The two-dimensional (2D) superconducting state is a fragile state of matter susceptible to quantum phase fluctuations. Although superconductivity has been observed in ultrathin metal films down to a few layers 1-10 , it is still not known whether a single layer of ordered metal atoms, which represents the ultimate 2D limit of a crystalline film, could be superconducting. Here we report scanning tunnelling microscopy measurements on single atomic layers of Pb and In grown epitaxially on Si(111) substrate, and demonstrate unambiguously that superconductivity does exist at such a 2D extreme. The film shows a superconducting transition temperature of 1.83 K for an atom areal density n = 10.44 Pb atoms nm −2 , 1.52 K for n = 9.40 Pb atoms nm −2 and 3.18 K for n = 9.40 In atoms nm −2 , respectively. We confirm the occurrence of superconductivity by the presence of superconducting vortices under magnetic field. In situ angle-resolved photoemission spectroscopy measurements reveal that the observed superconductivity is due to the interplay between the Pb-Pb (In-In) metallic and the Pb-Si (In-Si) covalent bondings.The one-atomic-layer films of Pb and In studied here were grown with atomic precision on bulk-terminated Si(111) substrate using molecular beam epitaxy. The one-atomic-layer films of Pb have two different structural phases depending on the coverage (for sample preparation, see the Methods section). Figure 1a,d shows the schematic structure and scanning tunnelling microscopy (STM) topograph of the so-called striped incommensurate (SIC) phase, which has a Pb coverage of 4/3 monolayers (ML;. Here 1 ML is defined as the surface atomic density of the Si(111) with areal density n = 7.84 atoms nm −2 . In a unit cell of the SIC-Pb phase, there are four Pb atoms per three surface Si atoms. Three of the four Pb atoms each form a covalent bond with an underlying Si atom, leaving one Pb atom without bonding to the Si substrate. Besides the covalent bonds with the Si substrate, the metal atoms also form metallic bonds within the metal overlayer. As all Pb atoms are located exactly in the same atomic-layer sheet (see the large-scale STM image and cross-section height profiles in Supplementary Fig. S1), the resulting areal density of Pb atoms is 10.44 nm −2 . Compared with the bulk Pb(111) plane, the lattice of the SIC phase is compressed by 5%.Ultralow-temperature (down to 0.40 K) scanning tunnelling spectroscopy (STS) on the SIC phase reveals a clear signature of superconductivity. Figure 2a shows the tunnelling spectra taken on the SIC phase using a superconducting Nb tip. At 0.42 K,

The universal enveloping algebra of W 1+∞ is isomorphic to the affine Yangian of gl 1. We study the N = 2 supersymmetric version of this correspondence, and identify the full set of defining relations of the supersymmetric affine Yangian. These relations can be deduced by demanding that the algebra has a representation on twin-plane-partitions, which we construct by gluing pairs of plane partitions. We define the action of the algebra on these twin-plane-partitions explicitly.

In the weak coupling limit of SU(N ) Yang-Mills theory and the O(N ) vector model, explicit state counting allows us to demonstrate the existence of a partially deconfined phase: M of N colors deconfine, and M N gradually grows from zero (confinement) to one (complete deconfinement). We point out that the mechanism admits a simple interpretation in the form of spontaneous breaking of gauge symmetry. In terms of the dual gravity theory, such breaking occurs during the formation of a black hole. We speculate whether the breaking and restoration of gauge symmetry can serve as an alternative definition of the deconfinement transition in theories without center symmetry, such as QCD. We also discuss the role of the color degrees of freedom in the emergence of the bulk geometry in holographic duality.

We compute the asymptotic symmetry of the higher-spin supergravity theory in AdS 3 and obtain an infinite-dimensional non-linear superalgebra, which we call the super-W ∞ [λ] algebra.According to the recently proposed supersymmetric duality between higher-spin supergravity in an AdS 3 background and the 't Hooft limit of the N = 2 CP n Kazama-Suzuki model on the boundary, this symmetry algebra should agree with the 't Hooft limit of the chiral algebra of the CFT, SW n . We provide two nontrivial checks of the duality. By comparing the algebras, we explicitly match the lowest-spin commutation relations in the super-with the corresponding commutation relations in the 't Hooft limit on the CFT side. We also consider the degenerate representations of the two algebras and find that the spectra of the chiral primary fields are identical.