We provide a model of a decentralized, dynamic auction market platform (e.g., eBay) in which a large number of buyers and sellers participate in simultaneous, singleunit auctions each period. Our model accounts for the endogenous entry of agents and the impact of intertemporal optimization on bids. Solving our model with a finite number of bidders is computationally intractable due to the curse of dimensionality, so we prove that a continuum version of our model provides a good approximation of an equilibrium in the finite model. We use the approximation to estimate the structural primitives of our model using Kindle sales on eBay. We find that just over one third of Kindle auctions on eBay result in an inefficient allocation with deadweight loss amounting to 13.5% of total possible market surplus. We also find that partial centralization - for example, runnng half as many 2-unit, uniform price auctions each day - would eliminate a large fraction of the inefficiency, but yield lower seller revenues. Our results highlight the importance of understanding platform composition effects—selection of agents into the market—in assessing the implications of market design.

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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.

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.