5,879 research outputs found
Ideal webs, moduli spaces of local systems, and 3d Calabi-Yau categories
A decorated surface S is an oriented surface with punctures and a finite set
of marked points on the boundary, such that each boundary component has a
marked point. We introduce ideal bipartite graphs on S. Each of them is related
to a group G of type A, and gives rise to cluster coordinate systems on certain
spaces of G-local systems on S. These coordinate systems generalize the ones
assigned to ideal triangulations of S. A bipartite graph on S gives rise to a
quiver with a canonical potential. The latter determines a triangulated 3d CY
category with a cluster collection of spherical objects. Given an ideal
bipartite graph on S, we define an extension of the mapping class group of S
which acts by symmetries of the category. There is a family of open CY 3-folds
over the universal Hitchin base, whose intermediate Jacobians describe the
Hitchin system. We conjecture that the 3d CY category with cluster collection
is equivalent to a full subcategory of the Fukaya category of a generic
threefold of the family, equipped with a cluster collection of special
Lagrangian spheres. For SL(2) a substantial part of the story is already known
thanks to Bridgeland, Keller, Labardini-Fragoso, Nagao, Smith, and others. We
hope that ideal bipartite graphs provide special examples of the
Gaiotto-Moore-Neitzke spectral networks.Comment: 60 page
Orthogonal polynomials on generalized Julia sets
We extend results by Barnsley et al. about orthogonal polynomials on Julia
sets to the case of generalized Julia sets. The equilibrium measure is
considered. In addition, we discuss optimal smoothness of Green functions and
Parreau-Widom criterion for a special family of real generalized Julia sets.Comment: We changed the second part of the article a little bit and gave
sharper results in this versio
Quantum geometry of moduli spaces of local systems and representation theory
Let G be a split semi-simple adjoint group, and S an oriented surface with
punctures and special boundary points. We introduce a moduli space P(G,S)
parametrizing G-local system on S with some boundary data, and prove that it
carries a cluster Poisson structure, equivariant under the action of the
cluster modular group M(G,S), containing the mapping class group of S, the
group of outer automorphisms of G, and the product of Weyl / braid groups over
punctures / boundary components. We prove that the dual moduli space A(G,S)
carries a M(G,S)-equivariant cluster structure, and the pair (A(G,S), P(G,S))
is a cluster ensemble. These results generalize the works of V. Fock & the
first author, and of I. Le.
We quantize cluster Poisson varieties X for any Planck constant h s.t. h>0 or
|h|=1. First, we define a *-algebra structure on the Langlands modular double
A(h; X) of the algebra of functions on X. We construct a principal series of
representations of the *-algebra A(h; X), equivariant under a unitary
projective representation of the cluster modular group M(X). This extends works
of V. Fock and the first author when h>0.
Combining this, we get a M(G,S)-equivariant quantization of the moduli space
P(G,S), given by the *-algebra A(h; P(G,S)) and its principal series
representations. We construct realizations of the principal series
*-representations. In particular, when S is punctured disc with two special
points, we get a principal series *-representations of the Langlands modular
double of the quantum group Uq(g).
We conjecture that there is a nondegenerate pairing between the local system
of coinvariants of oscillatory representations of the W-algebra and the one
provided by the projective representation of the mapping class group of S.Comment: 199 pages. Minor correction
Orthogonal polynomials for the weakly equilibrium Cantor sets
Let be the weakly equilibrium Cantor type set introduced in [10].
It is proven that the monic orthogonal polynomials with respect to
the equilibrium measure of coincide with the Chebyshev polynomials
of the set. Procedures are suggested to find of all degrees and the
corresponding Jacobi parameters. It is shown that the sequence of the Widom
factors is bounded below
Two Measures on Cantor Sets
We give an example of Cantor type set for which its equilibrium measure and
the corresponding Hausdorff measure are mutually absolutely continuous. Also we
show that these two measures are regular in Stahl-Totik sense
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