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 $$\mathcal {M}_{0,n}$$ 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 $$T^*_D\mathcal {M}_{0,n}$$ T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and $$\mathcal {K}_n$$ 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 $$n-3$$ n - 3 -forms on $$\mathcal {K}_n$$ K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral $$n-3$$ n - 3 -planes in $$\mathcal {K}_n$$ K n introduced by ABHY.

Translation-Invariant Excitons in a Phonon Field

Large-radius excitons in polar crystals are considered. It is shown that translation invariant description of excitons interacting with a phonon field leads to a nonzero contribution of phonons into the exciton ground state energy only in the case of weak or intermediate electron-phonon coupling. A conclusion is made that self-trapped excitons cannot exist in the limit of strong coupling. Peculiarities of the absorption and emission spectra of translation invariant excitons in a phonon field are discussed. Conditions when the hydrogen-like exciton model remains valid in the case of electron-phonon interaction are found.

Notes on cluster algebras and some all-loop Feynman integrals

We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $$ {D}_2\simeq {A}_1^2 $$ D 2 ≃ A 1 2 , we show that penta-box ladder has an alphabet of D3 ≃ A3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D4 cluster algebra. We relate the symbol letters to the u variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D5 and D6 cluster functions respectively.

True and Fake Lax Pairs: How to Distinguish Them

The gauge-invariant description of zero-curvature representations of evolution equations is applied to the problem of how to distinguish the fake Lax pairs from the true Lax pairs. The main difference between the true Lax pairs and the fake ones is found in the structure of their cyclic bases.