Title&Abstract

Juliet Cooke : The Skein Theory of Surfaces and Factorisation Homology

Skein algebras are classical invariants of surfaces based on considering ribbon graphs in thickened surfaces subject to some skein relations which are determined by the representations of quantum groups. In these lectures, I will discuss how to fit the skein theory of surfaces within the framework of factorisation homology. To do this I shall consider the classical skein algebra as an endomorphism algebra in Walker--Johnson-Freyd's skein category. I will describe my work showing that the skein category functor is the k-linear factorisation homology of surfaces with coefficient system given by the representations of the quantum group which determines the skein relations. By combining this with the work of Ben-Zvi--Brouchier--Jordan--Gunningham--Safronov, you get that the internal skein algebra for any quantum group with generic parameter is the Alekseev moduli algebra, which is constructed out of copies of the reflection equation algebra depending on the topology of the surface. Finally, I will sketch how Le's stated skein algebra of a punctured sphere with generic parameters is isomorphic to the higher rank Askey-Wilson algebra (joint work with Abel Lacabanne).

The tentative plan of the lectures is as follows:

Lecture 1: Factorisation homology and categorical background

Lecture 2: Skein categories and internal skein categories

Lecture 3: Skein categories as factorisation homology and Alekseev moduli algebras

Lecture 4: Stated skein algebras and Askey-Wilson algebras

References:

[Co19] J. Cooke, Excision of Skein Categories and Factorisation Homology, arXiv:1910.02630.

[BBJ18] D. Ben-Zvi, A. Brochier, and D. Jordan, Integrating quantum groups over surfaces, Journal of Topology 11 (2018), no.4, 874--917.

[GJS19] S. Gunningham, D. Jordan, and P. Safronov, The finiteness conjecture for skein modules, arXiv:1908.05233.

[DC19] H. De Clercq, Higher rank relations for the Askey-Wilson and q-Bannai-Ito algebra, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications 15 (2019), Paper No. 099, 32 pp.

Julien Korinman : Representations of skein and stated skein algebras

In these lectures, I will introduce skein and stated skein algebras in the SL_2 case, and review some recent results towards the classification of their finite dimensional weight modules.

Lecture 1: Stated skein algebras

In the first lecture, I will introduce skein and stated skein algebras and motivate the study of their representations by applications in topological and hyperbolic quantum field theories. I will present the basic properties of stated skein algebras which are their behaviour for the gluing and fusion operations and review Bonahon--Wong's quantum trace.

Lecture 2: Azumaya loci and Poisson orders

In the second lecture, I will present some tools towards the classifications of skein representations which are the Azumaya locus and the theory of Poisson orders. These tools works efficiently to study the representation theory of quantum enveloping algebras, quantum groups, quantum tori and stated skein algebras.

Lecture 3: Character varieties

In the third lecture, I will introduce Bonahon--Wong's Frobenius morphism for (stated) skein algebras which, together with the theory of Poisson orders, relates the classification of weight representations of (stated) skein algebras to the classifications of the symplectic leaves of (relative) character varieties. We will thus study the Poisson geometry of these moduli spaces.

Lecture 4: Towards the classification of representations of (stated) skein algebras

In the last lecture, we will put all previous results together to obtain some partial results towards the classification of skein algebras. In particular, we will review recent results of Frohman--Le--Kania-Bartoszynska and of Ganev--Jordan--Safronov. I will end the lectures by listing some open problems on the subject.

References:

[BG03] K. Brown and I. Gordon, Poisson orders, symplectic reflection algebras and representation theory, Journal fur die Reine und Angewandte Mathematik 559 (2003), 193--216.

[BW11] F. Bonahon and H. Wong, Quantum traces for representations of surface groups in SL_2(C), Geometry & Topology 15 (2011),1569--1615.

[BW16] F. Bonahon and H. Wong, Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations, Inventiones Mathematicae 204 (2016), 195--243.

[FKL19] C. Frohman, J. Kania-Bartoszynska, and T.T.Q. Le, Unicity for representations of the Kauffman bracket skein algebra, Inventiones Mathematicae 215 (2019), no. 2, 609--650.

[GJS19] I. Ganev, D. Jordan, and P. Safranov, The quantum Frobenius for character varieties and multiplicative quiver varieties, arXiv:1901.11450.

[Le18] T.T.Q. Le, Triangular decomposition of skein algebras, Quantum Topology 9 (2018), 591--632.

Daniel Douglas : Quantum traces in higher Teichmüller theory

The quantum trace map is a bridge between quantum topology and quantum geometry. More precisely, this mapping relates two quantizations of the SL(n) character variety of a punctured surface: the first, more topological, is the skein algebra of the surface; and the other, more geometric, is the quantum higher Teichmüller space due to Fock--Goncharov.

In this mini-course, we begin by discussing SL(n) character varieties and how to compute higher rank “classical traces”—traces of certain monodromies of curves in surfaces—via Fock-Goncharov coordinates. We then introduce the two quantizations mentioned above, and show how to compute higher rank “quantum traces” of links/webs in thickened surfaces, generalizing the strategy of Bonahon--Wong for SL(2). If time permits, we will touch on relations to Fock--Goncharov duality and Gaiotto--Moore--Neitzke spectral networks.

References:

[BW11] F. Bonahon and H. Wong, Quantum traces for representations of surface groups in SL_2(C), Geom. Topol. 15 (2011), no. 3, 1569–1615.

[Dou21] D. C. Douglas, Quantum traces for SL_n(C): the case n=3. arXiv:2101.06817.

[FG06] V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. No. 103 (2006), 1–211.

[FG09] V. Fock and A. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 6, 865–930.

[GMN14] D. Gaiotto, G. W. Moore, and A. Neitzke, Spectral networks and snakes, Ann. Henri Poincaré 15 (2014), no. 1, 61–141.

[Kim20] H. Kim, SL_3-laminations as bases for PGL_3 cluster varieties for surfaces, arXiv:2011.14765.

[NY21] A. Neitzke and F. Yan, The quantum UV-IR map for line defects in gl(3)-type class S theories. arXiv:2112.03775.