Matteo Smerlak

Bubble Divergences from Cellular Cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang–Mills theory, the Ponzano–Regge model of 3d quantum gravity and discrete BF theory, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined in general, because of a phenomenon called ‘bubble divergences’. A common expectation is that the degree of these divergences is given by the number of ‘bubbles’ of the 2-complex. In this note, we show that this expectation, although not realistic in general, is met in some special cases: when the 2-complex is simply connected, or when the structure group is Abelian – in both cases, the divergence degree is given by the second Betti number of the 2-complex.

The predator-prey power law: Biomass scaling across terrestrial and aquatic biomes

A general scaling law for ecology Despite the huge diversity of ecological communities, they can have unexpected patterns in common. Hatton et al. describe a general scaling law that relates total predator and prey biomass in terrestrial and aquatic animal communities (see the Perspective by Cebrian). They draw on data from many thousands of population counts of animal communities ranging from plankton to large mammals, across a wide range of biomes. They find a ubiquitous pattern of biomass scaling, which may suggest an underlying organization in ecosystems. It seems that communities follow systematic changes in structure and dynamics across environmental gradients. Science, this issue 10.1126/science.aac6284; see also p. 1053 The number and size of predators and their prey scale across a broad range of terrestrial and aquatic animal communities. [Also see Perspective by Cebrian] INTRODUCTION A surprisingly general pattern at very large scales casts light on the link between ecosystem structure and function. We show a robust scaling law that emerges uniquely at the level of whole ecosystems and is conserved across terrestrial and aquatic biomes worldwide. This pattern describes the changing structure and productivity of the predator-prey biomass pyramid, which represents the biomass of communities at different levels of the food chain. Scaling exponents of the relation between predator versus prey biomass and community production versus biomass are often near ¾, which indicates that very different communities of species exhibit similar high-level structure and function. This recurrent community growth pattern is remarkably similar to individual growth patterns and may hint at a basic process that reemerges across levels of organization. RATIONALE We assembled a global data set for community biomass and production across 2260 large mammal, invertebrate, plant, and plankton communities. These data reveal two ecosystem-level power law scaling relations: (i) predator biomass versus prey biomass, which indicates how the biomass pyramid changes shape, and (ii) community production versus community biomass, which indicates how per capita productivity changes at a given level in the pyramid. Both relations span a wide range of ecosystems along large-scale biomass gradients. These relations can be linked theoretically to show how pyramid shape depends on flux rates into and out of predator-prey communities. In order to link community-level patterns to individual processes, we examined community size structure and, particularly, how the mean body mass of a community relates to its biomass. RESULTS Across ecosystems globally, pyramid structure becomes consistently more bottom-heavy, and per capita production declines with increasing biomass. These two ecosystem-level patterns both follow power laws with near ¾ exponents and are shown to be robust to different methods and assumptions. These structural and functional relations are linked theoretically, suggesting that a common community-growth pattern influences predator-prey interactions and underpins pyramid shape. Several of these patterns are highly regular (R2 > 0.80) and yet are unexpected from classic theories or from empirical relations at the population or individual level. By examining community size structure, we show these patterns emerge distinctly at the ecosystem level and independently from individual near ¾ body-mass allometries. CONCLUSION Systematic changes in biomass and production across trophic communities link fundamental aspects of ecosystem structure and function. The striking similarities that are observed across different kinds of systems imply a process that does not depend on system details. The regularity of many of these relations allows large-scale predictions and suggests high-level organization. This community-level growth pattern suggests a systematic form of density-dependent growth and is intriguing given the parallels it exhibits to growth scaling at the individual level, both of which independently follow near ¾ exponents. Although we can make ecosystem-level predictions from individual-level data, we have yet to fully understand this similarity, which may offer insight into growth processes in physiology and ecology across the tree of life. African large-mammal communities are highly structured. In lush savanna, there are three times more prey per predator than in dry desert, a pattern that is unexpected and systematic. [Photo: Amaury Laporte] Ecosystems exhibit surprising regularities in structure and function across terrestrial and aquatic biomes worldwide. We assembled a global data set for 2260 communities of large mammals, invertebrates, plants, and plankton. We find that predator and prey biomass follow a general scaling law with exponents consistently near ¾. This pervasive pattern implies that the structure of the biomass pyramid becomes increasingly bottom-heavy at higher biomass. Similar exponents are obtained for community production-biomass relations, suggesting conserved links between ecosystem structure and function. These exponents are similar to many body mass allometries, and yet ecosystem scaling emerges independently from individual-level scaling, which is not fully understood. These patterns suggest a greater degree of ecosystem-level organization than previously recognized and a more predictive approach to ecological theory.

Bubble Divergences: Sorting out Topology from Cell Structure

We conclude our analysis of bubble divergences in the flat spinfoam model. In Bonzom and Smerlak, Comm. Math. Phys., (submitted), we showed that the divergence degree of an arbitrary 2-complex Γ can be evaluated exactly by means of twisted cohomology. Here, we specialize this result to the case where Γ is the 2-skeleton of the cell decomposition of a pseudomanifold, and sharpen it with a careful analysis of the cellular and topological structures involved. Moreover, we explain in detail how this approach reproduces all the previous powercounting results for the Boulatov–Ooguri (colored) tensor models, and sheds light on algebraic-topological aspects of Gurau’s 1/N expansion.

Bubble Divergences from Twisted Cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory and 3d Riemannian quantum gravity, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined because of a phenomenon known as ‘bubble divergences’. In this paper, we extend recent results of the authors to the cases where these divergences cannot be understood in terms of cellular cohomology. We introduce in its place the relevant twisted cohomology, and use it to compute the divergence degree of the partition function. We also relate its dominant part to the Reidemeister torsion of the complex, thereby generalizing previous results of Barrett and Naish-Guzman. The main limitation to our approach is the presence of singularities in the representation variety of the fundamental group of the complex; we illustrate this issue in the well-known case of two-dimensional manifolds.

Towards the graviton from spinfoams: The complete perturbative expansion of the 3d toy model

Abstract We consider an exact expression for the 6j-symbol for the isosceles tetrahedron, involving SU ( 2 ) group integrals, and use it to write the two-point function of 3d gravity on a single tetrahedron as a group integral. The perturbative expansion of this expression can then be performed with respect to the geometry of the boundary using a simple saddle-point analysis. We derive the complete expansion in inverse powers of the length scale and evaluate explicitly the quantum corrections up to second order. Finally, we use the same method to provide the complete expansion of the isosceles 6j-symbol with the explicit phases at all orders and the next-to-leading correction to the Ponzano–Regge asymptotics.

In quantum gravity, summing is refining

In perturbative QED, the approximation is improved by summing more Feynman graphs, while in non-perturbative QCD, by refining the lattice. Here we observe that in quantum gravity, the two procedures may well be the same. We outline the combinatorial structure of spinfoam quantum gravity, define the continuum limit and show that under general conditions, refining foams is the same as summing over them. The conditions bear on the cylindrical consistency of the spinfoam amplitudes and on the presence of appropriate combinatorial factors, related to the implementation of diffeomorphism invariance. Intuitively, the sites of the lattice are points of space: these are themselves quanta of the gravitational field, and thus a lattice discretization is also a Feynman history of quanta.

Melonic Phase Transition in Group Field Theory

Group field theories have recently been shown to admit a 1/N expansion dominated by so-called ‘melonic graphs’, dual to triangulated spheres. In this note, we deepen the analysis of this melonic sector. We obtain a combinatorial formula for the melonic amplitudes in terms of a graph polynomial related to a higher-dimensional generalization of the Kirchhoff tree-matrix theorem. Simple bounds on these amplitudes show the existence of a phase transition driven by melonic interaction processes. We restrict our study to the Boulatov–Ooguri models, which describe topological BF theories and are the basis for the construction of 4-dimensional models of quantum gravity.

Entanglement entropy and negative energy in two dimensions

It is well known that quantum effects can produce negative energy densities, though for limited times. Here we show in the context of two-dimensional CFT that such negative energy densities are present in any non-trivial conformal vacuum and can be interpreted in terms of the entanglement structure of such states. We derive an exact identity relating the outgoing energy flux and the entanglement entropy in the in-vacuum. When applied to two-dimensional models of black hole evaporation, this identity implies that unitarity is incompatible with monotonic mass loss.

New perspectives on Hawking radiation

We develop an adiabatic formalism to study the Hawking phenomenon from the perspective of Unruh-DeWitt detectors moving along non-stationary, non-asymptotic trajectories. When applied to geodesic trajectories, this formalism yields surprising results: (i) though they have zero acceleration, the temperature measured by detectors on circular orbits is higher than that measured by static detectors at the same distance from the hole, and diverges on the photon sphere, (ii) in the near-horizon region, both outgoing and incoming modes excite infalling detectors, and, for highly bound trajectories (E ≪ 1), the latter actually dominate the former, (iii) in this region, the relationship between the temperature of Hawking radiation and the relative velocity between the detector and the hole is not of Doppler type. We confirm the apparent perception of high-temperature ingoing Hawking radiation by infalling observers with E ≪ 1 by a flux computation. We close by a discussion of the role played by spacetime curvature on the near-horizon Hawking radiation.

Entanglement entropy production in gravitational collapse: covariant regularization and solvable models

A bstractWe study the dynamics of vacuum entanglement in the process of gravitational collapse and subsequent black hole evaporation. In the first part of the paper, we introduce a covariant regularization of entanglement entropy tailored to curved spacetimes; this regularization allows us to propose precise definitions for the concepts of black hole “exterior entropy” and “radiation entropy.” For a Vaidya model of collapse we find results consistent with the standard thermodynamic properties of Hawking radiation. In the second part of the paper, we compute the vacuum entanglement entropy of various spherically-symmetric spacetimes of interest, including the nonsingular black hole model of Bardeen, Hayward, Frolov and Rovelli-Vidotto and the “black hole fireworks” model of Haggard-Rovelli. We discuss specifically the role of event and trapping horizons in connection with the behavior of the radiation entropy at future null infinity. We observe in particular that (i) in the presence of an event horizon the radiation entropy diverges at the end of the evaporation process, (ii) in models of nonsingular evaporation (with a trapped region but no event horizon) the generalized second law holds only at early times and is violated in the “purifying” phase, (iii) at late times the radiation entropy can become negative (i.e. the radiation can be less correlated than the vacuum) before going back to zero leading to an up-down-up behavior for the Page curve of a unitarily evaporating black hole.