Infinitesimal sl(2)-symmetries on the equivariant skein lasagna module
We construct an sl(2)-action on the equivariant skein lasagna module.
math.GT↗
arXiv subjects
Joshua Sussan
Publications and source records attributed to Joshua Sussan.
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 $α$, 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 $α$ where the computational subspace remains positive-definite. We identify special values of $α$ 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 $α$ is not required for efficient gate compilation, significantly enhancing the physical plausibility of non-semisimple anyonic architectures.
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Symmetries of equivariant Khovanov-Rozansky homology
We construct an $\mathfrak{sl}_2$-action on equivariant $\mathfrak{gl}_N$-link homologies. As a consequence we obtain an action of $\mathfrak{sl}_2$ on these homologies as well as a $p$-DG structures for $p$ a prime number. We explore topological applications of these structures.
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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.
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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 $ν= 5/2$ 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.
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Remarks on some infinitesimal symmetries of Khovanov--Rozansky homologies in finite characteristic
We give a new proof of a theorem due to Shumakovitch and Wang on base point independence of Khovanov--Rozansky homology in characteristic $p$. Some further symmetries of $\mathfrak{gl}(p)$-homology in characteristic $p$ are also discussed.
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Symmetries of $\mathfrak{gl}_N$-foams
We give an action of a Lie subalgebra of the Witt algebra on foams. This action is compatible with the $\mathfrak{gl}_N$-foam evaluation formula. In particular, this endows states spaces associated with $\mathfrak{gl}_N$-webs with an $\mathfrak{sl}_2$-action. When working in positive characteristic, this can be used to define a $p$-DG structure on these state spaces.
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A braid group action on an $A_\infty$-category for zigzag algebras
We construct differential graded enhancements of the zigzag algebras which were used by Khovanov, Seidel and Thomas to produce categorical braid group actions. These enhancements are related to $p$-differential graded structures by a version of Koszul duality. We prove that the minimal model $A_\infty$-structure on the zigzag algebras is {\em not} formal. We construct a braid group action in this setting and suggest a symplectic interpretation.
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On some $p$-differential graded link homologies
We show that the triply graded Khovanov-Rozansky homology of knots and links over a field of positive odd characteristic $p$ descends to an invariant in the homotopy category finite-dimensional $p$-complexes. A $p$-extended differential on the triply graded homology discovered by Cautis is compatible with the $p$-DG structure. As a consequence we get a categorification of the Jones polynomial evaluated at an odd prime root of unity
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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.
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A braid group action on a $p$-DG homotopy category
We construct a braid group action on a homotopy category of $p$-DG modules of a deformed Webster algebra.
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p-DG cyclotomic nilHecke algebras
We categorify a tensor product of two Weyl modules for quantum sl(2) at a prime root of unity.
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p-DG cyclotomic nilHecke algebras II
We categorify tensor products of the fundamental representation of quantum $\mathfrak{sl}_2$ at prime roots of unity building upon earlier work where a tensor product of two Weyl modules was categorified.
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A categorification of the colored Jones polynomial at a root of unity
There is a $p$-differential on the triply-graded Khovanov--Rozansky homology of knots and links over a field of positive characteristic $p$ that gives rise to an invariant in the homotopy category finite-dimensional $p$-complexes. A differential on triply-graded homology discovered by Cautis is compatible with the $p$-differential structure. As a consequence we get a categorification of the colored Jones polynomial evaluated at a $2p$th root of unity.
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A Lie theoretic categorification of the coloured Jones polynomial
We use the machinery of categorified Jones-Wenzl projectors to construct a categorification of a type A Reshetikhin-Turaev invariant of oriented framed tangles where each strand is labeled by an arbitrary finite-dimensional representation. As a special case, we obtain a categorification of the coloured Jones polynomial of links.
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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↗
On some $p$-differential graded link homologies II
In arXiv:2009.06498, a link invariant categorifying the Jones polynomial at a $2p$th root of unity, where $p$ is an odd prime, was constructed. This categorification utilized an $N=2$ specialization of a differential introduced by Cautis. Here we give a family of link homologies where the Cautis differential is specialized to a positive integer of the form $N=kp+2$. When $k$ is even, all these link homologies categorify the Jones polynomial evaluated at a $2p$th root of unity, but they are non-isomorphic invariants.
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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.
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