arXiv subjects

Giulio Salvatori

Publications and source records attributed to Giulio Salvatori.

13 recordsLinked to original sources

arXiv2026

Configuration Spaces of Finite Representation Type Algebras

To every finite-dimensional $\mathbb C$-algebra $Λ$ 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↗

arXiv2026

SubTropica

We present SubTropica, a Mathematica package that performs symbolic integration of multi-polylogarithmic integrals using recent advances in tropical geometry. It focuses on the class of linearly-reducible Euler integrals, such as Feynman integrals, and expands them using a tropical subtraction scheme. The engine behind it is HyperIntica, a native Mathematica package for hyperlogarithm integration that can be used independently. This paper documents both packages and illustrates their usage on examples from across different physics applications. Additionally, we introduce an AI-driven library of Feynman integrals, which catalogs diagrams discussed in the literature and serves as a database for computed results. Its online version is available at: https://subtropi.ca and features a graphical user interface for diagram input and retrieval of records.

hep-th↗

arXiv2026

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↗

arXiv2025

All Loop Scattering As A Sampling Problem

How to turn the flip of a coin into a random variable whose expected value equals a scattering amplitude? We answer this question by constructing a numerical algorithm to evaluate curve integrals - a novel formulation of scattering amplitudes - by a Monte Carlo strategy. To achieve a satisfactory accuracy we take advantage of tropical importance sampling. The crucial result is that the sampling procedure can be realized as a stochastic process on surfaces which can be simulated efficiently on a computer. The key insight is to let go of the Feynman-bias that amplitudes should be presented as a sum over diagrams, and instead re-arrange the sum as suggested by a dual triangulation of curve integrals. We attach an implementation of this algorithm as an ancillary file, which we have used to evaluate amplitudes for the massive $\mathrm{Tr}(ϕ)^3$ theory in $D=2$ space-time dimensions, up to 10-loops. Interestingly, we observe experimentally that the number of sample points required to achieve a fixed accuracy remains significantly smaller than what the number of diagrams would suggest. Finally we propose an extension of our method which is inspired by ideas from artificial intelligence. We use the stochastic process to define a parametrization for a space of distributions, where we formulate importance sampling for an arbitrary curve integrand as a convex optimization problem.

hep-th↗

arXiv2024

The Tropical Geometry of Subtraction Schemes

We study the construction of local subtraction schemes through the lenses of tropical geometry. We focus on individual Feynman integrals in parametric presentation, and think of them as particular instances of Euler integrals. We provide a necessary and sufficient condition for a combination of Euler integrands to be locally finite, i.e. to be expandable as a Taylor series in the exponent variables directly under sign of integration. We use this to construct a local subtraction scheme that is applicable to a class of Euler integrals that satisfy a certain geometric property. We apply this to compute the Laurent expansion in the dimensional regulator $ε$ of various Feynman integrals involving both UV and IR singularities, as well as to generalizations of Feynman integrals that arise in effective field theories and in phase-space integrations, for which we provide new analytic results.

hep-th↗

arXiv2024

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$ϕ^3$ 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↗

arXiv2024

Tropical Amplitudes For Colored Lagrangians

Recently a new formulation for scattering amplitudes in Tr($Φ^3$) 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↗

arXiv2022

Causal Diamonds, Cluster Polytopes and Scattering Amplitudes

The "amplituhedron" for tree-level scattering amplitudes in the bi-adjoint $ϕ^3$ theory is given by the ABHY associahedron in kinematic space, which has been generalized to give a realization for all finite-type cluster algebra polytopes, labelled by Dynkin diagrams. In this letter we identify a simple physical origin for these polytopes, associated with an interesting (1+1)-dimensional causal structure in kinematic space, along with solutions to the wave equation in this kinematic "spacetime" with a natural positivity property. The notion of time evolution in this kinematic spacetime can be abstracted away to a certain "walk", associated with any acyclic quiver, remarkably yielding a finite cluster polytope for the case of Dynkin quivers. The ${\cal A}_{n{-}3},{\cal B}_{n{-}1}/{\cal C}_{n{-}1}$ and ${\cal D}_n$ polytopes are the amplituhedra for $n$-point tree amplitudes, one-loop tadpole diagrams, and full integrand of one-loop amplitudes. We also introduce a polytope $\bar{\cal D}_n$, which chops the ${\cal D}_n$ polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.

hep-th↗

arXiv2020

Scattering Amplitudes and Simple Canonical Forms for Simple Polytopes

We provide an efficient recursive formula to compute the canonical forms of arbitrary $d$-dimensional simple polytopes, which are convex polytopes such that every vertex lies precisely on $d$ facets. For illustration purposes, we explicitly derive recursive formulae for the canonical forms of Stokes polytopes, which play a similar role for a theory with quartic interaction as the Associahedron does in planar bi-adjoint $ϕ^3$ theory. As a by-product, our formula also suggests a new way to obtain the full planar amplitude in $ϕ^4$ theory by taking suitable limits of the canonical forms of constituent Stokes polytopes.

hep-th↗

arXiv2020

The complete set of two-loop master integrals for Higgs + jet production in QCD

In this paper we complete the computation of the two-loop master integrals relevant for Higgs plus one jet production initiated in arXiv:1609.06685, arXiv:1907.13156, arXiv:1907.13234. We compute the integrals by defining differential equations along contours in the kinematic space, and by solving them in terms of one-dimensional generalized power series. This method allows for the efficient evaluation of the integrals in all kinematic regions, with high numerical precision. We show the generality of our approach by considering both the top- and the bottom-quark contributions. This work along with arXiv:1609.06685, arXiv:1907.13156, arXiv:1907.13234 provides the full set of master integrals relevant for the NLO corrections to Higgs plus one jet production, and for the real-virtual contributions to the NNLO corrections to inclusive Higgs production in QCD in the full theory.

hep-ph↗

arXiv2019

1-loop Amplitudes from the Halohedron

We recently proposed the Halohedron to be the 1-loop Amplituhedron for planar $ϕ^3$ theory. Here we prove this claim by showing how it is possible to extract the integrand for the partial amplitude $m^1_n(1,\dots,n|1,\dots,n)$ from the canonical form of an Halohedron which lives in an abstract space. This space is just a step away from ordinary kinematical space at 1-loop, because it is composed by abstract variables associated to propagators of 1-loop Feynman diagrams. Such variables, however, are unbound from momentum conservation relations that would give problems such as double poles. As an application of our construction, we exploit a well known recursion formula for the canonical form of a polytope in order to produce an expression for the 1-loop integrand which would not be evident starting from Feynman diagrams.

hep-th↗

arXiv2018

Hyperbolic Geometry and Amplituhedra in 1+2 dimensions

Recently, the existence of an Amplituhedron for tree level amplitudes in the bi-adjoint scalar field theory has been proved by Arkhani-Hamed et al. We argue that hyperbolic geometry constitutes a natural framework to address the study of positive geometries in moduli spaces of Riemann surfaces, and thus to try to extend this achievement beyond tree level. In this paper we begin an exploration of these ideas starting from the simplest example of hyperbolic geometry, the hyperbolic plane. The hyperboloid model naturally guides us to re-discover the moduli space Associahedron, and a new version of its kinematical avatar. As a by-product we obtain a solution to the scattering equations which can be interpreted as a special case of the two well known solutions in terms of spinor-helicity formalism. The construction is done in $1+2$ dimensions and this makes harder to understand how to extract the amplitude from the dlog of the space time Associahedron. Nevertheless, we continue the investigation accommodating a loop momentum in the picture. By doing this we are led to another polytope called Halohedron, which was already known to mathematicians. We argue that the Halohedron fulfils many criteria that make it plausible to be understood as a 1-loop Amplituhedron for the cubic theory. Furthermore, the hyperboloid model again allows to understand that a kinematical version of the Halohedron exists and is related to the one living in moduli space by a simple generalisation of the tree level map.

hep-th↗

arXiv2015

Quantum metrology in Lipkin-Meshkov-Glick critical systems

The Lipkin-Meshkov-Glick (LMG) model describes critical systems with interaction beyond the first-neighbor approximation. Here we address the characterization of LMG systems, i.e. the estimation of anisotropy, and show how criticality may be exploited to improve precision. In particular, we provide exact results for the Quantum Fisher Information of small-size LMG chains made of $N=2, 3$ and $4$ lattice sites and analyze the same quantity in the thermodynamical limit by means of a zero-th order approximation of the system Hamiltonian. We then show that the ultimate bounds to precision may be achieved by tuning the external field and by measuring the total magnetization of the system. We also address the use of LMG systems as quantum thermometers and show that: i) precision is governed by the gap between the lowest energy levels of the systems, ii) field-dependent level crossing provides a resource to extend the operating range of the quantum thermometer.

quant-ph↗