Categorified Coulomb Branches

Workshop on Algebra and Representation Theory, Held on Oregonian Grounds

July 10-14, 2023
University of Oregon
Eugene, OR

Coulomb branches are certain moduli spaces arising in supersymmetric field theory. They include as special cases many spaces of independent interest in geometry and representation theory. In the four-dimensional case, one expects that their coordinate rings should be categorified (lifted) to tensor categories that carry a lot of interesting structure. The physics literature, in particular work of Kapustin-Saulina and Gaiotto-Moore-Neitzke, provides various clues about these categories, but not a precise definition. However, recent advances have led to a satisfactory mathematical definition in an important special case (gauge theories of cotangent type), building on work of Braverman-Finkelberg-Nakajima.

In this workshop we will explore the structure of these categorified Coulomb branches. The simplest (!) example of a categorified Coulomb branch is the coherent Satake category, that is, the category of perverse coherent sheaves on the affine Grassmannian of a complex reductive group (such as GL_n). This is the coherent analogue of the more classical (constructible) Satake category which plays a fundamental role in the geometric Langlands program. We don't expect participants to know what many of those words mean, but they will by the end of the workshop!

The basic goal of the workshop is to explain and illustrate with specific examples the ingredients which go into constructing and studying these categories. This includes discussing the basic geometry of the affine Grassmannian, the theory of perverse coherent sheaves, certain t-structures related to Koszul duality and perhaps some other related topics such as cluster algebras (depending on time and interest). We hope to highlight various tools and ideas from the subject that are of broader, independent interest.

The workshop will be led by Sabin Cautis and Harold Williams, and will consist of a combination of lectures and problem sessions.
It will be organized by Ben Elias, under the watchful eye of Nicholas Proudfoot. In 2023 it is supported by NSF grant DMS-2039316.

Background reading

If you plan to attend WARTHOG, please read this page which describes the background we expect you to acquire before the workshop begins. The link also contains many helpful references.

Schedule, Exercises, and Lecture Notes

Please follow this link to see the schedule of lectures. Exercises and lecture notes for each lecture will be posted there after each lecture.

Housing information for participants

Participants will be staying in Global Scholars Hall, at 1710 E 15th Ave, Eugene, OR 97403. This is also where you check in with your photo id. Unless you've told me otherwise, check in is after 4PM, and check out is before noon. If you're checking in after 9PM, you should call the Service Center phone number, which is 541-346-9114.

Logistics for Participants

General Schedule: There will be lectures and exercises Monday through Friday, from approximately 9AM to 5PM, with gaps between and a space for lunch. Wednesday is a half-day, with the afternoon being left open for an organized hiking trip to nearby Spencer Butte. Wednesday evening we will have a pizza (+ salads) party at a local park, sponsored by the workshop. On Monday, Tuesday, and Thursday evening there will be an optional exercise session in the evening.

Our typical daily routine is: hour of lecture, 45 minutes of exercise, 15 minute break, repeat four times, with a longer break for lunch.

Food: Starting at 830AM each day of the workshop, coffee and pastries will be catered to the lecture room. Aside from this and the pizza party, participants will have to pay for their own food. Several nearby residence halls have a cafeteria with more substantial breakfast for purchase, and there are many restaurants in the EMU (student center). There will be time to eat lunch at many local restaurants, and here are some very opinionated restaurant suggestions.

Accommodations: Participants will be staying in a dormitory on campus, in single occupancy rooms which have both shared bathrooms and single-occupancy gender-neutral bathrooms. Linens and towels are provided. The cost of this will be covered by the workshop. Details will be emailed shortly before arrival. Participants with special accommodation needs are encouraged to email Ben.

Accommodations and arrival/departure timing: It is assumed that participants will arrive in Eugene on Sunday before the workshop, and leave on either Friday night or Saturday after the workshop. Participants arriving earlier or leaving later should contact me for possible options, but housing is not guaranteed.

Transportation costs: Unless you have been told otherwise, we will not be covering your transportation costs. However, if other options have not availed you, please do email me (Ben Elias) and we can discuss options. We have funds to cover travel for a decent number of participants, but not for everyone. If we do end up covering your transportation, make sure to fly to EUG during the expected dates, or contact me to discuss price comparisons.

Flying to Eugene: You should fly to the Eugene airport, code EUG. There are direct flights connecting Eugene to Seattle, Portland, Salt Lake City, Denver, San Francisco, Los Angeles, and recently Chicago. In my personal experience, if you fly through San Francisco then your flight will be delayed and you will miss your connection, so try to find another option if possible. It is also possible to fly to Portland, which is a little over a two hour drive from Eugene - there are shuttles and a train which can get you from Portland to Eugene, but this is typically more complicated then just flying directly to Eugene.

Airport to Campus: There is no public transportation from the airport, so you'll have to take a cab, which should take about twenty minutes and cost about $30. Cabs in Eugene are run by a host of private companies, some better than others. Avoid Oregon Taxi if you can. If the driver asks which route to take, just say to take 99. Since a lot of you will be arriving around the same time, you might consider trying to find each other at the airport and sharing a cab. A google group will be set up closer to the time of the conference, which helps with coordination. Cabs back to the airport will be organized later in the conference. UPDATE: You might want to try this new shuttle service.

Getting around: During the conference you will be able to walk everywhere you need to go. If you are staying longer in Eugene or in Oregon for recreational purposes, it is easy to rent a bicycle, and car rentals (within the city, not from the airport) are decently priced.