684 Categorification in Geometry, Topology, and Physics Anna Beliakova Aaron D. Lauda Editors AmericanMathematicalSociety Categorification in Geometry, Topology, and Physics Anna Beliakova Aaron D. Lauda Editors 684 Categorification in Geometry, Topology, and Physics Anna Beliakova Aaron D. Lauda Editors AmericanMathematicalSociety Providence,RhodeIsland EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 81R50, 57M25,14F05, 18D10, 58J28, 17B81, 20C08, 17B55, 17B67. Library of Congress Cataloging-in-Publication Data LibraryofCongressCataloging-in-PublicationData Names: Beliakova,Anna,1968–editor. —Lauda,Aaron,1981–editor. Title: Categorification in geometry, topology, and physics / Anna Beliakova, Aaron Lauda, editors. Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2017]|Series: Con- temporarymathematics;volume684|Includesbibliographicalreferences. Identifiers: LCCN2016042028|ISBN9781470428211(alk. paper) Subjects: LCSH:Categories(Mathematics)|Mathematicalanalysis. |Topology. |Geometry. |Physics. |AMS:Quantumtheory–Groupsandalgebrasinquantumtheory–Quantumgroups and related algebraic methods. msc | Manifolds and cell complexes – Low-dimensional topology – Knots and links in S3. msc | Algebraic geometry – (Co)homology theory – Sheaves, derived categories of sheaves and related constructions. msc | Category theory; homological algebra – Categorieswith structure–Monoidalcategories(multiplicative categories),symmetricmonoidal categories,braided categories. msc | Global analysis, analysis on manifolds – Partial differential equations on manifolds; differential operators – Eta-invariants, Chern-Simons invariants. msc | Nonassociativeringsandalgebras–LiealgebrasandLiesuperalgebras–Applicationstophysics. msc|Grouptheoryand generalizations– Representationtheoryofgroups –Hecke algebrasand their representations. msc | Nonassociative rings and algebras – Lie algebras and Lie superalge- bras – Homological methods in Lie (super)algebras. msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Kac-Moody (super)algebras; extended affine Lie algebras; toroidalLiealgebras. msc Classification: LCCQA169.C37462017|DDC512/.62–dc23LCrecordavailableat https://lccn.loc.gov/2016042028 ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online) DOI:http://dx.doi.org/10.1090/conm/684 Color graphic policy. 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VisittheAMShomepageathttp://www.ams.org/ 10987654321 222120191817 Dedicated to Christian Blanchet on the occasion of his sixtieth birthday Contents Preface ix Geometry and categorification Ben Webster 1 A geometric realization of modified quantum algebras Yiqiang Li 23 The cube and the Burnside category Tyler Lawson, Robert Lipshitz, and Sucharit Sarkar 63 Junctions of surface operators and categorification of quantum groups Sungbong Chun, Sergei Gukov, and Daniel Roggenkamp 87 Khovanov-Rozansky homology and 2-braid groups Rapha¨el Rouquier 147 DAHA approach to iterated torus links Ivan Cherednik and Ivan Danilenko 159 vii Preface Theemergentmathematicalphilosophyofcategorificationisreshapingourview of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. This philosophy has led to a number of shocking new results and has ushered in a new area of interaction between algebra, geometry, and theoretical physics. Categorification is a powerful tool for relating various branches of math- ematics and exploiting the commonalities between fields. It provides a language emphasizing essential features and allowing precise relationships between vastly different fields. Categorification exemplifies the duality between pure mathematics and theo- retical physics, with ideas from one area inspiring innovation in the other. Key concepts are exchanged between fields and transported to other areas using the powerfuldescriptiveabilityenabledbycategorification. Indeed,theterm“categori- fication”originatedinworkofCraneandFrenkelwhowerestudyingtechniquesfor extending 3-dimensional topological quantum field theories to 4-dimensions. This areahasrapidlydevelopedoverthelasttwentyyearsfarbeyondtheseoriginalcon- siderations, blossoming intoanexcitinganddynamic areaof modernmathematics, with deep relations to geometric representation theory, low-dimensional topology, algebraic geometry, and mathematical physics. Classical structures in representation theory, such as simple Lie algebras and their representations, allow deformations into quantum groups and modules over them, as discovered by Drinfeld and Jimbo. In turn, quantum groups lead to a representation-theoretical interpretation of the Jones, Kauffman, HOMFLY-PT andother link polynomials. These theories are closely connectedtophysics via the Chern-Simons pathintegral, asshown by Witteninhis seminal paperontheJones polynomial. More recently, it became clear that quantum deformation was only a prelude into deeper structural beauty and meaning. In the work of Khovanov, Ozsvath, Szabo, Rasmussen, Rozansky, and many others, quantum link invariants were interpreted as Euler characteristics of various bigraded and multi-graded link homology theories. These theories carry four-dimensional information, providing invariants of link cobordisms. The resulting link homology theories appear to be a nexus between some of the most sophisticated directions in modern mathematics and theoretical physics. The discovery of link homology gave rise to new advances in representation theory, as quantum groups and related algebras were categorified. The resulting higher representation theory created a new bridge between representation theory andthelinkhomologytheories,whichdespitetheincreasingnumberofpublications in this area is still far from being fully explored. ix