We prove that for spacetimes solving the EinsteinMaxwell (EM) equations, the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In Nonlinear nature of gravitation and gravitational-wave experiments, Christodoulou showed that gravitational waves have a nonlinear memory. He discussed how this effect can be measured as a permanent displacement of test masses in a laser interferometer gravitational-wave detector. Christodoulou derived a precise formula for this permanent displacement in the Einstein vacuum (EV) case. We prove in Theorem 2.6 that for the EM equations this permanent displacement exhibits a term coming from the electromagnetic field. This term is at the same highest order as the purely gravitational term that governs the EV situation. On the other hand, in Section 3, we show that to leading order, the presence of the electromagnetic field

In recent years, fermionic topological phases of quantum matter has attracted a lot of attention. In a pioneer work by Gu, Wang and Wen, the concept of equivalence classes of fermionic local unitary(FLU) transformations was proposed to systematically understand non-chiral topological phases in 2D fermion systems and an incomplete classification was obtained. On the other hand, the physical picture of fermion condensation and its corresponding super pivotal categories give rise to a generic mathematical framework to describe fermionic topological phases of quantum matter. In particular, it has been pointed out that in certain fermionic topological phases, there exists the so-called q-type anyon excitations, which have no analogues in bosonic theories. In this paper, we generalize the Gu, Wang and Wen construction to include those fermionic topological phases with q-type anyon excitations. We argue that all non-chiral fermionic topological phases in 2+1D are characterized by a set of tensors (N^{ij}_k,F^{ij}_k,F^{ijm,αβ}_{kln,χδ},n_i,d_i), which satisfy a set of nonlinear algebraic equations parameterized by phase factors Ξ^{ijm,αβ}_{kl}, Ξ^{ij}_{kln,χδ}, Ω^{kim,αβ}_{jl} and Ω^{ki}_{jln,χδ}. Moreover, consistency conditions among algebraic equations give rise to additional constraints on these phase factors which allow us to construct a topological invariant partition for an arbitrary triangulation of 3D spin manifold. Finally, several examples with q-type anyon excitations are discussed, including the Fermionic topological phase from Tambara-Yamagami category for \mathbb{Z}_{2N}, which can be regarded as the \mathbb{Z}_{2N} parafermion generalization of Ising fermionic topological phase.

We define a notion of stability for chiral ring of four dimensional N= 1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N= 1 field theory derived from D3 branes probing a three-fold singularity X, and show that the K stability which implies the existence of Ricci-flat conic metric on X is equivalent to the stability of chiral ring of the corresponding field theory.

In the past decade, tremendous efforts have been made towards understanding fermionic symmetry protected topological (FSPT) phases in interacting systems. Nevertheless, for systems with continuum symmetry, e.g., electronic insulators, it is still unclear how to construct an exactly solvable model with a finite dimensional Hilbert space in general. In this paper, we give a lattice model construction and classification for 2D interacting electronic insulators. Based on the physical picture of U(1)_f-charge decorations, we illustrate the key idea by considering the well known 2D interacting topological insulator. Then we generalize our construction to an arbitrary 2D interacting electronic insulator with symmetry Gf=U(1)_f ⋊_{ρ_1,ω_2} G, where U(1)_f is the charge conservation symmetry and ρ_1,ω_2 are additional data which fully characterize the group structure of G_f. Finally we study more examples, including the full interacting classification of 2D crystalline topological insulators.

We consider for <i>j</i>=, a spherically symmetric, static system of (2<i>j</i>+1) Dirac particles, each having total angular momentum <i>j</i>. The Dirac particles interact via a classical gravitational and electromagnetic field.