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Aaron D. Lauda

Publications and source records attributed to Aaron D. Lauda.

At least 19 recordsLinked to original sources
arXiv2026

Spectral geometry of Khovanov Laplacians

For an oriented link diagram $D$, the Khovanov cochain complex carries a canonical Hermitian inner product that defines a combinatorial Hodge Laplacian. Its kernel is naturally isomorphic to rational Khovanov cohomology, while its positive spectrum depends on the chosen diagram. Building on the numerical work of Jones and Wei, we develop a structural study of this diagram-dependent higher spectrum. In the minimal and maximal cube degrees, we identify the Khovanov Laplacian with signless Laplacians of explicit weighted graphs, up to diagonal sign conjugation in maximal degree. The graph model describes rational Khovanov classes and harmonic representatives through its bipartite components and gives inverse-polynomial gap bounds in fixed-width bands near the extremal $q$-degrees when the corresponding rational cohomology vanishes. It also yields exact $\Theta(N^{-2})$ bidegree gaps for twisted unknots, odd $(2,N)$-torus knots, and even twist knots. The exact gap calculations are motivated in part by the spectral-resolution requirement in a recent quantum algorithm for Khovanov homology. We also prove a finite-dimensional analytic--combinatorial torsion correspondence for the Khovanov complex showing that an alternating product of nonzero Laplacian pseudodeterminants recovers integral Khovanov torsion in rationally acyclic $q$-degrees and differs from it by an explicit regulator in general. Finally, we transfer Lee's deformation to a filtered differential on the harmonic Khovanov subspace, giving a canonical harmonic model for the Lee spectral sequence.

math.GT↗
arXiv2026

Analytic Torsion and Spectral Gap Capture Persistent-Laplacian Performance

While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales. We propose a compact spectral representation that distills the persistent Laplacian into three mathematically grounded invariants: Betti numbers, the spectral gap, and analytic torsion. Across benchmark datasets including MNIST, QM-3D, and SKEMPI WT, we demonstrate that this reduced feature space captures the essential predictive signal of the full spectrum, and in some cases outperforms it, while significantly reducing computational overhead and preventing the noise introduced by higher-frequency eigenvalues. Our results suggest that these invariants provide a principled, fixed-length interface between spectral geometry and topological learning.

cs.LG↗
arXiv2025

Robust Universal Braiding with Non-semisimple Ising Anyons

Non-semisimple extensions of the Ising anyon model developed in our previous work enable universal topological quantum computation via braiding alone, overcoming the Clifford-only limitation of semisimple theories. The non-semisimple theory provides new anyon types indexed by a real parameter $\alpha$, the neglecton. Braiding acts unitarily with respect to an indefinite Hermitian form, while the computational subspace sits in a positive-definite sector. We demonstrate that this universality is robust, persisting over an open interval of the neglecton parameter $\alpha$ where the computational subspace remains positive-definite. We identify special values of $\alpha$ where the physical subspace decouples exactly from negative-norm components, ensuring fully unitary evolution and suppressed leakage. We further present an alternative encoding supporting exact single-qubit Clifford gates alongside a non-Clifford phase gate. We show that high-precision tuning of $\alpha$ is not required for efficient gate compilation, significantly enhancing the physical plausibility of non-semisimple anyonic architectures.

quant-ph↗
arXiv2025

Action of the Witt algebra on categorified quantum groups

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 $\mathfrak{gl}_n$-foams, recovering the action of Qi, Robert, Sussan, and Wagner. We also show that this construction is compatible with the trace decategorification, inducing the action of the positive Witt algebra on the current algebra.

math.QA↗
arXiv2024

Universal quantum computation using Ising anyons from a non-semisimple Topological Quantum Field Theory

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. The conventional theory of Ising anyons, which is believed to describe excitations in the $\nu = 5/2$ fractional quantum Hall state, is not universal for quantum computation via braiding of quasiparticles. However, we show that the non-semisimple theory introduces new anyon types that extend the Ising framework. By adding just one new anyon type, universal quantum computation can be achieved through braiding alone. This result opens new avenues for realizing fault-tolerant quantum computing in topologically ordered systems.

quant-ph↗
arXiv2024

Density and unitarity of the Burau representation from a non-semisimple TQFT

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. We prove that the image of the braid group in the space of possibly indefinite unitary representations is dense. We also argue for the potential applications of non-semisimple TQFTs toward topological quantum computation.

math.GT↗
arXiv2023

Derived superequivalences for spin symmetric groups and odd sl(2)-categorifications

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 affine Hecke-Clifford algebra, our equivalences imply that blocks related by a certain affine Weyl group action are derived equivalent. By recent results of Kleshchev and Livesey, we show this implies Broue's abelian defect conjecture for the modular representations of the spin symmetric group.

math.RT↗
arXiv2023

Current algebras and categorified quantum groups

We identify the trace, or 0th Hochschild homology, of type ADE categorified quantum groups with the corresponding current algebra of the same type. To prove this, we show that 2-representations defined using categories of modules over cyclotomic (or deformed cyclotomic) quotients of KLR-algebras correspond to local (or global) Weyl modules. We also investigate the implications for centers of categories in 2-representations of categorified quantum groups.

math.QA↗
arXiv2023

Strands algebras and the affine highest weight property for equivariant hypertoric categories

We show that the equivariant hypertoric convolution algebras introduced by Braden-Licata-Proudfoot-Webster are affine quasi hereditary in the sense of Kleshchev and compute the Ext groups between standard modules. Together with the main result of arXiv:2009.03981, this implies a number of new homological results about the bordered Floer algebras of Ozsvath-Szabo, including the existence of standard modules over these algebras. We prove that the Ext groups between standard modules are isomorphic to the homology of a variant of the Lipshitz-Ozsvath-Thurston bordered strands dg algebras.

math.RT↗
arXiv2022

Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups

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 representation theory of quantum groups and quantum superalgebras. The Hermitian theory developed here for the modified Turaev-Viro TQFT is applied to define new pseudo-Hermitian topological phases that can be considered as non-semisimple analogs of Levin-Wen models.

math.QA↗
arXiv2021

Pseudo-Hermitian Levin-Wen models from non-semisimple TQFTs

We construct large classes of exactly solvable pseudo-Hermitian 2D spin Hamiltonians. The ground states of these systems depend only on the spatial topology of the system. We identify the ground state system on a surface with the value assigned to the surface by a non-semisimple TQFT generalizing the Turaev-Viro model. A non-trivial example arises from a non-semisimple subcategory of representations of quantum sl(2) where the quantum parameter is specialized to a root of unity.

cond-mat.str-el↗
arXiv2021

A Hermitian TQFT from a non-semisimple category of quantum sl(2)-modules

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 of the mapping class group in the group of indefinite unitary matrices.

math.QA↗
arXiv2021

Super rewriting theory and nondegeneracy of odd categorified sl(2)

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 constructing bases and normal forms for 2-supercategories given by generators and relations. We then employ this newly developed theory to prove the non-degeneracy conjecture for the odd categorification of quantum sl(2) from arXiv:1307.7816 and arXiv:1701.04133. As a corollary, this gives a classification of dg-structures on the odd 2-category conjectured in arXiv:1808.04924.

math.QA↗
arXiv2020

DG structures on odd categorified quantum sl(2)

We equip Ellis and Brundan's version of the odd categorified quantum group for sl(2) with a differential giving it the structure of a graded dg-2-supercategory. The presence of the super grading gives rise to two possible decategorifications of the associated dg-2-category. One version gives rise to a categorification of quantum sl(2) at a fourth root of unity, while the other version produces a subalgebra of quantum gl(1|1) defined over the integers. Both of these algebras appear in connection with quantum algebraic approaches to the Alexander polynomial.

math.QA↗
arXiv2020

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

We give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement, using the algebras defined in arXiv:0905.1335 from the equivariant cohomology of toric varieties. We prove this conjecture for cyclic arrangements by showing that these algebras are isomorphic to algebras appearing in work of Ozsvath-Szabo arXiv:1603.06559 in bordered Heegaard Floer homology arXiv:0810.0687. The proof of our conjecture in the cyclic case extends work of Karp-Williams arXiv:1608.08288 on sign variation and the combinatorics of the m=1 amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of gl(1|1).

math.GT↗
arXiv2019

Ozsvath-Szabo bordered algebras and subquotients of category O

We show that Ozsváth-Szabó's bordered algebra used to efficiently compute knot Floer homology is a graded flat deformation of the regular block of a $\mathfrak{q}$-presentable quotient of parabolic category $\mathcal{O}$. We identify the endomorphism algebra of a minimal projective generator for this block with an explicit quotient of the Ozsváth-Szabó algebra using Sartori's diagrammatic formulation of the endomorphism algebra. Both of these algebras give rise to categorifications of tensor products of the vector representation $V^{\otimes n}$ for $U_q(\mathfrak{gl}(1|1))$. Our isomorphism allows us to transport a number of constructions between these two algebras, leading to a new (fully) diagrammatic reinterpretation of Sartori's algebra, new modules over Ozsváth-Szabó's algebra lifting various bases of $V^{\otimes n}$, and bimodules over Ozsváth-Szabó's algebra categorifying the action of the quantum group element $F$ and its dual on $V^{\otimes n}$.

math.QA↗
arXiv2019

Parameters in categorified quantum groups

In this note we give explicit isomorphisms of 2-categories between various versions of the categorified quantum group associated to a simply-laced Kac-Moody algebra. These isomorphisms are convenient when working with the categorified quantum group. They make it possible to translate results from the gl(n) variant of the 2-category to the sl(n) variant and transfer results between various conventions in the literature. We also extend isomorphisms of finite type KLR algebras for different choices of parameters to the level of 2-categories.

math.QA↗