Amplitude recursions with an extra marked point

The recursive calculation of Selberg integrals by Aomoto and Terasoma using the Knizhnik-Zamolodchikov equation and the Drinfeld associator makes use of an auxiliary point and facilitates the recursive evaluation of string amplitudes at genus zero: open-string N-point amplitudes can be obtained from those at N-1 points. We establish a similar formalism at genus one, which allows the recursive calculation of genus-one Selberg integrals using an extra marked point in a differential equation of Knizhnik-Zamolodchikov-Bernard type. Hereby genus-one Selberg integrals are related to genus-zero Selberg integrals. Accordingly, N-point open-string amplitudes at genus one can be obtained from (N+2)-point open-string amplitudes at tree level. The construction is related to and in accordance with various recent results in intersection theory and string theory

Relations between elliptic multiple zeta values and a special derivation algebra

We investigate relations between elliptic multiple zeta values and describe a method to derive the number of indecomposable elements of given weight and length. Our method is based on representing elliptic multiple zeta values as iterated integrals over Eisenstein series and exploiting the connection with a special derivation algebra. Its commutator relations give rise to constraints on the iterated integrals over Eisenstein series relevant for elliptic multiple zeta values and thereby allow to count the indecomposable representatives. Conversely, the above connection suggests apparently new relations in the derivation algebra. Under https://tools.aei.mpg.de/emzv we provide relations for elliptic multiple zeta values over a wide range of weights and lengths.Comment: 43 pages, v2:replaced with published versio

Engineering orthogonal dual transcription factors for multi-input synthetic promoters

Synthetic biology has seen an explosive growth in the capability of engineering artificial gene circuits from transcription factors (TFs), particularly in bacteria. However, most artificial networks still employ the same core set of TFs (for example LacI, TetR and cI). The TFs mostly function via repression and it is difficult to integrate multiple inputs in promoter logic. Here we present to our knowledge the first set of dual activator-repressor switches for orthogonal logic gates, based on bacteriophage λ cI variants and multi-input promoter architectures. Our toolkit contains 12 TFs, flexibly operating as activators, repressors, dual activator–repressors or dual repressor–repressors, on up to 270 synthetic promoters. To engineer non cross-reacting cI variants, we design a new M13 phagemid-based system for the directed evolution of biomolecules. Because cI is used in so many synthetic biology projects, the new set of variants will easily slot into the existing projects of other groups, greatly expanding current engineering capacities

Deformed one-loop amplitudes in N = 4 super-Yang-Mills theory

We investigate Yangian-invariant deformations of one-loop amplitudes in N = 4 super-Yang-Mills theory employing an algebraic representation of amplitudes. In this language, we reproduce the deformed massless box integral describing the deformed four-point one-loop amplitude and compare different realizations of said amplitude.Comment: 19 page

Polylogarithms, Multiple Zeta Values and Superstring Amplitudes

A formalism is provided to calculate tree amplitudes in open superstring theory for any multiplicity at any order in the inverse string tension. We point out that the underlying world-sheet disk integrals share substantial properties with color-ordered tree amplitudes in Yang-Mills field theories. In particular, we closely relate world-sheet integrands of open-string tree amplitudes to the Kawai-Lewellen-Tye representation of supergravity amplitudes. This correspondence helps to reduce the singular parts of world-sheet disk integrals -including their string corrections- to lower-point results. The remaining regular parts are systematically addressed by polylogarithm manipulations.Comment: 79 pages, LaTeX; v2: final version to appear in Fortschritte der Physik; for additional material, see: http://mzv.mpp.mpg.d

Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral

We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure math- ematics and string theory. We then focus on the equal-mass and non-equal-mass sunrise integrals, and we develop a formalism that enables us to compute these Feynman integrals in terms of our iterated integrals on elliptic curves. The key idea is to use integration-by-parts identities to identify a set of integral kernels, whose precise form is determined by the branch points of the integral in question. These kernels allow us to express all iterated integrals on an elliptic curve in terms of them. The flexibility of our approach leads us to expect that it will be applicable to a large variety of integrals in high-energy physics.Comment: 22 page

Towards single-valued polylogarithms in two variables for the seven-point remainder function in multi-Regge-kinematics

We investigate single-valued polylogarithms in two complex variables, which are relevant for the seven-point remainder function in N=4 super-Yang-Mills theory in the multi-Regge regime. After constructing these two-dimensional polylogarithms, we determine the leading logarithmic approximation of the seven-point remainder function up to and including five loops.Comment: 20 pages, 2 figures; v2: replaced with published versio

A dictionary between R-operators, on-shell graphs and Yangian algebras

We translate between different formulations of Yangian invariants relevant for the computation of tree-level scattering amplitudes in N=4 super-Yang--Mills theory. While the R-operator formulation allows to relate scattering amplitudes to structures well known from integrability, it can equally well be connected to the permutations encoded by on-shell graphs.Comment: 44 pages; replaced with published versio

All order alpha'-expansion of superstring trees from the Drinfeld associator

We derive a recursive formula for the alpha'-expansion of superstring tree amplitudes involving any number N of massless open string states. String corrections to Yang-Mills field theory are shown to enter through the Drinfeld associator, a generating series for multiple zeta values. Our results apply for any number of spacetime dimensions or supersymmetries and chosen helicity configurations.Comment: 6 pages, LaTeX; v2: Final version to appear in PR