This conference will consist of SGU lectures for graduate students, mini-courses and talks.
Invited Speakers:
Place: Room 127, Math Department, Kyoto Univeristy.
Program:
July 21 (Tue)
10:30--11:30 Duncan: K3 Surfaces, Mock Modularity and Moonshine I
13:30--14:30 Linshaw: Orbifolds and cosets via invariant theory
15:00--17:30 Malikov: Introduction to algebras of chiral differential operators (SGU lecture) I
July 22 (Wed)
09:50--10:50 Duncan: K3 Surfaces, Mock Modularity and Moonshine II
11:00--12:00 Matsuo: Classification of vertex operator algebras of class $\mathcal{S}^4$ with minimal conformal weight one
13:30--14:30 Zhu: Automorphic Forms on Loop Groups I
15:00-17:30 Malikov: Introduction to algebras of chiral differential operators (SGU lecture) II
July 23 (Thu)
09:50--10:50 Duncan: K3 Surfaces, Mock Modularity and Moonshine III
11:00--12:00 Zhu: Automorphic Forms on Loop Groups II
13:30--16:00 Malikov: Introduction to algebras of chiral differential operators (SGU lecture) III
16:30--17:30 Kuwabara: Sheaves of asymptotic chiral differential operators on symplectic resolutions
18:30 --- conference dinner ----waten---
July 24 (Fri)
10:30--11:30 Zhu: Automorphic Forms on Loop Groups III
13:30--14:30 Kawasetsu: W-algebras with non-admissible levels and the Deligne exceptional series
15:00--17:30 Malikov: Introduction to algebras of chiral differential operators (SGU lecture) IV
Abstracts:
John Duncan, K3 Surfaces, Mock Modularity and Moonshine.
Perhaps the most surprising aspect of the 2010 Mathieu moonshine observation of Eguchi--Ooguri--Tachikawa is its suggestion of deep connections between K3 surfaces, mock modular forms, and sporadic simple groups. The goal of these lectures is to explain some recent results which concretely intertwine these topics.
In the first lecture we will introduce the basic features of the various moonshine phenomena involved, including Mathieu moonshine, umbral moonshine, and moonshine for the Conway group, as formulated by Conway--Norton.
In the second lecture we will explain joint work with M. Cheng which uses a certain version of the Shimura lift to establish a genus zero classification of the mock modular forms of umbral moonshine.
In the third lecture we will explain joint work with S. Mack-Crane which illuminates the role of sporadic groups, and moonshine, in K3 surface geometry, by reinterpreting certain data on derived categories of coherent sheaves in vertex algebraic terms.
Kazuya Kawasetsu, W-algebras with non-admissible levels and the Deligne exceptional series.
In this talk, structure (branching rule) of certain simple $\mathcal{W}$-algebras assocated with the Deligne exceptional Lie algebras and non-admissible levels are described as the simple current extensions of certain vertex operator algebras.
As an application, the $C_2$-cofiniteness and $\mathbb{Z}_2$-rationality of the algebras are proved.
Since $\mathcal{W}$-algebras has been conjectured to be $C_2$-cofinite and rational only if the level $k$ is admissible, they are new example of $C_2$-cofinite rational $\mathcal{W}$-algebras.
Toshiro Kuwabara, Sheaves of asymptotic chiral differential operators on symplectic resolutions.
In this talk, we discuss sheaves of (h-adic) vertex algebras on symplectic manifolds, which give quantization of vertex Poisson algebras of their Jet bundles. On each formal coordinate, these sheaves are isomorphic to the vertex algebra of a formal beta-gamma system and we can determine the Lie algebra of derivations. Using Harish-Chandra extensions, we consider the classification of such sheaves. Such sheaves include localization of affine W-algebras which were constructed by Arakawa, Malikov and the speaker. Moreover, they include quantization of Jet bundles of hypertoric varieties and Nakajima quiver varieties. We also discuss construction of such quantization by semi-infinite reduction.
Andrew Linshaw, Orbifolds and cosets via invariant theory.
The orbifold and coset constructions are standard ways to create new vertex algebras from old ones. It is believed that orbifolds and cosets will inherit nice properties such as strong finite generation, C_2-cofiniteness, and rationality, but few general results of this kind are known. I will discuss how these problems can be studied systematically using ideas from classical invariant theory. This is based on joint work with T. Creutzig.
Fedor Malikov, Introduction to algebras of chiral differential operators (SGU lecture).
In these lectures we shall cover the following.
(1) Ordinary algebras of differential operators. Applications to representation theory.
(2) The Beilinson-Drinfeld categories: Lie* algebras, coisson algebras, chiral algebras. The purpose of this part will be to establish a link to the conventional vertex algebra theory.
(3) Algebras of chiral differential operators.
(4) Applications.
Atsushi Matsuo, Classification of vertex operator algebras of class $\mathcal{S}^4$ with minimal conformal weight one.
Yongchang Zhu, Automorphic Forms on Loop Groups.
The purpose of these lectures is to survey some of results about autmorphic forms on loop groups and explain our recent work on the theta lifting for loop groups.
The study of automorphic forms on loop groups was initiated by H. Garland in 1980s. The first lecture will start from his early results on arithmetic quotients in 1979 and the results about Eisenstein series in 2006. Then we will survey some of recent results in the subject.
The second lecture will concentrate on the Weil representation and theta functional of loop symplectic groups. We will discuss its connection with $\beta-\gamma$ vertex operator algebras.
The third lecture we will explain our on-going work on theta lifting of classical automorphic forms to loop groups.
Acknowledgments: This conference is supported by KYOTO UNIVESITY SGU, JSPS KAKENHI Grant Numbers 26610006.
organizer: Tomoyuki Arakawa