Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization

Attached to both Yang-Mills and General Relativity about Minkowski spacetime are distinguished gauge independent objects known as the on-shell tree scattering amplitudes. We reinterpret and rigorously construct them as $L_\infty$ minimal model brackets. This is based on formulating YM and GR as differential graded Lie algebras. Their minimal model brackets are then given by a sum of trivalent (cubic) Feynman tree graphs. The amplitudes are gauge independent when all internal lines are off-shell, not merely up to $L_\infty$ isomorphism, and we include a homological algebra proof of this fact. Using the homological perturbation lemma, we construct homotopies (propagators) that are optimal in bringing out the factorization of the residues of the amplitudes. Using a variant of Hartogs extension for singular varieties, we give a rigorous account of a recursive characterization of the amplitudes via their residues independent of their original definition in terms of Feynman graphs (this does neither involve so-called BCFW shifts nor conditions at infinity under such shifts). Roughly, the amplitude with $N$ legs is the unique section of a sheaf on a variety of $N$ complex momenta whose residues along a finite list of irreducible codimension one subvarieties (prime divisors) factor into amplitudes with less than $N$ legs. The sheaf is a direct sum of rank one sheaves labeled by helicity signs. To emphasize that amplitudes are robust objects, we give a succinct list of properties that suffice for a dgLa so as to produce the YM and GR amplitudes respectively.


Introduction
Gauge theories in physics are redundant in their description of physical processes; to extract actual physical information one must quotient by a large gauge group. Tree scattering amplitudes, specifically the on-shell tree scattering amplitudes in Yang-Mills theory (YM) and General Relativity (GR), are interesting because they are gauge independent, devoid of any redundancy. These objects are extensively studied in the physics literature, see for example the Parke-Taylor formulas for YM amplitudes in [14] exposing significant cancellations among Feynman graphs for certain helicity configurations 1,2 ; the discussion of the factorization of residues along collinear and multiparticle singularities in [15] and earlier S-matrix literature; the Britto-Cachazo-Feng-Witten (BCFW) recursion relations in [17,18]; the monograph [19] and a textbook [20]. Tree scattering amplitudes are so named because they can be defined as a sum of Feynman tree graphs, and they are associated with classical physics, whereas graphs with loops are associated with quantum physics.
An interesting open question for mathematics, not answered in this paper but an important motivation, is how these gauge independent amplitudes are rigorously related to solutions to the partial differential field equations; for GR these are near-Minkowski solutions to the Einstein equations for vacuum, Ricci = 0 3 . One would like to see rigorously how the tree scattering amplitudes relate past incoming to future outgoing asymptotic data for actual solutions to the semilinear (YM) respectively quasilinear (GR) hyperbolic partial differential field equations, not an easy task. This would clarify to what extent these amplitudes 'describe the nonlinear interaction of gravitational waves' as modeled by GR. This paper is about the tree scattering amplitudes themselves, developed rigorously from their definition as L ∞ minimal model brackets implemented using Feynman graphs to their recursive characterization independent of Feynman graphs. It includes details often taken for granted in the literature on amplitudes, and departs conceptually from it because the development is consistently homological and geometrical. Here is a summary: Part I comprising Sections 4, 5, 6. To prove gauge independence and to treat YM and GR in a unified way, we define the tree scattering amplitudes in a non-standard manner as the L ∞ minimal model brackets of a differential graded Lie algebra (dgLa) with an additional 'momentum' grading. In the mathematics literature on the homotopy transfer theorem, e.g. [7], it is shown that the minimal model brackets of a dgLa can be defined as a sum of trivalent (cubic) Feynman tree graphs of the caricatural form where each node stands for an application of the dgLa bracket [−, −] whose inputs are the two lines coming in on the bottom and whose output is the line leaving on the top, whereas the lines (edges) are decorated by a contraction for the dgLa differential d encoding a choice of gauge 4 . We prove gauge independence in this homological language, namely that individual trees depend on the gauge but the sum of trees does not. Only then do we show that YM and GR can actually be formulated as the Maurer-Cartan equations du + 1 2 [u, u] = 0 in a suitable dgLa 5 . That these dgLa yield the partial differential field equations and gauge transformations of YM and GR suffices for our purpose; we omit a more direct and pedantic proof that the tree amplitudes as we define them coincide with those in the physics literature, defined in terms of the (non-trivalent) Feynman tree graphs associated to the YM and GR Lagrangians. (The details of the construction of these dgLa do not permeate this paper. It is an important by-product 4 In physics terminology, the on-shell quasi-isomorphisms i and p choose polarization vectors, the off-shell homotopy H is the propagator. The differential d is, as a Fourier multiplier, a matrix with entries polynomial in the momentum k ∈ 4 , with d 2 = 0, with homology on-shell but no homology off-shell. The shell is the light cone in momentum space 4 . 5 The unknown u stands for, roughly, a Lie algebra valued connection one-form and curvature two-form in YM, respectively an orthonormal frame and connection in GR. These are formulations where the partial differential field equations are first order. of the recursive characterization in Part II that all dgLa with the right homology, nontrivial bracket, Lorentz invariance and suitable homogeneity, yield the same amplitudes.) The homotopy H is not unique and is the direct homological analog of the propagator in the physics literature, a matrix whose entries are rational functions of the momentum k ∈ 4 that is necessarily singular when k is on the light cone. We construct 'optimal homotopies' H using the homological perturbation lemma, whose residue at the light cone factor in a specific way (namely with an i factoring out on the left and a p factoring out on the right) and which at once implies that the residues of the corresponding poles of the scattering amplitude factor, key for Part II. These homotopies are defined locally, on Zariski open sets, and hence so are individual trees, but the amplitudes can be glued to global sections of a sheaf by gauge independence; gluing is in Part II. The main results of Part I are Theorems 4.6, 5.1, 6.2.
Part II comprising Sections 7,8,9,10 and to show that these recursions characterize the amplitudes uniquely, given the homogeneity degrees of the amplitudes, given permutation invariance, and given the amplitudes with three legs, {−, −}, which in turn admit a simple characterization using Lorentz invariance. (This detaches tree scattering amplitudes from Feynman graphs, and indeed Part I enters Part II only to prove the existence of a solution to these recursions. One can go ahead and try to find other constructions without Feynman graphs that satisfy the recursions; literature along these lines includes [16,19]. Interesting as such approaches are, they are no replacement for Part I for those interested in studying the connection between amplitudes and the partial differential field equations, namely the Maurer-Cartan equations, in more depth, a key motivation for this paper.) To make sense of (1) we study the complex algebraic variety of kinematically admissible momenta; we classify the irreducible codimension one subvarieties, labeled by prime divisors p, along which amplitudes can have poles 7 ; we clarify what we mean by the residue along p; we introduce sheaves suitable for YM respectively GR amplitudes; we construct a codimension two subset where amplitudes need not be defined; and we obtain a variant of Hartogs extension for these sheaves 8 . Hartogs is used to prove the recursive char- 6 So the residues factor into products of amplitudes with fewer legs, that is, compositions of minimal model brackets with fewer arguments. The case of four legs, that is, minimal model brackets with three arguments, can involve an extra sum on the right. For details see (47). 7 Poles can only occur when an internal momentum (a sum of a subset of momenta excluding trivial subsets and those equal to a single momentum) goes on-shell, but this is not always an irreducible variety. The irreducible components are labeled by height one prime ideals p. For some helicity configurations, some potential poles are absent due to cancellations. 8 Recall that the classical Hartogs extension theorem says that a holomorphic function of two complex variables z, w defined on say 0 < |z| 2 + |w| 2 < 1 extends holomorphically to the origin, so there can be no singularities in codimension two. For singular varieties or sheaves, an analogous Hartogs phenomenon may or may not hold. It holds for the structure sheaf of complete intersections, sometimes said to have mild singularities, and we reduce our statement to this case. It can fail in the presence of wilder singularities, see Remark 7.2. acterization. A hint of a proof of the recursive characterization, glossing over geometric details, is at the end of [17]. (This discussion at the end of [17] is separate from, and not to be confused with, the main result of that paper known as BCFW recursion, see below.) The recursive characterization implies qualitative properties that are not clear from Feynman graphs, such as the pole structure of the Parke-Taylor formulas. In our non-Lagrangian setup it is not immediate that the amplitudes {−, . . . , −}, which are invariant under permuting the inputs, are also invariant under exchanging the output with an input relative to suitable bilinear forms, but we prove this also using Hartogs. The main result of Part II are Theorems 7. 14, 9.3, 9.4, 10.11, 10.16. The following things appear to be new: the interpretation of on-shell tree scattering amplitudes as L ∞ minimal model brackets and the unified treatment of YM and GR made possible by this, including a homological proof of gauge independence; the purely trivalent tree Feynman graphs particularly in GR 9 ; the classification of prime divisors along which amplitudes can have poles; the construction of optimal homotopies using the homological perturbation lemma from which the factorization of residues easily follows; a detailed statement and proof of the recursive characterization, using a Hartogs extension argument.
Let us discuss how BCFW recursion [17] compares. In its original form, this effectively restricts amplitudes to suitable 2 subspaces contained in the variety of kinematically admissible momenta (there are enough such subspaces) and by homogeneity one passes to the Riemann sphere È( 2 ) hence one complex variable. The north pole is a point of the form P = (0, . . . , 0, q, 0, . . . , 0, −q, 0, . . . , 0) with on-shell q ∈ 4 − 0. One uses the familiar theorem from complex analysis by which a meromorphic function on È( 2 ) with simple poles is determined by the location and residues of the poles up to an additive constant 10 . The residues come from restricting (1) but one also needs control at the point at infinity P , which is not immediate [17,18]. By comparison, the recursive characterization in this paper does not require checking conditions at points such as P because it exploits the Hartogs phenomenon. It requires (1) only away from a codimension two subset Z of the variety; Z contains the entire singular locus and in particular P ∈ Z. Roughly, the amplitudes are the unique sections given away from Z that satisfy the recursions (1) away from Z. No conditions need to be checked along Z. This seems to be a simple and natural recursive characterization, hence we give a rigorous account of this, with Hartogs as a simple proof.

An informal overview
The technical sections of this paper are, at least to some extent, logically independent units. This entails that the connections and the bigger picture may not always be clear. The only two places where things are tied together is this technical but informal overview, where we have taken the freedom to reorganize the material to simplify the discussion, and then rigorously in Section 10.
Homological framework. Both YM and GR about Minkowski spacetime admit a differential graded Lie algebra (dgLa) formulation. The partial differential classical field equations are the Maurer-Cartan equations for the unknown u, an element of degree one in the dgLa. In particular, in the case of GR the MC-equations are equivalent to the vanishing of the Ricci curvature. Elements of degree zero act as infinitesimal gauge transformations. References are [9,10] for YM and [11] for GR. The only nonlinearity is the bracket, which will give rise to trivalent Feynman tree graphs. For amplitudes, we use a variant of these dgLa, set in complex momentum space, k ∈ 4 . So k is formally the variable appearing in the Fourier transform. This dgLa g comes with an additional 'momentum' grading, g = k∈ 4 g k , that is respected by the operations, dg k ⊆ g k and [g k1 , g k2 ] ⊆ g k1+k2 : We have g k ≃ V for a finite-dimensional graded vector space V . Relative to a basis of V , the differential and bracket are arrays whose entries are polynomials in the momenta k and k 1 , k 2 respectively. Actually, they are first order polynomials, corresponding to first order partial differential operators. Elements of g may be interpreted as finite linear combination of plane waves, namely interpret v ∈ g k ≃ V as the plane wave Ê 4 ∋ x → ve ikx . However, we use g not because we are interested in finite linear combinations of plane waves, but because it is convenient for the definition of amplitudes.
The homology of YM and GR. The differential d describes the linearized problem. Its homology is a distinguishing feature of YM and GR.
Let Q be the Minkowski square of k, so Q = 0 is the complex light cone. It is convenient to let k be complex 2 × 2 matrices and to set Q = det k. Both YM and GR have homology only on-shell, meaning on the cone, h = k:Q=0 h k where h k = H(g k ), and only in degrees i = 1, 2. These split into helicities, The h i,± k define rank one sheaves on the cone away from the origin, times a Lie algebra in the case of YM that we refer to as the internal Lie algebra. Amplitudes as minimal model brackets. Gauge independence. We define the amplitudes to be the L ∞ minimal model brackets 11 with all internal lines off-shell, meaning for all J ⊆ {1, . . . , n} with 1 < |J| < n, the momentum k J = i∈J k i is not on the cone. The minimal model brackets are defined as a sum of trivalent tree graphs, implementing L ∞ homotopy transfer [2,3,4,5]. A typical such tree is . One needs the following objects to define trees, informally: • An off-shell homotopy for every internal line: A matrix H with entries rational in k such that H 2 = 0 and Hd + dH = ½. Singular along Q = 0.
Its evaluation at k is a map H k : g k → g k of degree minus one.
• An on-shell contraction for every external line: A contraction (h, i, p) meaning hdh = h, h 2 = 0, ip = ½−dh−hd, pi = ½. Further dhd = d along Q = 0. They are matrices with entries rational in k, regular including along Q = 0. Here i k : h k → g k and p k : g k → h k of degree zero.
Then (2) is the sum of all trivalent trees decorated with i at every input, p at the output, H at every internal line, and [−, −] at each node. The homotopies and contractions encode gauge choices. But while individual trees depend on these choices, we prove that the sum of all trees does not if all internal lines are off-shell. The gauge independence is in the strong sense, and not merely up to L ∞ isomorphism which holds for abstract reasons [2].
Gauge independence is a prerequisite to constructing global objects. In fact the homotopies and contractions are only required to be regular on some open patch of momenta k 1 , . . . , k n . Gauge independence allows one to glue these locally valid tree expansions to construct a global object.
The amplitudes are invariant under the group S n that permutes the n inputs, but invariance under S n+1 is not by construction, since our setting is without a Lagrangian and it distinguishes inputs and outputs. We prove invariance under S n+1 at a late stage, using residues and the recursive characterization.
Optimal homotopies via the homological perturbation lemma. The requirement Hd + dH = ½ means that the homotopy H is an inverse of d in a homological sense, and as such it is highly non-unique. Sticking to momentum 11 These are all minimal model brackets given that h is concentrated in degrees one and two, because the minimal model bracket h ⊗n → h has degree 2 − n.
conserving homotopies, H is a matrix whose entries are rational functions of k. However, we never write down explicit matrix entries. In fact the H that we use would not necessarily be convenient for explicit by-hand calculations, instead they distinguish themselves by their structure.
A homotopy necessarily has a singularity near the cone, of type 1/Q. By an optimal homotopy we mean one where 1/Q is multiplied by a matrix of the lowest possible rank. We show that every point on the cone has a Zariski open neighborhood on which there is a homotopy of the form Here h Q is a two-by-two matrix 12 so that ih Q p has the lowest possible rank. The restriction h Q | Q=0 is a canonical isomorphism h 2 → h 113 . Intuitively, p receives and i emits an on-shell particle.
Our construction of H is conceptual, without reference to YM or GR. It is based on a nested application of the homological perturbation lemma.
The momenta variety and the helicity sheaf for N = n + 1 particles. The amplitudes are naturally sections of a sheaf on a variety that we refer to as the variety of kinematically admissible complex momenta, a direct product of N cones, intersected with a codimension 4 plane. We discuss its geometry. It has two irreducible components when N = 3, but it is irreducible when N ≥ 4. It is a complete intersection and therefore has the property S 2 of Serre so that a variant of Hartogs extension holds for the structure sheaf.
The amplitudes will be sections of a sheaf that is a product of N sheaves, one for every input and output, times the sheaf of an effective divisor to accommodate 1/Q singularities in the homotopies. The tree expansion yields sections on open patches. Gluing yields amplitudes on the complement of a subset Z of codimension two, a union of various sets where one does not have a valid tree expansion. Hartogs can limit the behavior along the undefined locus Z.
Prime divisors and the factorization of residues. For every subset J ⊆ {1, . . . , N } such that both J and its complement J c have at least two elements, define the internal momentum k J = i∈J k i with Minkowski square Q J . Homotopies generally introduce 1/Q J singularities in amplitudes, known as collinear or multiparticle singularities. If |J|, |J c | ≥ 3 then the codimension one subvariety Q J = 0 is irreducible, but in general it decomposes. Each component is a prime divisor p and we classify them. Associated 12 Actually a block matrix that is zero except for a two-by-two block going from homological degree two to one. Times the dimension of the internal Lie algebra in the case of YM. 13 We describe in words how this map arises, see Section 6 for details. The derivative of d transversal to Q = 0 induces a differential on the homology h along Q = 0. It has a canonical normalization, using Q. For YM and GR one obtains a complex 0 → h 1 → h 2 → 0 at each point of the cone away from the origin. It is exact, hence induces a map h 2 → h 1 .
to each p is a residue condition for minimal model brackets, schematically 14 There is actually a sum on the right hand side over all J such that p is a minimal prime over (Q J ), but it always degenerates to a single term if N > 4. There are interesting exceptions in the case N = 4, where the sum is over three terms. The residue condition can be thought of as a factorization. It follows immediately if, exploiting gauge independence, one uses optimal homotopies (4). We illustrate this for a small tree and a J for which (Q J ) is itself a prime divisor: The subtree below h QJ is terminated by a p, and the subtree above is entered with an i. Hence, crucially, the subtrees are again of the general form (3).
If we sum over all trees, then only those containing k J as an internal momentum contribute to this particular residue. On the right hand side we get a sum that factors into a double sum over all trees below and all trees above the low-rank operator h QJ . They are reassembled into two brackets, giving (5).
We prove that these recursive conditions uniquely determine the amplitudes. This is what we refer to as the unique recursive characterization.
Recursive characterization. Hartogs extension. Local cohomology. Let X be the variety of kinematically admissible momenta, R its coordinate ring. The amplitudes are only constructed away from a closed subset Z ⊆ X of codimension two; Z is the union of all pairwise intersections of the zero loci of distinct prime divisors. For every helicity configuration there is a sheaf M associated to some finitely generated graded R-module M 15 . We refer to the sheaf that does not allow poles, so the amplitudes are not in M (X − Z) because they have poles, but the difference of two amplitudes with identical residues is in M (X − Z). Consider the restriction map of graded R-modules Its failure to be injective or surjective is measured by respectively the 0th and 1st local cohomology modules 16 . Though M is not locally free, a variant of Hartogs extension shows that (6) is an isomorphism. The recursive characterization follows since M (X) = M is empty in the relevant degree, for YM and GR 17 .

Preliminaries
Here we collect a few definitions and facts that are used in several sections, the reader may want to at least skim this section before moving on.
Ground field. All vector spaces and algebras and varieties are over the complex numbers . Tensor products of vector spaces are over .
Differential. Homotopy. Contraction. Homotopy equivalence. On a -graded vector space or module, by a differential we mean an endomorphism d of degree one that satisfies d 2 = 0. A space with a differential is called a complex. We use the following terminology: • A homotopy for d is an endomorphism h of degree minus one that satisfies Every homotopy yields three mutually orthogonal projections • A contraction for d is what some authors call a strong deformation retract with side conditions. Namely a triple (h, i, p) where h is a homotopy as above and where i, p are linear maps such that Note that hi = ph = 0. The codomain of p and domain of i is a second graded module, as in the non-commutative contraction diagram: Using the differential pdi, the maps p and i are a homotopy equivalence.
• A homotopy equivalence between two complexes (C, d) and (C ′ , d ′ ) are maps R ∈ Hom 0 (C, where R, L are chain maps and LR = ½−du−ud and We usually consider all four maps to be part of the data. Note that R, L are quasi-isomorphisms. The composition of two homotopy equivalences C ↔ C ′ and C ′ ↔ C ′′ is a homotopy equivalence C ↔ C ′′ . We often require that a homotopy satisfy dhd = d. Then the images of the projections in (7) are respectively im d, a complement of im d in ker d, and a complement of ker d. Conversely, for every choice of two such graded complements there is a unique corresponding such homotopy 18 . For a contraction, amplitudes near various singular points of the variety. Essentially one asks how constraining the structure of the sheaf itself is, when combined with permutation symmetry and Lorentz invariance and homogeneity. In this case, in (6) one must replace M by a module (just) big enough to accommodate poles, so that the amplitude is a section on X − Z. 18 In particular every differential on a vector space admits a homotopy with dhd = d. On infinite-dimensional vector spaces, the existence of complementary subspaces is not immediate, but it is standard and convenient to adopt axioms that imply the existence. dhd = d is equivalent to di = 0 or pd = 0 or pdi = 0. In this case the space on the right in (8) is canonically isomorphic, via i and p, to the homology of d.
Homological perturbation lemma. Given a contraction and a perturbation of the differential, the HPL produces a perturbed contraction. Explicitly, if the perturbed differential is called d ′ , and if we abbreviate δ = d ′ − d, then is the new contraction if δ is suitably small so that the inverses are defined. The HPL keeps the spaces fixed and only perturbs the arrows in (8). Beware that [1] for an exposition. One may think h as analogous to Hilbert's resolvent identity.
Differential graded Lie algebra and MC-elements. A graded Lie algebra or gLa is a -graded vector space g with a bracket [−, −] ∈ Hom 0 (g ⊗ g, g) that respects the grading, is graded antisymmetric, and satisfies the graded Jacobi identity. Explicitly for all homogeneous elements, the degree of [x, y] is the sum of the degrees of x and y and [ A differential graded Lie algebra or dgLa g is a gLa with d ∈ End 1 (g) a differential, d 2 = 0, compatible with the bracket in the sense of the Leibniz rule Formally, the Lie algebra g 0 acts on this set, and MC(g)/∼ is the moduli space of interest, a rigorous variant of which is the deformation functor [2].
The Lie algebra of the Lorentz group and its representations. Given a 4-dimensional complex vector space with a nondegenerate quadratic form Q, the automorphism Lie algebra is sl 2 ⊕ sl 2 with sl 2 the complex Lie algebra of traceless 2 × 2 matrices. If the vector space is that of 2 × 2 complex matrices 19 k = a b c d and Q = det k = ad−bc, then left-multiplication and right-multiplication by matrices with determinant one yield all automorphisms. Define right-multiplication with a transpose to get a left-action. At the Lie algebra level, sl 2 ⊕ sl 2 . The finite-dimensional irreducible representations are where p, q ≥ 0 are half-integers, ( 1 2 , 0) ≃ 2 and (0, 1 2 ) ≃ 2 are the fundamental representations of left and right sl 2 respectively, and S is the symmetric tensor product. So dim(p, q) = (2p + 1)(2q + 1). As sl 2 ⊕ sl 2 representations, 19 A minor clash of notations, the letter d is also used for differentials.
where p ′′ and q ′′ increase in steps of one. The Lie algebra sl 2 ⊕ sl 2 is the complexification of the real Lie algebra of the Lorentz group, as follows. On the real subspace of Hermitian 2×2 matrices, Q is a real quadratic form of signature +−−− whose automorphism Lie algebra is the real subalgebra of sl 2 ⊕ sl 2 of elements of the form A ⊕ A where a bar means element-wise conjugation.
The Lorentz equivariant complexes Γ ±h . For every half-integer 20 h ≥ 1 2 called helicity and for every momentum k ∈ ( 1 2 , 1 2 ) ≃ 4 define complexes where, by definition, the three terms are in homological degrees 1, 2, 3 respectively and where the last term is dropped when h = 1 2 . The dependence on k is implicit in the differential. By definition, the differential is linear in k ∈ ( 1 2 , 1 2 ) and it is the unique sl 2 ⊕ sl 2 equivariant map for Γ h , analogous for Γ −h . The uniqueness is by (10) and is up to an irrelevant multiplicative constant. This is a differential because its square is an equivariant 1) and (10). Explicitly, for Γ ±h the first part of the differential is given by and the second is given by where k = ( a b c d ) and ǫ = ( 0 1 −1 0 ) and k + = k and k − = k T , splitting is defined by z ⊗p → z ⊗p−1 ⊗ z for all z ∈ 2 , and ǫk ± ǫ : 2 ⊗ 2 → , x ⊗ y → x T ǫk ± ǫy 21 . More explicitly still, there are bases for which the differential for Γ 2 is given by  Switch b, c for Γ −2 . Similar for Γ ±h . The homologies are in Lemmas 6.6, 8.5. 20 A minor clash of notations, the letter h is also used for homotopies. 21 The composition of the maps (12) is zero by construction. To check it directly, it suffices to show that it annihilates all elements of the form z ⊗2h with z ∈ 2 since they span S 2h 2 . One obtains (z T ǫk ± ǫk ∓ ǫz)z ⊗(2h−2) which is zero since ǫk ± ǫk ∓ ǫ is antisymmetric.

Part I
This concerns the construction of tree scattering amplitudes using Feynman graphs. This contains, if at a rather formal level, the link between the classical partial differential field equations and amplitudes. 4 The L ∞ minimal model is gauge independent Homotopy transfer refers generally to the transfer of certain algebraic structures through quasi-isomorphisms, see e.g. [7]. In the special case of a dgLa g and a contraction to the homology h, one obtains an L ∞ algebra structure on h called the L ∞ minimal model, unique up to L ∞ isomorphisms. There are explicit formulas for the L ∞ minimal model brackets as a sum of trees, see below. We refer to the literature for proofs that this actually defines the L ∞ minimal model, including formulas for suitable L ∞ quasi-isomorphisms [2,3,4,5,7].
We show that for a dgLa with a momentum grading, and homotopies that respect the momentum grading, the L ∞ minimal model brackets defined using a tree expansion are independent of the homotopy in the strong sense (not merely up to L ∞ isomorphism) when all internal lines are off-shell. Namely, individual trees may depend on the homotopy, but the sum of all trees does not. The results are for all homological degrees, so amplitudes are a special case.
Definition 4.1 (Momentum grading). Suppose K is an Abelian group that we call momentum space. By a dgLa g = i∈ g i with momentum grading we mean one that carries a compatible K-grading, with algebraic direct sum g = k∈K g k Compatibility means that the K-grading respects the -grading and that Then the homology of g also decomposes, h = k∈K h k where h k is the homology of g k . A momentum k will be called on- We define the minimal model brackets as a sum of trivalent trees. Let T n be the set of tree graphs with n + 1 labeled leaves, n − 2 unlabeled internal lines, n−1 internal nodes of degree 3 (known as trivalent or cubic). The leaves 1, . . . , n are called inputs, and n + 1 the output. Let P n be such trees with a planar embedding. The map P n → T n that forgets the embedding is surjective 22 .
Definition 4.2 (Trees). For a dgLa g and a homotopy h that satisfies dhd = d, let p : g → h and i : h → g be the induced contraction. For every P ∈ P n define m P,h ∈ Hom 2−n (h ⊗n , h) as follows: • Decorate each input leaf by i, the output leaf by p.
• Given x 1 ⊗ · · · ⊗ x n ∈ h ⊗n one inserts each x j at the input labeled j.
• Multiply by the sign needed to permute x 1 , . . . , x n into place, where an even (odd) x j is considered odd (even) for the purpose of this permutation 23 .
Then m P,h is independent of the planar embedding 25 . So for every T ∈ T n we can set m T,h = m P,h where P ∈ P n is any planar embedding of T .
Example. The set P n is in bijection with full parenthesizations of any permutation of the elements 1, . . . , n. With this understanding, Note that if all inputs have odd degree, one always gets a plus sign.

Definition 4.3 (The minimal model brackets associated to a homotopy).
For a dgLa g and a homotopy h as above, the n-slot minimal model bracket . , x, y, . . .} h . In this section, g has a momentum grading and we assume hg k ⊆ g k for all k.
One may want to take a more cavalier attitude towards signs. By contrast, momentum conservation at each node is essential. Pictorially, a bracket such as h k1 ⊗ · · · ⊗ h k5 → h k1+...+k5 uses trees like the following where h k = h| g k .
Concretely, the permutation is given by reading off the input labels of P counter-clockwise, starting just to the left of the output. The sign is equal to the ordinary permutation signature for permuting only the x j with even degree. 24 This sign, independent of P , contributes to the simple Koszul sign rule in Definition 4.3. 25 To see this, note that the building block h −, − is homogeneous of degree zero and graded symmetric as a map g[1] ⊗2 → g [1]. Here g [1] is obtained from g by shifting the degree by one.
Warning 4.4 (Discontinuous nature of h). The space K has no topology and no continuity in k is assumed. But even if K = 4 as in other sections, even if the g k are finite-dimensional and isomorphic and d k = d| g k is polynomial in k, the homotopy h k must still be discontinuous because the projection i k p k = ½ − d k h k − h k d k is zero off-shell but nonzero on-shell. When we apply this section, h will be separately defined off-shell and on-shell, see Section 3.
with all internal lines off-shell, meaning for all (k 1 , . . . , k n ) ∈ K n such that h kJ = 0 for all subsets J ⊆ {1, . . . , n} with 1 < |J| < n and k J = i∈J k i .
Proof. By Theorem 4.6, M h : g ⊗n → g is a chain map so dM h = M h d tot since dπ = πd = 0, it induces the minimal model bracket h ⊗n → h on homology, and M h and M h ′ are homotopy equivalent when internal lines off-shell. Therefore they induce the same map on homology.
Theorem 4.6 (Gauge independence II). Let M h ∈ Hom 2−n (g ⊗n , g) be defined like the minimal model brackets but with input and output leaves decorated by π = ½ − dh − hd, replacing i and p. Let d tot be the differential on g ⊗n , so where ±½ is the sign map. Then for all homotopies h and h ′ there exists a linear map E : g ⊗n → g (that can be given as a sum of trees built using only h and h ′ and d and the bracket) such that when evaluated on g k1 ⊗ · · · ⊗ g kn → g k1+...+kn with all internal lines off-shell, where off-shell means homology-free just like in Theorem 4.5.
The following lemma, which is for any complex of vector spaces, connects any two homotopies h and h ′ by three transformations. We will use it to connect the homotopies by three curves polynomial in a parameter s. Pictorially, Given is a complex with differential d. In this lemma we only consider homotopies h that satisfy dhd = d, and we denote π = ½ − dh − hd. If h is a homotopy then another homotopy h ′ is given by for all a, c ∈ End 0 (g) or b ∈ End −2 (g) respectively, subject to the constraints. And any two homotopies h and h ′ are related by a composition of A, B, C. then Before proving the second part of the theorem, we derive another characterization of A, B, C. Note Say in the case A, the given a has degree zero, satisfies the constraints in the table, and h( Though one can proceed at the level of equations, we switch to a geometric argument. Recall the bijection between homotopies h with dhd = d and pairs (X, Y ) of graded subspaces where X is a complement of im d in ker d, Y a complement of ker d. The bijection is established by X = im π, Y = im hd. The last paragraph shows that A connects any two homotopies with the same Y , B those with the same X and Y ⊕ im d, C those with the same X and Y ⊕ X. Now, given any two homotopies h, h ′ corresponding to (X, Y ), One can see that they are given by (14). By construction, h C = h ′ .
Proof (of Theorem 4.6). It suffices to prove the theorem in the special cases A, B, C of Lemma 4.7. Since hg k ⊆ g k , (14) implies ag k , bg k , cg k ⊆ g k . The brackets are polynomial in h and π, so if we consider polynomial curves a(s), b(s), c(s) then the brackets are polynomial in s. It suffices to show that we get the desired result when differentiating with respect to s, namely thaṫ M = dĖ +Ėd tot for someĖ where a dot denotes a derivative at s = 0 26 : Since all internal lines are off-shell hence annihilated by π, we effectively haveḣ = 0. So only inputs and outputs are affected. One can takeĖ = hȧM in this case.
With this setup, it suffices to show separately for every nonempty J ⊆ {1, . . . , n} that the following infinitesimal variations, affecting internal lines with momentum k J = i∈J k i , yield zero after summation over all trees: • Variations of type dḃ at an internal line if 1 < |J| < n respectively dhȧπ at the input if |J| = 1. Note the d on the left.
• Variations of type −ḃd at an internal line if 1 < |J| < n, respectively πċhd at the output if |J| = n. Note the d on the right.
All cases reduce to Lemma 4.8. Given the results forĖ, one can see that E has the claimed form. For A take E = haM , for B take E = 0, similar for C.
The map g k1 ⊗· · ·⊗g kn → g k1+...+kn defined just like M but with the output leaf decorated by N d (rather than π) is identically zero when all internal lines are off-shell. Likewise if one input leaf is decorated by dN . Here N is any momentum conserving operator of degree −1 to guarantee that N d respectively dN have degree zero 27 .
Proof. When d is the output, and all inputs are odd, then this is the lemma in dgLa-based deformation theory that says that obstructions are cocycles [6]. In general, when the output is decorated by N d, the proof is by repeatedly moving occurrences of d down the trees using this algorithm: If d hits a bracket as in d[−, −], use the Leibniz rule.
Terms from ½ are put in a basket, to be dealt with later.
We must show that the terms in the basket add to zero. These terms are one-to-one with T ′ n , the set of trees like T n but with a distinguished internal line. The distinguished line is the one decorated by ½, corresponding to a direct nesting of two brackets as in [[−, −], −]. Define an equivalence relation on T ′ n that identifies trees that differ only by a permutation of the four lines adjacent to the distinguished line. Each equivalence class has three elements as in ½ and so they add to zero by the definition of −, − and the Jacobi identity. The relative signs are due to the permutation sign in Definition 4.2, since A+1, B+1, C + 1 are equal to the number of even elements entering t A , t B , t C respectively, mod 2. No relative sign was produced by algorithm (15), in particular the last application of the Leibniz rule is without sign since ½ is left-adjacent to t R .
Analogous if dN decorates an input leaf, in this case one repeatedly moves occurrences of d away from that input, a modification of (15). Suppose that input is in t A . We get zero again since the relative signs are because the number of even elements entering t A is now A mod 2 due to the presence of the operator N , and the underlined relative signs are introduced by the final application of the Leibniz rule.

YM and GR admit a homological formulation
We define two dgLa whose associated Maurer-Cartan equations are the ordinary nonlinear classical field equations of YM respectively GR about Minkowski spacetime Ê 4 . In particular, for GR the solutions are Ricci-flat metrics.
Logically, this section is organized around an existence theorem that says that for h = 1, 2 there exist dgLa with nontrivial bracket whose homology is globally isomorphic for k ∈ 4 − 0 to that of the complexes Γ −h ⊕ Γ +h 28 , plus some Lorentz invariance and homogeneity. The point is that the amplitudes 29 are the same for all such dgLa, by the unique recursive characterization in Section 10. That is, the amplitudes will be seen to be insensitive to properties of the dgLa other than those listed in the existence theorem below.
This section is in part based on [9,10] for YM and [11] for GR.
Theorem 5.1 (Existence of a dgLa for YM and GR about Minkowski). Let h = 1 or h = 2, that we refer to as YM and GR respectively. For h = 1 suppose we are given a finite-dimensional non-Abelian Lie algebra u. Then there exists a dgLa g with the following properties: • Momentum grading: It has a 4 momentum grading as in Definition 4.1, There is a graded vector space V of finite dimension d V and isomorphisms g k ≃ V such that the differential g k → g k and bracket g k1 ⊗ g k2 → g k1+k2 are given by arrays of size and [k 1 , k 2 ] respectively 30 .
• Global structure of the homology: Recall the definition of the Γ-complexes in (11). There is a collection of isomorphisms, one for every k = 0, 28 Tensored with the internal Lie algebra in the YM case h = 1. 29 Which we recall are the minimal model brackets with all internal lines off-shell. 30 Of course, many entries vanish due to the -grading.
between the homology of g k ≃ V and the homology of regular in the sense that: Every k = 0 has a Zariski open neighborhood on which this isomorphism is induced by a homotopy equivalence (Section 3) given by four matrices whose entries are regular rational functions in k. 31 (17) • Homogeneity and Lorentz equivariance: (16). The minimal model bracket of g, viewed as a map with k 1 , . . . , k n , k 1 + . . . + k n = 0 and assuming all internal lines off-shell, so that it is well-defined by Theorem 4.5, satisfies: • It is homogeneous of degree 3 − 2n 32 .
• For n = 2 it is Lorentz invariant, with u the trivial representation. For YM this n = 2 bracket is proportional to the Lie bracket of u, namely an antisymmetric map times the Lie bracket of u.
Proof. The remainder of this section. We omit straightforward checks that the dgLa constructed below yield minimal model brackets that are homogeneous and, for n = 2, Lorentz invariant and nontrivial as claimed. For Lorentz invariance and homogeneity, note that the Lorentz group and × act as dgLa automorphisms. A sloppy calculation of the degree of homogeneity is (−1) · #inputs + 0 · #nodes + (−1) · #internallines + (+1) · #outputs = 3 − 2n where inputs and output contribute the degree of homogeneity of the connecting morphisms for the short exact sequences (22) respectively (26).
The elements of g may be interpreted as finite linear combinations of plane waves with complex momenta. In fact, in this section we freely transition between the momentum space dgLa g and the position space or C ∞ -variant g ∞ : . This will be a dgLa whose differential and bracket are constant coefficient differential operators. 31 These homotopy equivalences on Zariski open sets need not coincide on overlaps, but they induce the same isomorphism on homology. The matrices are relative to a basis of V ≃ g k .
One determines the other by requiring that the map g → g ∞ given by be a dgLa morphism. Here (x → e ikx ) ∈ C ∞ is a plane wave, kx the componentwise dot product. Note that g ∞ is a free C ∞ -module of rank equal to the dimension of V , but the differential and bracket will not be C ∞ -linear.
Remark 5.2 (On constructing dgLa). A dgca is a differential graded commutative associative algebra over . There are natural products dgca⊗dgca = dgca and dgca ⊗ dgLa = dgLa and if one forgets the algebra structure, they correspond to the standard tensor product of complexes 33 .
Let Ω be the dgca of complex de Rham differential forms on Ê 4 with the de Rham differential. So Ω = C ∞ ⊗ ∧ 4 where the second factor ∧ 4 is the unital gca freely generated in degree one by the symbols dx 0 , dx 1 , dx 2 , dx 3 . As a representation of the Lorentz group, where the arrows indicate the essentially unique differential that depends linearly on k ∈ ( 1 2 , 1 2 ) and is Lorentz equivariant, which is nothing but the de Rham differential in momentum space. It is exact if k = 0. Decompose where Ω 2 ± is the C ∞ -submodule generated by all dx 0 dx a ± idx b dx c with a, b, c a cyclic permutation of 1, 2, 3. That is Set Ω ≤2 ± = Ω 0 ⊕ Ω 1 ⊕ Ω 2 ± and Ω ≥2 ± = Ω 2 ± ⊕ Ω 3 ⊕ Ω 4 . Note that Ω ≥2 ± coincide with the complexes Γ ±1 up to a shift of the homological degree by one. Proposition 5.3 (YM dgLa). Let ⊕ ǫ be the dgca with ǫ a symbol of degree −1, product given by ǫ 2 = 0, and differential z ⊕ wǫ → w ⊕ 0ǫ. Then the tensor product of dgca ( ⊕ ǫ) ⊗ Ω ≃ Ω ⊕ ǫΩ has a dgca subquotient 34 As a C ∞ -module, a ∞ ≃ Ω ≤2 + ⊕ ǫΩ ≥2 + . The tensor product of this dgca with any finite-dimensional complex Lie algebra u yields a dgLa, with u ungraded, The associated MC-equation yields ordinary YM with 'internal Lie algebra' u.
Proof. The dgca axioms hold for ⊕ ǫ. In (21), the numerator is a subcomplex, and a subalgebra using ΩΩ 2 + ⊆ Ω ≥2 + . The denominator is a subcomplex 33 Incidentally if A is a dgca then A ⊗ A can also be given the structure of a dgLa by setting 34 That is, the numerator is a sub-dgca, and the denominator is a dgca ideal in the numerator.
of the numerator, and an algebra ideal using Ω 2 in Ω 3 ⊗ u with d Ω the de Rham differential and the bracket is the product of forms and the bracket in u. These are the YM equations 35 . See also First order Yang-Mills theory in Costello [10].
Using the standard basis for Ω generated by dx 0 , dx 1 , dx 2 , dx 3 and the basis for Ω 2 + given before, we get and with summands in homological degrees 0, 1, 2, 3 respectively. The differential is a constant coefficient first order differential operator, the bracket is bilinear over C ∞ meaning it does not involve derivatives.
Proposition 5.4 (YM homology). Let a = k∈ 4 a k be the algebraic variant of a ∞ . Then for k = 0 there is a global canonical isomorphism between the homology of a k and that of Γ −1 ⊕ Γ 1 . This extends trivially to g = a ⊗ u.
Proof. Abbreviate I ± = Ω ≥2 ± viewed as a complex with the de Rham differential. Let Ω ⊕ ǫI + be the complex with differential a ⊕ ǫb → (da + b) ⊕ (−ǫdb). There is a short exact sequence of complexes The left term is a direct sum of complexes. The middle term has a differential that in 2×2 block form is lower triangular with the lower left term Ω⊕ǫI + → I + , a ⊕ ǫb → b, which yields a differential. The first map in the sequence is the direct sum of the two inclusion maps, hence a chain map. We get a short exact sequence inducing the correct differential on a ∞ ≃ (Ω ⊕ ǫI + )/I − . Passing from C ∞ to the algebraic level, analogous to (18), we get a short exact sequence of complexes for every k ∈ 4 . If k = 0 then the middle term is exact because if we reorganize as (ǫI + ⊕ I + ) ⊕ Ω then the differential is lower block triangular as a 2 × 2 matrix and exact since both diagonal entries are if k = 0, namely ǫI + ⊕ I + is exact since it is the mapping cone for the identity map, and Ω is exact if k = 0. So for k = 0, the associated long exact sequence in homology yields an isomorphism of the homology of a k with the homology of I − ⊕ I + with a degree shift by one, which is Γ −1 ⊕ Γ 1 by (19), (20).
To define the dgLa for GR we use Minkowski spacetime as auxiliary background structure. This account obscures invariance properties, and the more invariant account [11] does not use Ω = C ∞ ⊗ ∧ 4 the way we do here. We define directly the complex version, but there is an obvious real structure at every step so nothing is lost. Consider the direct sum of Lie algebras v = 4 ⊕ so 1,3 which in particular is not the Lie algebra of the Poincare group. We have the following Lie algebra representations: • The Abelian Lie algebra 4 acts on Ω by differentiation on C ∞ by iden- are the usual partial derivatives, and by acting trivially on ∧ 4 .
• The Lie algebra so 1,3 ≃ sl 2 ⊕ sl 2 , which is the complexification of the Lie algebra of the Lorentz group, acts on Ω by acting trivially on C ∞ , and as the fundamental representation on dx 0 , dx 1 , dx 2 , dx 3 which extends uniquely to an action as derivations of degree zero on ∧ 4 .
They combine to an action of v on Ω denoted v → (ω → v(ω)). Then the tensor product of vector spaces Ω ⊗ v is a gLa with grading from Ω and bracket 36 for all ω, ω ′ ∈ Ω and v, v ′ ∈ v. Note that Ω ⊗ v is naturally a module over Ω by left-multiplication, but the bracket is not bilinear over Ω nor even C ∞37 . Elements of degree one sometimes define a metric, as follows.
Definition 5.5 (Associated frame and metric). Given a real element of Ω 1 ⊗ v, drop the so 1,3 part, which leaves something of the form e µ a dx a ⊗ ∂ µ with summation implicit, and e µ a ∈ C ∞ . Consider the real vector fields e a = e µ a ∂ µ . If linearly independent, we get a metric by g(e a , e b ) a,b=0,1,2,3 = diag (−1, 1, 1, 1).
An example is the real element of Ω 1 ⊗v given by (24), which is in MC(Ω⊗v) and whose associated metric is the Minkowski metric. But not every solution to the Einstein equations of GR comes from a real nondegenerate element of MC(Ω ⊗ v), so Ω ⊗ v is not yet the right object.
Proof (Sketch). As representations ∧ 2 4 ≃ so 1,3 ≃ (1, 0) ⊕ (0, 1) so by (10) their product contains (2, 0) ⊕ (0, 2) once, and I is isomorphic to C ∞ times 36 This bracket is well-defined, bilinear, and satisfies the graded Jacobi identity. More generally, if X is a graded commutative algebra, P a Lie algebra and P → Der 0 (X) a Lie algebra map, so a representation as -linear derivations of degree zero, then on X ⊗ P we get a gLa bracket by setting [xp, 37 So the bracket involves differentiation. The language of algebroids captures this structure. 38 Since m has a nondegenerate frame, so do all elements close to m.
Using standard bases, we get with summands in homological degrees 0, 1, 2, 3, 4. The stabilizer Lie algebra of m given by {x ∈ (g ∞ ) 0 | [x, m] = 0} acts as dgLa automorphisms, corresponding to infinitesimal translations and Lorentz transformations. The differential and the bracket are constant coefficient first order differential operators.
Proposition 5.7 (GR homology). Let g = k∈ 4 g k be the algebraic variant of g ∞ . Then for k = 0 there is a global canonical isomorphism between the homology of g k and that of Γ −2 ⊕ Γ 2 .
Proof. Recall that m given by (24) satisfies [m, m] = 0 both in Ω ⊗ v and in g ∞ , and therefore defines a differential on both by [m, −], and by restriction on the ideal I. Therefore we have a short exact sequence of complexes Passing from C ∞ to the algebraic level, see (18), we get a short exact sequence of complexes for every k ∈ 4 . If k = 0 then the middle term is exact because if we reorganize as (Ω ⊗ so 1,3 ) ⊕ (Ω ⊗ 4 ) then the differential is lower block triangular as a 2 × 2 matrix and exact because the diagonal entries are exact, namely the differential on both Ω ⊗ so 1, 3 and Ω ⊗ 4 is simply the de-Rham differential on Ω, and Ω is exact when k = 0. So for k = 0, the associated long exact sequence in homology yields an isomorphism of the homology of g k with the homology of I with a degree shift by one, which is Γ −2 ⊕ Γ 2 by (25).
We now construct the homotopy equivalence required for Theorem 5.1.
Lemma 5.8 (A zig-zag lemma with homotopy equivalence). Given is a short exact sequence of complexes of vector spaces 40 with exact middle term, ℓ ∈ Hom(C, C ′ ) witness the exactness of the short exact sequence, r ′ ℓ ′ + ℓr = rℓ = ℓ ′ r ′ = ½ 41 . Then a homotopy equivalence C ′′ ↔ C, with a degree shift by one, is given by R = rh ′ r ′ ∈ Hom(C ′′ , C), L = ℓ ′ d ′ ℓ ∈ Hom(C, C ′′ ), rh ′ ℓ ∈ End(C), ℓ ′ h ′ r ′ ∈ End(C ′′ ), and L induces the usual connecting morphism. 39 The paper [11] uses a more general variant with v replaced by 4 ⊕ (so 1,3 ⊕ ). This generalization is not needed for this paper, but it can provide flexibility in other applications. 40 Or modules over a unital commutative ring. 41 Note that ℓ, ℓ ′ need not be chain maps. Proof. Apply Lemma 5.8 to the short exact sequences (22) and (26). Here r ′ and r are matrices with constant meaning k-independent complex entries, hence ℓ ′ and ℓ can be taken to be constant matrices. The middle term's differential has polynomial entries, and it is exact at every k = 0. A homotopy exists pointwise since we are over . Pick a homotopy at the given point k = 0. Use the HPL (9) to extend the homotopy to a neighborhood, with rational entries in k. See the proof of Theorem 6.2 for an application of the HPL, with more details.

Proof. For example, Ld
Remark 5.10 (Color-ordered amplitudes). We expect that the A ∞ minimal model for the dgca a gives the so-called color-ordered amplitudes. We have not pursued interesting relations known as Bern-Carrasco-Johansson or BCJ, and Kawai-Lewellen-Tye or KLT relations, in the physics literature.

Optimal homotopies
We construct homotopies for complexes C of vector spaces that depend on a parameter k ∈ m . We assume that the differential d is a matrix with entries in the polynomial ring [k] = [k 1 , . . . , k m ] and that there is homology only along the zero locus of an irreducible polynomial Q ∈ [k]. Under some assumptions, we construct homotopies with entries rational in k that degenerate just as much as they have to along Q = 0, formula (27) below. The construction is an iterated application of the homological perturbation lemma, HPL.
This section applies in particular to the complexes Γ ±h that depend parametrically on the momentum k ∈ 4 for which the homology carrying subvariety Q = 0 is the light cone, see Lemmas 6.6, 6.7. Recall that homotopies play the role of propagators in YM and GR, and they encode gauge choices.
Consider this situation: Start from a smooth point q of the homology carrying variety Q = 0. To construct a homotopy in a neighborhood of q we need an assumption that we paraphrase as the homology disappears to first order transversal to the variety at q. To state this precisely, pick any vector ξ transversal to the variety at q. Differentiate d 2 = 0 to find, with a dot denoting a derivative along ξ, dd + dḋ = 0 2ḋ 2 +dd + dd = 0 The first implies thatḋ induces a linear map on the homology h q at q. The second implies that this induced map is a differential. The assumption will be that h q , as a complex with differential induced byḋ, is exact.
Definition 6.1 (Regular homology point). Let C be a finite-dimensional vector space with a differential d with entries in the polynomial ring [k]. Let h k be the homology at k ∈ m . Suppose Q ∈ [k] is irreducible and that h k = 0 implies Q(k) = 0. We say that q ∈ m is a regular homology point if: • Q(q) = 0 and q is a smooth point of the variety Q = 0.
• dim h k = dim h q for all k with Q(k) = 0 in a Zariski neighborhood of q.
• There exists a ξ ∈ T q m such that, with a dot denoting a derivative along ξ at q, we haveQ = 0 and the differential on h q induced byḋ is exact.
Example. Set m = 1 and consider C : 0 → k a −→ → 0 and Q = k. Then q = 0 is regular homology point if a = 1, but not if a ≥ 2.
Theorem 6.2 (Optimal homotopy). Given a regular homology point q, there exist matrices h, i, p, d Q , h Q with entries in the field of fractions (k) whose denominators do not vanish at q, such that over (k) we have: Furthermore H 2 = 0 and Hd + dH = ½ where One can freely choose h, i, p at q provided they satisfy the first bullet at q. If C is graded, everything can be made compatible with the grading.
In summary, h, i, p is a contraction regular at and near q; the contraction is to h q ; the induced differential is of the form Qd Q with a regular differential d Q that is exact as witnessed by a regular h Q ; and C is exact over (k) as witnessed by a homotopy H that is regular except for an explicit 1/Q. The proof below is in two stages. With reference to (8): • 1st stage. If k is a point close to q, then the differential d is a small perturbation of d q . Therefore starting from a contraction diagram at q, it extends to a Zariski open neighborhood using the HPL: • 2nd stage. The factorization Qd Q is proved using the Nullstellensatz, and it is natural to study d Q . The assumptions imply that d Q is exact at q, hence in a neighborhood by the HPL: Contraction diagrams can be composed, and this gives H.
Proof. Denote by d q the differential at q, a matrix with complex entries. Such a differential always admits a contraction h q , i q , p q with d q h q d q = d q , given by matrices with complex entries. Set δ = d − d q and Since δ has entries in [k] and vanishes at q, Cramer's rule implies that h, i, p are matrices with entries in (k) with denominators that do not vanish at q. The HPL (9) implies that h, i, p is a homotopy of d over (k) but not necessarily dhd = d, though this does hold at q by construction. The homologies of d and pdi coincide at each point. The assumption that k → dim h k be constant along Q = 0 near q implies that pdi vanishes along Q = 0 because the dimension of the homology drops at points where pdi = 0. So Hilbert's Nullstellensatz and the irreducibility of Q imply pdi = Qd Q for some matrix d Q with entries in (k) whose denominators do not vanish at q. Necessarily (d Q ) 2 = 0 and we now show that d Qq is exact. Differentiating pdi = Qd Q and evaluating at q yields using p q d q = d q i q = 0. The differential p qḋ i q is that on h q induced byḋ which was assumed exact, so nowQ = 0 implies that d Qq is exact. Hence it admits a homotopy h Qq with (h Qq ) 2 = 0 and h Qq d Qq + d Qq h Qq = ½. Set By Cramer's rule, h Q has entries in (k) with denominators that do not vanish at q, and the HPL (9) implies (h Q ) 2 = 0 and It is now immediate that it has all the required properties. Lemma 6.3 (Canonical exact differential on homology). The map id Q p induces an exact differential on h q , at all regular homology points q where it is regular. It is equivalently induced byḋ/Q for all transversal ξ ∈ T q m .
Lemma 6.4 (Trivial homotopy away from Q = 0). If Q(q) = 0 so that h q = 0, then there exists a matrix h with entries in (k) whose denominators do not vanish at q, such that over (k) we have h 2 = 0 and dh + hd = ½.
Proof. Again, an application of the HPL. Lemma 6.5 (Regular homology point stable under homotopy equivalence). Suppose C, C ′ are two complexes with homology only along Q = 0 with Q irreducible. Suppose Q(q) = 0 and there exists a homotopy equivalence C ↔ C ′ given by four matrices with entries in (k), regular at q. Then q is a regular homology point of C if and only if q is a regular homology point of C ′ .
Proof. We work with matrices whose entries are in (k) with denominators that do not vanish at q. If q is a regular homology point for C, apply Theorem 6.2 to it. Composition yields a homotopy equivalence (h q , Qd Q ) ↔ (C ′ , d ′ ) by Since R, L are quasi-isomorphisms, the differential thatḋ ′ induces on homology is isomorphic toQd Q hence exact. So q is a regular homology point of C ′ . Lemma 6.7 (Single particle homology for the dgLa). Let h = 1, 2. Take any dgLa as in Theorem 5.1, viewed as a vector space V with differential polynomial in k. Use Γ as a shorthand for (16). Let Q = ad − bc and pick any q = 0 with Q(q) = 0. Then q is a regular homology point of V and Γ, and there exists a diagram of homotopy equivalences, commutative up to homotopy equivalence, by matrices with entries in (k) whose denominators are nonzero at q: Here: • The homotopy equivalence R, L is as in (17).
• The vertical contractions are two separate applications of Theorem 6.2.
• The homotopy equivalence R ′ , L ′ is defined by composition, as indicated.
The matrices R ′ , L ′ are also a homotopy equivalence of the exact complexes Proof. The first part is by Lemmas 6.5, 6.6. Since composing homotopy equivalences yields a homotopy equivalence, Since there is nothing outside degrees 1, 2 by Lemma 6.6, conclude h Γ The homotopy equivalence also yields matrices u, u ′ such that

Part II
This concerns characterizing tree scattering amplitudes without reference to Feynman graphs, in algebro-geometric language.

The variety of kinematically admissible complex momenta
The variety for N = n + 1 momenta is a direct product of N light cones, intersected with a codimension 4 plane that implements momentum conservation. We discuss its geometry emphasizing the Hartogs phenomenon, and list the irreducible codimension one subvarieties (prime divisors) where amplitudes can have residues. By convention, variety does not mean irreducible variety. There is a corresponding map on coordinate rings. Namely the variable replacement rule k → vw T induces an injective map of -algebras Here k is a shorthand for four variables and v, w are shorthands for two variables each. This map embeds a prototypical non-UFD into a UFD. The image consists of precisely the elements that are invariant under the algebra automorphisms v → λv and w → λ −1 w for all λ ∈ × . The next lemma, that we will not actually use, introduces Hartogs extension as a theme.
Counting equations, V (I N ) and V (I ′ N ) ought to have dimensions 3N − 4 and 4N − 4 respectively. To make this rigorous one must show that the defining equations are suitably independent, a consequence of the next lemma. Proof. Sufficient for nonzero r 1 , . . . , r n in a polynomial ring to be a regular sequence 42 is that there exists a monomial order such that the leading monomials of r 1 , . . . , r n are pairwise coprime 43 . For 3 works where, abusing notation, no dash or a dash indicate the first or second component respectively. The leading monomials are For I 3 the given generators are a regular sequence if and only if, after eliminating k 3 using momentum conservation, That is to say, every section of the structure sheaf away from Y extends uniquely to a global section, and is therefore the restriction of a polynomial to X.

Proof.
A complete intersection is S 2 which implies Hartogs. More in detail, a complete intersection, meaning a polynomial ring modulo a regular sequence, is a local complete intersection, meaning all its local rings are complete intersections. Hence it is a Gorenstein ring and a Cohen-Macaulay ring by Theorems 21.3 and 18.1 of [23], and it has the property S k of Serre for all k, see Section 23 of [23]. Equation (31) now follows from Proposition 1.11 of [25]. The elements of O X (X) are well-known to be restrictions of polynomials. 42 In a Noetherian ring R, a sequence r 1 , . . . , rn ∈ R is called a regular sequence iff multiplication by r i+1 is an injective map on R/(r 1 , . . . , r i ) for all i = 0, . . . , n − 1. If R is graded and the r i are homogeneous of positive degree, then the notion of a regular sequence is independent of the ordering, and replacing r i by r i + pr j for some j = i and p ∈ R maps regular sequences to regular sequences, like elementary row operations in linear algebra.
43 Coprimality implies that every subsequence r 1 , . . . , r i is a Gröbner basis for the ideal that it generates, by Buchberger's criterion. Hence the well-ordered sequence of monomials that are not divisible by the leading monomial of any of r 1 , . . . , r i is a -basis of R/(r 1 , . . . , r i ). In this basis, one can see that multiplication by r i+1 is injective from R/(r 1 , . . . , r i ) to itself. 44 Degrevlex, or degree reverse lexicographic, is a well-known admissible monomial order. We will mainly use: If four symbols satisfy a, b, c > d then degrevlex implies ab > cd.  We now decompose these varieties into irreducible components and determine the prime ideal of each component. I + = k 1 + k 2 + k 3 , all maximal minors of the 2 × 6 matrix (k 1 k 2 k 3 ) Proof. The ring [k 1...3 ]/I ± is isomorphic to the coordinate ring of the determinantal variety of 2×4 matrices of rank < 2, hence an integral domain and Cohen-Macaulay. Alternatively, use Macaulay2 [8] to check that I + is the kernel    This is a finite set by a theorem of Noether, and each p has height one by Krull's principal ideal theorem. So P N is a finite set of prime divisors.  p is generated by the maximal minors of p is minimal over If N ≥ 5 then every prime divisor p ∈ P N lies over a unique (Q J ): • If J and J c both have at least three elements, then (Q J ) is itself prime.
• For i = j there are exactly two minimal primes over (Q ij ), namely p + ij and p − ij generated by the maximal minors of (k i k j ) respectively (k T i k T j ).
Proof. For N = 4 one can make a computer check. For N ≥ 5 we write these varieties as fiber products, see Appendix B: Here the first fiber product is defined using and the second is analogous. Irreducibility and primality follow from Lemmas B.2, B.3 given that I |J|+1 , I ± are prime. We have used |J|, |J c | ≥ 3. The V (p ± ij ) have equal codimension by symmetry, and their union is V (Q ij ), hence their codimension is one. Hence p ± ij are minimal primes over (Q ij ) and there are no others. We have listed all p ∈ P N . There are no duplicates since the V (p) are pairwise different. So every p lies over a unique (Q J ).
Definition 7.15 (The P and Z subsets). For N ≥ 4 set which are closed subsets of V (I N ). Let viewed as closed subsets of one of V (I ± ).
Note that P N is equivalently the union of all V (Q J ), but there is no analogous definition of Z N . Note that Z 3 has codimension two. The k i ∈ 4 are pairwise linearly independent.
• The variety V (I N ) is smooth of dimension 3N − 4.
• Q J has nonzero derivative tangent to V (I N ), for all internal J.
In particular V (Q J ) is smooth and has codimension one.
• P N is a smooth codimension one subvariety. If a point in V (p) also lies in V (Q J ), then necessarily p ⊇ (Q J ) and locally V (p) ≃ V (Q J ).
Proof. Every intersection of two distinct irreducible codimension one subvarieties has codimension ≥ 2, so Z N has codimension ≥ 2. Concerning Z c N , several claims follow from V (p + ij ) ∩ V (p − ij ) which contains all points with k i = 0, all points with k {i,j} = k i +k j = 0, and all points where k i and k j are linearly dependent. Points where Lemma 7.6. The tangent derivative of Q J = det k J is nonzero since k J = 0 and the Jacobian of Z c N → 4 , (k 1 , . . . , k N ) → k J has rank four, which one sees using pairwise linear independence of the k i . Every 8 The helicity sheaf for one particle Here we discuss certain rank one sheaves on the cone X = V (ad − bc) and show how they arise as the homology of the complexes Γ ±h .
In this section we denote R = [k]/(ad − bc) and k = ( a b c d ). It is convenient to set k + = k and k − = k T . For every half-integer h ≥ 0 let be the (2h + 1) × (2h + 1) matrix with entries in R where multiplication of an entry by b a = d c gives the next entry to the right, multiplication by c a = d b gives the next entry below. It is the symmetrized Kronecker product of matrices, using S 2h R 2 ≃ R 2h+1 . The matrix S 2h k − is the transpose.
Remark 8.1. The Lorentz group acts as graded ring automorphisms on R. The degree subspaces of R are irreducible, R ≃ h (h, h) where a -basis of (h, h) is given by the entries of (32). There is a category of graded R-modules whose objects M are also Lorentz modules with Lorentz equivariant scalar multiplication R × M → M , and whose morphisms are Lorentz equivariant. In this category, S 2h k + is the unique morphism R ⊗ (0, h) → R ⊗ (h, 0) of degree 2h, up to normalization. Hence its image is also a Lorentz module, and its degree subspaces are seen to be irreducible 47 , im S 2h k + ≃ p (p, p + h).
Lemma 8.2 (Locally free of rank one). For every maximal ideal m corresponding to a point in X − 0, the complement of the origin, there exist column vectors i, r and row vectors ℓ, s with entries in R m such that S 2h k ± = is and ℓi = sr = 1. The sheaf (im S 2h k ± ) | X−0 is locally free of rank one 48 .
See Appendix A for the module to sheaf functor .
Proof. Over R m such a 'split monomorphism • split epimorphism' factorization always exists for k, for example if a / ∈ m then k = ( a c )(1, b/a) and (1/a, 0)( a c ) = (1, b/a)( 1 0 ) = 1. Hence one exists for S 2h k ± . Such a factorization is also valid over R f where f is a common multiple of the denominators. The locally free rank one claim follows from (im S 2h k ± ) f ≃ im((S 2h k ± ) f ) ≃ R f using respectively the exactness of localization and the factorization. Remark 8.3. As rank one vector bundles on X −0, these are subbundles of a trivial bundle with fibers at k = vw T given by v ⊗2h and w ⊗2h respectively. The fiberwise pairing v ⊗2h ⊗ w ⊗2h → 1 is independent of the factorization k = vw T , hence it is globally well-defined. This is the next lemma.

Lemma 8.4 (Inverse sheaf).
There is an isomorphism of sheaves It is induced by taking powers of the identity which holds over R, using the matrix Kronecker product on the left hand side.
Proof. Set K ± = S 2h k ± and L = (2h + 1) 2 . By (33b), K + ⊗ K − = is where i is a column vector and s a row vector with entries in R. Their entries are a -basis of the degree 2h subspace R 2h ⊆ R, they have the same entries as the matrix (32). Since s : R L → R ≥2h is surjective, i : R ≥2h → R L is injective, they induce an isomorphism φ : im(K + ⊗ K − ) → R ≥2h . Define the canonical map ψ : im K + ⊗im K − → im(K + ⊗K − ). Then φ•ψ : im K + ⊗im K − → R ≥2h induces (33a). In fact, every point in X − 0 has an open neighborhood of the standard form D f such that (R ≥2h ) f ≃ R f , and ψ f is an isomorphism using factorizations of K ± over R f as in Lemma 8.2.
Lemma 8.5 (Single particle homology, cf. Lemma 6.6). View Γ ±h as a complex of free R-modules, so the differential d is a matrix with entries in R.
• Local claim: For every m corresponding to a point in X − 0 there exists a contraction of the differential by matrices h, i, p with entries in R m , with dhd = d 49 . We have i = i 1 ⊕ i 2 and p = p 1 ⊕ p 2 where i 1 , i 2 are column vectors and p 1 , p 2 are row vectors. In particular H j (Γ ±h ) | X−0 is locally free of rank one if j = 1, 2 and zero otherwise.
• Global claim: There are isomorphisms of locally free rank one sheaves induced by im S 2h k ± ֒→ S 2h R 2 ≃ Γ 1 ±h resp. the differential in Lemma 6.3.
Proof. The contraction is by Theorem 6.2, available by Lemma 6.6. The contraction is valid over some localization is exact. This implies the local claim. The map im S 2h k + ֒→ Γ 1 h induces a map φ : im S 2h k + → H 1 (Γ h ), by (12) and since k − ǫk + = 0 over R. For all m corresponding to a point in X − 0 the map φ m is an isomorphism. To prove this use a factorization as in Lemma 8.2 and a contraction as in this lemma to get a map R m → R m and check that it is an isomorphism, which comes down to a statement over R m /mR m ≃ namely checking that at every point k = vw T the element v ⊗2h is nonzero in homology. This proves the first isomorphism in (34). For the second, note that id Q p induces an isomorphism H 1 (Γ h ) m → H 2 (Γ h ) m whose inverse is induced by ih Q p. These matrices from Theorem 6.2 have entries in R m . These local isomorphisms are induced by a global isomorphism on X − 0 by Lemma 6.3.

The helicity sheaf
We show that a certain rank one sheaf on V (I N ) satisfies a variant of Hartogs extension if N ≥ 4, while for N = 3 there is local cohomology along Z 3 . These results are used respectively to prove the uniqueness of amplitudes and to define the 2-to-1 amplitude, in the next section.
In this section, R denotes one of R ± if N = 3 respectively R N if N ≥ 4, and X denotes one of X ± if N = 3 respectively X N if N ≥ 4. Here, Note that X is irreducible, R is its coordinate ring and it is Cohen-Macaulay. Set k i = ( ai bi ci di ) and define the matrices S 2h k ± i as in Section 8. • If N ≥ 4, let M σ h be the finitely generated R-module that is the image of the following Kronecker product of matrices: • If N = 3, make the same definition over R ± , denoted M ±,σ h . We often use the shorthand M . Note that the module M inherits a grading from the ambient free module, and as such is generated in degree 2hN . Remark 9.2. Analogous to Remark 8.1, here the Lorentz group acts as automorphisms on R separately on each k i , preserving k 1 + . . . + k N = 0. Hence M is also a Lorentz module.
One always has M (X) ≃ M , see Appendix A. Theorem 9.3 (Hartogs extension for the helicity sheaf). If N ≥ 4 then for all Zariski closed subsets Y ⊆ X of codimension ≥ 2, the restriction map from X to X − Y induces an isomorphism The proof below contains elements of the proof of Lemma 7.1. Lemma 7.11. Let L = (2h + 1) N and let K be the L × L matrix (35). The definition of φ implies a matrix factorization φ(K) = IS where I is a column vector and S is a row vector with entries in R ′ . And I : R ′ → R ′L is injective since R ′ is an integral domain. Note that φ L K = ISφ L hence the injectivity of φ and I yield a well-defined injective φ-linear map 50 α : M → R ′ , Kx → Sφ L x. The image of α is precisely the -subspace of all y ∈ R ′ that for all (λ 1 , . . . , λ N ) ∈ ( × ) N transform like under the algebra automorphism of R ′ given by We have the following commutative diagram, where ρ and ρ ′ are restriction maps, and β = α(X − Y ) is induced by α and is injective 51 : Crucially, ρ ′ is bijective by Proposition 7.5 for X ′ and because Y ′ has codimension ≥ 2 by Lemma 7.7 and N ≥ 4. The diagram implies that ρ is injective. There are obvious ( × ) N -actions on the spaces in the diagram, given on R ′ as above, that make the diagram equivariant. Geometrically, because Y ′ is ( × ) Ninvariant. Hence elements in the image of (ρ ′ ) −1 β transform like (36) and are contained in the image of α. This implies that ρ is also surjective. Lemma 7.7 is not available for N = 3 and in fact then there is a different result as the next lemma shows. It is worked out in detail since it will give the 2-to-1 amplitude for YM and GR. The codimension two subset Z 3 is chosen with this application in mind. There is an analogous proposition for X − .
The module on the left is generated in degree 6h. The module on the right is generated in degree 6h if σ = +++, −−− and in degree 4h if σ = ++−, +−−. As a Lorentz representation, see Remark 9.2, the degree 4h subspace is: Analogous for permutations of σ.
50 By φ-linear we mean α(rm) = φ(r)α(m) for all r ∈ R, m ∈ M . 51 An injective module map induces an injective map on sections since and the sections functor are left-exact. Apply this to α ∈ Hom R (M, R R ′ ) and use Proposition II.5.2 (d) [24]. 52 The cokernel is the 1st local cohomology module [22] of this sheaf along Z 3 .
Proof. This proof follows closely that of Theorem 9.3. Here M = M +,σ h and X = X + and R ′ = [v, w 1 , w 2 , w 3 ]/(w 1 + w 2 + w 3 ) an integral domain and φ : R + → R ′ , k i → vw T i an injective ring map. The map α : M → R ′ , defined analogously to the one in the proof of Theorem 9.3, is injective. The image of α is the -subspace of R ′ spanned by all elements of the schematic form 53 for all n 1 , n 2 , n 3 ∈ ≥0 . Here and has codimension 2. We again have the commutative diagram (37), in particular Hartogs on X ′ ≃ 6 implies that ρ ′ is bijective. The following are equal: 1) The set of all y ∈ R ′ that for λ ∈ × transform like y → λ −2h(σ1+σ2+σ3) y under the algebra automorphism v → λv and w i → λ −1 w i .
is proved using the fact that on X − Z 3 one can locally factor 54 k i = vw T i , and 3) ⊆ 1) by × -equivariance of (37). Using 2), the lowest degree pieces in the image of (ρ ′ ) −1 β are all elements w 6h for +++, w 2h for ++−, v 2h for +−−, v 6h for −−−. For ++− and +−− this is below the lowest degree of the image of α, by 4h in R ′ -degree, or 2h in R + -degree. Given the quotient by w 1 + w 2 + w 3 , the Lorentz modules of all w 2h respectively v 2h are as claimed using the natural Lorentz action on R ′ , since (37) is Lorentz equivariant using the Lorentz module structure on M in Remark 9.2.
Lemma 9.6 (2-to-1 amplitude to-be). If h ∈ ≥0 then M +,++− h (X + − Z 3 ) contains a unique Lorentz invariant element of degree 4h, up to normalization. It is given for any local factorization k i = vw T i with w 1 + w 2 + w 3 = 0 by Analogous for M +,σ if σ has two plus, and for M −,σ if σ has two minus.
Proof. The expression is independent of the local factorization, so we get a section. It is Lorentz invariant. It is unique by Remark 9.5.
53 For example, w 5 1 are all monomials in the two components of w 1 of degree 5. 54 For example, use v T = (a 1 , c 1 ) and w T i = (a i /a 1 , b i /a 1 ) over the ring (R + )a 1 .
The next statement is a strict aside, we will not actually use it. Our use of base change along [k i ]/Q i → R will be straightforward.   • If N = 3, make the same definition over R ± , denoted C ±,σ h . Remark 9.9. As a matrix, the differential d i of Γ i,±h is that of Γ ±h but with entries reinterpreted in R. The differential of Γ 1,±h ⊗ R · · · ⊗ R Γ N −1,±h is given, using the Kronecker product of matrices, by Then chain maps are matrices A with entries in R with Ad tot = d N A, modulo the trivial chain maps of the form A = Bd tot + d N B. The homological grading restricts all matrices to a specific block structure. If f ∈ R, f = 0 then C(D f ) = C f is the corresponding quotient space of matrices with entries in R f . 55 See Proposition II.5.2 (e) [24]. 56 For i = N , use the -basis of R associated to a Gröbner basis of I N with leading monomials Lemma 7.4). The main point is that at the level of leading monomials we have a product situation, all but one only involving k 1 , . . . , k N−1 , the last one only involving k N . 57 Flatness at ring level fails for N = 3 due to relations such as a 3 c 1 − c 3 a 1 in I + . Starting from (a 3 , c 3 ) ֒→ [k 3 ]/Q 3 , if we apply R + ⊗ [k 3 ]/Q 3 − we fail to get an injection.
58 Actually use −k N for i = N . Since Γ h (and Γ −h ) is isomorphic to itself with k replaced by −k simply by sign reversal in homological degree 2, we gloss over this distinction.
• Local claim: For every maximal ideal m corresponding to a point in X −Y , there is over R m a factorization of S 2h k ± i analogous to the one in Lemma 8.2, respectively a contraction of the differential of Γ i,±h analogous to the one in Lemma 8.5. Both are obtained by applying [k i ]/Q i → R. The sheaves C and M are locally free of rank one on X − Y .
• Global claim: There is an isomorphism of locally free rank one sheaves induced by (33a) and (34). For N = 3 there are additional signs. Proof.
Recall that for maps f, f ′ or complexes E, E ′ of finite dimensional vector spaces one has im and H(Hom(E, E ′ )) ≃ Hom(H(E), H(E ′ )). To prove this one can choose factorizations of f, f ′ analogous to Lemma 8.2 and contractions for E, E ′ analogous to Lemma 8.5 and construct explicit factorizations of f ⊗ f ′ and contractions of E ⊗ E ′ and Hom(E, E ′ ). Clearly, the same arguments apply over R m , given factorizations and contractions over R m . In particular C and M are locally free on X − Y . The fact that C has rank one is because homological degree 3 − N implies that the only contribution comes from a product of degree 1 elements giving a degree 2 element. The global claim is now clear given (33a), (34).

The recursive characterization of amplitudes
This section combines all previous sections to construct amplitudes. We introduce the notion of a recursive amplitude, Definition 10.8. Then there is a uniqueness result, the recursive characterization in Theorem 10.11. Finally the existence result, namely a proof that the minimal model brackets for YM and GR are recursive amplitudes, Theorem 10.16.
This section uses notation from all previous sections. Often implicit are an integer N , a tuple of signs σ ∈ {−, +} N , and the helicity h. Often the length N σ = |σ| is understood to determine N . In this section h = 1 or h = 2, though some things immediately generalize to other h. We use these shorthands: We start with a preliminary for the internal Lie algebra for YM.
Definition 10.1 (Internal Lie algebra). For YM, u, is a finite-dimensional non-Abelian Lie algebra together with an invariant nondegenerate symmetricbilinear form u ⊗ u → . It could be the Killing form if u is semisimple.
To keep the notation uniform, we introduce a corresponding but trivial object for GR. Both are denoted u, a finite-dimensional vector space 59 together with a nondegenerate symmetric -bilinear form u ⊗ u → and an element of u ⊗3 : The isomorphism u ≃ u * induced by u ⊗ u → is implicitly used in this section. With this understanding, for YM the element in u ⊗3 is the Lie algebra bracket ∧ 2 u → u and it is totally antisymmetric. For GR, u ⊗3 is totally symmetric.
Introduce the additional notation One must think of each factor u as associated to one momentum, hence one factor in (35), (41). For example, it is implicit that the action of the permutation group S N on sections also permutes the u factors. For C one might want to use u * for the first N − 1 factors, but we are already using the u ≃ u * convention.
As R-modules, M and C are direct sums of finitely many copies of M and C.
See Appendix A for the module to sheaf functor .
Recall that the minimal model brackets are defined when all internal lines are off-shell, which is X −P . But given how 1/Q singularities appear in homotopies, the minimal model brackets will be seen to extend uniquely to sections on X −Z, of a sheaf constructed using the effective divisor D that allows first order poles along the prime divisors in P 60 . That is, they will be seen to be in the image of the injective (see Remark 10.3) restriction map We will directly define and construct amplitudes on the left hand side. This is suitable for the recursive characterization which is all about residues across P .  To define what a recursive amplitude is, we have to define residues and a fusion operation. These are only used away from Z, where the variety is smooth, 59 Viewed as a trivial representation of the Lorentz group. D is smooth, and M is locally free, by Proposition 7.16. So there is no point in being overly technical when defining residues and fusion.
The residue along p ∈ P is the failure of a section of O X (p) to be a section of O X . Defined this way, the residue does not depend on the choice of an equation that locally defines V (p), but it requires introducing another sheaf: This short exact sequence of sheaves defines N p , the normal sheaf along V (p)−Z. It is a locally free sheaf of rank one on V (p) − Z.
Definition 10.4 (Residue). If F is an O X -sheaf, locally free on X − Z, then is defined in the obvious way. The common kernel of all Res p is F (X − Z).
Remark 10.5. Recall that p ++++ is a minimal prime over all three (Q ij ). The relative residues of the Q ij will play an interesting role. They are where k i = vw T i and w 1 + w 2 + w 3 + w 4 = 0 and ǫ ij = w T i ǫw j and ǫ = ( 0 1 −1 0 ) 61 .
Let N ≥ 4, let J be an internal momentum and p ⊇ (Q J ) a prime divisor. Fusion is a bilinear operation that multiplies two amplitudes, for |J| + 1 and 1 + |J c | particles respectively, and produces a new amplitude for N particles that is however only defined along a fiber product. Here are the details: • Geometrically, fusion is based on the isomorphism of varieties The fiber product and its identification with V (p) ⊆ X N are defined by This is for |J|, |J c | ≥ 3, otherwise see Remark 10.7. Both factors restrict the distinguished momentum k = (a, b, c, d) to the cone V (ad − bc) ⊆ 4 .
Given an amplitude on each factor, pull them back along the projection morphisms X |J|+1 ← V (p) → X 1+|J c | and take their tensor product, then reorder the im S 2h k ± i ⊗ u factors. The two superfluous im S 2h k ± ⊗ u associated to the distinguished momentum k, one from each amplitude, are annihilated: • The im S 2h k ± factors are annihilated using (33a). So fusion is only defined if one is im S 2h k + and the other is im S 2h k − . Effectively one cancels Fiberwise v ⊗2h ⊗ w ⊗2h → 1 at k = vw T , in the sense of Remark 8.3.
• The u factors are annihilated using u ⊗ u → . with J and J c sorted in ascending order, σ J = (σ i ) i∈J , σ J c = (σ i ) i / ∈J , ζ = ±.
Since the output is taken in V (p) − Z, and by the definition of Z, it follows immediately that the input only requires sections on X −P . When we use fusion in (47), restriction of the arguments from X − Z to X − P is implicit.
Remark 10.7 (Special rules). If |J| = 2 replace the lower left factor in (44) by one of X ± , if |J c | = 2 replace the upper right factor by one of X ± . Which of X ± is determined by requiring that the result be in V (p). So if J = {1, 2}: • For p = p ++++ use X + × 4 X + .
All other cases are analogous.
Definition 10.8 (Recursive amplitudes). For h = 1 or h = 2, by a recursive amplitude we mean a collection of objects (B σ ), one for every σ ∈ {−, +} Nσ for all N σ ≥ 3, such that: Nonzero are only B +,σ with σ any permutation of ++−, and B −,σ with σ any permutation of −−+. They are up to normalization the elements in Lemma 9.6 times the given element in u ⊗3 in (43). Furthermore is a homogeneous element whose degree is 2h + 2(N σ − 3) lower than the generators of M σ . Furthermore, for all prime divisors p ∈ P: 62 If |J| = 2 or |J c | = 2 one must observe the special rules in Remark 10.7.
• For all N ≥ 3, the collection (B σ ) Nσ=N is S N permutation invariant.
Remark 10.9 (Normalization). If (B σ ) is a recursive amplitude then so is for all λ, µ ∈ × , where N ± σ is the number of plus and minus signs in σ. Up to such transformations, any two recursive amplitudes have identical (B σ ) Nσ =3 .
Proof. By (47) the amplitude B −+++ can have residue only along p ++++ , so it is zero by Lemma 10.12 since its degree is 2h+2 below the generators of M. This implies the first claim for N σ = 4, for all N σ by induction using (47). The second claim is also by (47), since here the sum over J, ζ always degenerates to at most one term and fusion vanishes iff both fusion factors vanish.
In Theorem 10.16 below we construct recursive amplitudes using the minimal model brackets. Hence S N −1 permutation invariance is by construction, but by the next argument this implies S N permutation invariance.
Remark 10.14 (Recursive pre-amplitudes are recursive amplitudes). Define a recursive pre-amplitude just like a recursive amplitude in Definition 10.8, but with S N permutation invariance relaxed to S N −1 permutation invariance in the first N − 1 factors, indicated by a vertical bar below. In particular, (47) remains in force verbatim; the fact that N is always the rightmost element of J c is now critical. For N = 3, permutation invariance can only fail due to a mismatch of normalizations. For the (B +,σ ) Nσ=3 define normalization constants n ± ∈ by where the B on the left are those of a given pre-amplitude, those on the right are S 3 permutation invariant and nonzero and used as a reference. By (46) for the pre-amplitude we have n + = 0. We must show n − = n + . Abbreviate where p = p ++++ . By (47) for the pre-amplitude and Lemma 10.12 we have B −++|+ = 0, hence n 2 + A + n 2 + A ′ + n − n + A ′′ = 0 by (47). On the other hand we have A + A ′ + A ′′ = 0 by direct calculation, using the relative residues in Remark 10.5, the expressions in Lemma 9.6, the given U ∈ u ⊗3 , and: • For YM, the Jacobi identity for the Lie algebra bracket U .
The upcoming theorem finally connects amplitudes and minimal model brackets. It can be viewed as an existence theorem for recursive amplitudes, but beware that if existence is the only goal, there can be more direct constructions, perhaps using [16] or [19], but this is not a direction we pursue.
One can now check all properties of a recursive pre-amplitude. The N = 3 pre-amplitudes are as required, compatible with Lemma 9.6, since the degrees match and by Lorentz invariance in Theorem 5.1. In particular we have (46) by Theorem 5.1. For N ≥ 4, the residue (47) at p is checked at a given maximal m ⊇ p by choosing homotopy data and using: • The structure of optimal homotopies (27) 71,72 .
The sum over ζ = ± in (47) is because Γ is a direct sum of two complexes. Permutation invariance in all but the last factor is by construction.
It would be interesting to study qualitative properties of the amplitudes near Z in codimension ≥ 2. Perhaps local cohomology calculations yield interesting constraints only based on the structure of the sheaves. The case where one k i approaches zero is known as the soft gluon respectively soft graviton limit.
A On the module to sheaf functor Throughout this appendix: R is the coordinate ring of an irreducible affine -variety X. Hence R is a reduced affine -algebra, Noetherian, and an integral domain.
We also have M | D f = M f .
The second half of Lemma A.1 can alternatively be taken to define the tensor product, the Hom , and via exactness the kernel and cokernel and image of a morphism, for such sheaves. These operations are local in the sense that they commute with restriction to D f , say

B Irreducibility of a fiber product
The goal of this appendix is Lemma B.3. We give two proofs, a longer proof via Lemmas B.1 and B.2 that conveys a geometric picture, and a direct proof. Proof. The morphism g is flat since Y is Cohen-Macaulay, m is smooth, and the fibers of g have dimension dim Y − m; see Exercise III.10.9 in [24]. This is sometimes referred to as 'miracle flatness'. The morphism g is locally of finite presentation, because the coordinate ring of Y admits a finite presentation as (via g) a [z 1 , . . . , z m ]-algebra. Flat and locally of finite presentation implies that every base change of g is open, by [21] IV.2, 2.4.6. So base change by f yields an open map X ← X × m Y . By topology, if X ← A is continuous and open, if X is irreducible, and if there is an open dense subset of X such that the corresponding fibers are irreducible, then A is irreducible.
In the following, each k i has four components and each v i and w i has two components. We denote k i = (a i , b i , c i , d i ) and Q i = a i d i − b i c i . Lemma B.2 (Irreducibility). Suppose X is an irreducible affine -variety, and f : X → 4 a morphism of varieties. Let n ≥ 3 be an integer and if n = 3 then demand that f not be identically zero. Consider .n ] be the result of Gröbner reduction applied to p i . Then p i − P i ∈ q, and P i = 0 if x ∈ U i . Expand P i = M P iM M where P iM ∈ [x 1...d ] and M runs over the monomials in k 1...n . Then P 1M1 P 2M2 = 0 for x ∈ U = U 1 ∪ U 2 for all M 1 and M 2 , therefore P 1M1 P 2M2 ∈ p using primality of p, hence P 1M1 ∈ p or P 2M2 ∈ p. If P 1M ∈ p for all M then p 1 ∈ q and we are done. If P 1M / ∈ p for one M then P 2M ∈ p for all M and then p 2 ∈ q.