Let : (M1) (M2) be a bijection (not assumed affine nor continuous) between the sets of normal states of two quantum systems, modelled on the self-adjoint parts of von Neumann algebras M1 and M2, respectively. This paper concerns with the situation when preserves (or partially preserves) one of the following three notions of transition probability on the normal state spaces: the transition probability PU introduced by Uhlmann [Rep. Math. Phys. 9, 273-279 (1976)], the transition probability PR introduced by Raggio [Lett. Math. Phys. 6, 233-236 (1982)], and an asymmetric transition probability P0 (as introduced in this article). It is shown that the two systems are isomorphic, i.e., M1 and M2 are Jordan -isomorphic, if preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, in the sense that for any normal states and , we have P(),()=0 if and only if P(,

Using canonical 1-parameter family of Hermitian connections on the tangent bundle, we provide invariant solutions to the Strominger system on certain complex Lie groups and their quotients. Both flat and non-flat cases are discussed in detail. This paper answers a question proposed by Andreas and Garcia-Fernandez in Comm Math Phys 332(3):13811383, 2014.

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra M(p), p≥2. The first category consists of all finite-length M(p)-modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra W(p). We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers.

A bstract 6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by 2 curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactification. Adding the affine node of the ADE quiver to the base geometry, we connect to recent results on SYZ mirror symmetry for the A case and provide a physical interpretation in terms of little string theory. Our results, however, go beyond this case as our construction naturally covers the D and E cases as

We consider the Einstein/Yang-Mills equations in 3+ 1 space time dimensions with SU (2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang/Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.