S-duality and 2d Topological QFT

We study the superconformal index for the class of N=2 4d superconformal field theories recently introduced by Gaiotto. These theories are defined by compactifying the (2,0) 6d theory on a Riemann surface with punctures. We interpret the index of the 4d theory associated to an n-punctured Riemann surface as the n-point correlation function of a 2d topological QFT living on the surface. Invariance of the index under generalized S-duality transformations (the mapping class group of the Riemann surface) translates into associativity of the operator algebra of the 2d TQFT. In the A_1 case, for which the 4d SCFTs have a Lagrangian realization, the structure constants and metric of the 2d TQFT can be calculated explicitly in terms of elliptic gamma functions. Associativity then holds thanks to a remarkable symmetry of an elliptic hypergeometric beta integral, proved very recently by van de Bult.Comment: 25 pages, 11 figure

Export diversification and resource-based industrialization: the case of natural gas

For resource-rich economies, primary commodity specialization has often been considered to be detrimental to growth. Accordingly, export diversification policies centered on resource-based industries have long been advocated as effective ways to moderate the large variability of export revenues. This paper discusses the applicability of a mean-variance portfolio approach to design these strategies and proposes some modifications aimed at capturing the key features of resource processing industries (presence of scale economies and investment lumpiness). These modifications help make the approach more plausible for use in resource-rich countries. An application to the case of natural gas is then discussed using data obtained from Monte Carlo simulations of a calibrated empirical model. Lastly, the proposed framework is put to work to evaluate the performances of the diversification strategies implemented in a set of nine gas-rich economies. These results are then used to formulate some policy recommendations

Making Good on LSTMs' Unfulfilled Promise

LSTMs promise much to financial time-series analysis, temporal and cross-sectional inference, but we find that they do not deliver in a real-world financial management task. We examine an alternative called Continual Learning (CL), a memory-augmented approach, which can provide transparent explanations, i.e. which memory did what and when. This work has implications for many financial applications including credit, time-varying fairness in decision making and more. We make three important new observations. Firstly, as well as being more explainable, time-series CL approaches outperform LSTMs as well as a simple sliding window learner using feed-forward neural networks (FFNN). Secondly, we show that CL based on a sliding window learner (FFNN) is more effective than CL based on a sequential learner (LSTM). Thirdly, we examine how real-world, time-series noise impacts several similarity approaches used in CL memory addressing. We provide these insights using an approach called Continual Learning Augmentation (CLA) tested on a complex real-world problem, emerging market equities investment decision making. CLA provides a test-bed as it can be based on different types of time-series learners, allowing testing of LSTM and FFNN learners side by side. CLA is also used to test several distance approaches used in a memory recall-gate: Euclidean distance (ED), dynamic time warping (DTW), auto-encoders (AE) and a novel hybrid approach, warp-AE. We find that ED under-performs DTW and AE but warp-AE shows the best overall performance in a real-world financial task

Normal Factor Graphs and Holographic Transformations

This paper stands at the intersection of two distinct lines of research. One line is "holographic algorithms," a powerful approach introduced by Valiant for solving various counting problems in computer science; the other is "normal factor graphs," an elegant framework proposed by Forney for representing codes defined on graphs. We introduce the notion of holographic transformations for normal factor graphs, and establish a very general theorem, called the generalized Holant theorem, which relates a normal factor graph to its holographic transformation. We show that the generalized Holant theorem on the one hand underlies the principle of holographic algorithms, and on the other hand reduces to a general duality theorem for normal factor graphs, a special case of which was first proved by Forney. In the course of our development, we formalize a new semantics for normal factor graphs, which highlights various linear algebraic properties that potentially enable the use of normal factor graphs as a linear algebraic tool.Comment: To appear IEEE Trans. Inform. Theor

Duality between Spin networks and the 2D Ising model

The goal of this paper is to exhibit a deep relation between the partition function of the Ising model on a planar trivalent graph and the generating series of the spin network evaluations on the same graph. We provide respectively a fermionic and a bosonic Gaussian integral formulation for each of these functions and we show that they are the inverse of each other (up to some explicit constants) by exhibiting a supersymmetry relating the two formulations. We investigate three aspects and applications of this duality. First, we propose higher order supersymmetric theories which couple the geometry of the spin networks to the Ising model and for which supersymmetric localization still holds. Secondly, after interpreting the generating function of spin network evaluations as the projection of a coherent state of loop quantum gravity onto the flat connection state, we find the probability distribution induced by that coherent state on the edge spins and study its stationary phase approximation. It is found that the stationary points correspond to the critical values of the couplings of the 2D Ising model, at least for isoradial graphs. Third, we analyze the mapping of the correlations of the Ising model to spin network observables, and describe the phase transition on those observables on the hexagonal lattice. This opens the door to many new possibilities, especially for the study of the coarse-graining and continuum limit of spin networks in the context of quantum gravity.Comment: 35 page

Unitary Chern-Simons matrix model and the Villain lattice action

We use the Villain approximation to show that the Gross-Witten model, in the weak- and strong-coupling limits, is related to the unitary matrix model that describes U(N) Chern-Simons theory on S^3. The weak-coupling limit corresponds to the q->1 limit of the Chern-Simons theory while the strong-coupling regime is related to the q->0 limit. In the latter case, there is a logarithmic relationship between the respective coupling constants. We also show how the Chern-Simons matrix model arises by considering two-dimensional Yang-Mills theory with the Villain action. This leads to a U(1)^N theory which is the Abelianization of 2d Yang-Mills theory with the heat-kernel lattice action. In addition, we show that the character expansion of the Villain lattice action gives the q deformation of the heat kernel as it appears in q-deformed 2d Yang-Mills theory. We also study the relationship between the unitary and Hermitian Chern-Simons matrix models and the rotation of the integration contour in the corresponding integrals.Comment: 17 pages, Minor corrections to match the published versio

Quantum dynamics of long-range interacting systems using the positive-P and gauge-P representations

We provide the necessary framework for carrying out stochastic positive-P and gauge-P simulations of bosonic systems with long range interactions. In these approaches, the quantum evolution is sampled by trajectories in phase space, allowing calculation of correlations without truncation of the Hilbert space or other approximations to the quantum state. The main drawback is that the simulation time is limited by noise arising from interactions. We show that the long-range character of these interactions does not further increase the limitations of these methods, in contrast to the situation for alternatives such as the density matrix renormalisation group. Furthermore, stochastic gauge techniques can also successfully extend simulation times in the long-range-interaction case, by making using of parameters that affect the noise properties of trajectories, without affecting physical observables. We derive essential results that significantly aid the use of these methods: estimates of the available simulation time, optimized stochastic gauges, a general form of the characteristic stochastic variance and adaptations for very large systems. Testing the performance of particular drift and diffusion gauges for nonlocal interactions, we find that, for small to medium systems, drift gauges are beneficial, whereas for sufficiently large systems, it is optimal to use only a diffusion gauge. The methods are illustrated with direct numerical simulations of interaction quenches in extended Bose-Hubbard lattice systems and the excitation of Rydberg states in a Bose-Einstein condensate, also without the need for the typical frozen gas approximation. We demonstrate that gauges can indeed lengthen the useful simulation time.Comment: 19 pages, 11 appendix, 3 figure