Knot invariants and higher representation theory

Abstract

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl 2 and sl 3 and by Mazorchuk-Stroppel and Sussan for sl n .Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is sl n , we show that these categories agree with certain subcategories of parabolic category O for gl k .We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory.The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius.In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

“…(Such affine braid group actions have played a central role in constructions of knot homology, both in mathematics and physics, cf. [54][55][56][57][58][59].) In the 2d reductions of 3d gauge theories that we study in section 7, two commuting braid-group actions will appear.…”

“…In the SQED example of section 7.4.1, with charge matrices (6.4), the three vacua lead to matrices 58) and central charges…”

“…the basic idea in [136], all ultimately tracing back to the physics of M2-M5 and related M5-M5 brane systems from [94,95].) In the mathematics literature, braid actions on categories D b O H and D b O C have also been used to construct knot homology [57,58], cf. the related [54][55][56].…”

Transitive $2$-representations of finitary $2$-categories

“…More specifically, we study finitary 2-categories over an algebraically closed field which include the 2-category of Soergel bimodules associated to a finite Coxeter system (see [BG, So, EW]), an exhaustive family of quotients of 2-Kac-Moody algebras (see [BFK,KL,Ro1,CL,We]), quiver 2-categories constructed in [Xa] and the 2-category of projective functors on the module category of a finite dimensional algebra (see [MM1]). We define a new class of 2-representations for such 2-categories which we call simple transitive 2-representations and which we believe serves as the correct 2-analogue for the class of irreducible representations of an algebra.…”

On generalized category $$\mathcal {O}$$ for a quiver variety

“…A]; it is generated by the skyscraper sheaves F i on E 0 α i {0}/C * . Since this action can also be interpreted as convolution with Harish-Chandra bimodules by [Weba,3.3], they preserve the set of modules supported on any system of subvarieties closed under convolution with Nakajima's Hecke correspondence Z. Thus, applying this to M λ µ;0 , we have:…”