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.

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In pattern-forming developmental systems, cells commonly interpret graded input signals, known as morphogens. Morphogens often pattern tissues through cascades of sequential gene expression steps. Such a multi-tiered structure appears to constitute suboptimal use of the positional information provided by the input morphogen because noise is added at each tier. However, the conventional theory neglects the role of the format in which information is encoded. We argue that the relevant performance measure is not solely the amount of information carried by the morphogen, but the amount of information that can be accessed by the downstream network. We demonstrate that quantifying the information that is accessible to the system naturally explains the prevalence of multi-tiered network architectures as a consequence of the noise inherent to the control of gene expression. We support our argument with empirical observations from patterning along the major body axis of the fruit fly embryo.

In this paper, we study the relaxation limit of the relaxing Cauchy problem for non-isentropic compressible Euler equations with damping in multi-dimensions. We prove that the velocity of the relaxing equations converges weakly to the velocity of the relaxed equations, while other variables of the relaxing equations converge strongly to the corresponding variables of the relaxed equations. We prove that as relaxation time approaches 0, there exists an initial layer for the ill-prepared data, the convergence of the velocity is strong outside the layer; while there is no initial layer for the well-prepared data, the convergence of the velocity is strong near t = 0. The strong convergence rates of all variables are also estimated.

In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semiclassical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex saddle points, even when the parameters in the action are real. We find new exact complex saddles, and show that without their contribution the semi-classical expansion is in conflict with basic properties such as positive-semidefiniteness of the spectrum, and constraints of supersymmetry. Generic saddles are not only complex, but also possibly multi-valued, and even singular. This is in contrast to instanton solutions, which are real, smooth, and single-valued. The multi-valuedness of the action can be interpreted as a hidden topological angle, quantized in units of π in supersymmetric theories. The general ideas also apply to non-supersymmetric theories.

The fermionic signature operator is constructed in Rindler space-time. It is shown to be an unbounded self-adjoint operator on the Hilbert space of solutions of the massive Dirac equation. In two-dimensional Rindler space-time, we prove that the resulting fermionic projector state coincides with the Fulling-Rindler vacuum. Moreover, the fermionic signature operator gives a covariant construction of general thermal states, in particular of the Unruh state. The fermionic signature operator is shown to be well-defined in asymptotically Rindler space-times. In four-dimensional Rindler space-time, our construction gives rise to new quantum states.

We discuss Nakamaye's Theorem and its recent extension to compact complex manifolds, together with some applications.

We initiate the study of M-strings in the thermodynamic limit. In this limit the BPS partition function of M5 branes localizes on configurations with a large number of strings which leads to a reformulation of the partition function in terms of a matrix model. We solve this matrix model and obtain its spectral curve which can be interpreted as the Seiberg-Witten curve associated to the compactified M5 brane theory.

Most of classical theoretical ecology is based on the assumption that organisms in a community can be naturally partitioned into groups of individuals that can be treated as identical. At the same time, mounting experimental evidence from studies of microbial communities raises the intriguing question whether this intuition is an accurate description of the microbial world. This work builds on Mac Arthur’s model of competitive coexistence on multiple resources to construct a framework that does not rely on postulated existence of species as fundamental ecological variables. In one parameter regime, effective “species” with a core and accessory genome naturally appear in this model as emergent concepts. However, the same model allows a smooth transition to a highly diverse regime where the species formalism becomes inadequate. An alternative description is proposed based on the dynamical modes of population fluctuations. This approach provides a naturally hierarchical description of community dynamics which is well-defined even when the species description breaks down. The relevance of this framework for understanding the complexity of naturally observed microbial communities is discussed.

We prove that the punctured generalized conifolds and punctured orbifolded conifolds are mirror symmetric under the SYZ program with quantum corrections. This mathematically confirms the gauge-theoretic prediction by Aganagic–Karch–L¨ust–Miemiec, and also provides a supportive evidence to Morrison’s conjecture that geometric transitions are reversed under mirror symmetry.

In this paper we study segmented strings in AdS3 coupled to a background two-form whose field strength is proportional to the volume form. By changing the coupling, the theory interpolates between the Nambu-Goto string and the SL(2) Wess-Zumino-Witten model. In terms of the kink momentum vectors, the action is independent of the coupling and the classical theory reduces to a single discrete-time Toda-type theory. The WZW model is a singular point in coupling space where the map into Toda variables degenerates.

We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.

We study chiral algebras associated with Argyres-Douglas theories engineered from M5 brane. For the theory engineered using 6d (2,0) type J theory on a sphere with a single irregular singularity (without mass parameter), its chiral algebra is the minimal model of W algebra of J type. For the theory engineered using an irregular singularity and a regular full singularity, its chiral algebra is the affine Kac-Moody algebra of J type. We can obtain the Schur index of these theories by computing the vacua character of the corresponding chiral algebra.

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.

Information theory is gaining popularity as a tool to characterize performance of biological systems. However, information is commonly quantified without reference to whether or how a system could extract and use it; as a result, information-theoretic quantities are easily misinterpreted. Here we take the example of pattern-forming developmental systems which are commonly structured a cascades of sequential gene expression steps. Such a multi-tiered structure appears to constitute sub-optimal use of the positional information provided by the input morphogen because noise is added at each tier. However, the conventional theory fails to distinguish between the total information in a morphogen and information that can be usefully extracted and interpreted by downstream elements. We demonstrate that quantifying the information that is accessible to the system naturally explains the prevalence of multi-tiered network architectures as a consequence of the noise inherent to the control of gene expression. We support our argument with empirical observations from patterning along the major body axis of the fruit fly embryo. Our results exhibit the limitations of the standard information-theoretic characterization of biological signaling and illustrate how they can be resolved.

We revisit the “state-dependence” of the map that we proposed recently between bulk operators in the interior of a large AdS black hole and operators in the boundary CFT. By refining recent versions of the information paradox, we show that this feature is necessary for the CFT to successfully describe local physics behind the horizon — not only for single-sided black holes but even in the eternal black hole. We show that state-dependence is invisible to an infalling observer who cannot differentiate these operators from those of ordinary quantum effective field theory. Therefore the infalling observer does not observe any violations of quantum mechanics. We successfully resolve a large class of potential ambiguities in our construction. We analyze states where the CFT is entangled with another system and show that the ER=EPR conjecture emerges from our construction in a natural and precise form. We comment on the possible semi-classical origins of state-dependence.

We prove a compactness principle for the anisotropic formulation of the Plateau’s problem in codimension one, along the same lines of previous works of the authors [DGM14, DPDRG15]. In particular, we perform a new strategy for proving the rectifiability of the minimal set, avoiding the Preiss’ Rectifiability Theorem.

We continue our study of the chemical (graph) distance inside large critical percolation clusters in dimension two. We prove new estimates, which involve the three-arm probability, for the point-to-surface and point-to-point distances. We show that the point-to-point distance in Z 2 between two points in the same critical percolation cluster has infinite second moment. We also give quantitative versions of our previous results comparing the length of the shortest crossing to that of the lowest crossing of a box.

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In this article we study a differential algebra of modular-type functions attached to the periods of a one-parameter family of Calabi–Yau varieties which is mirror dual to the universal family of quintic threefolds. Such an algebra is generated by seven functions satisfying functional and differential equations in parallel to the modular functional equations of classical Eisenstein series and the Ramanujan differential equation. Our result is the first example of automorphic-type functions attached to varieties whose period domain is not Hermitian symmetric. It is a reformulation and realization of a problem of Griffiths from the seventies on the existence of automorphic functions for the moduli of polarized Hodge structures.

We propose a construction of K¨ahler and non-K¨ahler Calabi-Yau manifolds by branched double covers of twistor spaces. In this construction we use the twistor spaces of four-manifolds with self-dual conformal structures, with the examples of connected sum of n Ps. We also construct K3-fibered Calabi-Yau manifolds from the branched double covers of the blow-ups of the twistor spaces. These manifolds can be used in heterotic string compactifications to four dimensions. We also construct stable and polystable vector bundles. Some classes of these vector bundles can give rise to supersymmetric grand unified models with three generations of quarks and leptons in four dimensions.

In this paper we study the support recovery problem for single index models Y=f(X⊺β,ε), where f is an unknown link function, X∼Np(0,𝕀p) and β is an s-sparse unit vector such that βi∈{±1s√,0}. In particular, we look into the performance of two computationally inexpensive algorithms: (a) the diagonal thresholding sliced inverse regression (DT-SIR) introduced by Lin et al. (2015); and (b) a semi-definite programming (SDP) approach inspired by Amini & Wainwright (2008). When s=O(p1−δ) for some δ>0, we demonstrate that both procedures can succeed in recovering the support of β as long as the rescaled sample size κ=nslog(p−s) is larger than a certain critical threshold. On the other hand, when κ is smaller than a critical value, any algorithm fails to recover the support with probability at least 12 asymptotically. In other words, we demonstrate that both DT-SIR and the SDP approach are optimal (up to a scalar) for recovering the support of β in terms of sample size. We provide extensive simulations, as well as a real dataset application to help verify our theoretical observations.

We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.

We show that there exists an infinite tower of fermionic symmetries in pure d=4, N=1 supergravity on an asymptotically flat background. The Ward identities associated with these symmetries are equivalent to the soft limit of the gravitino and to the statement of supersymmetry at every angle. Additionally, we show that these charges commute into charges associated with the (unextended) BMS group, providing a supersymmetrization of the BMS translations.

The operators with large scaling dimensions can be labeled by Young diagrams. Among other bases, the operators using restricted Schur polynomials have been known to have a large N but nonplanar limit under which they map to states of a system of harmonic oscillators. We analyze the oscillator algebra acting on pairs of long rows or long columns in the Young diagrams of the operators. The oscillator algebra can be reached by a Inonu– Wigner contraction of the u(2) algebra inside of the u(p) algebra of p giant gravitons. We present evidences that integrability in this case can persist at higher loops due to the presence of the oscillator algebra which is expected to be robust under loop corrections in the nonplanar large N limit.