Aaron D. Lauda
YOU?
Author Swipe
View article: Density and unitarity of the Burau representation from a non-semisimple TQFT
Density and unitarity of the Burau representation from a non-semisimple TQFT Open
We study the density of the Burau representation from the perspective of a non-semisimple topological quantum field theory (TQFT) at a fourth root of unity. This gives a TQFT construction of Squier’s Hermitian form on the Burau representat…
View article: Action of the Witt algebra on categorified quantum groups
Action of the Witt algebra on categorified quantum groups Open
We construct an action of the positive Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. In the type A case, we show that this action induces an action of the positive Witt algebra on $\mathfrak{gl}_n…
View article: A quantum algorithm for Khovanov homology
A quantum algorithm for Khovanov homology Open
Khovanov homology is a topological knot invariant that categorifies the Jones polynomial, recognizes the unknot, and is conjectured to appear as an observable in $4D$ supersymmetric Yang--Mills theory. Despite its rich mathematical and phy…
View article: Universal quantum computation using Ising anyons from a non-semisimple Topological Quantum Field Theory
Universal quantum computation using Ising anyons from a non-semisimple Topological Quantum Field Theory Open
We propose a framework for topological quantum computation using newly discovered non-semisimple analogs of topological quantum field theories in 2+1 dimensions. These enhanced theories offer more powerful models for quantum computation. T…
View article: Super rewriting theory and nondegeneracy of odd categorified $\mathfrak{sl}_{2}$
Super rewriting theory and nondegeneracy of odd categorified $\mathfrak{sl}_{2}$ Open
We develop the rewriting theory for monoidal supercategories and 2 -supercategories. This extends the theory of higher-dimensional rewriting established for (linear) 2-categories to the super setting, providing a suite of tools for constru…
View article: Categorification of the internal braid group action for quantum groups, I: 2-functoriality
Categorification of the internal braid group action for quantum groups, I: 2-functoriality Open
We define 2-functors on the categorified quantum group of a simply-laced Kac-Moody algebra that induce Lusztig's internal braid group action at the level of the Grothendieck group.
View article: Density and unitarity of the Burau representation from a non-semisimple TQFT
Density and unitarity of the Burau representation from a non-semisimple TQFT Open
We study the density of the Burau representation from the perspective of a non-semisimple TQFT at a fourth root of unity. This gives a TQFT construction of Squier's Hermitian form on the Burau representation with possibly mixed signature. …
View article: Spectral 2-actions, foams, and frames in the spectrification of Khovanov arc algebras
Spectral 2-actions, foams, and frames in the spectrification of Khovanov arc algebras Open
Leveraging skew Howe duality, we show that Lawson-Lipshitz-Sarkar's spectrification of Khovanov's arc algebra gives rise to 2-representations of categorified quantum groups over $\mathbb{F}_2$ that we call spectral 2-representations. These…
View article: Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups
Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups Open
We develop the categorical context for defining Hermitian non-semisimple TQFTs. We prove that relative Hermitian modular categories give rise to modified Hermitian WRT-TQFTs and provide numerous examples of these structures coming from the…
View article: Categorification of the internal braid group action for quantum groups I: 2-functoriality
Categorification of the internal braid group action for quantum groups I: 2-functoriality Open
We define 2-functors on the categorified quantum group of a simply-laced Kac-Moody algebra that induce Lusztig's internal braid group action at the level of the Grothendieck group.
View article: Derived superequivalences for spin symmetric groups and odd sl(2)-categorifications
Derived superequivalences for spin symmetric groups and odd sl(2)-categorifications Open
We show that actions of the odd categorification of sl(2) induce derived superequivalences analogous to those introduced by Chuang and Rouquier. Using Kang, Kashiwara, and Oh's action of the odd 2-category on blocks of the cyclotomic affin…
View article: An Invitation to Categorification
An Invitation to Categorification Open
What is Categorification?Categorification is the process of promoting an algebraic object to one with more structure.All of the structure of the original object is retained and can be accessed via a decategorification procedure that forget…
View article: A Hermitian TQFT from a non-semisimple category of quantum sl(2)-modules
A Hermitian TQFT from a non-semisimple category of quantum sl(2)-modules Open
We endow a non-semisimple category of modules of unrolled quantum sl(2) with a Hermitian structure. We also prove that the TQFT constructed in arXiv:1202.3553 using this category is Hermitian. This gives rise to projective representations …
View article: Super rewriting theory and nondegeneracy of odd categorified sl(2)
Super rewriting theory and nondegeneracy of odd categorified sl(2) Open
We develop the rewriting theory for monoidal supercategories and 2-supercategories. This extends the theory of higher-dimensional rewriting established for (linear) 2-categories to the super setting, providing a suite of tools for construc…
View article: DG structures on odd categorified quantum $sl(2)$
DG structures on odd categorified quantum $sl(2)$ Open
We equip Ellis and Brundan's version of the odd categorified quantum group\nfor sl(2) with a differential giving it the structure of a graded\ndg-2-supercategory. The presence of the super grading gives rise to two\npossible decategorifica…
View article: Curved Rickard complexes and link homologies
Curved Rickard complexes and link homologies Open
Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to…
View article: Categorification in Geometry, Topology, and Physics
Categorification in Geometry, Topology, and Physics Open
We describe a number of geometric contexts where categorification appears\nnaturally: coherent sheaves, constructible sheaves and sheaves of modules over\nquantizations. In each case, we discuss how "index formulas" allow us to easily\nper…
View article: Current algebras and categorified quantum groups
Current algebras and categorified quantum groups Open
We identify the trace, or 0th Hochschild homology, of type ADE categorified\nquantum groups with the corresponding current algebra of the same type. To\nprove this, we show that 2-representations defined using categories of modules\nover c…
View article: An odd categorification of $U_q (\mathfrak{sl}_2)$
An odd categorification of $U_q (\mathfrak{sl}_2)$ Open
We define a 2-category that categorifies the covering Kac–Moody algebra for \mathfrak{sl}_2 introduced by Clark and Wang. This categorification forms the structure of a super-2-category as formulated by Kang, Kashiwara, and Oh. The super-2…