QM Masterclass: VOA – Vertex Operator Algebras

Vertex operator algebras (VOAs) first appeared in the 1980's as the rigorous notion of chiral algebras (the symmetry algebras) of two-dimensional conformal quantum field theories. Since then, they have been employed as key ingredients in many modern problems of mathematical physics and pure mathematics, ranging from monstrous moonshine to knot theory and geometry. The older problems have been mostly concerned with the simplest type of VOAs, so‐called rational theories.

In the last few years, it has been realized that VOAs and their representation theories yield rich invariants of three and four‐dimensional supersymmetric quantum field theories. This provides new insights into low‐dimensional topology and the quantum geometric Langlands program. Involved VOAs are however not rational (often called logarithmic) and so their representation theory is rich and exciting.

These lectures will be a very modern introduction to the theory of VOAs. We will use techniques from representation theory (especially Lie theory), geometry and topology; no knowledge of VOAs is needed. The lectures will be a mix of general theory and illustrating it with the most important examples, that is free field theories, affine and W‐algebras; and the school will end with an exposition of the very recent use and appearance of VOAs in physics, geometry, and low‐dimensional topology.

Monday 1 August

10:15‐11:00	Formal calculus and definition of VOA, modules, fields and intertwiners (1/3)
11:15‐12:00	Formal calculus and definition of VOA, modules, fields and intertwiners (2/3)
14:15‐15:00	Formal calculus and definition of VOA, modules, fields and intertwiners (3/3)
15:15‐16:00	Illustrating these in the simplest examples of free field theories ‐ free boson, free fermion, symplectic boson and symplectic fermion (1/2)

Tuesday 2 August

10:15‐11:00	Illustrating these in the simplest examples of free field theories ‐ free boson, free fermion, symplectic boson and symplectic fermion (2/2)
11:15‐12:00	Finiteness conditions (Zhu's algebra and associated varieties) and their implications to representation theory (1/2)
14:15‐15:00	Finiteness conditions (Zhu's algebra and associated varieties) and their implications to representation theory (2/2)
15:15‐16:00	Affine Lie algebras and WZW theories with a focus on admissible levels (1/5)

Wednesday 3 August

10:15‐11:00	Affine Lie algebras and WZW theories with a focus on admissible levels (2/5)
11:15‐12:00	Affine Lie algebras and WZW theories with a focus on admissible levels (3/5)
14:15‐15:00	Affine Lie algebras and WZW theories with a focus on admissible levels (4/5)
15:15‐16:00	Affine Lie algebras and WZW theories with a focus on admissible levels (5/5)

Thursday 4 August

10:15‐11:00	W‐algebras (1/2)
11:15‐12:00	W‐algebras (2/2)
14:15‐15:00	Cosets and Orbifolds (1/2)
15:15‐16:00	Cosets and Orbifolds (2/2)

Friday 5 August

10:15‐11:00	VOAs in physics, geometry and topology (1/4)
11:15‐12:00	VOAs in physics, geometry and topology (2/4)
14:15‐15:00	VOAs in physics, geometry and topology (3/4)
15:15‐16:00	VOAs in physics, geometry and topology (4/4)

Admittance
You have to be admitted to participate in the event. If we cannot verify your affiliation and that you will scientifically benefit from the event, you will be asked to supply documentation.

Included
There is no participant fee.
Included in the event is Monday-Friday lunch + 2 x coffee/tea (mornings/afternoons).

Financial support
You can apply for support for the accommodation. Apply via the registration form below. We will prioritize junior researchers without other means of funding. We are unfortunately not able to offer support for the travel expenses.

Accommodation
We recommend the affordable Hotel Cabinn, which is located right next to the train station (SDU agreement is DKK 412 per night + breakfast. This price may increase by 1-3%).
Alternatively there are many Airbnb options in Odense.

Travel restrictions
There are currently no covid-19 restrictions to enter Denmark. You can see the official updated information here: https://en.coronasmitte.dk/travel-rules/covidtravelrules

Directions
To Odense: You can fly to Copenhagen Airport (CPH) or Billund Airport (BLL) both have 1½ to 2 hours travel time to Odense. From CPH there is a train option directly from the airport to Odense and from BLL all travel to Odense is with min. 1 transfer (bus + train). You can find the connections at: https://rejseplanen.dk/

To SDU: from Odense train station (20 min. by light rail ‘Letbane’ / 5 km): the interval between the rides is 7,5 min. during rush hours and 15 min.  the rest of the day. Tickets can be purchased at the ticket automat at Odense train station & online https://webshop.fynbus.dk/ (choose from “Odense Banegaard” to “SDU”).

At SDU: download the app 'SDU Maps' or search SDU Maps online and enter 'DIAS Auditorium' or 'V24-501a-0'. After admittance to the event you will also receive a pdf-version of the SDU map via email.