Fractionalization and Anomalies in Symmetry-Enriched U(1) Gauge Theories

We classify symmetry fractionalization and anomalies in a (3+1)d U(1) gauge theory enriched by a global symmetry group $G$. We find that, in general, a symmetry-enrichment pattern is specified by 4 pieces of data: $\rho$, a map from $G$ to the duality symmetry group of this $\mathrm{U}(1)$ gauge theory which physically encodes how the symmetry permutes the fractional excitations, $\nu\in\mathcal{H}^2_{\rho}[G, \mathrm{U}_\mathsf{T}(1)]$, the symmetry actions on the electric charge, $p\in\mathcal{H}^1[G, \mathbb{Z}_\mathsf{T}]$, indication of certain domain wall decoration with bosonic integer quantum Hall (BIQH) states, and a torsor $n$ over $\mathcal{H}^3_{\rho}[G, \mathbb{Z}]$, the symmetry actions on the magnetic monopole. However, certain choices of $(\rho, \nu, p, n)$ are not physically realizable, i.e. they are anomalous. We find that there are two levels of anomalies. The first level of anomalies obstruct the fractional excitations being deconfined, thus are referred to as the deconfinement anomaly. States with these anomalies can be realized on the boundary of a (4+1)d long-range entangled state. If a state does not suffer from a deconfinement anomaly, there can still be the second level of anomaly, the more familiar 't Hooft anomaly, which forbids certain types of symmetry fractionalization patterns to be implemented in an on-site fashion. States with these anomalies can be realized on the boundary of a (4+1)d short-range entangled state. We apply these results to some interesting physical examples.Comment: are welcome; v2 references adde

Topics in pricing American type financial contracts

In this thesis we study three pricing problems related to American type financial contracts: firstly, we derive a closed form upper bound for American put options. This upper bound can be used in conjunction with traditional Monte Carlo simulation, which usually generates a lower bound, to obtain a better estimate for the option price; secondly, we solve an optimal control problem and derive an optimal strategy for the owner of a stock which is subject to default risk; thirdly, we prove an ordering result for American options with a piecewise linear payoff under a family of equivalent martingale measures used in stochastic volatility models

Application of rough sets technique in resource selection problem

In this paper a resource selection problem is studied. We analyze the properties of the resource selection problem and propose some criteria for a good resource selection model. A simple-minded model, a traditional rough set model, a generalized rough set (GRS) model, and a high-order rough set (HORS) model are introduced and the advantages and disadvantages of each model are compared. It is shown that the HORS model is superior to other models except that it is much more time consuming. Some methods to reduce the complexity of the HORS model are also proposed

Exploring the low redshift universe: two parametric models for effective pressure

Astrophysical observations have put unprecedentedly tight constraints on cosmological theories. The $\Lambda$CDM model, mathematically simple and fits observational data-sets well, is preferred for explaining the behavior of universe. But many basic features of the dark sectors are still unknown, which leaves rooms for various nonstandard cosmological hypotheses. As the pressure of cosmological constant dark energy is unvarying, ignoring contributions from radiation and curvature terms at low redshift, the effective pressure keeps constant. In this paper, we propose two parametric models for non-constant effective pressure in order to study the tiny deviation from $\Lambda$CDM at low redshift. We recover our phenomenological models in the scenarios of quintessence and phantom fields, and explore the behavior of scalar field and potential. We constrain our model parameters with SNe Ia and BAO observations, and detect subtle hints of $\omega_{de}<-1$ from the data fitting results of both models, which indicates possibly a phantom dark energy scenario at present.Comment: 11 pages, 24 figure