Coaction and double-copy properties of configuration-space integrals at genus zero

Abstract

We investigate configuration-space integrals over punctured Riemann spheres from the viewpoint of the motivic Galois coaction and double-copy structures generalizing the Kawai-Lewellen-Tye (KLT) relations in string theory. For this purpose, explicit bases of twisted cycles and cocycles are worked out whose orthonormality simplifies the coaction. We present methods to efficiently perform and organize the expansions of configuration-space integrals in the inverse string tension α′ or the dimensional-regularization parameter ϵ of Feynman integrals. Generating-function techniques open up a new perspective on the coaction of multiple polylogarithms in any number of variables and analytic continuations in the unintegrated punctures. We present a compact recursion for a generalized KLT kernel and discuss its origin from intersection numbers of Stasheff polytopes and its implications for correlation functions of two-dimensional conformal field theories. We find a non-trivial example of correlation functions in ($$ \mathfrak{p} $$ p , 2) minimal models, which can be normalized to become uniformly transcendental in the $$ \mathfrak{p} $$ p → ∞ limit.

Open-string integrals with multiple unintegrated punctures at genus one

“…Using the same change of variables, the remaining differential form in the integral Z τ n,p 18 The factors of (−2πizm) s ij...r cancel the nonanalytic behavior of limz m →0 e −s ij...r Γ 1 0…”

“…These methods have been used with great success for tree-level and one-loop string integrals, as can be seen in refs. [9][10][11][12][13][14][15][16][17][18].…”

“…In doing this, we are also generalizing some of the results of ref. [18] into a genus-one setting. In this last reference, the authors studied open-and closed-string integrals with multiple unintegrated punctures and wrote them in terms of generating functions of multiple polylogarithms.…”

KLT factorization of winding string amplitudes

“…10 There is also a different representation of KLT relations as noted in [1], which can be obtained by evaluating the integrals in (4.11) directly without performing any further deformation of the integration contours. This representation has a simple combinatorial interpretation [20]. In the winding case, this representation has the benefit of making the resulting KLT relation manifestly symmetric with respect to the subscripts "L" and "R", However, this KLT which is exactly the expression in (4.27) but with the subscripts "L" and "R" swapped.…”

Co-homology of Differential Forms and Feynman diagrams

“…The geometric origin of the analytic properties of Feynman integrals finds its roots in the application of topology to the S-matrix theory [50,51,52]. In more recent studies, co-homology played an important role for identifying relations among Feynman integrals and to expose deeper properties of scattering amplitudes [39,41,53,54,55,56,45,46,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73] In this editorial, we elaborate on the recently understood vector space structure of Feynman integrals [59,61,60,62,63,64,65,66] and the role played by the intersection theory for twisted de Rham (co)-homology to access it.…”