Geometric Recursion from Polytope Triangulations and Twisted Homology

A geometric approach to understanding recursion relations for scattering amplitudes is developed. We achieve this by studying intersection numbers of triangulated accordiohedra presented as hyperplane arrangements. The cancellation of spurious divergences is subsequently realized as a topological no-boundary condition.


I. INTRODUCTION
In a number of quantum field theories of practical interest, such as pure Yang-Mills, calculations using Feynman diagrams at high multiplicity are often prohibitive. The inefficiency of the Feynman expansion owing to the profusion of algebraic and combinatorial complexity indicates often that a simpler picture should be at work, especially considering the simplicity of the final answers involved. Some of the earliest evidence in support of this was supplied by the recursion relations of Berends and Giele [1], followed a couple of decades later by the on shell recursion derived by Britto, Cachazo, Feng and Witten [2,3].
Since the development of these techniques, a variety of practical problems could be solved with relative ease. Applications and extensions have been made to gravitational theories [4][5][6][7][8], generic theories [9][10][11], string models [12,13] and one loop integrands [14]. For the special case of Yang-Mills theory with maximal supersymmetry, the insight provided by the idea of recursion has led to the discovery of the integrand to all loop orders in the planar limit [15][16][17].
Attempts to find the geometric structures underlying amplitudes have also led to alternate interpretations of recursion relations. In particular, there is the realization that amplitudes are often volumes of certain polytopes [18][19][20][21][22][23][24][25][26][27]. This allows for recasting recursion relations as specific triangulations of the polytopes in question. In order to clarify this point, let us see how exactly this plays out.
In this approach to geometrizing amplitudes, known as the positive geometry program, the scattering amplitudes are recast in two dual pictures. On the one hand, certain polytopes (associahedra for planar φ 3 for example) are realized as convex objects in kinematical space K n . A canonical form Ω P , which can be defined uniquely for a given polytope P [28], is shown to encode the amplitude as a residue evaluated on the polytope, Alternatively, the same amplitude is realized as the volume of the dual polytope P * , * nikhilkaly@gmail.com Here, the recursion relation is realized as a triangulation of the dual polytope. It can be shown that a triangulation of the dual polytope leads to a recursive definition of the canonical form.
The positive geometry program has traditionally focused on massless particles. More recently, the tools of twisted intersection theory [29][30][31][32][33][34][35][36][37][38][39] have been employed to provide a generalization of the positive geometry program to massive internal states and more generally for external particles with generic kinematics off the mass shell [40,41] in terms of polytopes known as accordiohedra. In light of these developments, one can ask if recursion relations can be understood by leveraging the ideas of positive geometries and intersection theory. It is our objective in this letter to address this question.
The intersection theory of n dimensional polytopes discussed in [40,41] entailed the realization of these polytopes as hyperplane arrangements CP n − {f 1 , ..., f N }, where the f i 's are the N facets of the polytopes, namely Stokes polytopes or accordiohedra [42]. The intersection theory is then defined by considering cycles (domains of intersection) and cocycles (top-dimensional differential forms), which are then twisted by a connection defined by logarithmic singularities as these hyperplanes are approached.
Crucially, these works have focused on the intersection theory of cycles or cocycles with themselves. These can be computed by certain localization techniques (see [33,34] for further details) and give rise to exotic quasistring theory amplitudes and field theory amplitudes of scalar field theory with polynomial interactions respectively. What has not been discussed so far is the pairing between a cycle and cocycle associated to the accordiohedra A n defined in CP n . As it turns out, these quantities serve to provide the ideal arena to discuss a geometric form of recursion relations.
In order to define this pairing, the data required is as follows. First, a function F α on CP n , with the property that it vanishes as the hyperplanes are approached. Secondly, one requires a differential form ϕ An , which diverges logarithmically near the hyperplanes. The pairing is then given by the integral, Here, α is a regulator that is present for formal purposes (aspects of this were discussed in [40]). For suitable choices of this defining data, we will show that a large class of field theory amplitudes can be recast as the leading contributions of such intersection numbers in the limit of vanishing α . In this light, we will then be able to realize recursion relations as a triangulation of the domain of integration A n , which is a cycle by virtue of the removal of hyperplanes. This gives a purely geometrical meaning to the idea of recursion in scattering amplitudes for a very large class of theories. Additionally, the meaning of spurious poles is also made transparent -they occur as a consequence of the individual simplices of the triangulations having nontrivial boundary. The cancellation of spurious singularities in the final results is then recast as a requirement that the sum over boundaries of the triangulating simplices add up to a trivial boundary.

II. CYCLES AND COCYCLES FOR ACCORDIOHEDRA
We start by describing how cycles and cocycles are defined for an accordiohedron constructed out of a hyperplane arrangement. To keep the discussion short, we restrict ourselves to a relatively simple, yet nontrivial example, namely the case of 6 particles scattering in φ 3 +φ 4 theory. Some of the tools tacitly used, namely the specific convex realizations of these polytopes have been discussed previously in [40][41][42].
Accordiohedra are defined for a collection of dissections of an n-gon. For our case, we need to focus on dissections of a hexagon with two triangulations and one quadrangulation. Here, we have four accordiohedra. For our purposes, we focus on the one that arises out of the dissection (13,46). The accordiohedron is a square, with vertices being given as in figure 1. Using now the rules for obtaining a convex realization, we can derive the following hyperplane arrangement for this polytope in CP 2 as follows, In order to define a twisted homology and cohomology, we have to work on a space where the interior of this hyperplane arrangement can be realized as a cycle, namely a domain of integration with vanishing boundary. This is possible if we define X (13,46) as our ambient space, always making sure to take into account the removal of the hyperplanes. Now the region bounded by the f i s is trivially a cycle on X (13,46) . Given this construction, a twist is a differential oneform that has logarithmic singularities as the hyperplanes are approached. The respective residues encode kinematical data. For our purposes, the desired twist is Here, the variables X ij should not be confused with the space X (13,46) ; they simply represent planar Mandelstam variables defined by Additionally, we have used m 2 to denote the mass of the interacting scalars. Working at tree level, there is no issue, but these are bare masses if one wants to start handling loops. Having these definitions now allows us to define the notion of twisted cycles and cocyles. A cycle is topologically a region of zero boundary, which in our case will be the region bounded by the hyperplanes. Now, a twisted cycle is a cycle that is loaded, or in other words, multiplied by a function that vanishes as the hyperplanes are removed 1 For our purposes, we have the following twisted cycle, where, Again here, as in the analysis done in [40], an auxiliary parameter α rears its head. It is introduced here mainly as a regulating parameter, and too much should not be thought of it. We will ultimately be interested in the final limit of α → 0, so we won't discuss this particular point further, but will make some comments on this construction later in the letter. Now, twisted cocycles are differential forms that diverge logarithmically as the removed hyperplanes are approached. For our purposes, the most useful choice is, There are two things we now have to clarify before moving on -the sense in which these things are cycles and cocyles in the sense of a homology theory and how they give rise to intersection numbers. To do this, we observe that the operator ∇ (13,46) is nilpotent. Accordingly, it defines a chain complex and a corresponding cohomology H n (X (13,46) , ∇ (13,46) ) on the space X (13,46) . It can be shown that the form ϕ (13,46) belongs to the highest cohomology group (n = 2) of the chain complex. Additionally, it can be shown that the twisted cycle Acc (13,46) is an element of the twisted homology. Further details and a number of worked out examples for associahedra can be found in [33].
(13) Proving this equation amounts to a rederivation of the localization formula proved in Appendix A of [33]. For completeness nowever, we note that this is proven by noting that the foregoing integral in equation 12 is evaluated for generic exponents by regularization on a Pochhammer contour. As the Pochhammer contour encircles a vertex comprehended by hyperplanes f i and f j , it picks up a factor (e 2πiα Xi − 1) −1 (e 2πiα Xj − 1) −1 . In the limit of vanishing α , the denominator structure in equation 13 is recovered as these constitute the most divergent contributions. The (2πi) factors are compensated for by contour integrals around the vertices.

III. GEOMETRIC RECURSION VIA TRIANGULATIONS
Drawing on the ideas set forth to understand recursion in [19,20,26], our focus now will be to derive recursion relations for generic amplitudes with polynomial interactions. To do this, we note two things. First, using intersection theory, amplitudes for a very large class of theories, including polynomial scalar field theories with masses and even some loop integrands can be derived [41].
Furthermore, as we illustrated with a simple example in the foregoing section, the cocycle-cocycle intersection numbers which gave rise to these amplitudes can actually be rederived as cycle-cocycle intersection numbers. The advantage here is that a cycle is a polytope, which can be triangulated, and this triangulation gives rise to the desired recursion.
Let us be quantitative about this. For generality, suppose we have a hyperplane arrangement f 1 , ..., f N in CP n bounding an accordiohedron A n , which is a cycle in the space CP n with the hyperplanes removed. More precisely, ∂A n = 0 (14) in CP n − {f 1 , ..., f N }. Now we consider a specific triangulation of this domain of integration into m disjoint simplices, namely a decomposition, where the arrow indicates that the simplices are oriented. This is due to the fact that the no-boundary condition 14 place conditions on the boundaries on the simplices. In order to satisfy equation 14, we require, Having said this, the recursion relation for the amplitude simply becomes the following formula, We note here that we have used D to denote the particular dissection giving rise to the accordiohedron in question. The twisted pairing on the left side of the equation is defined in analogy to 13. A couple of things can be said about the foregoing formula. Firstly, the individual terms on the right side are not genuine twisted pairings. Only their sum is an intersection number. Consequently, an evaluation of the individual terms should be expected to give spurious poles which cancel in the sum. As we will see in a specific example, these arise on account of the fact that the simplices have nontrivial boundary. Accordingly, the requirement that spurious poles cancel is just the realization at the level of the amplitude of the no-boundary condition 14.
We also note that equation 17 holds to all orders in the parameter α . Indeed, what we have just described would hold equally well as a recursion relation for string amplitudes described by intersection numbers, if α is identified with the string tension.
In the next section, we will focus on a specific example to illustrate our result more effectively. We have written the right side of the equation in a manner that makes the specific orientations of the simplices manifest. To compute the intersection number this way, we note the following, and where we have defined and where B(x 1 ; a 1 , b 1 ) is the incomplete Beta function. In our discussion of the square accordiohedron, we encounter integrals of the following type, Making use of equations 19 and 20, we can evaluate the first integral in this formula to obtain the following, which reduces to Here we note that the cancellation of the incomplete Beta function terms is the cancellation of spurious divergences. One can check (by somewhat tedious expansion) that these terms individually supply terms such as (a + c) −1 .
Once we use this formula to compute the amplitude, we will note that this is indeed a spurious divergence, which quite gratifyingly drops out. The y integral can be carried out by simply taking the y = 1 limit of equation 19, to give us the final answer as, This is a 'squared' Veneziano amplitude. This is not unexpected; a square is simply the product of two lines, namely a product of two copies of M 0,4 .
With this formula, we can proceed to evaluate the intersection number 12. The final answer devolves upon the evaluation of four terms, I (13,46) where we have defined The expansion of this formula can be carried out by use of the well known expansion of the Gamma function near x = 0, Using this, we have the following as the leading contribution to the amplitude as we take the limit of vanishing α which evidently reproduces the correct divergence structure of equation 13. We point out parenthetically that we have not obtained the 2πi factors. Since we are working with real momenta, we had no trouble directly performing the integrals Now we comment on the appearance of spurious poles. Observing that the evaluation on each triangulation gives denominators (a + c) −1 , for say a = 2πiα (X 35 − m 2 ) and c == 2πiα (X 13 − m 2 ), we see that this is indeed a spurious pole. Fortunately, we did not have to deal with them, as they nicely cancelled out when the contributions from the two triangulations were summed together.
Finally, we make some comments on how this formalism actually tells us how to perform this kind of recursion for theories with more generic numerator structure. For a familiar example, suppose we work with four-fermi theory. This is a theory with only quartic vertices, so it would be well described by accordiohedra. A square accordiohedron in this theory would describe 8 particle scattering. Here, we can define the scattering form ϕ by requiring that while it diverges logarithmically as a hyperplane is reached, the residue near a vertex would not be 1, but rather the numerical kinematical factor derived from the Feynman expansion.

V. DISCUSSION
Our focus in this letter has been to present a description of recursion relations in quantum field theory in a fashion that exploits modern geometric approaches to scattering amplitudes, namely the theory of intersection numbers and polytopes. Since we are able to leverage the fact that we can now describe quantum field theories with arbitrarily complicated interactions systematically using polytopes, the approach to recursion given here can be applied to nearly any theory of interest.
In [43][44][45][46], among other things, rules were formulated for evaluating integrals that showed up in the moduli space formalism of Cachazo, He and Yuan [47][48][49][50][51][52]. These were then used to provide a string-based model to describe scattering amplitudes for scalar theories with arbitrary interactions. It is hard not to suspect that this bears some relationship to the perspective outlined in [41] and this work, the pursuing of which could be of value.
It should be emphasized that in this letter, we have taken the right side of the duality expressed in 2 as our approach to understanding recursion, by realizing recursion relations as triangulations of cycles. More traditional however is the point of view view indicated by the left side, in which the canonical form viz. the amplitude itself is recursed by some kind of complex shift. The understanding of recursion relations applied to accordiohedra developed in the papers [26,27] follows this latter perspective. Investigating possible connections between these ideas and the techniques developed in this letter would merit further research.
Lastly, the study of intersection theory has also shed new light recently on the phenomenon of the duality between color and kinematics [53]. Exploring the implications of these ideas in the context of accordiohedra and the possibility of realizing recursion relations for amplitudes which can be expressed as double copies may also be interesting as a topic of further research.