Higher Gauge Theory and Discrete Geometry

In four dimensions, gravity can be seen as a constrained topological model. This provides a natural way to construct quantum gravity models, since topological models are relatively straightforward to quantize. Difficulty arises in the implementation of the constraints at the quantum level. Different procedures have generated so-called spin foam models. Following the dimensional/categorical ladder, the natural structure to quantize 4d topological models are 2-categories, augmenting the gauge group symmetries of the model into 2-group symmetries. One can study these models classically by examining their phase space. At the quantum level, one attempts to construct a partition function. As there are no local degrees of freedom in topological theories, it is convenient to characterize its phase space in terms of a discretization, providing insights to the quantum theory. A key question is understanding how these topological models defined in terms of 2-categories can be related to gravity. A first hint that 2-categories are relevant to describe quantum gravity models comes when we introduce a cosmological constant in the theory. As we will recall, this can be done at the classical (discrete) level in a consistent manner, only if we use 2-group symmetries. This thesis focuses on understanding the symmetry aspects, the different possible discretizations and the quantization of four dimensional topological theories for some skeletal 2-group symmetries. When we discuss the quantum aspects using some field theory techniques to generate the quantum amplitudes, we extend the construction to non-skeletal 2-groups

Reweighting and the complex Langevin Algorithm

The complex Langevin algorithm was developed to evade the sign problem by extending the degrees of freedom into a complex space. The validity of the results depends on the fall off of the resulting probability distribution. To explore the falloff, we attempted to reweight the probability distribution by introducing an acceptance probability at each step in the algorithm, and by direct manipulation of the Fokker-Planck equation. The acceptance probability did not lead to the target reweighted distribution because the complex Langevin algorithm does not follow detailed balance. Manipulating the Fokker-Planck equation gave a differential equation for the reweighted distribution, but we could not find a way to simulate it.L'algorithme complexe de Langevin a été développé pour échapper la problme de la signe en étendant les degrés de liberté dans un espace complexe. La validité des résultats de l'algorithme dépend de la décroissance de la distribution de probabilité résultante. Pour analyser cette décroissance, nous avons tenté de modifier la distribution de probabilité en introduisant une probabilité d'approbation à chaque étape de l'algorithme et en manipulant directement l'équation de Fokker-Planck. La probabilité d'approbation n'a pas reproduit la distribution desirée puisque l'algorithme Langevin complexe ne suit pas le principe la balance détaillé. La manipulation de l'équation de Fokker-Planck a donnée une équation differentielle pour la distribution modifier, mais nous n'avons pas trouvé un moyen de la simuler

Diffeomorphisms as quadratic charges in 4d BF theory and related TQFTs

International audienceWe present a Sugawara-type construction for boundary charges in 4d BF theory and in a general family of related TQFTs. Starting from the underlying current Lie algebra of boundary symmetries, this gives rise to well-defined quadratic charges forming an algebra of vector fields. In the case of 3d BF theory (i.e. 3d gravity), it was shown in [PRD 106 (2022), arXiv:2012.05263 [hep-th]] that this construction leads to a two-dimensional family of diffeomorphism charges which satisfy a certain modular duality. Here we show that adapting this construction to 4d BF theory first requires to split the underlying gauge algebra. Surprisingly, the space of well-defined quadratic generators can then be shown to be once again two-dimensional. In the case of tangential vector fields, this canonically endows 4d BF theory with a $\mathrm{diff}(S^2)\times\mathrm{diff}(S^2)$ or $\mathrm{diff}(S^2)\ltimes\mathrm{vect}(S^2)_\mathrm{ab}$ algebra of boundary symmetries depending on the gauge algebra. The prospect is to then understand how this can be reduced to a gravitational symmetry algebra by imposing Plebański simplicity constraints

Simulated evolutionary pathway (label <i>ftz</i> on Fig 2A) from <i>Drosophila</i> to <i>Anopheles</i>, including <i>ftz</i>.

<p>Conventions of <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g003" target="_blank">Fig 3</a> for <i>eve</i> and <i>ftz</i> stripes are used.</p

Summary of our predictions.

<p>A Predicted evolutionary pathways from different simulations detailed in <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g003" target="_blank">Fig 3</a> (label F1), <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g004" target="_blank">Fig 4</a> (label F2), <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g005" target="_blank">Fig 5</a> (label Ftz), <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g007" target="_blank">Fig 7</a> (label LC). The times shown for the intermediates are only schematic. Gap and <i>eve</i> patterns in three insect species and the inferred last common ancestor (LCA) are indicated. B Summary of homology between Eve modules in different species predicted by our evolutionary simulations.</p

Simulated evolutionary pathway (label F1 on Fig 2A) from <i>Drosophila</i> to <i>Anopheles</i>, with salient changes discussed in the main text.

<p>For each transcriptional <i>eve</i> module only the gap genes that regulate it are shown with the same color scheme as Figs <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g001" target="_blank">1</a> and <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g002" target="_blank">2A</a>. <i>eve</i> stripe 1 is not shown and the maximum expression of each module is normalized to 1 except when it dips beneath a threshold equivalent to its loss.</p

Predicting Ancestral Segmentation Phenotypes from Drosophila to Anopheles Using <i>In Silico</i> Evolution

<div><p>Molecular evolution is an established technique for inferring gene homology but regulatory DNA turns over so rapidly that inference of ancestral networks is often impossible. <i>In silico</i> evolution is used to compute the most parsimonious path in regulatory space for anterior-posterior patterning linking two Dipterian species. The expression pattern of gap genes has evolved between <i>Drosophila</i> (fly) and <i>Anopheles</i> (mosquito), yet one of their targets, <i>eve</i>, has remained invariant. Our model predicts that stripe 5 in fly disappears and a new posterior stripe is created in mosquito, thus <i>eve</i> stripe modules 3+7 and 4+6 in fly are homologous to 3+6 and 4+5 in mosquito. We can place <i>Clogmia</i> on this evolutionary pathway and it shares the mosquito homologies. To account for the evolution of the other pair-rule genes in the posterior we have to assume that the ancestral Dipterian utilized a dynamic method to phase those genes in relation to <i>eve</i>.</p></div

A model for the LCA that imparts stable phase relations among the pair-rule genes and remains consistent with the evolutionary pathway from fly to mosquito.

<p>(A) Schematic of the model showing gap input to eve only. The intensity of repression among the remaining genes (chosen arbitrarily as the primary pair-rule genes in fly) is shown by the line intensity and defines their relative phase. (B) Behavior of the model in A in response to imposed temporal oscillations of Eve, showing phase relationships between different pair-rule genes. Viewed within a cell, one cycle of temporal oscillations would result from the forward shift of the entire Eve pattern by one period. (C,D) If we implement a forward shift of <i>eve</i> by one stripe (left to right panels), by suitably scaling the maternal gradients, then an arbitrary initial arrangement of the three remaining genes is reset to the proper phasing for fly. We show the gap gene configuration for fly in (C) and for mosquito in (D). <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.s003" target="_blank">S3</a> and <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.s004" target="_blank">S4</a> Videos show the evolution from the left to right panels respectively for panels (C) and (D). For simplicity only, the model is applied across the entire embryo, though in reality the anterior gap gene input to the primary pair rule genes can remain invariant.</p

Simplified model of <i>Drosophila</i> network.

<p>A-B: simulated maternal and gap gene profiles. C: simulated pair-rule gene profiles. A’-C’ Summary of interactions used to generate these profiles. Equations and references for the interactions are given in the <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.s005" target="_blank">S1 Text</a>. A generic spatially uniform activator is assumed where needed.</p