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Braided tensor categories give rise to 4dimensional topological field theories, extending constructions of CraneYetterKauffman in the case of modular categories. In this paper we apply the mechanism of factorization homology in order to construct and compute the categoryvalued invariants of surfaces which form the twodimensional part of the theory. Starting from modules for the DrinfeldJimbo quantum group Uq(g) we obtain in this way a form of topologically twisted 4dimensional N=4 super YangMills theory, the setting introduced by KapustinWitten for the geometric Langlands program.
For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of Glocal systems) on the surface; these give uniform constructions of a variety of wellknown algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to Uq(g), and from the punctured torus we recover the algebra of quantum differential operators associated to Uq(g). From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for wellknown mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum Dmodules.
For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of Glocal systems) on the surface; these give uniform constructions of a variety of wellknown algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to Uq(g), and from the punctured torus we recover the algebra of quantum differential operators associated to Uq(g). From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for wellknown mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum Dmodules.
Original language  English 

Pages (fromto)  873916 
Number of pages  44 
Journal  Journal of Topology 
Volume  11 
Issue number  4 
Early online date  3 Aug 2018 
DOIs  
Publication status  Published  Dec 2018 
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Dive into the research topics of 'Integrating quantum groups over surfaces'. Together they form a unique fingerprint.Projects
 1 Finished

QuantGeomLangTFT  The Quantum Geometric Langlands Topological Field Theory
1/06/15 → 31/05/21
Project: Research
Profiles

David Jordan
 School of Mathematics  Personal Chair of Categorical Symmetry
Person: Academic: Research Active