We study certain six dimensional theories arising on (p, q) brane webs living on R × S. These brane webs are dual to toric elliptically fibered Calabi-Yau threefolds. The compactification of the space on which the brane web lives leads to a deformation of the partition functions equivalent to the elliptic deformation of the Ding-Iohara algebra. We compute the elliptic version Dotsenko-Fateev integrals and show that they reproduce the instanton counting of the six dimensional theory.
Tian LanPerimeter Institute for Theoretical Physics, University of WaterlooLiang KongUniversity of New Hampshire, Harvard CMSAXiao-Gang WenMassachusetts Institute of Technology, Perimeter Institute for Theoretical Physics
Publications of CMSA of Harvardmathscidoc:1702.38027
In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local topological order, which is a (potentially anomalous) topological order defined on an open disk. Using this uniqueness, we show that the notion of “bulk” is equivalent to the notion of center in mathematics. We achieve this by first introducing the notion of a morphism between two local topological orders of the same dimension, then proving that the bulk satisfying the same universal property as that of the center in mathematics. We propose a classification (formulated as a macroscopic definition of n+1D local topological orders by unitary multi-fusion n-categories, and explain that the notion of a morphism between two local topological orders is compatible with that of a unitary monoidal n-functor in a few low dimensional cases. We also explain in some low dimensional cases that this classification is compatible with the result of “bulk = center”. In the end, we explain that above boundary-bulk relation is only the first layer of a hierarchical structure which can be summarized by the functoriality of the bulk (or center). This functoriality also provides the physical meanings of some well-known mathematical results on fusion 1-categories. This work can also be viewed as the first step towards a systematic study of the category of local topological orders, and the boundary-bulk relation actually provides a useful tool for this study.
Lauren CohenHarvard Business School, National Bureau of Economic ResearchUmit G. GurunNational Bureau of Economic Research, University of Texas at DallasScott KominersHarvard Business School, Harvard CMSA, National Bureau of Economic Research
Publications of CMSA of Harvardmathscidoc:1702.38025
New, “big data” sources allow measurement of city characteristics and outcomevariables at higher collection frequencies and more granular geographic scales thanever before. However, big data will not solve large urban social science questionson its own. Big urban data has the most value for the study of cities when it allowsmeasurement of the previously opaque, or when it can be coupled with exogenous shocksto people or place. We describe a number of new urban data sources and illustrate howthey can be used to improve the study and function of cities. We rst show how GoogleStreet View images can be used to predict income in New York City, suggesting thatsimilar imagery data can be used to map wealth and poverty in previously unmeasuredareas of the developing world. We then discuss how survey techniques can be improved tobetter measure willingness to pay for urban amenities. Finally, we explain how Internetdata is being used to improve the quality of city services.