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Register a new userPoints of quantum $mathrm{SL}_n$ coming from quantum snakes
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Authors
Daniel C. Douglas
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We show that the quantized Fock-Goncharov monodromy matrices satisfy the relations of the quantum special linear group $mathrm{SL}_n^q$. The proof employs a quantum version of the technology invented by Fock-Goncharov called snakes. This relationship between higher Teichmuller theory and quantum group theory is integral to the construction of a $mathrm{SL}_n$-quantum trace map for knots in thickened surfaces, developed in a companion paper (arXiv:2101.06817).
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We generalize Bonahon and Wongs $mathrm{SL}_2(mathbb{C})$-quantum trace map to the setting of $mathrm{SL}_3(mathbb{C})$. More precisely, for each non-zero complex number $q$, we associate to every isotopy class of framed oriented links $K$ in a thickened punctured surface $mathfrak{S} times (0, 1)$ a Laurent polynomial $mathrm{Tr}_lambda^q(K) = mathrm{Tr}_lambda^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov coordinates $X_i$ for a higher Teichm{u}ller space, depending on the choice of an ideal triangulation $lambda$ of the surface $mathfrak{S}$. Along the way, we propose a definition for a $mathrm{SL}_n(mathbb{C})$-version of this invariant.
For a finite-type surface $mathfrak{S}$, we study a preferred basis for the commutative algebra $mathbb{C}[mathcal{X}_{mathrm{SL}_3(mathbb{C})}(mathfrak{S})]$ of regular functions on the $mathrm{SL}_3(mathbb{C})$-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface $mathfrak{S}$. We show that this basis can be naturally indexed by positive integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock-Goncharov, to the tropical points at infinity of the dual version of the character variety.
We construct smooth concordance invariants of knots which take the form of piecewise linear maps from [0,1] to R, one for each n greater than or equal to 2. These invariants arise from sl(n) knot cohomology. We verify some properties which are analogous to those of the invariant Upsilon (which arises from knot Floer homology), and some which differ. We make some explicit computations and give some topological applications. Further to this, we define a concordance invariant from equivariant sl(n) knot cohomology which subsumes many known concordance invariants arising from quantum knot cohomologies.
Let $p$ be a prime number. We prove that the $P=W$ conjecture for $mathrm{SL}_p$ is equivalent to the $P=W$ conjecture for $mathrm{GL}_p$. As a consequence, we verify the $P=W$ conjecture for genus 2 and $mathrm{SL}_p$. For the proof, we compute the perverse filtration and the weight filtration for the variant cohomology associated with the $mathrm{SL}_p$-Hitchin moduli space and the $mathrm{SL}_p$-twisted character variety, relying on Grochenig-Wyss-Zieglers recent proof of the topological mirror conjecture by Hausel-Thaddeus. Finally we discuss obstructions of studying the cohomology of the $mathrm{SL}_n$-Hitchin moduli space via compact hyper-Kahler manifolds.
The ribbon cocycle invariant is defined by means of a partition function using ternary cohomology of self-distributive structures (TSD) and colorings of ribbon diagrams of a framed link, following the same paradigm introduced by Carter, Jelsovsky, Kamada, Langfor and Saito in Transactions of the American Mathematical Society 2003;355(10):3947-89, for the quandle cocycle invariant. In this article we show that the ribbon cocycle invariant is a quantum invariant. We do so by constructing a ribbon category from a TSD set whose twisting and braiding morphisms entail a given TSD $2$-cocycle. Then we show that the quantum invariant naturally associated to this braided category coincides with the cocycle invariant. We generalize this construction to symmetric monoidal categories and provide classes of examples obtained from Hopf monoids and Lie algebras. We further introduce examples from Hopf-Frobenius algebras, objects studied in quantum computing.
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