aa r X i v : . [ h e p - t h ] F e b Learning scattering amplitudes by heart
Severin Barmeier ∗ Hausdorff Research Institute for Mathematics, Poppelsdorfer Allee 45,53115 Bonn, GermanyAlbert-Ludwigs-Universit¨at Freiburg, Ernst-Zermelo-Str. 1, 79104Freiburg im Breisgau, Germany
Koushik Ray † Indian Association for the Cultivation of Science,Calcutta 700 032, India
Abstract
The canonical forms associated to scattering amplitudes of planar Feynman dia-grams are interpreted in terms of masses of projectives, defined as the modulus oftheir central charges, in the hearts of certain t -structures of derived categories ofquiver representations and, equivalently, in terms of cluster tilting objects of thecorresponding cluster categories. ∗ email: [email protected] † email: [email protected] Introduction
The amplituhedron program of N = 4 supersymmetric Yang–Mills theories [1] culminat-ing in the ABHY construction [2] has provided renewed impetus to the study of compu-tation of scattering amplitudes in quantum field theories, even without supersymmetry,using geometric ideas. While homological methods for the evaluation of Feynman dia-grams have been pursued for a long time [3], the ABHY program associates the geometryof Grassmannians to the amplitudes [4]. Relations of amplitudes to a variety of mathe-matical notions and structures have been unearthed [5–7]. For scalar field theories theamplitudes are expressed in terms of Lorentz-invariant Mandelstam variables. The spaceof Mandelstam variables is known as the kinematic space. The combinatorial structureof the amplitudes associated with the planar Feynman diagrams of the cubic scalar fieldtheory is captured by writing those as differential forms associated to a polytope in thekinematic space, called the associahedron, and their integrals. The form is named canon-ical form.In an attempt to categorify the ideas we extract the canonical form from clustertilting objects in the cluster category of a quiver of type A and, equivalently, from torsionpairs for the category of quiver representations and their intermediate t -structures. Sucha connection may be anticipated from the existing literature, but here we highlight thecategorical framework available in the representation-theoretic literature, which we expectto prove instructive in generalizing the computation of scattering amplitudes to higherorders. We restrict our attention to the planar tree level Feynman diagrams in a cubicscalar field theory. These diagrams can be thought of as rooted binary trees. The Feynmandiagrams are dual to triangulated polygons in the sense that they are obtained by drawinglines intersecting the edges of a triangulated polygon as indicated in Fig. 1.= dual Figure 1: Binary tree, Feynman diagram and triangulation of polygonThe various triangulations of a plane polygon, and hence the Feynman diagrams, arein one-to-one correspondence with the vertices of an associahedron, which also appearsin the representation theory of quivers [8] in the form of exchange graphs [9–11]. Thecanonical form is identified as the volume of the polytope dual to the associahedron [2, 5].The vertices of the associahedron are also associated to the generators of cluster algebras[12], two adjacent vertices being related by a mutation [13]. In a categorical frameworkcluster algebras are associated to cluster categories which can be realized as triangulatedorbit categories of the bounded derived categories of quiver representations. We showthat the canonical form is directly obtained from the central charges of the projectiveindecomposable objects of hearts of intermediate t -structures. In other words, we obtainthe red arrows in Fig. 2.Let us describe the proposed interpretation at the outset. We shall then work out twoexamples, which are easily generalized. In a perturbative treatment of the scattering of N particles in quantum field theory, the conservation of total momenta carried by the1eynmandiagramTree Quiver Q Clustercategory C Q Clustertilting objectsDerivedcategory D Q Intermediate t -structuresAssociahedron CanonicalformTriangulationFigure 2: Flow chartparticles as well as the nature of interactions are encoded in Feynman diagrams. Thecontribution to the scattering amplitude corresponding to the diagrams are expressed interms of Lorentz-invariants formed out of the momenta of the particles. Consideration ofthe diagrams beyond tree level, which involve integration over momenta, will be postponedto future work. A set of such invariants, called planar variables, are denoted X ij , with i and j running over the labels of N particles. The indices are defined modulo N . This,in addition to the symmetry of the planar variables under the exchange of the indices,gives the set of planar variables a periodic structure. A certain combination of the planarvariables, namely, the discrete Laplacian operating on X ij , relates to the Mandelstamvariables. This combination may be presented as a diamond-like relation with arrowsindicating the sign of terms in the discrete Laplacian. Considered seriatim for all theparticles, the diamond can be knit into a mesh. If the mesh is continued ad infinitum ityields the Auslander–Reiten (AR) quiver of the bounded derived category D Q = D b (rep Q )of representations of a quiver Q of type A [14,15], the planar variables corresponding one-to-one with indecomposable objects and the arrows in the mesh taken to represent themorphisms in D Q . The periodicity of the planar variables further restricts the structure bydictating an identification of objects in the AR quiver. We observe that this identificationhappens to correspond precisely to the passage from the derived category D Q to the clustercategory C Q , which is the triangulated category of orbits under a certain autoequivalence[13].Planar tree level Feynman diagrams of N particles are dual to triangulations of an N -gon, which in turn can be related to the cluster algebra or the cluster category of the A N − quiver [16, 17], the combinatorial structure of the set of all triangulations being describedby mutations in the cluster algebra or cluster category. In the categorical setting, this isthe combinatorics of cluster tilting objects, which on the level of the derived category D Q also admit a description via t -structures obtained from torsion pairs [9]. Our goal is to2se the correspondencesFeynman diagrams ←→ cluster tilting objects in C Q ←→ torsion pairs in rep Q to obtain the explicit expression of the canonical form.The similarity between the mesh and the AR quiver is formal so far. While the objectsin the cluster category are orbits of quiver representations in the derived category, theplanar variables are real numbers. In order to relate these we remark that the centralcharge of objects are complex numbers associated to equivalence classes of the objectsin the Grothendieck group of the derived category. These are related through the meshrelations, which coincide with the mesh relations among the planar variables X ij . Byidentifying the modulus of the central charge of an object in the AR quiver of the clustercategory with the corresponding planar variable X ij we re-derive, not surprisingly, therelations among the latter from the mesh relations of the derived category. The modulusof central charge is called the mass [18]. In here it relates to the squared invariantmass of a collection of particles. In the categorical parlance X ij are interpreted as themasses of indecomposable objects in the derived category. Their relations as derived frommomentum conservations descend from the central charges on the stability lines. Eachcluster tilting object in the cluster category corresponds to a Feynman diagram. Thecontribution of each Feynman diagram to the amplitude is thus given by a term in thecanonical form. Each term is a logarithmic ( N − t -structures.Categorical formulation facilitates organizing computations. The present formulation,while not as geometric as the associahedron or the Grassmannians, has the virtue of beingalgebraic and seems to be of use in developing computer algorithms for the evaluation ofthe tree diagrams. We now exemplify the program chalked out above for two examples, N = 5 and N = 6, the latter being completely generic. In the next section we review thederivation of the mesh relations among the planar variables from momentum conservation.The two examples are described in the following two sections. In each case we firstidentify the cluster category from the mesh relations of the planar variables. We thenobtain the cluster tilting objects and identify their direct summands, which may be viewedequivalently as projective objects of hearts of intermediate t -structures. Their masses arethen shown to correspond to terms of the canonical form. Let us recall the definition of kinematic variables for the scattering of a system of N scalarparticles [2]. Their momenta are vectors in R , , denoted p i , i = 1 , , . . . , N , satisfyingthe conservation equation N X i =1 p i = 0 . (1)This is solved by writing the momenta in terms of another set of N four-vectors x as p i = x i +1 − x i , (2)3here from now on we define the indices modulo N , in particular, x N +1 = x . Mandelstamvariables are quadratic invariants for a pair of particles s ij = ( p i + p j ) (3)where the norm of a four-vector p = ( p , p , p , p ) is defined as p = − p + p + p + p .Entities defined similarly with the x ’s as X ij = ( x i − x j ) (4)are called planar variables. These are symmetric with respect to exchange of indices bydefinition, X ji = X ij . Using (2) and the periodicity of the indices the planar variables arerelated to the Mandelstam variables as s ij = p i + p j + X i,j +1 + X i +1 ,j − X i,j − X i +1 ,j +1 , (5)If we now assume that the particles are massless, that is, the momentum vectors are null, p i = 0 for each i , then s ii = 2 p i = 0 , X i,i +1 = 2 p i = 0 (6)and the relation (5) becomes s ij = X i,j +1 + X i +1 ,j − X ij − X i +1 ,j +1 . (7)The right hand side is the negative discrete Laplacian operating on the planar variables.In particular, we have X N,N +1 = X N, = X ,N = 0 ,X ,N +1 = X , = X , = 0 . (8)Equation (7) can be pictorially presented as s ij = X i,j +1 X ij X i +1 ,j +1 X i +1 ,j (9)For any given value of N this unit can be used to weave a mesh [14], which, upon usingthe periodicity of the indices, the symmetry of X ’s and equation (8), gives rise to thecluster category of the A N − quiver. In the next two sections we work out the examplesof N = 5 and N = 6 and obtain the canonical forms from the central charges.4 Example of N = 5 The mesh diagram knit from (9) is X X X X X X X X X X X X X X X X X X X X = X = X = X = X = X = X = X = X = X = X (10)Using the periodicity of the indices modulo 5 and the symmetry X ji = X ij , for example X = X = X , the ones in red blocks are null by (6) and (8). Omitting them we obtaina part of the Auslander–Reiten (AR) quiver of the derived category D A = D b (rep A )of the A quiver 1 −→
2. Furthermore, due to the periodicity of the indices we mayrestrict ourselves to the portion of the AR quiver in the green block and identify the nodelabelled X with the one labelled X and similarly identify X with X , as indicated.The canonical form associated to the N = 5 amplitude is [2]Ω = d log X ∧ d log X − d log X ∧ d log X + d log X ∧ d log X − d log X ∧ d log X + d log X ∧ d log X . (11)The scattering amplitude is obtained from the canonical form using relations among theplanar variables [2]. These relations are derived from the mesh relations in the derivedcategory.Let us indicate the combinatorial scheme for fixing the relative signs of the terms. Theset of planar variables appearing in each term of the canonical form are identified firstand one term is fixed, say, the first one in (11) with a positive sign. We then replace oneof the planar variables with a new one. The new term is given a negative sign, as in thesecond term in (11), where X of the first term is replaced with X . Repeating this weobtain the full canonical form. Below we identify the set of planar variables of every termas the masses of projective objects of the hearts in the derived category. We now describe the cluster category of the A quiver and with it we relate the canon-ical form (11). The A quiver is 1 −→
2. The Abelian category of finite dimensionalrepresentations rep A contains three indecomposable representations C −→ , −→ C and C −→ C . (12)These representations are also denoted 1, 2 and , respectively. Here a single number i denotes the simple representation with a one-dimensional vector space at the vertex5abelled i in the quiver and ij denotes an extension of i by j . These three representationsfit into a short exact sequence 0 −→ −→ −→ −→ A . Including the AR translation τ indicating the existence of an extension of 1 by τ (1) = 2 furnishes the following diagrammatic picture of rep A τ (14)Repeating this unit on both left and right to include the shift functors (given diagram-matically by a glide reflection) yields the AR quiver corresponding to the bounded derivedcategory D A = D b (rep A ) whose objects are bounded complexes of representations inrep A . Identifying the AR translation τ with the shift functor [1] in D A one obtains thecluster category C A [17, 19]. It is a triangulated category [13] depicted as F (2)2 1 [1] 2[2] = F ( ) ττ τττX (0 , X (1 , X (2 , X (0 , X (1 , X (2 , X (3 , (15)where F denotes the cluster automorphism F = τ − ◦ [1] : D A −→ D A so that in thecluster category C A , 2 is isomorphic to F (2) = 1[1] and is isomorphic to F ( ) = 2[2].Here the nodes are labelled by indecomposable representations in rep A and their shifts.Thus in (15) M refers to the complex with M placed in degree 0 and M [ n ] denotes itsshift, i.e. − − M ( · · · M · · · ) M [1] ( · · · M · · · ). (16)In (15) we have also indicated the AR label above and the corresponding planar variablebelow each node. Objects in the cluster category satisfy mesh relations. The meshrelations written in terms of the AR labels are [20, 21] r ( p,i ) = ( p, i − −→ ( p, i ) −→ ( p + 1 , i ) . (17)The objects in the cluster category are orbits of complexes of quiver representationsunder the cluster automorphism, but we just need to work with the indecomposableobjects, which we continue to denote simply by the indecomposable representations andtheir shifts. The planar variables are obtained as the modulus of central charges of these ( p, i ) referring to the i th vertex in the p th copy of Q in the translation quiver of Q Z : K ( D A ) −→ C . We label the central chargeby the objects as well as the AR labels, for example, Z = Z (0 , etc. and use the notationsinterchangeably. The mesh relations then give rise to relations among central charges as Z ( p,i ) = Z ( p,i − + Z ( p +1 ,i ) . (18)For the present instance, in D A , these are, with middle node (0 , , , , , Z = Z + Z ,Z = Z + Z ,Z = Z + Z [1] ,Z [1] = Z + Z ,Z = Z [1] + Z , (19)repectively. We have five relations among seven central charges. Thus two of them are“independent” and the rest can be expressed as their linear combinations. The choice ofthe two independent ones correspond to a choice of intermediate t -structure, whose hearthas in the case of A two indecomposable projective objects. Once the independent onesare chosen, the assignment of central charges is made according to Z A [1] = − Z A (20)for any object A .The reason for defining the charge on the Grothendieck group of the derived categoryis that the Grothendieck group of the cluster category with its standard triangulatedstructure is too small, as was brought to our attention by Yann Palu. This may beremedied by working with so-called extriangulated structures on the cluster category [22].For the present article we shall freely use the two equivalent perspectives of projectiveshearts of intermediate t -structures on the one hand and direct summands of cluster tiltingobjects on the other hand, and content ourselves with defining the charge only from theprojectives of hearts. We now briefly recall the notion intermediate t -structures obtained by tilting with respectto torsion pairs and the corresponding cluster tilting objects in the cluster category C A .The standard heart in D A is given by the Abelian category rep A (14) and the inde-composable projective objects of the heart are the representations 2 and correspondingto the planar variables X and X , respectively, as indicated in (15). The correspondingcontribution to the canonical form (11) is d log X ∧ d log X .Let ( D ≤ A , D ≥ A ) denote the standard t -structure on D A with heart D ≤ A ∩ D ≥ A =rep A . Given a torsion pair ( T , F ) in the Abelian category rep A one can associatean “intermediate” t -structure ( D (cid:22) A , D (cid:23) A ), one that satisfies D ≤ A ⊂ D (cid:22) A ⊂ D ≤ A [1] [9].The heart of this new t -structure is again an Abelian category which may or may not7 Figure 3: The 5 torsion pairs ( T , F ) of rep A with torsion class T (blue) and torsion-freeclass F (red)
13 14 24 25 35 [13][14] [14][24][24][25] [25][35] [13][35]
Figure 4: The 5 intermediate t -structures of D A obtained by tilting with respect to atorsion pair from the standard heart (filled vertices), their hearts (shaded), projectiveobjects (circled vertices) and their contribution to the canonical formbe equivalent to rep A . Practically speaking, one may choose T to be the Abeliansubcategory generated by any collection of indecomposable representations which is closedunder extensions and quotients and then let F = T ⊥ = { Y ∈ rep A | Hom (
X, Y ) =0 , ∀ X ∈ T } be its right orthogonal complement. The Abelian category rep A hasfive torsion pairs which are presented in Fig. 3 and the corresponding intermediate t -structures of D A in Fig. 4 where we omitted the arrows (cf. (14) and (15)). The blackdots correspond to the indecomposable objects of rep A . In each diagram of Fig. 4the blue part corresponds to D (cid:23) A and the red part to D (cid:22) A [ −
1] of the t -structure. Thevertices in the shaded part correspond to the indecomposable objects in the heart of the t -structure, which is given by D (cid:22) A ∩ D (cid:23) A . Diagrammatically, the heart is obtained as theintersection of the blue part and the shift (given by glide reflection to the right) of thered part.Given a t -structure ( D (cid:22) , D (cid:23) ) on a triangulated category D with heart H = D (cid:22) ∩D (cid:23) , an object P ∈ D is called a projective of the heart if for all M ∈ H and all k = 0 onehas Hom D ( P, M [ k ]) = 0. For type A quivers the dimension of the Hom space betweenindecomposable objects can be read off the AR quiver [8, § P, M ) = 0precisely when M lies in the maximal slanted (possibly degenerate) rectangle R ( P ) whoseleft-most point is P as illustrated for an A quiver in Fig. 5.For the intermediate t -structures of D A the projective objects of their hearts aremarked in Fig. 4 as the circled vertices. From the point of view of cluster categories,the direct sums of the images of the projectives of the hearts in the cluster category areprecisely the cluster tilting objects of the cluster category C A . The corresponding planarvariables contribute to the canonical form as follows.8 R ( P ) P R ( P ) P R ( P ) Figure 5: Three maximal slanted rectangles in the AR quiver of D A indicating the nonzeroHom spaces from the objects marked by filled vertices to the objects in the rectangles • For the standard t -structure corresponding to the torsion pair ( T , F ) = (rep A , A = h , , i with(indecomposable) projectives 2 and so that 2 ⊕ is a cluster tilting object of C A , where we have written also 2 and for their images in the cluster category.Their central charges are chosen to be the independent ones. All the others can beexpressed in terms of them using (19). We have, for example, Z = Z − Z . (21)Using (20) all the equations (19) coincide with this, leaving only two independentcentral charges. Writing Z A = X A e iπφ , for objects and choosing X ’s according to(19), we can write this as a relation among the X ’s, X = X − X , (22)where we have ignored the common phases of the central charges, tantamount tochoosing a line of stability. All the other X ’s can be similarly expressed in terms of X and X . Contribution to the canonical form is d log X ∧ d log X . • The heart of the t -structure obtained by tilting with respect to the torsion pair( T , F ) with T = h , i and F = h i is given by h , , i whose projectives are 1and so that 1 ⊕ is the corresponding cluster tilting object. (Note that here 2 is nolonger a projective of the heart, since 2[1] is in the heart and Hom D A (2 , − , ≃ C = 0.) In this case the independent charges are Z and Z . Corre-spondingly, the independent X ’s are X and X , the rest being expressed in termsof these. Contribution to the canonical form is d log X ∧ d log X . • The contribution to the canonical form is determined completely analogously forthe remaining three hearts, whose cluster tilting objects are 1 ⊕ ⊕ [1]and 2 ⊕ [1]. Matching their direct summands with the planar variables, theircontributions to the canonical form are given in Fig. 4, where we have omitted ∧ andwritten [ ij ] for d log X ij . Note that the heart of the t -structure of the last diagram inFig. 4 is not equivalent to rep A and that one of the two indecomposable projectivesof the heart does not lie inside the heart. Indeed, this kind of phenomenon was partof the motivation for developing τ -tilting theory [23,24] which generalizes the notionof tilting modules in a way that is compatible with mutations.Collecting all these contributions we recover the canonical form (11). The signs of theterms are fixed, up to an overall factor, by demanding invariance under simultaneousscaling of the planar variables [2]. 9 Example of N = 6 We proceed similarly for the case of N = 6 particles. We identify the mesh diagram of themomenta with a portion of the AR quiver of D A corresponding to the cluster category C A which can be obtained by identifying the indecomposable objects 3 ≃ F (3) = 2[1], ≃ F ( ) = [1] and ≃ F (cid:0) (cid:1) = 3[2] in the green block2[ −
1] 1[ − [ −
23 12 23 [1] [1] [ −
1] 3 2 1 [1] 3[2] X X X X X X X X X X X X (23)where the shaded part is the heart of the standard t -structure of D A . There are now14 torsion pairs, giving 14 intermediate t -structures whose hearts have three projectiveobjects each, which in turn correspond to the 14 cluster tilting objects. The canonicalform has 14 termsΩ (6) = − [13][14][15] + [15][25][24] − [15][14][24] + [24][25][26] + [26][25][35] − [26][36][35] + [26][36][46] − [15][25][35] + [13][14][46] − [13][15][35]+ [46][14][24] − [13][36][46] + [24][26][46] − [35][36][13] (24)where we denoted [ ij ] = d log X ij . The hearts correspond to the fourteen vertices of anassociahedron [10, 11]. The torsion pairs for rep A and their corresponding t -structures for D A are illustrated inFig. 6. (In each diagram the torsion class is the Abelian category generated by the blackvertices in the blue part and the torsion-free class the one generated by the black verticesin the red part.) Let us remark that the torsion pairs of rep A can be “pasted” fromthe torsion pairs of rep A using the following rule. First, a choice of a torsion pair onthe three lower left vertices and a torsion pair on the three lower right vertices has to bemade such that the overlap agrees. Since both T and F are closed under extensions andthere can be no nonzero morphisms from objects in T to objects in F . This determinesin most cases uniquely whether the top vertex belongs to T , to F , or to neither. Whenthere is a choice, both choices (i.e. including the top vertex in T or in F ) define torsionpairs. In this way one can obtain torsion pairs for rep A n +1 recursively from “pasting”torsion pairs of rep A n and rep A (or equivalently from pasting n torsion pairs of rep A ).There are nine mesh relations ensuing from (17) among the twelve objects of the10 Figure 6: The 14 intermediate t -structures of D A obtained by tilting from the standardheart (filled vertices), their hearts (shaded), projective objects (circled vertices) and theircontribution to the canonical formcluster category, namely,3 , , [1] [1] , [1] [1] 3[2] ,
123 12 , [1] 2[1] ,
23 123 ⊕ , ⊕ [1] , [1] 2[1] ⊕ [1] [1] . (25)Accordingly, the twelve central charges are related by nine equations similar to (19).The general rule of assignment of charges are Z A [1] = − Z A Z A ⊕ B = Z A + Z B , (26)leaving three of them independent. As before, choosing the independent ones as thecentral charges of the projectives of the hearts from Fig. 6 their masses furnish the termsof the canonical form (24). The signs are again fixed by the scheme described in section 3,which guarantees its invariance under the scaling of the planar variables.11 Conclusion
In this note we have considered the canonical form appearing in the computation of planartree level Feynman diagrams of a cubic scalar field theory. The canonical form is a meansto encode the contribution of these Feynman diagrams to the scattering amplitude. Itsrelation to various mathematical structures have been studied earlier. In here we interpretthe terms of the canonical form as arising from the cluster tilting objects of the clustercategories of quivers of type A which correspond to projectives of hearts of intermediate t -structures. This approach is categorical and makes no allusion to associahedrons andtriangulations of polygons.In a nutshell the scheme is as follows. In order to obtain the canonical form of N -particle scattering in a cubic scalar field theory one writes the cluster category of the A N − quiver. The planar variables are obtained as the mass (modulus) of the centralcharges of the projectives of the hearts or equivalently of the direct summands of clustertilting objects in the cluster category.The different Feynman diagrams thus correspond to different stability regimes. Wehave illustrated this in two examples. The first one is for N = 5 particles, which issimpler, if somewhat restricted. The second example of N = 6 particles correspondingto the cluster category of A quivers is generic. These considerations can be generalizedto arbitrary number of particles. The present treatment also generalizes to quadraticand higher order scalar field theories [25] in terms of higher cluster categories. Thecategorification is expected to help the organization of the canonical form, especially intheir evaluation using computer programs. Acknowledgement
We warmly thank Hipolito Treffinger for very helpful discussions and Yann Palu for severalhelpful comments and for bringing the reference [22] to our attention.
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