitudes efficiently arXiv:1512.04954 Volumes of Polytopes Without Triangulations [ (2010) 020 (2015) 030 (2010) 016 883 super Yang-Mills theory 03 08 09 = 4 arXiv:1408.0932 arXiv:1312.7878 arXiv:1212.5605 arXiv:1312.2007 Grassmannians JHEP N (2016) 014 [ for Scattering Amplitudes JHEP [ Grassmannian Geometry of Scattering Amplitudes [ [ Physics JHEP Theoretical Advanced Study Institute inU.S.A., Elementary June Particle 4-30, Physics, 1995 TASI-95, Boulder, N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, J.M. Drummond, J.M. Henn and J. Plefka, L. Ferro, T. Lukowski, A. Orta and M. Parisi, M. Enciso, L.J. Mason and D. Skinner, N. Arkani-Hamed, A. Hodges and J. Trnka, A. Postnikov, N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Hodges and J. Trnka, N. Arkani-Hamed and J. Trnka, N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, N. Arkani-Hamed, F. Cachazo and J. Kaplan, N. Arkani-Hamed and J. Trnka, H. 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