Cluster Varieties and Mathematical Physics

Toric degenerations of Grassmannians

Abstract: Toric degenerations arise in variuos contexts such as tropical geometry, from cluster algebras, and representation theory. In this talk I will compare for Gr(2,n) two construction of such: the first from the tropical Grassmannian as introduced by Speyer and Sturmfels and the second from the cluster structure realized combinatrially through plabic graphs as studied by Rietsch and Williams. The main result is that toric degenerations from the cluster algebra approach can be realized in the tropical geometry setting. This talk is based on joint work with X.Fang, G.Fourier, M.Hering and M.Lanini (arXiv:1612.03838)

Quiver representations and theta functions

Abstract: Scattering diagrams theta functions and broken lines were developed in order to describe toric degenerations of Calabi-Yau varieties and construct mirror pairs. Later, Gross-Hacking-Keel-Kontsevich unravel the relation of those objects with cluster algebras. In the talk, we will discuss how we can combine the representation theory with these objects. We will also see how the broken lines on scattering diagram give a stratification of quiver Grassmannians using this setting.

Cluster structures on Poisson-Lie groups

Abstract: The goal of the mini-course is to use Poisson geometry approach to construct a family of non-isomorphic (generalized) cluster structures on GL(n). To this end we will

• briefly recall the basic facts on Poisson brackets and Hamiltonian flows, review the notion of a Poisson bracket compatible with a cluster algebra structure and illustrate its utility by describing a Hamiltonian approach to verify dilogarithm identities in cluster algebras;
• provide a motivation for and various versions and examples of generalized cluster structures;
• review a construction of a generalized cluster structure in the Drinfeld double of GL(n) and of exotic cluster structures in GL(n) compatible with Poisson-Lie brackets covered by the Belavin-Drinfeld classification.

Toric degenerations of X-cluster varieties

Abstract:In this talk I’ll describe how scattering diagrams give rise to partial compactifications of X-cluster varieties, and toric degenerations of these partial compactifications. Scattering diagrams are an important tool in the Gross-Siebert mirror symmetry program. They were introduced to the cluster community by Gross, Hacking, Keel, and Kontsevich, who used scattering diagram techniques to prove several conjectures about cluster algebras (e.g. positivity). A portion of the scatte-ring diagram (called the cluster complex) has a fan structure in which each maximal cone is spanned by the g-vectors of the A-cluster variables of some seed. Fock and Goncharov used this structure to define a partial compactification of the dual X-variety (which they call the special completion of X) and gave the boundary components a clean Teichmüller theory interpretation. By introducing coefficients carefully, for each seed we get a toric degeneration of the special completion of the X-variety, where the toric central fiber is given by the fan associated to the cluster complex by the choice of seed. X-variables extend canonically to functions on the total space. The total space has a natural torus action, and the extended X-variables are eigenfunctions of this action whose weight is precisely the c-vector of the X-variable. Time permitting, I will also discuss how this construction relates to Nakanishi and Zelevinsky’s tropical duality of g- and c-vectors. This is based on ongoing work with Lara Boßinger, Juan Bosco Frías Medina, and Alfredo Nájera Chávez.

Cluster algebras, SL(N,C) flat connections and 3-manifolds

Abstract: SL(N,C) Chern-Simons theory have been getting some increasing interest in math and physics in the last years. I'll describe some recently developed techniques inspired by string theoretical constructions, that makes use of the connection between higher Teichmuller theory and cluster algebras to produce topological invariants of hyperbolic 3-manifolds. As time allows I'll talk about connections with other subjects in this context. They include volume conjecture, defects in quantum field theory and spectral networks.

Cluster structures on Poisson-Lie groups

Abstract: The goal of the mini-course is to use Poisson geometry approach to construct a family of non-isomorphic (generalized) cluster structures on GL(n). To this end we will

• briefly recall the basic facts on Poisson brackets and Hamiltonian flows, review the notion of a Poisson bracket compatible with a cluster algebra structure and illustrate its utility by describing a Hamiltonian approach to verify dilogarithm identities in cluster algebras;
• provide a motivation for and various versions and examples of generalized cluster structures;
• review a construction of a generalized cluster structure in the Drinfeld double of GL(n) and of exotic cluster structures in GL(n) compatible with Poisson-Lie brackets covered by the Belavin-Drinfeld classification.

On Cluster Duality

Cluster Duality I: Grassmannian and Cyclic Sieving. The Grassmannian Gr(k,n) parametrizes k-dimensional subspaces in Cⁿ. Due to work of Scott, its homogenous coordinate ring C[Gr(k,n)] is a cluster algebra of geometric type. In the first talk, we introduce a periodic configuration space X(k,n) equipped with a natural potential function W. We prove that the topicalization of (X(k,n), W) canonically parametrizes a linear basis of C[Gr(k,n)], as expected by the Duality Conjecture of Fock-Goncharov. We identify the tropical set of (X(k,n), W) with the set of plane partitions. As an application, we show a cyclic sieving phenomenon involving the latter. This is joint work with Jiuzu Hong and Daping Weng.

Cluster Duality II: Configuration space and representation theory. In the second talk, we will focus on the cluster theory of configuration spaces of decorated flags. The tropical sets of the latter spaces parametrize top components of the affine Grassmannian convolution varieties. By the geometric Satake Correspondence, they parametrize bases in the tensor invariants of representations of the Langlands dual groups. This is joint work with Alexander Goncharov.

Cluster Duality III: Donaldson-Thomas transformation of Higher Teichmuller space. In the third talk, we will focus on the cluster theory of higher Teichmuller spaces. We determine the Donaldson-Thomas transformations for higher Teichmuller spaces. Combining with the work of Gross-Hacking-Keel-Kontsevich, we prove the Duality Conjecture for higher Teichmuller spaces. This is joint work with Alexander Goncharov.

Theta functions on moduli spaces of local systems

Abstract: Moduli spaces of local systems on Riemann surfaces have a cluster structure due to Fock and Goncharov. Further they carry a canonical basis of so-called theta functions by work of Gross-Hacking-Keel-Kontsevich.

We will study this theta function basis from both the representation theoretic and geometric viewpoints. We will focus on the lowest dimensional examples of such moduli spaces, where we can identify theta functions with natural functions of the monodromy data of the local systems.

Renormalized R-matrices and Chiral Categories

Abstract: A unifying structure in monoidal categorifications of cluster algebras is the notion of a renormalized r-matrix, which generalizes the notion of a braiding on a monoidal category. These r-matrices first appeared in work of Kashiwara on finite-dimensional representations of quantum affine algebras, and later in the setting of KLR algebras in the proof of the dual canonical basis conjecture by Kang-Kashiwara-Kim-Oh. We explain how renormalized r-matrices exist in any monoidal category which appears as a fiber of a monoidal chiral category over the affine line. We discuss how this construction leads to an identification of the coherent Satake category (the category of perverse coherent sheaves on the affine Grassmannian) as a monoidal quantum cluster categorification. We further explain how the construction leads to a new conceptual explanation for the appearance of cluster structures on algebras of loop operators in 4d N=2 theories and their holomorphic-topological twists.