Title and Abstract

Title and Abstract

Magdalena Boos (Ruhr University Bochum)

An introduction to symmetric quiver representations

Abstract: When thinking about quiver representations, the fact that the theory only deals with type A setups (i.e. general linear groups and their Lie algebras) can be viewed as a drawback.

Therefore, Derksen and Weyman introduced the notion of a symmetric quiver in 2002 which allows considering classical analogues, so-called symplectic or orthogonal representations (or symmetric quiver representations). Symmetric quiver representations are collected in so-called symmetric representation varieties which are acted on by reductive groups via change of basis.

We give an introduction to the subject, develop and motivate the theory and discuss certain known results, focussing on the orbits of the aforementioned group actions as well as their closures (which correspond to degenerations). In particular, we present results obtained joint with G. Cerulli Irelli.

Pierrick Bousseau(U. Georgia)

Mirror symmetry and enumerative geometry of cluster varieties

Abstract: There are several viewpoints on mirror symmetry for cluster varieties. One of them is the Fock-Goncharov duality, defined combinatorially at the level of cluster seeds. Another one is the Gross-Siebert algebro-geometric construction of mirrors for arbitrary Calabi-Yau varieties and which can be in particular applied to cluster varieties. In the first two lectures, I will review these two perspectives and explain how to relate them. The third lecture will be devoted to enumerative geometry: I will describe how log Gromov—Witten invariants of cluster varieties, which play an important role in the Gross-Siebert mirror construction, are related to Donaldson-Thomas invariants of quivers with potentials. This is joint work with Hulya Arguz.

Roger Casals (UC Davis)

Title Lecture 1: “Cluster algebras and symplectic topology I: How cluster algebras arise in 4D symplectic topology”

Abstract Lecture 1: We will introduce the questions and geometric problems in symplectic topology that lead to considering cluster algebras. The goal in this first lecture is both to provide intuition for why cluster algebras arise in this area and to give a rigorous description of the commutative rings that appear. We will use motivating examples to help with the former objective. For the latter, we will translate rather elaborate categorical matters in symplectic topology into explicit descriptions of affine varieties, using standard Lie-theoretic language. This lecture will include a precise breakdown of the results known to date and a series of applications that follow from this interaction between 4D symplectic topology and cluster algebras.

Title Lecture 2: “Cluster algebras and symplectic topology II: Construction of cluster algebras via the microlocal theory of sheaves”

Abstract Lecture 2: The main goal of this lecture is to show that the coordinate ring of any braid variety is a cluster algebra. The geometry motivated in the first lecture will guide the steps proof, with weaves and their sheaf quantizations being the central ingredient. We present the argument with the following structure. First, we explain how to use weaves to construct an upper cluster algebra structure on open Bott-Samelson varieties. Second, we construct upper cluster algebra structures on braid varieties from open Bott-Samelson varieties through a localization procedure. Finally, we show that these upper cluster algebras are locally acyclic and argue that cyclic rotation induces a quasi-cluster isomorphism.

Title Lecture 3: “Cluster algebras and symplectic topology III: Every cluster seed is induced by a Lagrangian filling”

Abstract Lecture 3: The focus of this last lecture is to show that any cluster seed in the cluster algebras featured in the first two lectures can actually be constructed symplectic geometrically, i.e. it is induced by an embedded exact Lagrangian filling. The main new ingredient for this talk will be the construction of a new quiver with potential. This quiver with potential will be associated to certain collections of curves in surfaces. We will prove several invariance properties as well as its rigidity. In a sense, this talk establishes that the current map from 4D symplectic geometry, studying Lagrangian fillings, to cluster algebras in braid varieties, is surjective. We will also mention how many elements of the cluster modular group, including interesting braid group actions and the square of the Donaldson-Thomas transformation, can be constructed entirely through symplectic geometric means. This series of lectures will end with a list of natural follow-up questions on this topic.

Naoki Fujita (Kumamoto)

Toric degenerations and Newton-Okounkov bodies arising from cluster algebras

Abstract: A toric degeneration is a flat degeneration of a projective variety to a toric variety, which can be used to apply the theory of toric varieties to other projective varieties. The theory of cluster algebras is useful to construct toric degenerations. Indeed, introducing the notion of positive polytopes, Gross-Hacking-Keel-Kontsevich [GHKK] gave a general framework to construct toric degenerations of compactified A-cluster varieties.

A different approach is given by the theory of Newton-Okounkov bodies. A Newton-Okounkov body is a convex body constructed from a projective variety with an ample line bundle and with a higher rank valuation on the function field, which gives a systematic method of constructing toric degenerations of projective varieties. The notions of g-vectors and c-vectors in cluster algebras induce higher rank valuations, and we can systematically construct Newton-Okounkov bodies of compactified A-cluster varieties [FO] and of compactified X-cluster varieties [RW, BCMNC].

In this lecture series, we study Newton-Okounkov bodies arising from cluster structures and discuss their relation with Gross-Hacking-Keel-Kontsevich’s positive polytopes, following Bossinger-Cheung-Magee-Nájera Chávez [BCMNC] and my joint work [FO] with Hironori Oya.

References:

[BCMNC] L. Bossinger, M.-W. Cheung, T. Magee, and A. N'{a}jera Ch'{a}vez, Newton-Okounkov bodies and minimal models for cluster varieties, preprint 2023, arXiv:2305.04903v1.

[FO] N. Fujita and H. Oya, Newton-Okounkov polytopes of Schubert varieties arising from cluster structures, preprint 2020, arXiv:2002.09912v2.

[GHKK] M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 497-608.

[RW] K. Rietsch and L. Williams, Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), no. 18, 3437-3527.

Tsukasa Ishibashi (Tohoku)

Cluster $K_2$-structure on the moduli space of decorated twisted $G$-local systems

Abstract: The moduli space $\mathcal{A}_{G,\Sigma}$ of decorated twisted $G$-local systems on a marked surface $\Sigma$ has been introduced by Fock—Goncharov [FG06] as an ``algebro-geometric avatar” of the higher Teichmüller space. It admits a natural cluster $K_2$-structure: explicit cluster charts are given by Fock—Goncharov [FG06] for type $A_n$; by Le [Le19] for all classical types and $G_2$; and finally by Goncharov—Shen [GS19] for general types, unifying the previous type-by-type constructions. The resulting cluster $K_2$-variety is one of important examples that motivates the Fock—Goncharov duality conjecture.

In the 1st lecture, I will review the geometry of the moduli space $\mathcal{A}_{G,\Sigma}$ and the construction of its cluster $K_2$-structure, following [GS19] and [IOS22]. “Wilson line” is a key geometric notion for the description of its function ring, which is an “arc version” of the monodromy (Wilson loop).

In the 2nd lecture, I will focus on the polygon case, and explain how we can investigate the moduli space $\mathcal{A}_{G,\Sigma}$ via the interpolation of (decorated) flags with respect to a given reduced word of their w-distance. This is a common key notion to understand

  • the mutation sequence for a flip of decorated triangulation;
  • generalized minors of simple Wilson lines;
  • the relation to the braid varieties.

References:

  • [CGGLSS22] R. Casals, E. Gorsky, M. Gorsky, I. Le, L. Shen and J. Simental, Cluster structures on braid varieties, arXiv:2207.11607.
  • [FG06] V. V. Fock and A. B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes 'Etudes Sci., 103 (2006), 1–211.
  • [GS19] A. B. Goncharov and L. Shen, Quantum geometry of moduli spaces of local systems and representation theory, arXiv:1904.10491.
  • [IOS22] T. Ishibashi, H. Oya and L. Shen, $\mathscr{A}=\mathscr{U}$ for cluster algebras from moduli spaces of $G$-local systems, arxiv:2202.03168; to appear in Advances in Mathematics.
  • [Le19] I. Le, Cluster structures on higher Teichmüller spaces for classical groups, Forum of Mathematics, Sigma 7 (2019) e13.

Pages