Intersection theory rules symbology

This work introduces an explicit expression for the generating function for the reduction of an n-gon to an (n – k)-gon. A novel recursive relation of generating function is formulated based on Feynman Parametrization in projective space, involving a single ordinary differential equation. The explicit formulation of generating functions provides crucial insights into the complex analytic structure inherent in loop amplitudes.

Section: Jhep02(2024)158mentioning

confidence: 99%

An explicit expression of generating function for one-loop tensor reduction

Hu,

Li,

Shen

et al. 2024

“…We know where the intersection number localizes and don't have to go through the process of constructing IBP-forms. Before moving on to actually integrating our differential equation, we note that the ideas outlined here are applicable to the algorithm for computing the symbol in [69]. By promoting the external derivative to a derivative on the total space, the two-step algorithm for computing the intersection number could potentially be reduced to one-step!…”

Section: Jhep03(2024)156mentioning

confidence: 99%

Cosmology meets cohomology

De,

Pokraka

2024

The cosmological polytope and bootstrap programs have revealed interesting connections between positive geometries, modern on-shell methods and bootstrap principles studied in the amplitudes community with the wavefunction of the Universe in toy models of FRW cosmologies. To compute these FRW correlators, one often faces integrals that are too difficult to evaluate by direct integration. Borrowing from the Feynman integral community, the method of (canonical) differential equations provides an efficient alternative for evaluating these integrals. Moreover, we further develop our geometric understanding of these integrals by describing the associated relative twisted cohomology. Leveraging recent progress in our understanding of relative twisted cohomology in the Feynman integral community, we give an algorithm to predict the basis size and simplify the computation of the differential equations satisfied by FRW correlators.

A double copy from twisted (co)homology at genus one

Bhardwaj,

Pokraka,

Ren

et al. 2024

We study the twisted (co)homology of a family of genus-one integrals — the so called Riemann-Wirtinger integrals. These integrals are closely related to one-loop string amplitudes in chiral splitting where one leaves the loop-momentum, modulus and all but one puncture un-integrated. While not actual one-loop string integrals, they share many properties and are simple enough that the associated twisted (co)homologies have been completely characterized [1]. Using intersection numbers — an inner product on the vector space of allowed differential forms — we derive the Gauss-Manin connection for two bases of the twisted cohomology providing an independent check of [2]. We also use the intersection index — an inner product on the vector space of allowed contours — to derive a double-copy formula for the closed-string analogues of Riemann-Wirtinger integrals (one-dimensional integrals over the torus). Similar to the celebrated KLT formula between open- and closed-string tree-level amplitudes, these intersection indices form a genus-one KLT-like kernel defining bilinears in meromorphic Riemann-Wirtinger integrals that are equal to their complex counterparts.