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Register a new userQuantum traces for $mathrm{SL}_n(mathbb{C})$: the case $n=3$
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Daniel C. Douglas
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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.
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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).
We provide an explicit set of algebraically independent generators for the algebra of invariant differential operators on the Riemannian symmetric space associated with $SL_n(R)$.
Generalizations of the AGT correspondence between 4D $mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${mathbb {C}}^2$ with $Omega$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories, $N = 2, 3, ldots$, on ${mathbb {C}}^2/{mathbb {Z}}_n$, $n = 2, 3, ldots$, with $Omega$-deformation and 2D conformal field theories with $mathcal{W}^{, para}_{N, n}$ ($n$-th parafermion $mathcal{W}_N$) symmetry and $widehat{mathfrak{sl}}(n)_N$ symmetry. In this work, we trivialize the factor with $mathcal{W}^{, para}_{N, n}$ symmetry in the 4D $SU(N)$ instanton partition functions on ${mathbb {C}}^2/{mathbb {Z}}_n$ (by using specific choices of parameters and imposing specific conditions on the $N$-tuples of Young diagrams that label the states), and extract the 2D $widehat{mathfrak{sl}}(n)_N$ WZW conformal blocks, $n = 2, 3, ldots$, $N = 1, 2, ldots, .$
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.
Let (N, G), where N is a normal subgroup of G<SL_n(C), be a pair of finite groups and V a finite-dimensional fundamental G-module. We study the G-invariants in the symmetric algebra S(V) by giving explicit formulas of the Poincar{e} series for the induced modules and restriction modules. In particular, this provides a uniform formula of the Poincar{e} series for the symmetric invariants in terms of the McKay-Slodowy correspondence. Moreover, we also derive a global version of the Poincare series in terms of Tchebychev polynomials in the sense that one needs only the dimensions of the subgroups and their group-types to completely determine the Poincare series.
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