We show that the Lieb-Liniger model for one-dimensional bosons with repulsive $\delta$-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length $a$ and the radius $r$ of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant $g \sim a/r^2$ is derived. Our bounds are uniform in $g$ in the whole parameter range $0\leq g\leq \infty$, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size $\sim r^{-2}$ above the ground state energy.

We study the limit of quasilocal energy defined in [7] and [8] for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi-Sachs energy-momentum at a retarded time. The quasilocal mass in [7] and [8] is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases.

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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

A bstract M5 branes probing D-type singularities give rise to 6d (1, 0) SCFTs with SO SO flavor symmetry known as D-type conformal matter theories. Gauging the diagonal SO-flavor symmetry leads to a little string theory with an intrinsic scale which can be engineered in F-theory by compactifying on a doubly-elliptic Calabi-Yau manifold. We derive Seiberg-Witten curves for these little string theories which can be interpreted as mirror curves for the corresponding Calabi-Yau manifolds. Under fiber-base duality these models are mapped to D-type quiver gauge theories and we check that their Seiberg-Witten curves match. By taking decompactification limits, we construct the curves for the related 6d SCFTs and connect to known results in the literature by further taking 5d and 4d limits.