Factorization homology: sigma-models as state-sum TQFTs.

Geometry Topology Seminar
Friday, May 31, 2019 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
David Ayala – Montana State University
Kirsten Wickelgren
Roughly, factorization homology pairs an n-category and an n-manifold to produce a vector space.  Factorization homology is to state-sum TQFTs as singular homology is to simplicial homology: the former is manifestly well-defined (ie, independent of auxiliary choices), continuous (ie, beholds a continuous action of diffeomorphisms), and functorial; the latter is easier to compute.  
Examples of n-categories to input into this pairing arise, through deformation theory, from perturbative sigma-models.  For such n-categories, this state sum expression agrees with the observables of the sigma-model — this is a form of Poincare’ duality, which yields some surprising dualities among TQFTs.  A host of familiar TQFTs are instances of factorization homology; many others are speculatively so.  
The first part of this talk will tour through some essential definitions in what’s described above.  The second part of the talk will focus on familiar manifold invariants, such as the Jones polynomial, as instances of factorization homology, highlighting the Poincare’/Koszul duality result.  The last part of the talk will speculate on more such instances.