Loewner’s equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function of its boundary. The first main result of the present paper is that the Dirichlet energy of this driving function (also known as the Loewner energy) is equal to the Dirichlet energy of the log-derivative of the (appropriately defined) uniformizing conformal map. This description of the Loewner energy then enables to tie direct links with regularized determinants and Teichmüller theory: We show that for smooth simple loops, the Loewner energy can be expressed in terms of the zeta-regularized determinants of a certain Neumann jump operator. We also show that the family of finite Loewner energy loops coincides with the Weil–Petersson class of quasicircles, and that the Loewner energy equals to a multiple of the universal Liouville action introduced by Takhtajan and Teo, which is a Kähler potential for the Weil–Petersson metric on the Weil–Petersson Teichmüller space.

In this paper, we establish the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two-phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh--Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and the Cauchy formula (1882) for the magnetic field. This enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and on the other hand, it allows us to define a special good unknown of the magnetic field to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh--Taylor sign condition required by Morando, Tarkhinin and Trebeschi (\emph{J. Differential Equations} \textbf{258} (2015), no. 7, 2531--2571; \emph{Arch. Ration. Mech. Anal.} \textbf{228} (2018), no. 2, 697--742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is {\it no loss of derivatives} in our well-posedness theory. The solution is constructed as {\it the inviscid limit} of solutions to well-chosen nonlinear approximate problems for the two-phase compressible viscous non-resistive MHD.

Recently Freed and Hopkins [11] proved that there is no parity anomaly in M-theory on pin+ manifolds in the low-energy field theory approximation, and they also developed an algebraic theory of cubic forms. Earlier Witten [33] proved the anomaly cancellation for spin manifolds by introducing the E8-bundle technique. Motivated by the cubic forms and the anomaly cancellation formulas of Witten-Freed-Hopkins, we give some new cubic forms on spin, spinc, spinω2 and orientable 12- manifolds respectively. We relate them to η-invariants when the manifolds are with boundary, and mod 2 indices on 10 dimensional characteristic submanifolds when the manifolds are spinc or spinω2 . Our method of producing these cubic forms is a combination of (generalized) Witten classes and the character of the basic representation of affine E8.

We use entangled multimode coherent states to produce entangled giant graviton states, in the context of gauge/gravity duality. We make a smeared distribution of the entangled multimode coherent states on the circle, or on the five-sphere, in the higher dimensional view. In gauge/gravity duality, we analyze the superposition of giant graviton states, and the entangled pairs of giant graviton states. We map a class of angular distribution functions to unitary operations on the pairs. We also use Young tableau states to construct cat states and qudit states. Various bipartite quantum states involving Young tableau states are analyzed, including micro-macro entangled states. Mixed states of Young tableau states are generated, by using ensemble mixing using angular distribution functions, and also by going through noisy quantum channels. We then produce mixed entangled pair of giant graviton states, by including interaction with the environment and using noisy quantum channels.

We focus on two types of coherent states, the coherent states of multi graviton states and the coherent states of giant graviton states, in the context of gauge/gravity correspondence. We conveniently use a phase shift operator and its actions on the superpositions of these coherent states. We find N-state Schrodinger cat states which approach the one-row Young tableau states, with fidelity between them asymptotically reaches 1 at large N. The quantum Fisher information of these states is proportional to the variance of the excitation energy of the underlying states, and characterizes the localizability of the states in the angular direction in the phase space. We analyze the correlation and entanglement between gravitational degrees of freedom using different regions of the phase space plane in bubbling AdS. The correlation between two entangled rings in the phase space plane is related to the area of the annulus between the two rings. We also analyze two types of noisy coherent states, which can be viewed as interpolated states that interpolate between a pure coherent state in the noiseless limit and a maximally mixed state in the large noise limit.