Topics in Mathematical Physics (520-1-41)
Instructors
Eric Zaslow
8474676447
Lunt 302
Meeting Info
Lunt Hall 101: Mon, Wed, Fri 1:00PM - 1:50PM
Overview of class
Math 520 -- Topics in Mathematical Physics
I will begin by giving an overview of the area (more on that below), but after that my goal for this course is that the students learn by preparing presentations and by actively querying one another. The general area was determined in coversation with some students, but if you have a strong desire for a nearby topic we may be able to accommodate. Rules of the game: student presentations must provide context and must contain concrete examples to illustrate the main principles (speakers will not be allowed to "not have time" to get to the examples!).
The subjects will be cluster varieties, knots, and their relation to physics.
Some promising models of string theory reduce in four dimensions to N=1 or N=2 supersymmetric theories. These theories have "BPS states" with enhanced symmetries, whose enumeration and interaction can often be described mathematically, and often in more ways than one, leading to dualities. Notable BPS states are "branes." Categories of branes and their moduli of objects play a key role in elucidating the mathematical structure from physical theoreis. For us, the mathematical stand-in for brane categories will be categories of constructible sheaves with singular support defined by a Legendrian variety --- for example, the boundary of a knot conormal. These categories are invariants of the Legendrian up to Legendrian isotopy.
Foundational work on the above mathematics was provided by Kashiwara, Schapira and Guillermou. We will be more concerned with various applications, where moduli of objects can be identified with various cluster varieties, especially those of Fock and Goncharov.
Here are some topics that students might explore. A main goal of the course will be to thread them together.
* DT invariants of quivers (mathematically and as an effective theory of intersecting branes -- various papers, e.g. Kontsevich-Soibelman [COHA] and Cordoba-Cecotii-Vafa, others )
* Cluster varieties and their quantization, especially those of Fock-Goncharov: framed local systems on surfaces. (Goncharov-Shen survery paper)
* Cluster varieties and "curves on surfaces." (Shende-Treumann-Williams)
* Clusters and braids (Casals-Gorsky-Gorsky-Shen-Simental)
* The chromatic Lagrangian inside the cluster variety of framed local systems on surfaces. (Dijkgraaf-Gabella-Goncharov and other references)
* HOMFLY-PT invariants of knots as augmentations, and wavefunctions (Aganagic-Ekholm-Ng-Vafa)
* Skeins and branes (Ekholm-Shende)
* Goncharov's representation of the cluster groupoid (Goncharov).
Class Materials (Required)
No required or suggested textbooks
Class Materials (Suggested)
No required or suggested textbooks