While two-dimensional symmetry-enriched topological phases (𝖲𝖤𝖳s) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry Gs on gauge theories (denoted by 𝖦𝖳) with gauge group Gg. The resulting symmetric gauge theories are dubbed "symmetry-enriched gauge theories" (𝖲𝖤𝖦), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on 𝖲𝖤𝖦s with gauge group Gg=ℤN1×ℤN2×⋯ and on-site unitary symmetry group Gs=ℤK1×ℤK2×⋯ or Gs=U(1)×ℤK1×⋯. Each 𝖲𝖤𝖦(Gg,Gs) is described in the path integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground state properties (i.e., 𝖲𝖤𝖳 orders) of 𝖲𝖤𝖦s in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the \emph{mixed} multi-loop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from 𝖲𝖤𝖦s to 𝖲𝖤𝖳s. By giving full dynamics to background gauge fields, 𝖲𝖤𝖦s may be eventually promoted to a set of new gauge theories (denoted by 𝖦𝖳∗). Based on their gauge groups, 𝖦𝖳∗s may be further regrouped into different classes each of which is labeled by a gauge group G∗g. Finally, a web of gauge theories involving 𝖦𝖳, 𝖲𝖤𝖦, 𝖲𝖤𝖳 and 𝖦𝖳∗ is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.

We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by Li [1991]. Under mild conditions, the asymptotic ratio ρ = lim p/n is the phase transition parameter and the SIR estimator is consistent if and only if ρ = 0. When dimension p is greater than n, we propose a diagonal thresholding screening SIR (DT-SIR) algorithm. This method provides us with an estimate of the eigen-space of the covariance matrix of the conditional expectation var(E[x|y]). The desired dimension reduction space is then obtained by multiplying the inverse of the covariance matrix on the eigen-space. Under certain sparsity assumptions on both the covariance matrix of predictors and the loadings of the directions, we prove the consistency of DT-SIR in estimating the dimension reduction space in high dimensional data analysis. Extensive numerical experiments demonstrate superior performances of the proposed method in comparison to its competitors.

For the physical vacuum free boundary problem with the sound speed being C 1/2 - H¨older continuous near vacuum boundaries of the compressible Euler equations with damping, the global existence of solutions and convergence to Barenblatt self-similar solutions of the porous media equation was recently proved in [34] for 1-d motions by Luo and the author. This paper generalizes the results for 1-d motions to 3-d spherically symmetric motions. Compared with the 1-d theory, besides the high degeneracy of the equations near the physical vacuum boundary, the analytical difficulties lie in the complexity of equations and the coordinates singularity in the center of symmetry which is resolved by constructing suitable weights. The results obtained in this work contribute to the theory of global solutions to free boundary problems of compressible inviscid fluids in the presence of vacuum states, for which the currently available results are mainly for the local-in-time well-posedness theory, also to the theory of global smooth solutions of dissipative hyperbolic systems which fail to be strictly hyperbolic.

This paper is concerned with the classical motion of a string in global AdS3. The initially static string stretches between two antipodal points on the boundary circle. Both endpoints are perturbed which creates cusps at a steady rate. The cusps propagate towards the interior where they collide. The behavior of the string depends on the strength of forcing. Three qualitatively different phases can be distinguished: transparent, gray, and black. The transparent region is analogous to a standing wave. In the black phase, there is a horizon on the worldsheet and cusps never reach the other endpoint. The string keeps folding and its length grows linearly over time. In the gray phase, the string still grows linearly. However, cusps do cross to the other side. The transparent and gray regions are separated by a transition point where a logarithmic accumulation of cusps is numerically observed. A simple model reproduces the qualitative behavior of the string in the three phases.

We consider the mixed q-Gaussian algebras introduced by Speicher which are generated by the variables Xi = li + l ∗ i , i = 1, . . . , N, where l ∗ i lj − qij lj l ∗ i = δi,j and −1 < qij = qji < 1. Using the free monotone transport theorem of Guionnet and Shlyakhtenko, we show that the mixed q-Gaussian von Neumann algebras are isomorphic to the free group von Neumann algebra L(FN ), provided that maxi,j |qij | is small enough. Similar results hold in the reduced C ∗ -algebra setting. The proof relies on some estimates which are generalizations of Dabrowski’s results for the special case qij ≡ q.

A compactness framework is established for approximate solutions to subsonicsonic flows governed by the steady full Euler equations for compressible fluids in arbitrary dimension. The existing compactness frameworks for the two-dimensional irrotational case do not directly apply for the steady full Euler equations in higher dimensions. The new compactness framework we develop applies for both non-homentropic and rotational flows. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and the entropy relation, through a more delicate analysis on the phase space. As direct applications, we establish two existence theorems for multidimensional subsonic-sonic full Euler flows through infinitely long nozzles.

Segmented strings in flat space are piecewise linear classical string solutions. Kinks between the segments move with the speed of light and their worldlines form a lattice on the worldsheet. This idea can be generalized to AdS3 where the embedding is built from AdS2 patches. The construction provides an exact discretization of the non-linear string equations of motion. This paper computes the area of segmented strings using cross-ratios constructed from the kink vectors. The cross-ratios can be expressed in terms of either left-handed or right-handed variables. The string equation of motion in AdS3 is reduced to that of an integrable time-discretized relativistic Toda-type lattice. Positive solutions yield string embeddings that are unique up to global transformations. In the appendix, the Poincare target space energy is computed by integrating the worldsheet current along kink worldlines and a formula is derived for the integrated scalar curvature of the embedding.

Any Calabi–Yau threefold X with infinite fundamental group admits an ´etale Galois covering either by an abelian threefold or by the product of a K3 surface and an elliptic curve. We call X of type A in the former case and of type K in the latter case. In this paper, we provide the full classification of Calabi–Yau threefolds of type K, based on Oguiso and Sakurai’s work [24]. Together with a refinement of their result on Calabi–Yau threefolds of type A, we finally complete the classification of Calabi–Yau threefolds with infinite fundamental group.

We study evolutionary games on graphs. The individuals of a population occupy the vertices of the graph and interact with their neighbors to receive payoff. We consider finite population size, regular graphs, probabilistic death-birth updating and weak selection. There are two types of strategies, A and B, and a payoff matrix [(a, b),(c, d)]. The initial condition is given by an arbitrary configuration where each vertex is occupied by either A or B. The conjugate initial condition is obtained by swapping A and B. We ask: when is the fixation probability of A for the original configuration greater than the fixation probability of B for the conjugate configuration? The answer is a linear condition of the form σa+b > c+σd. We calculate σ for any initial condition. For large population size we obtain the well known result σ = (k + 1)/(k − 1), but now this result extends to any mixed initial condition. As a specific example we study evolution of cooperation. We calculate the critical benefit-to-cost ratio for natural selection to favor the fixation of cooperators for any initial condition. We obtain results that specify which initial conditions reduce and which initial conditions increase the critical benefit-to-cost ratio. Adding more cooperators to the initial condition does not necessarily favor cooperation. But strategic placing of cooperators in a network can enhance the takeover of cooperation.

Topological phases of matter are usually realized in deconfined phases of gauge theories. In this context, confined phases with strongly fluctuating gauge fields seem to be irrelevant to the physics of topological phases. For example, the low-energy theory of the two-dimensional (2D) toric code model (i.e. the deconfined phase of ℤ2 gauge theory) is a U(1)×U(1) Chern-Simons theory in which gauge charges (i.e., e and m particles) are deconfined and the gauge fields are gapped, while the confined phase is topologically trivial. In this paper, we point out a new route to constructing exotic 3D gapped fermionic phases in a confining phase of a gauge theory. Starting from a parton construction with strongly fluctuating compact U(1)×U(1) gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and U(1)×U(1) magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-H\"uckel-like screening cloud formed by the condensed composite particles. Within our general framework, we explore two aspects of symmetry-enriched 3D Abelian topological phases. First, we construct a new fermionic state of matter with time-reversal symmetry and Θ≠π, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry (𝖢𝗁𝖺𝗋𝗅𝖾𝗌) of 3D Abelian topological phases. We show that line twist defects, which realize 𝖢𝗁𝖺𝗋𝗅𝖾𝗌 transformations, exhibit non-Abelian fusion properties.