Given a Hopf algebra $H$, we study modules and bimodules over an algebra $A$ that carry an $H$-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For quasitriangular Hopf algebras and bimodules with an extra quasi-commutativity property we induce connections on the tensor product over $A$ of two bimodules from connections on the individual bimodules. This construction applies to arbitrary connections, i.e. not necessarily Hequivariant ones, and further extends to the tensor algebra generated by a bimodule and its dual. Examples of these noncommutative structures arise in deformation quantization via Drinfeld twists of the commutative differential geometry of a smooth manifold, where the Hopf algebra $H$ is the universal enveloping algebra of vector fields (or a finitely generated Hopf subalgebra).

We extend the Drinfeld twist deformation theory of modules and algebras to morphisms and connections that are not necessarily $H$-equivariant. The theory canonically lifts to the tensor product structure.

$F$-theory compactifications on elliptic Calabi-Yau manifolds may be related to IIb compactifications by taking a certain limit in complex structure moduli space, introduced by A. Sen. The limit has been characterized on the basis of $SL(2, \mathrm{Z})$ monodromies of the elliptic fibration. Instead, we introduce a stable version of the Sen limit. In this picture the elliptic Calabi-Yau splits into two pieces, a $\mathbf{P}^1$-bundle and a conic bundle, and the intersection yields the IIb space-time.We get a precise match between $F$-theory and perturbative type IIb. The correspondence is holographic, in the sense that physical quantities seemingly spread in the bulk of the $F$-theory Calabi-Yau may be rewritten as expressions on the log boundary. Smoothing the $F$-theory Calabi-Yau corresponds to summing up the $D(-1)$-instanton corrections to the IIb theory.

We analyze the transformation of the polarization of a photon propagating along an arbitrary null geodesic in Kerr geometry. The motivation comes from the problem of an observer trying to communicate quantum information to another observer in Kerr spacetime by transmitting polarized photons. It is essential that the observers understand the relationship between their frames of reference and also know how the photon’s polarization transforms as it travels through Kerr spacetime. Existing methods to calculate the rotation of the photon polarization (Faraday rotation) depend on choices of coordinate systems, are algebraically complex and yield results only in the weak-field limit.

We give a closed-form expression for a parallel propagated frame along an arbitrary null geodesic using Killing-Yano theory, and thereby solve the problem of parallel transport of the polarization vector in an intrinsic, geometrically-motivated fashion. The symmetries of Kerr geometry are utilized to obtain a remarkably compact expression for the geometrically induced phase of the photon’s polarization. We show that this phase vanishes on the equatorial plane and the axis of symmetry.

We prove area inequalities for stable marginally outer trapped surfaces in Einstein-Maxwell-dilaton theory. Our inspiration comes on the one hand from a corresponding upper bound for the area in terms of the charges obtained recently by Dain, Jaramillo and Reiris in the pure Einstein-Maxwell case without symmetries, and on the other hand from Yazadjiev's inequality in the axially symmetric Einstein-Maxwell-dilaton case. The common issue in these proofs and in the present one is a functional $\mathcal{W}$ of the matter fields for which the stability condition readily yields an upper bound. On the other hand, the step which crucially depends on whether or not a dilaton field is present is to obtain a lower bound for $\mathcal{W}$ as well. We obtain the latter by first setting up a variational principle for $\mathcal{W}$ with respect to the dilaton field $\phi$, then by proving existence of a minimizer $\psi$ as solution of the corresponding Euler-Lagrange equations and finally by estimating $\mathcal{W} (\psi)$. In the special case that the normal components of the electric and magnetic fields are proportional we obtain the area bound $A \geq 8\pi PQ$ in terms of the electric and magnetic charges. In the generic case our results are less explicit but imply rigorous 'perturbation' results for the above inequality. All our inequalities are saturated for a 2-parameter family of static, extreme solutions found by Gibbons. Via the Bekenstein-Hawking relation $A = 4S$ our results give positive lower bounds for the entropy $S$ which are particularly interesting in the Einstein-Maxwell-dilaton case.

We continue our investigation into intersections of closed paths on a torus, to further our understanding of the commutator algebra of Wilson loop observables in $2+1$ quantum gravity, when the cosmological constant is negative.We give a concise review of previous results, e.g. that signed area phases relate observables assigned to homotopic loops, and present new developments in this theory of intersecting loops on a torus. We state precise rules to be applied at intersections of both straight and crooked/rerouted paths in the covering space $\mathbb{R}^2$. Two concrete examples of combinations of different rules are presented.

We study the quantum phase transition of light of a system when atom trapped in microcavities and interacting through the exchange of virtual photons. We predict the quantum phase transition between the photonic Coulomb blocked induce insulating phase and anisotropic exchange induced photonic superfluid phase in the system due to the existence of two Rabi frequency oscillations. The renormalization group equation shows explicitly that for this system there is no self-duality. The system also shows two Berezinskii-Kosterlitz-Thouless (BKT) transitions for the different physical situation of the system. The presence of single Rabi frequency oscillation in the system leads to the BKT transition where system shows the quantum phase transition from photonic metallic state to the Coulomb blocked induced insulating phase. For the other BKT transition when the z-component of exchange interaction is absent, the system shows the transition from the photonic metallic state to the photonic superfluid phase. We also predict the commensurate to incommensurate transition under the laser field detuning.