Twisted supergravity is supergravity in a background where the bosonic ghost field takes a non-zero value. This is the supergravity counterpart of the familiar concept of twisting supersymmetric field theories. In this paper, we give conjectural descriptions of type IIA and IIB supergravity in 10 dimensions. Our conjectural descriptions are in terms of the closed-string field theories associated to certain topological string theories, and we conjecture that these topological string theories are twists of the physical string theories. For type IIB, the results of [CL15] show that our candidate twisted supergravity theory admits a unique quantization in perturbation theory. This is despite the fact that the theories, like the original physical theories, are non-renormalizable. Although we do not prove our conjectures, we amass considerable evidence. We find that our candidates for the twisted supergravity theories contain the residual supersymmetry one would expect. We also prove (using heavily a result of Baulieu [Bau10]) the open string version of our conjecture: the theory living on a brane in the topological string theory is a twist of the maximally supersymmetric gauge theory liv- ing on the brane in the physical string theory.

We quantize the interaction of gravity with a Yang-Mills and Higgs field using canonical quantization. Similar to the approach in a previous paper we discard the Wheeler-DeWitt equation and express the Hamilton constraint by the evolution equation of the mean curvature of the hypersurfaces in the foliation defined by the Hamiltonian setting. Expressing the time derivative of the mean curvature with the help of the Poisson brackets the canonical quantization of this equation leads to a wave equation in $Q=(0,\infty)\times \so$, where $\so$ is one of the Cauchy hypersurfaces in the Hamiltonian setting. The wave equation describes the interaction of an arbitrary Riemannian metric in $\so$ and a given Yang-Mills and Higgs field. If the metric is complete $Q$ is globally hyperbolic. In case $\so$ is compact we also prove a spectral resolution of the wave equation and establish sufficient conditions guaranteeing a mass gap.

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.

In this note, we study the phase transitions arising in a modified Smoluchowski equation on the sphere with dipolar potential. This equation models the competition between alignment and diffusion, and the modification consists in taking the strength of alignment and the intensity of the diffusion as functions of the order parameter. We characterize the stable and unstable equilibrium states. For stable equilibria, we provide the exponential rate of convergence. We detail special cases, giving rise to second order and first order phase transitions, respectively. We study the hysteresis diagram, and provide numerical illustrations of this phenomena.

We prove uniqueness of ground state solutions$Q$=$Q$(|$x$|) ≥ 0 of the non-linear equation $$(-\Delta)^s Q+Q-Q^{\alpha+1}= 0 \quad {\rm in} \, \mathbb{R},$$ where 0 <$s$< 1 and 0 <$α$< 4$s$/(1−2$s$) for $${s<\frac{1}{2}}$$ and 0 <$α$<$∞$for $${s\geq \frac{1}{2}}$$ . Here (−Δ)^{$s$}denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $${s=\frac{1}{2}}$$ and$α$= 1 in [5] for the Benjamin–Ono equation.