We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmüller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston’s stretch maps and prove the following: (1) On the Teichmüller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and define a geodesic metric on this space. (2) On the same space, the arc and the curve metrics coincide when the length of the boundary component is ≤4arcsinh(1), but differ when the boundary length is large. We further apply our stretch map generalization to construct novel Thurston geodesics on the Teichmüller spaces of closed hyperbolic surfaces, and use these geodesics to show that the sum-symmetrization of the Thurston metric fails to exhibit Gromov hyperbolicity.

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The quantitative adiabatic condition (QAC), or quantitative condition, is a convenient (a priori) tool for estimating the adiabaticity of quantum evolutions. However, the range of the applicability of QAC is not well understood. It has been shown that QAC can become insufficient for guaranteeing the validity of the adiabatic approximation, but under what conditions the QAC would become necessary has become controversial. Furthermore, it is believed that the inability for the QAC to reveal quantum adiabaticity is due to induced resonant transitions. However, it is not clear how to quantify these transitions in general. Here we present a progress to this problem by finding an exact relation that can reveal how transition amplitudes are related to QAC directly. As a posteriori condition for quantum adiabaticity, our result is universally applicable to any (nondegenerate) quantum system and gives a clear picture on how QAC could become insufficient or unnecessary for the adiabatic approximation, which is a problem that has gained considerable interest in the literature in recent years

In these notes, we provide an introduction to the hemisphere partition function of 2d (2, 2) supersymmetric gauge theories, and discuss its relation to the “D-brane central charge” which were studied in superstring theory, in 2d supersymmetric quantum field theory, and in topological string theory. We also discuss relation to “macroscopic loop” in matrix models. They are mostly reviews of the work by the authors, but contains some new results such as the partition function for a rotated supersymmetry as well as the differential equations.

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