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An extension of the notion of classical equivalence of equivalence in the Batalin–Vilkovisky (BV) and Batalin–Fradkin–Vilkovisky (BFV) frameworks for local Lagrangian field theory on manifolds possibly with boundary is discussed. Equivalence is phrased in both a strict and a lax sense, distinguished by the compatibility between the BV data for a field theory and its boundary BFV data, necessary for quantisation. In this context, the first- and second-order formulations of nonabelian Yang–Mills and of classical mechanics on curved backgrounds, all of which admit a strict BV–BFV description, are shown to be pairwise equivalent as strict BV–BFV theories. This in particular implies that their BV complexes are quasi-isomorphic. Furthermore, Jacobi theory and one-dimensional gravity coupled with scalar matter are compared as classically equivalent reparametrisation-invariant versions of classical mechanics, but such that only the latter admits a strict BV–BFV formulation. They are shown to be equivalent as lax BV–BFV theories and to have isomorphic BV cohomologies. This shows that strict BV–BFV equivalence is a strictly finer notion of equivalence of theories., Letters in Mathematical Physics, 113 (1), ISSN:0377-9017, ISSN:1573-0530

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases., 44 pages, no figures. The proof of the main result has been simplified and generalized. Furthermore, we included the derivation of the time-dependent relativistic Hartree equation, for all macroscopic times

We consider a system of $N$ bosons, in the two-dimensional unit torus. We assume particles to interact through a repulsive two-body potential, with a scattering length that is exponentially small in $N$ (Gross-Pitaevskii regime). In this setting, we establish the validity of the predictions of Bogoliubov theory, determining the ground state energy of the Hamilton operator and its low-energy excitation spectrum, up to errors that vanish in the limit $N \to \infty$., 51 pages, typos corrected

Recent strides in economic complexity have shown that the future economic development of nations can be predicted with a single "economic fitness" variable, which captures countries' competitiveness in international trade. The predictions by this low-dimensional approach could match or even outperform predictions based on much more sophisticated methods, such as those by the International Monetary Fund (IMF). However, all prior works in economic complexity aimed to quantify countries' fitness from World Trade export data, without considering the possibility to infer countries' potential for growth from alternative sources of data. Here, motivated by the long-standing relationship between technological development and economic growth, we aim to forecast countries' growth from patent data. Specifically, we construct a citation network between countries from the European Patent Office (EPO) dataset. Initial results suggest that the H-index centrality in this network is a potential candidate to gauge national economic performance. To validate this conjecture, we construct a two-dimensional plane defined by the H-index and GDP per capita, and use a forecasting method based on dynamical systems to test the predicting accuracy of the H-index. We find that the predictions based on the H-index-GDP plane outperform the predictions by IMF by approximately 35%, and they marginally outperform those by the economic fitness extracted from trade data. Our results could inspire further attempts to identify predictors of national growth from different sources of data related to scientific and technological innovation.

We prove an upper bound for the ground state energy of a Bose gas consisting of $N$ hard spheres with radius $\mathfrak{a}/N$, moving in the three-dimensional unit torus $\Lambda$. Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit $N \to \infty$. The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose-Einstein condensate and describing correlations on large scales., Comment: Final version, 58 pages

We construct a formal global quantization of the Poisson Sigma Model in the BV-BFV formalism using the perturbative quantization of AKSZ theories on manifolds with boundary and analyze the properties of the boundary BFV operator. Moreover, we consider mixed boundary conditions and show that they lead to quantum anomalies, i.e. to a failure of the (modified differential) Quantum Master Equation. We show that it can be restored by adding boundary terms to the action, at the price of introducing corner terms in the boundary operator. We also show that the quantum GBFV operator on the total space of states is a differential, i.e. squares to zero, which is necessary for a well-defined BV cohomology., Comment: 56 pages, 22 figures, glossary added, presentation improved for non experts. Section 2 is very similar to Section 2 of our companion paper arXiv:1807.11782. Also Appendix A.1-3, on the compactification of configuration spaces of manifolds with boundary, are taken from there but Section A.4, on the case of manifolds with corners, is added. To appear in Commun. Math. Phys

We discuss observables of an equivariant extension of the A-model in the framework of the AKSZ construction. We introduce the A-model observables, a class of observables that are homotopically equivalent to the canonical AKSZ observables but are better behaved in the gauge fixing. We discuss them for two different choices of gauge fixing: the first one is conjectured to compute the correlators of the A-model with target the Marsden-Weinstein reduced space; in the second one we recover the topological Yang-Mills action coupled with A-model so that the A-model observables are closed under supersymmetry., Comment: 16 pages; minor corrections

A construction of gauge-invariant observables is suggested for a class of topological field theories, the AKSZ sigma-models. The observables are associated to extensions of the target Q-manifold of the sigma model to a Q-bundle over it with additional Hamiltonian structure in fibers., Comment: 36 pages

The oriented graph complexes $${GC^{or}_n}$$ are complexes of directed graphs without directed cycles. They govern, for example, the quantization of Lie bialgebras and infinite dimensional deformation quantization. Similar to the ordinary graph complexes GC n introduced by Kontsevich they come in two essentially different versions, depending on the parity of n. It is shown that, surprisingly, the oriented graph complex $${GC^{or}_n}$$ is quasi-isomorphic to the ordinary commutative graph complex of opposite parity GC n-1, up to some known classes. This yields in particular a combinatorial description of the action of $${\mathfrak{grt}_{1} \cong H^0({\rm GC}_2)}$$ on Lie bialgebras, and shows that a cycle-free formality morphism in the sense of Shoikhet can be constructed rationally without reference to configuration space integrals. Curiously, the obstruction class in the oriented graph complex found by Shoikhet corresponds to the well known theta graph in the ordinary graph complex.

We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates, that are consistent with central limit theorems that have been established in the last years., 21 pages, small changes in the introduction

In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size $\varepsilon > 0$, a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least $O(\varepsilon^{-2})$. To the best of our knowledge, this is the first stability result of such type for periodic multi-solitons of large size of an integrable PDE., The title has been changed. Other minor changes in the text

Construct recursively a long string of words w1. .. wn, such that at each step k, w k+1 is a new word with a fixed probability p $\in$ (0, 1), and repeats some preceding word with complementary probability 1 -- p. More precisely, given a repetition occurs, w k+1 repeats the j-th word with probability proportional to j $\alpha$ for j = 1,. .. , k. We show that the proportion of distinct words occurring exactly times converges as the length n of the string goes to infinity to some probability mass function in the variable $\ge$ 1, whose tail decays as a power function when 1 -- p > $\alpha$/(1 + $\alpha$), and exponentially fast when 1 -- p < $\alpha$/(1 + $\alpha$).

General relativity in four dimensions can be reformulated as a gauge theory, referred to as Palatini-Cartan-Holst theory. This paper describes its reduced phase space using a geometric method due to Kijowski and Tulczyjew and its relation to that of the Einstein-Hilbert approach., Comment: Revised version comprising new results, a correction of Th 4.22 and the arguments leading to it. Manuscript accepted for publication in AHP. 31 pages

We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations., 72 pages, improved rate of condensation

We study interacting Bose gases in thermal equilibrium on a lattice. We establish convergence of the grand canonical Gibbs states of such gases to their mean-field (classical field) and large-mass (classical particle) limits. The former is a classical field theory for a complex scalar field with quartic self-interaction. The latter is a classical theory of point particles with two-body interactions. Our analysis is based on representations in terms of ensembles of interacting random loops, the Ginibre loop ensemble for Bose gases and the Symanzik loop ensemble for classical scalar field theories. For small enough interactions, our results also hold in infinite volume., Revised version

Binary random variables are the building blocks used to describe a large variety of systems, from magnetic spins to financial time series and neuron activity. In statistical physics the kinetic Ising model has been introduced to describe the dynamics of the magnetic moments of a spin lattice, while in time series analysis discrete autoregressive processes have been designed to capture the multivariate dependence structure across binary time series. In this article we provide a rigorous proof of the equivalence between the two models in the range of a unique and invertible map unambiguously linking one model parameters set to the other. Our result finds further justification acknowledging that both models provide maximum entropy distributions of binary time series with given means, auto-correlations, and lagged cross-correlations of order one. We further show that the equivalence between the two models permits to exploit the inference methods originally developed for one model in the inference of the other.

We consider systems of bosons trapped in a box, in the Gross-Pitaevskii regime. We show that low-energy states exhibit complete Bose-Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in \cite{BBCS1}, removing the assumption of small interaction potential., 99 pages, typos corrected

These notes give an introduction to the mathematical framework of the Batalin-Vilkovisky and Batalin-Fradkin-Vilkovisky formalisms. Some of the presented content was given as a mini course by the first author at the 2018 QSPACE conference in Benasque., 63 pages, 10 figures, minor changes made (added some references, corrected some typos and some small mistakes)

In this note the AKSZ construction is applied to the BFV description of the reduced phase space of the Einstein–Hilbert and of the Palatini–Cartan theories in every space-time dimension greater than two. In the former case one obtains a BV theory for the first-order formulation of Einstein–Hilbert theory, in the latter a BV theory for Palatini–Cartan theory with a partial implementation of the torsion-free condition already on the space of fields. All theories described here are BV versions of the same classical system on cylinders. The AKSZ implementations we present have the advantage of yielding a compatible BV–BFV description, which is the required starting point for a quantization in presence of a boundary., Communications in Mathematical Physics, 385, ISSN:1432-0916, ISSN:0010-3616

We compute partition functions of Chern-Simons type theories for cylindrical spacetimes $I \times \Sigma$, with $I$ an interval and $\dim \Sigma = 4l+2$, in the BV-BFV formalism (a refinement of the Batalin-Vilkovisky formalism adapted to manifolds with boundary and cutting-gluing). The case $\dim \Sigma = 0$ is considered as a toy example. We show that one can identify - for certain choices of residual fields - the "physical part" (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton-Jacobi action computed in the companion paper [arXiv:2012.13270], without any quantum corrections. This Hamilton-Jacobi action is the action functional of a conformal field theory on $\Sigma$. For $\dim \Sigma = 2$, this implies a version of the CS-WZW correspondence. For $\dim \Sigma = 6$, using a particular polarization on one end of the cylinder, the Chern-Simons partition function is related to Kodaira-Spencer gravity (a.k.a. BCOV theory); this provides a BV-BFV quantum perspective on the semiclassical result by Gerasimov and Shatashvili., Comment: 81 pages, 11 figures. v.2: updated references. v.3: small expository changes

We consider a construction of observables by using methods of supersymmetric field theories. In particular, we give an extension of AKSZ-type observables using the Batalin-Vilkovisky structure of AKSZ theories to a formal global version with methods of formal geometry. We will consider the case where the AKSZ theory is "split" which will give an explicit construction for formal vector fields on base and fiber within the formal global action. Moreover, we consider the example of formal global generalized Wilson surface observables whose expectation values are invariants of higher-dimensional knots by using $BF$ field theory. These constructions give rise to interesting global gauge conditions such as the differential Quantum Master Equation and further extensions., Comment: 46 pages. slightly corrected version. discussion on L-infinity algebras and some references added

While Hartree-Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree-Fock state given by plane waves and introduce collective particle-hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann-Brueckner-type upper bound to the ground state energy. Our result justifies the random phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials., 51 pages, 6 figures; v3: 2 new figures, some heuristic discussion added, typos fixed

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with two non degenerate isomorphic saddle has singular spectrum. More in general, singularity of the spectrum holds for special flows over a full measure set of interval exchange transformations with a hyperelliptic permutation (of any number of exchanged intervals), under a roof with symmetric logarithmic singularities. The result is proved using a criterion for singularity based on tightness of Birkhoff sums with exponential tails decay. A key ingredient in the proof, which is of independent interest, is a result on translation surfaces well approximated by single cylinders. We show that for almost every translation surface in any connected component of any stratum there exists a full measure set of directions which can be well approximated by a single cylinder of area arbitrarily close to one. The result, in the special case of the stratum $\mathcal{H}(1,1)$, yields rigidity sets needed for the singularity result., 32 pages, 7 figures

The triadic description of General Relativity in three dimensions is known to be a BF theory. Diffeomorphisms, as symmetries, are easily re- covered on shell from the symmetries of BF theory. This note describes an explicit off-shell BV symplectomorphism between the BV versions of the two theories, each endowed with their natural symmetries.

This paper introduces a general perturbative quantization scheme for gauge theories on manifolds with boundary, compatible with cutting and gluing, in the cohomological symplectic (BV-BFV) formalism. Explicit examples, like abelian BF theory and its perturbations, including nontopological ones, are presented., Ver. 2: Improved presentation, especially we added a detailed discussion of the space of states. To appear in Commun. Math. Phys

We consider a gas of $N$ bosons in a box with volume one interacting through a two-body potential with scattering length of order $N^{-1}$ (Gross-Pitaevskii limit). Assuming the (unscaled) potential to be sufficiently small, we show that the ground state of the system and all states with relatively small excitation energy exhibit complete Bose-Einstein condensation, with a uniform (i.e. $N$ independent) bound on the number of excitations., Comment: 55 pages. arXiv admin note: text overlap with arXiv:1702.05625

Random walks on binary scale-free networks have been widely studied. However, many networks in real life are weighted and directed, the dynamic processes of which are less understood. In this paper, we firstly present a family of directed weighted hierarchical scale-free networks, which is obtained by introducing a weight parameter [Formula: see text] into the binary (1, 3)-flowers. Besides, each pair of nodes is linked by two edges with opposite direction. Secondly, we deduce the mean first passage time (MFPT) with a given target as a measure of trapping efficiency. The exact expression of the MFPT shows that both its prefactor and its leading behavior are dependent on the weight parameter [Formula: see text]. In more detail, the MFPT can grow sublinearly, linearly and superlinearly with varied [Formula: see text]. Last but not least, we introduce a delay parameter p to modify the transition probability governing random walk. Under this new scenario, we also derive the exact solution of the MFPT with the given target, the result of which illustrates that the delay parameter p can only change the coefficient of the MFPT and leave the leading behavior of MFPT unchanged. Both the analytical solutions of MFPT in two distinct scenarios mentioned above agree well with the corresponding numerical solutions. The analytical results imply that we can get desired transport efficiency by tuning weight parameter [Formula: see text] and delay parameter p. This work may help to advance the understanding of random walks in general directed weighted scale-free networks.

We explain how to perform topological twisting of supersymmetric field theories in the language of factorization algebras. Namely, given a supersymmetric factorization algebra with a choice of a topological supercharge we construct an algebra over the operad of little disks. We also explain the role of the twisting homomorphism allowing us to construct an algebra over the operad of framed little disks. Finally, we give a complete classification of topological supercharges and twisting homomorphisms in dimensions $1$ through $10$., 56 pages

Weighted dependency graphs have been recently introduced by the second author, as a toolbox to prove central limit theorems. In this paper, we prove that spins in the $d$-dimensional Ising model display such a weighted dependency structure. We use this to obtain various central limit theorems for the number of occurrences of local and global patterns in a growing box., Comment: 24 pages, 1 figure, version 2 contains new results

We consider the problem of inferring a causality structure from multiple binary time series by using the Kinetic Ising Model in datasets where a fraction of observations is missing. We take our steps from a recent work on Mean Field methods for the inference of the model with hidden spins and develop a pseudo-Expectation-Maximization algorithm that is able to work even in conditions of severe data sparsity. The methodology relies on the Martin-Siggia-Rose path integral method with second order saddle-point solution to make it possible to calculate the log-likelihood in polynomial time, giving as output a maximum likelihood estimate of the couplings matrix and of the missing observations. We also propose a recursive version of the algorithm, where at every iteration some missing values are substituted by their maximum likelihood estimate, showing that the method can be used together with sparsification schemes like LASSO regularization or decimation. We test the performance of the algorithm on synthetic data and find interesting properties when it comes to the dependency on heterogeneity of the observation frequency of spins and when some of the hypotheses that are necessary to the saddle-point approximation are violated, such as the small couplings limit and the assumption of statistical independence between couplings.

A pivotal idea in network science, marketing research and innovation diffusion theories is that a small group of nodes -- called influencers -- have the largest impact on social contagion and epidemic processes in networks. Despite the long-standing interest in the influencers identification problem in socio-economic and biological networks, there is not yet agreement on which is the best identification strategy. State-of-the-art strategies are typically based either on heuristic centrality metrics or on analytic arguments that only hold for specific network topologies or peculiar dynamical regimes. Here, we leverage the recently introduced random-walk effective distance -- a topological metric that estimates almost perfectly the arrival time of diffusive spreading processes on networks -- to introduce a new centrality metric which quantifies how close a node is to the other nodes. We show that the new centrality metric significantly outperforms state-of-the-art metrics in detecting the influencers for global contagion processes. Our findings reveal the essential role of the network effective distance for the influencers identification and lead us closer to the optimal solution of the problem.

We prove the quantization of the Hall conductivity for general weakly interacting gapped fermionic systems on two-dimensional periodic lattices. The proof is based on fermionic cluster expansion techniques combined with lattice Ward identities, and on a reconstruction theorem that allows us to compute the Kubo conductivity as the analytic continuation of its imaginary time counterpart., Comment: 58 pages, 4 figures. v3: final version, accepted for publication on Comm. Math. Phys

We consider the many body quantum dynamics of systems of bosons interacting through a two-body potential $N^{3\beta-1} V (N^\beta x)$, scaling with the number of particles $N$. For $0< \beta < 1$, we obtain a norm-approximation of the evolution of an appropriate class of data on the Fock space. To this end, we need to correct the evolution of the condensate described by the one-particle nonlinear Schr\"odinger equation by means of a fluctuation dynamics, governed by a quadratic generator., Comment: 78 pages, acknowledgements added

Key traits of unicellular species, like cell size, often follow scale-free or self-similar distributions, hinting at the possibility of an underlying critical process. However, linking such empirical scaling laws to the critical regime of realistic individual-based model classes is difficult. Here we reveal new empirical scaling evidence associated with a transition in the population and chlorophyll dynamics of phytoplankton. We offer a possible explanation for these observations by deriving scaling laws in the vicinity of the critical point of a new universality class of non-local cell growth and division models. This `criticality hypothesis' can be tested through new scaling predictions derived for our model class, for the response of chlorophyll distributions to perturbations. The derived scaling laws may also be generalized to other cellular traits and environmental drivers relevant to phytoplankton ecology., Comment: Companion paper in BioRxiv https://doi.org/10.1101/679209

Efficiently controlling the trapping process is very significant in the study of trapping problem in diverse dynamic processes. In this paper, we explore the trapping efficiency on a family of scale-free tree networks with a deep trap positioned at an initial node, which is controlled by three different strategies. In the first technique, the transition probability is modified by an edge weight parameter. In the second method, the transition probability is controlled by a delay parameter. In the third approach, we use the delay parameter and weight parameter simultaneously to control the trapping process. For all the three control methods, the analytical results of average trapping time (ATT) exactly agree with the numerical results. The result of the first control strategy shows that the average trapping time can scale sublinearly, linearly or superlinearly by modifying the weight parameter. The analytic expression of the ATT in the second method shows that the delay parameter can only modify the main coefficient of ATT, but cannot change the dominant behavior of trapping efficiency. The explicit expression of average trapping time when random walk on scale-free tree network is controlled by the third method shows that it is a fine control. We can get desired trapping efficiency by changing the weight parameter and the delay parameter simultaneously. This work provides a better understanding of controlling the trapping process in a family of scale-free tree networks.

As new instances of nested organization --beyond ecological networks-- are discovered, scholars are debating around the co-existence of two apparently incompatible macroscale architectures: nestedness and modularity. The discussion is far from being solved, mainly for two reasons. First, nestedness and modularity appear to emerge from two contradictory dynamics, cooperation and competition. Second, existing methods to assess the presence of nestedness and modularity are flawed when it comes to the evaluation of concurrently nested and modular structures. In this work, we tackle the latter problem, presenting the concept of \textit{in-block nestedness}, a structural property determining to what extent a network is composed of blocks whose internal connectivity exhibits nestedness. We then put forward a set of optimization methods that allow us to identify such organization successfully, both in synthetic and in a large number of real networks. These findings challenge our understanding of the topology of ecological and social systems, calling for new models to explain how such patterns emerge., 16 pages, 5 figures, 1 Table

Complex networks are often used to represent systems that are not static but grow with time: people make new friendships, new papers are published and refer to the existing ones, and so forth. To assess the statistical significance of measurements made on such networks, we propose a randomization methodology---a time-respecting null model---that preserves both the network's degree sequence and the time evolution of individual nodes' degree values. By preserving the temporal linking patterns of the analyzed system, the proposed model is able to factor out the effect of the system's temporal patterns on its structure. We apply the model to the citation network of Physical Review scholarly papers and the citation network of US movies. The model reveals that the two datasets are strikingly different with respect to their degree-degree correlations, and we discuss the important implications of this finding on the information provided by paradigmatic node centrality metrics such as indegree and Google's PageRank. The randomization methodology proposed here can be used to assess the significance of any structural property in growing networks, which could bring new insights into the problems where null models play a critical role, such as the detection of communities and network motifs., 13 pages, 10 figures

‘Big’ high-dimensional data are commonly analyzed in low-dimensions, after performing a dimensionality reduction step that inherently distorts the data structure. For a similar analysis, clustering methods are also often used. These methods introduce a bias as well, either by starting from the assumption of a particular, often geometric, property of the clusters, or by using iterative schemes to enhance cluster contours, with consequences that are hard to control. The goal of data analysis should, however, be to encode and detect structural data features at all scales and densities simultaneously, without assuming a parametric form of data point distances, or modifying them. Here, we propose a novel approach that directly encodes data point neighborhood similarities as a sparse graph. Our non-iterative framework permits a transparent interpretation of data, without altering the original data dimension and metric. Several natural and synthetic data applications demonstrate the efficacy of our novel method.

We study the exponentiation of elements of the gauge Lie algebras ${\rm hs}(\lambda)$ of three-dimensional higher spin theories. Exponentiable elements generate one-parameter groups of finite higher spin symmetries. We show that elements of ${\rm hs}(\lambda)$ in a dense set are exponentiable, when pictured in certain representations of ${\rm hs}(\lambda)$, induced from representations of $SL(2,\mathbb{R})$ in the complementary series. We also provide a geometric picture of higher spin gauge transformations clarifying the physical origin of these representations. This allows us to construct an infinite-dimensional topological group $HS(\lambda)$ of finite higher spin symmetries. Interestingly, this construction is possible only for $0 \leq \lambda \leq 1$, which are the values for which the higher spin theory is believed to be unitary and for which the Gaberdiel-Gopakumar duality holds. We exponentiate explicitly various commutative subalgebras of ${\rm hs}(\lambda)$. Among those, we identify families of elements of ${\rm hs}(\lambda)$ exponentiating to the unit of $HS(\lambda)$, generalizing the logarithms of the holonomies of BTZ black hole connections. Our techniques are generalizable to the Lie algebras relevant to higher spin theories in dimensions above three., Comment: 34 pages. v3: references added. Added a discussion of the Euclidean higher spin symmetry group. Unlike what was claimed in a previous version, the formalism developed here can be applied to the Euclidean case as well

We develop a proposal by Freed to see anomalous field theories as relative field theories, namely field theories taking value in a field theory in one dimension higher, the anomaly field theory. We show that when the anomaly field theory is extended down to codimension 2, familiar facts about Hamiltonian anomalies can be naturally recovered, such as the fact that the anomalous symmetry group admits only a projective representation on the state space, or that the latter is really an abelian gerbe rather than an ordinary Hilbert space. We include in the discussion the case of non-invertible anomaly field theories, which is relevant to six-dimensional (2,0) superconformal theories. In this case, we show that the Hamiltonian anomaly is characterized by a degree 2 non-abelian group cohomology class, associated to the non-abelian gerbe playing the role of the state space of the anomalous theory. We construct Dai-Freed theories, governing the anomalies of chiral fermionic theories, and Wess-Zumino theories, governing the anomalies of Wess-Zumino terms, as extended field theories down to codimension 2., 50 pages, v7: References added

We derive the relativistic Vlasov equation from quantum Hartree dynamics for fermions with relativistic dispersion in the mean-field scaling, which is naturally linked with an effective semiclassic limit. Similar results in the non-relativistic setting have been recently obtained in [6]. The new challenge that we have to face here, in the relativistic setting, consists in controlling the difference between the quantum kinetic energy and the relativistic transport term appearing in the Vlasov equation., 30 pages

Hierarchical organization is an important, prevalent characteristic of complex systems; in order to understand their organization, the study of the underlying (generally complex) networks that describe the interactions between their constituents plays a central role. Numerous previous works have shown that many real-world networks in social, biologic and technical systems present hierarchical organization, often in the form of a hierarchy of community structures. Many artificial benchmark graphs have been proposed in order to test different community detection methods, but no benchmark has been developed to throughly test the detection of hierarchical community structures. In this study, we fill this vacancy by extending the Lancichinetti-Fortunato-Radicchi (LFR) ensemble of benchmark graphs, adopting the rule of constructing hierarchical networks proposed by Ravasz and Barab\'asi. We employ this benchmark to test three of the most popular community detection algorithms, and quantify their accuracy using the traditional Mutual Information and the recently introduced Hierarchical Mutual Information. The results indicate that the Ravasz-Barab\'asi-Lancichinetti-Fortunato-Radicchi (RB-LFR) benchmark generates a complex hierarchical structure constituting a challenging benchmark for the considered community detection methods., Comment: 9 pages, 9 figures

The relationship between the M-species stochastic Lotka-Volterra competition (SLVC) model and the M-allele Moran model of population genetics is explored via timescale separation arguments. When selection for species is weak and the population size is large but finite, precise conditions are determined for the stochastic dynamics of the SLVC model to be mappable to the neutral Moran model, the Moran model with frequency-independent selection and the Moran model with frequency-dependent selection (equivalently, a game-theoretic formulation of the Moran model). We demonstrate how these mappings can be used to calculate extinction probabilities and the times until a species' extinction in the SLVC model., Comment: 21 pages, 5 figures

We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the Gaussian Unitary Ensemble. In fact, the result is the by-product of a general limit theorem for the convergence of random entire functions whose zeros present a simple regularity property., Comment: 29 pages