arXiv subjects

Hadleigh Frost

Publications and source records attributed to Hadleigh Frost.

14 recordsLinked to original sources

arXiv2026

The Universe Fan

The wavefunction of the universe, as studied in perturbative quantum field theory, is a rational function whose singularities and factorization properties encode a rich underlying combinatorial structure. We define and study a broad generalization of such wavefunctions that can be associated to any lattice. We obtain these wavefunctions as the Laplace transform of a polyhedral fan, the universe fan, whose cones are defined by positivity conditions reflecting a notion of causality in the lattice, and we describe its face lattice. In the matroid case, the universe fan projects to the nested set fan, and the wavefunctions we define recover the matroid amplitudes introduced by Lam as residues. Moreover, in the case relevant for physics, the positivity conditions give a novel way to study the wavefunction, and we show how it is related to the cosmological polytopes of Arkani-Hamed, Benincasa, Postnikov. Finally, we study refinements of the universe fan induced by piecewise linear (tropical) functions. The resulting subdivisions project to refinements of the nested set fan and correspond dually to blow-ups of matroid polytopes, generalizing the cosmohedron polytope.

math.CO↗

arXiv2026

Dyck Paths, Configuration Spaces and Polytopes For Linear Nakayama algebras

We present a combinatorial model of configuration spaces and polytopes associated to the quotients of $\mathbb{C} A_n$, the path algebra of the linearly oriented $A_n$ quiver, i.e. the algebra of upper triangular matrices. These quotient algebras are known as linear Nakayama algebras. Such configuration spaces were recently introduced for more general algebras by the second author and collaborators. In this special setting, we provide elementary proofs and explicit combinatorial constructions. From a Dyck path we define three related objects: a finite-dimensional algebra, an affine algebraic variety, and a polytope. Moreover, our constructions are natural: each relation in the poset of Dyck paths gives a morphism between the corresponding objects.

math.CO↗

arXiv2025

Configuration Spaces of Finite Representation Type Algebras

To every finite-dimensional $\mathbb C$-algebra $\Lambda$ of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by "$u$ variables" satisfying "$u$-equations", first introduced in the context of open string theory and moduli space of ordered points on the real projective line by Koba and Nielsen, rediscovered by Brown as "dihedral co-ordinates", and recently generalized to any finite type hereditary algebras. We show that each such variety is irreducible and admits a rational parametrization. The assignment is functorial: algebra quotients correspond to monomial maps among the varieties. The non-negative real part of each variety has boundary strata that are controlled by Jasso reduction. These non-negative parts naturally define a generalization of open string integrals in physics, exhibiting factorization and splitting properties that do not come from a worldsheet picture. We further establish a family of Rogers dilogarithm identities extending results of Chapoton beyond the Dynkin case.

math.RT↗

arXiv2025

Kinematic Stratifications

We study stratifications of regions in the space of symmetric matrices. Their points are Mandelstam matrices for momentum vectors in particle physics. Kinematic strata in these regions are indexed by signs and rank two matroids. Matroid strata of Lorentzian quadratic forms arise when all signs are non-negative. We characterize the posets of strata, for massless and massive particles, with and without momentum conservation.

math.CO↗

arXiv2025

Deriving motivic coactions and single-valued maps at genus zero from zeta generators

Multiple polylogarithms are equipped with rich algebraic structures including the motivic coaction and the single-valued map which both found fruitful applications in high-energy physics. In recent work arXiv:2312.00697, the current authors presented a conjectural reformulation of the motivic coaction and the single-valued map via zeta generators, certain operations on non-commuting variables in suitable generating series of multiple polylogarithms. In this work, the conjectures of the reference will be proven for multiple polylogarithms that depend on any number of variables on the Riemann sphere.

hep-th↗

arXiv2024

The Cut Equation

Scattering amplitudes for colored theories have recently been formulated in a new way, in terms of curves on surfaces. In this note we describe a canonical set of functions we call surface functions, associated to all orders in the topological expansion, that are naturally suggested by this point of view. Surface functions are generating functions for all inequivalent triangulations of the surface. They generalize matrix model correlators, and in the planar limit, coincide with field theoretic loop integrands. We show that surface functions satisfy a universal recursion relation, the cut equation, that can be solved without introducing spurious poles, to all orders in the genus expansion. The formalism naturally extends to include triangulations with closed curves, corresponding to theories with uncolored particles. This new recursion is quite different from the topological recursion relations satisfied by matrix models. Applied to field theory, the new recursion efficiently computes all-order planar integrands for general colored theories, together with uncolored theories at tree-level. As an example we give the all-order recursion for the planar NLSM integrand. We attach a Mathematica notebook for the efficient computation of these planar integrands, with illustrative examples through four loops.

hep-th↗

arXiv2024

Tropical Amplitudes For Colored Lagrangians

Recently a new formulation for scattering amplitudes in Tr($\Phi^3$) theory has been given based on simple combinatorial ideas in the space of kinematic data. This allows all-loop integrated amplitudes to be expressed as ''curve integrals'' defined using tropical building blocks - the ''headlight functions''. This paper shows how the formulation extends to the amplitudes of more general Lagrangians. We will present a number of different ways of introducing tropical ''numerator functions'' that allow us to describe general Lagrangian interactions. The simplest family of these ''tropical numerators'' computes the amplitudes of interesting Lagrangians with infinitely many interactions. We also describe methods for tropically formulating the amplitudes for general Lagrangians. One uses a variant of ''Wick contraction'' to glue together numerator factors for general interaction vertices. Another uses a natural characterization of polygons on surfaces to give a novel combinatorial description of all possible diagrams associated with arbitrary valence interactions.

hep-th↗

arXiv2023

Amplitudes at strong coupling as hyperkähler scalars

Alday & Maldacena conjectured an equivalence between string amplitudes in AdS$_5 \times S^5$ and null polygonal Wilson loops in planar $\mathcal{N}=4$ super-Yang-Mills (SYM). At strong coupling this identifies SYM amplitudes with areas of minimal surfaces in AdS. For minimal surfaces in AdS$_3$, we find that the nontrivial part of these amplitudes, the \emph{remainder function}, satisfies an integrable system of nonlinear differential equations, and we give its Lax form. The result follows from a new perspective on `Y-systems', which defines a new psuedo-hyperkähler structure \emph{directly} on the space of kinematic data, via a natural twistor space defined by the Y-system equations. The remainder function is the (pseudo-)Kähler scalar for this geometry. This connection to pseudo-hyperkähler geometry and its twistor theory provides a new ingredient for extending recent conjectures for non-perturbative amplitudes using structures arising at strong coupling.

hep-th↗

arXiv2023

Motivic coaction and single-valued map of polylogarithms from zeta generators

We introduce a new Lie-algebraic approach to explicitly construct the motivic coaction and single-valued map of multiple polylogarithms in any number of variables. In both cases, the appearance of multiple zeta values is controlled by conjugating generating series of polylogarithms with Lie-algebra generators associated with odd zeta values. Our reformulation of earlier constructions of coactions and single-valued polylogarithms preserves choices of fibration bases, exposes the correlation between multiple zeta values of different depths and paves the way for generalizations beyond genus zero.

hep-th↗

arXiv2023

All Loop Scattering For All Multiplicity

This is part of a series of papers describing the new curve integral formalism for scattering amplitudes of the colored scalar tr$\phi^3$ theory. We show that the curve integral manifests a very surprising fact about these amplitudes: the dependence on the number of particles, $n$, and the loop order, $L$, is effectively decoupled. We derive the curve integrals at tree-level for all $n$. We then show that, for higher loop-order, it suffices to study the curve integrals for $L$-loop tadpole-like amplitudes, which have just one particle per color trace-factor. By combining these tadpole-like formulas with the the tree-level result, we find formulas for the all $n$ amplitudes at $L$ loops. We illustrate this result by giving explicit curve integrals for all the amplitudes in the theory, including the non-planar amplitudes, through to two loops, for all $n$.

hep-th↗

arXiv2021

The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes

We study the Kawai-Lewellen-Tye (KLT) relations for quantum field theory by reformulating it as an isomorphism between two Lie algebras. We also show how explicit formulas for KLT relations arise when studying rational functions on ${\mathcal M}_{0,n}$, and prove identities that allow for arbitrary rational functions to be expanded in any given basis. Via the Cachazo-He-Yuan formulas for, these identities also lead to new formulas for gauge and gravity tree amplitudes, including formulas for so-called Bern-Carrasco-Johansson numerators, in the case of non-linear sigma model and maximal-helicity-violating Yang-Mills amplitudes.

hep-th↗

arXiv2020

A Lie bracket for the momentum kernel

We develop new mathematical tools for the study of the double copy and colour-kinematics duality for tree-level scattering amplitudes using the properties of Lie polynomials. We show that the $S$-map that was defined to simplify super-Yang--Mills multiparticle superfields is in fact a new Lie bracket on the dual space of Lie polynomials. We introduce {\it Lie polynomial currents} based on Berends-Giele recursion for biadjoint scalar tree amplitudes that take values in Lie polynomials. Field theory amplitudes are obtained from the Lie polynomial amplitudes by numerators characterized as homomorphisms from the free Lie algebra to kinematic data. Examples are presented for the biadjoint scalar, Yang--Mills theory and the nonlinear sigma model. That these theories satisfy the Bern-Carrasco-Johansson amplitude relations follows from the identities we prove for the Lie polynomial amplitudes and the existence of BCJ numerators. A KLT map from Lie polynomials to their dual is obtained by nesting the S-map Lie bracket; the matrix elements of this map yield a recently proposed `generalized KLT matrix', and this reduces to the usual KLT matrix when its entries are restricted to a basis. Using this, we give an algebraic proof for the cancellation of double poles in the KLT formula for gravity amplitudes. We finish with some remarks on numerators and colour-kinematics duality from this perspective.

hep-th↗

arXiv2020

Lie Polynomials and a Twistorial Correspondence for Amplitudes

We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of $\mathcal{M}_{0,n}$, the moduli space of $n$ points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle $T^*_D\mathcal{M}_{0,n}$, the bundle of forms with logarithmic singularities on the divisor $D$ as the twistor space, and $\mathcal{K}_n$ the space of momentum invariants of $n$ massless particles subject to momentum conservation as the analogue of space-time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain $n-3$-forms on $\mathcal{K}_n$, introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral $n-3$-planes in $\mathcal{K}_n$ introduced by ABHY.}

hep-th↗

arXiv2018

Biadjoint scalar tree amplitudes and intersecting dual associahedra

We present a new formula for the biadjoint scalar tree amplitudes $m(α|β)$ based on the combinatorics of dual associahedra. Our construction makes essential use of the cones in 'kinematic space' introduced by Arkani-Hamed, Bai, He, and Yan. We then consider dual associahedra in 'dual kinematic space.' If appropriately embedded, the intersections of these dual associahedra encode the amplitudes $m(α|β)$. In fact, we encode all the partial amplitudes at $n$-points using a single object (a fan) in dual kinematic space. Equivalently, as a pleasant corollary of our construction, all $n$-point partial amplitudes can be understood as coming from integrals over subvarieties in a single toric variety. Explicit formulas for the amplitudes then follow by evaluating these integrals using the equivariant localisation (or Duistermaat-Heckman) formula. Finally, by introducing a lattice in kinematic space, we observe that our fan is also related to the inverse KLT kernel, sometimes denoted $m_{α'}(α|β)$.

hep-th↗