(3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces

We apply the recently suggested strategy to lift state spaces and operators for (2+1)-dimensional topological quantum field theories to state spaces and operators for a (3+1)-dimensional TQFT with defects. We start from the (2+1)-dimensional Turaev-Viro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects. This work has important applications for quantum gravity as well as the theory of topological phases in (3+1) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably. We in particular show that the fusion bases of the (2+1)-dimensional theory lead to a rich set of bases for the (3+1)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian. Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.Comment: 27 pages, many figures, v2: minor correction

Random field Ising model: dimensional reduction or spin-glass phase?

The stability of the random field Ising model (RFIM) against spin glass (SG) fluctuations, as investigated by M\'ezard and Young, is naturally expressed via Legendre transforms, stability being then associated with the non-negativeness of eigenvalues of the inverse of a generalized SG susceptibility matrix. It is found that the signal for the occurrence of the SG transition will manifest itself in free-energy {\sl fluctuations\/} only, and not in the free energy itself. Eigenvalues of the inverse SG susceptibility matrix is then approached by the Rayleigh Ritz method which provides an upper bound. Coming from the paramagnetic phase {\sl on the Curie line,\/} one is able to use a virial-like relationship generated by scaling the {\sl single\/} unit length $(D<6;$ in higher dimension a new length sets in, the inverse momentum cut off). Instability towards a SG phase being probed on pairs of {\sl distinct\/} replicas, it follows that, despite the repulsive coupling of the RFIM the effective pair coupling is {\sl attractive\/} (at least for small values of the parameter $g\bar \Delta ,$ $g$ the coupling and $\bar \Delta$ the effective random field fluctuation). As a result, \lq\lq bound states\rq\rq\ associated with replica pairs (negative eigenvalues) provide the instability signature. {\sl Away from the Curie line\/}, the attraction is damped out till the SG transition line is reached and paramagnetism restored. In $D<6,$ the SG transition always precedes the ferromagnetic one, thus the domain in dimension where standard dimensional reduction would apply (on the Curie line) shrinks to zero.Comment: te

Barnes Hospital Bulletin

https://digitalcommons.wustl.edu/bjc_barnes_bulletin/1084/thumbnail.jp

Influence of protein concentration and coagulation temperature on rennet-induced gelation characteristics and curd microstructure

peer-reviewedThis study characterized the coagulation properties and defined the cutting window (CW; time between storage modulus values of 35 and 70 Pa) using rheometry for milk standardized to 4, 5, or 6% protein and set at 28, 32, or 36°C. Milks were standardized to a protein-to-fat ratio of approximately 1 by blending ultrafiltration retentate, skim milk, and whole milk. The internal curd microstructure for selected curd samples was analyzed with transmission electron microscopy and scanning electron microscopy. Lowering the coagulation temperature caused longer rennet coagulation time and time to reach storage modulus of 35 Pa, translating into a wider CW. It also led to a lower maximum curd-firming rate (MCFR) with lower firmness at 40 min at a given protein level. Increasing protein levels resulted in the opposite effect, although without an effect on rennet coagulation time at a given temperature. On coagulation at 28°C, milk with 5% protein resulted in a similar MCFR (∼4 Pa/min) and CW (∼8.25 min) compared with milk with 4% protein at 32°C, which reflects more standard conditions, whereas increasing milk to 6% protein resulted in more than doubling of the curd-firming rate (MCFR = 9.20 Pa/min) and a shorter CW (4.60 min). Gels set at 28°C had lower levels of rearrangement of protein network after 40 min compared with those set at 36°C. Protein levels, on the other hand, had no influence on the levels of protein network rearrangement, as indicated by loss tangent values. The internal structure of curd particles, as investigated by both scanning electron microscopy and transmission electron microscopy, appeared to have less cross-linking and smaller casein aggregates when coagulated at 28°C compared with 36°C, whereas varying protein levels did not show a marked effect on aggregate formation. Overall, this study showed a marked interactive effect between coagulation temperature and protein standardization of milk on coagulation properties, which subsequently requires adjustment of the CW during cheesemaking. Lowering of the coagulation temperature greatly altered the curd microstructure, with a tendency for less syneresis during cutting. Further research is required to quantify the changes in syneresis and in fat and protein losses to whey due to changes in the microstructure of curd particles arising from the different coagulation conditions applied to the protein-fortified milk

Scaling behaviour of three-dimensional group field theory

Group field theory is a generalization of matrix models, with triangulated pseudomanifolds as Feynman diagrams and state sum invariants as Feynman amplitudes. In this paper, we consider Boulatov's three-dimensional model and its Freidel-Louapre positive regularization (hereafter the BFL model) with a ?ultraviolet' cutoff, and study rigorously their scaling behavior in the large cutoff limit. We prove an optimal bound on large order Feynman amplitudes, which shows that the BFL model is perturbatively more divergent than the former. We then upgrade this result to the constructive level, using, in a self-contained way, the modern tools of constructive field theory: we construct the Borel sum of the BFL perturbative series via a convergent ?cactus' expansion, and establish the ?ultraviolet' scaling of its Borel radius. Our method shows how the ?sum over trian- gulations' in quantum gravity can be tamed rigorously, and paves the way for the renormalization program in group field theory

Low-temperature behaviour of social and economic networks

Real-world social and economic networks typically display a number of particular topological properties, such as a giant connected component, a broad degree distribution, the small-world property and the presence of communities of densely interconnected nodes. Several models, including ensembles of networks also known in social science as Exponential Random Graphs, have been proposed with the aim of reproducing each of these properties in isolation. Here we define a generalized ensemble of graphs by introducing the concept of graph temperature, controlling the degree of topological optimization of a network. We consider the temperature-dependent version of both existing and novel models and show that all the aforementioned topological properties can be simultaneously understood as the natural outcomes of an optimized, low-temperature topology. We also show that seemingly different graph models, as well as techniques used to extract information from real networks, are all found to be particular low-temperature cases of the same generalized formalism. One such technique allows us to extend our approach to real weighted networks. Our results suggest that a low graph temperature might be an ubiquitous property of real socio-economic networks, placing conditions on the diffusion of information across these systems

An Introduction to Gauge Gravity Duality and Its Application in Condensed Matter

The past few years have witnessed a remarkable crossover of string theoretical ideas from the abstract world of geometrical forms to the concrete experimental realm of condensed matter physics. The basis for this --- variously known as holography, the AdS/CFT correspondence or gauge-gravity duality --comes from notions right at the cutting edge of string theory. Nevertheless, the insights afforded can often be expressed in ways very familiar to condensed matter physicists, such as relationships between response functions and new sum rules. The aim of this short, introductory review is to survey the ideas underpinning this crossover, in a way that -- as far as possible -- strips them of sophisticated mathematical formalism, whilst at the same time retaining their fundamental essence. I will sketch the areas in which progress has been made to date and highlight where the challenges and open questions lie. Finally, I will attempt to give a perspective upon these ideas. What contribution can we realistically expect from this approach and how might it be accommodated into the canon of condensed matter theory? Inevitably, any attempt to do this in such a rapidly evolving field will be superseded by events. Nevertheless, I hope that this will provide a useful way to think about gauge-gravity duality and the uncharted directions in which it might take us.Comment: Unedited version of article published in Contemporary Physics. Intended for advanced final-year undergraduate