Local charges in involution and hierarchies in integrable sigma-models

Integrable σ-models, such as the principal chiral model, ℤT-coset models for T∈ℤ≥2 and their various integrable deformations, are examples of non-ultralocal integrable field theories described by (cyclotomic) r/s-systems with twist function. In this general setting, and when the Lie algebra 픤 underlying the r/s-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space σ-model. In the present context, the local charges are attached to certain `regular' zeros of the twist function and have increasing degrees related to the exponents of the untwisted affine Kac-Moody algebra 픤ˆ associated with 픤. The Hamiltonian flows of these charges are shown to generate an infinite hierarchy of compatible integrable equations

Affine Gaudin models and hypergeometric functions on affine opers

We conjecture that quantum Gaudin models in affine types admit families of higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further conjecture that the Hamiltonians themselves are naturally expressed as such integrals. We go on to describe the space of meromorphic affine opers on an arbitrary Riemann surface. We prove that it fibres over the space of meromorphic connections on the canonical line bundle Ω. Each fibre is isomorphic to the direct product of the space of sections of the square of Ω with the direct product, over the exponents j not equal to 1, of the twisted cohomology of the jth tensor power of Ω

The Magic Renormalisability of Affine Gaudin Models

We study the renormalisation of a large class of integrable $\sigma$-models obtained in the framework of affine Gaudin models. They are characterised by a simple Lie algebra $\mathfrak{g}$ and a rational twist function $\varphi(z)$ with simple zeros, a double pole at infinity but otherwise no further restrictions on the pole structure. The crucial tool used in our analysis is the interpretation of these integrable theories as $\mathcal{E}$-models, which are $\sigma$-models studied in the context of Poisson-Lie T-duality and which are known to be at least one- and two-loop renormalisable. The moduli space of $\mathcal{E}$-models still contains many non-integrable theories. We identify the submanifold formed by affine Gaudin models and relate its tangent space to curious matrices and semi-magic squares. In particular, these results provide a criteria for the stability of these integrable models under the RG-flow. At one loop, we show that this criteria is satisfied and derive a very simple expression for the RG-flow of the twist function, proving a conjecture made earlier in the literature.Comment: 29 pages, 1 figur

The erratic mitochondrial clock: variations of mutation rate, not population size, affect mtDNA diversity across birds and mammals

<p>Abstract</p> <p>Background</p> <p>During the last ten years, major advances have been made in characterizing and understanding the evolution of mitochondrial DNA, the most popular marker of molecular biodiversity. Several important results were recently reported using mammals as model organisms, including (i) the absence of relationship between mitochondrial DNA diversity and life-history or ecological variables, (ii) the absence of prominent adaptive selection, contrary to what was found in invertebrates, and (iii) the unexpectedly large variation in neutral substitution rate among lineages, revealing a possible link with species maximal longevity. We propose to challenge these results thanks to the bird/mammal comparison. Direct estimates of population size are available in birds, and this group presents striking life-history trait differences with mammals (higher mass-specific metabolic rate and longevity). These properties make birds the ideal model to directly test for population size effects, and to discriminate between competing hypotheses about the causes of substitution rate variation.</p> <p>Results</p> <p>A phylogenetic analysis of cytochrome <it>b </it>third-codon position confirms that the mitochondrial DNA mutation rate is quite variable in birds, passerines being the fastest evolving order. On average, mitochondrial DNA evolves slower in birds than in mammals of similar body size. This result is in agreement with the longevity hypothesis, and contradicts the hypothesis of a metabolic rate-dependent mutation rate. Birds show no footprint of adaptive selection on cytochrome <it>b </it>evolutionary patterns, but no link between direct estimates of population size and cytochrome <it>b </it>diversity. The mutation rate is the best predictor we have of within-species mitochondrial diversity in birds. It partly explains the differences in mitochondrial DNA diversity patterns observed between mammals and birds, previously interpreted as reflecting Hill-Robertson interferences with the W chromosome.</p> <p>Conclusion</p> <p>Mitochondrial DNA diversity patterns in birds are strongly influenced by the wide, unexpected variation of mutation rate across species. From a fundamental point of view, these results are strongly consistent with a relationship between species maximal longevity and mitochondrial mutation rate, in agreement with the mitochondrial theory of ageing. Form an applied point of view, this study reinforces and extends the message of caution previously expressed for mammals: mitochondrial data tell nothing about species population sizes, and strongly depart the molecular clock assumption.</p

Testing whether major innovation capabilities are systemic design capabilities: analyzing rule-renewal design capabilities in a case-control study of historical new business developments

International audienceIn this paper, we empirically test the proposition that major innovation (MI) capabilities are systemic, dynamic capabilities. We rely on design theories and characterize the systemic, dynamic capabilities as design capabilities that renew a core of stabilized design rules. For the specific case of projects leading to new business development, we conducted a case-control study of 46 historical projects; 26 of these led to new business development, and 20 do not lead to new business development. Utilizing this sample, we show that our measurement model, based on rule-reuse vs. rule-renewal design capabilities, has a good fit. We find that rule-renewal design capabilities are positively related to new business development, whereas rule-reuse design capabilities (maintaining an invariant set of design rules) are independent of new business development. We discuss different combinations of rule-reuse and rule-renewal design capabilities. This paper contributes to the literature on MI capabilities. It also theoretically and methodologically contributes to the analysis of the dynamic capabilities of design activitie

Using design theory to characterize various forms of breakthrough R&D projects and their management: revisiting Manhattan & Polaris.

In this paper we propose to revisit two emblematic projects, Manhattan and Polaris, with the models developed by design theory. In particular we demonstrate, relying on recent advances in design theory, how these major projects, traditionally presented as radical innovations, are in fact quite different. We show that this explains the different managerial strategies of this two cases : whereas Polaris focuses on the control of the design process, Manhattan exhibit a very original strategy, characterized by the simultaneous exploration of different solutions, to manage unforeseeable uncertainties. We therefore hope to demonstrate the fruitfulness of the dialogue between design theory and project management

Single-photon entanglement generation by wavefront shaping in a multiple-scattering medium

We demonstrate the control of entanglement of a single photon between several spatial modes propagating through a strongly scattering medium. Measurement of the scattering matrix allows the wavefront of the photon to be shaped to compensate the distortions induced by multiple scattering events. The photon can thus be directed coherently to a single or multi-mode output. Using this approach we show how entanglement across different modes can be manipulated despite the enormous wavefront disturbance caused by the scattering medium.Comment: 4 pages, 3 figures, reference adde

Cubic hypergeometric integrals of motion in affine Gaudin models

© 2020 International Press of Boston, Inc. This is the accepted manuscript version of an article which has been published in final form at https://dx.doi.org/10.4310/ATMP.2020.v24.n1.a5.We construct cubic Hamiltonians for quantum Gaudin models of affine types $\hat{\mathfrak{sl}}_M$. They are given by hypergeometric integrals of a form we recently conjectured in arXiv:1804.01480. We prove that they commute amongst themselves and with the quadratic Hamiltonians. We prove that their vacuum eigenvalues, and their eigenvalues for one Bethe root, are given by certain hypergeometric functions on a space of affine opers.Peer reviewedFinal Accepted Versio