A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang. In this paper, we study not-necessarily-semisimple or rigid braided tensor categories C of modules for the fixed-point vertex operator subalgebra V^G of a vertex operator (super)algebra V with finite automorphism group G. The main results are that every V^G-module in C with a unital and associative V-action is a direct sum of g-twisted V-modules for possibly several g∈G, that the category of all such twisted V-modules is a braided G-crossed (super)category, and that the G-equivariantization of this braided G-crossed (super)category is braided tensor equivalent to the original category C of V^G-modules. This generalizes results of Kirillov and Müger proved using rigidity and semisimplicity. We also apply the main results to the orbifold rationality problem: whether V^G is strongly rational if V is strongly rational. We show that V^G is indeed strongly rational if V is strongly rational, G is any finite automorphism group, and V^G is C_2-cofinite.

In this article, we study the small sphere limit of the WangYau quasi-local energy defined in Wang and Yau (Phys Rev Lett 102(2):021101, 2009, Commun Math Phys 288(3):919942, 2009). Given a point <i>p</i> in a spacetime <i>N</i>, we consider a canonical family of surfaces approaching <i>p</i> along its future null cone and evaluate the limit of the WangYau quasi-local energy. The evaluation relies on solving an optimal embedding equation whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in Chen etal. (Commun Math Phys 308(3):845863, 2011). Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at <i>p</i>. For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding

The leading term of the ground state energy/particle of a dilute gas of bosons with mass m in the thermodynamic limit is $ 2\pi \hbar^2 a \varrho/m$ when the density of the gas is $ \varrho$, the interaction potential is non-negative and the scattering length $a$ is positive. In this paper, we generalize the upper bound part of this result to any interaction potential with positive scattering length, i.e, $a > 0$ and the lower bound part to some interaction potentials with shallow and/or narrow negative parts.

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Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface. We prove that there is a unique one-parameter extension. The 2D defects on the surfaces are quons, and surface tangles are transformations. We use quon language to simulate graphic states that appear in quantum information, and to simulate interesting quantities in modular tensor categories. This simulation relates the pictorial Fourier duality of surface tangles and the algebraic Fourier duality induced by the S matrix of the modular tensor category. The pictorial Fourier duality also coincides with the graphic duality on the sphere. For each pair of dual graphs, we obtain an algebraic identity related to the S matrix. These identities include well-known ones, such as the Verlinde formula; partially known ones, such as the 6j-symbol self-duality; and completely new ones.