# Spring 2015 GRASP Seminar, focusing on Class S theories, spectral networks, & related topics

As usual GRASP Seminar meets Fridays 3-4. This semester GRASP is in **939 Evans**.

## Topics to be covered

The goals of this semester are twofold: the first is to understand the integrable system associated to an N=2 supersymmetric field theory.
This integrable system was first described and used in [SW], where the authors were able to derive an exact formula for the metric on the moduli space of vacua in N=2 SUSY gauge theory with G=SU(2). The first few weeks of the semester will be spent on understanding the procedure which associates an integrable system to a field theory and on understanding the mathematical structure of this integrable system.

One interesting class of N=2 theories are the Class S theories, whose associated integrable systems are Hitchin systems. (The G=SU(2) super Yang-Mills theory mentioned above is a special case.) To help compute the BPS states in these theories, Gaiotto, Moore, and Neitzke introduced in [GMN] the notion of a *spectral network*. The second major goal of this seminar is to understand the definition of a spectral network, how it is used in class S theories, and how it relates to other mathematical structures of interest.
## Schedule

**19 February**: Introduction and organizational meeting (Ben G.)

**26 February** QFT Basics (Ryan T.)
["Recommended reading:" 1 2 3 4 5 6 7]

**4 March**: Integrable systems from Seiberg-Witten theory [W][SW] (Alex T.)

**11 March**: Integrable systems from Seiberg-Witten theory, Part II [W][SW] (Alex T.)

**16 March**: Integrable systems from Seiberg-Witten theory, Part III [W][SW] (Alex T.)

**18 March**: Singular integrable systems and monodromy [N][D] (Gus S.)

**25 March** (No meeting: Mirror symmetry conference)

**1 April** (No meeting: Cluster algebras conference)

**8 April**: BPS states in Class S theories [N2][GMN2][N] (Ben G.)

**15 April**: Spectral Networks [GMN] (Ammar H.)
## References

[D] R. Donagi, *Seiberg-Witten Integrable Systems*

[GMN] D. Gaiotto, G. Moore, A. Neitzke, *Spectral Networks*

[GMN2] ibid., *Wall-crossing, Hitchin Systems, and the WKB Approximation*

[H] N. Hitchin, *Stable Bundles and Integrable Systems*

[KS] M. Kontsevich and Y. Soibelman, *Stability structures, motivic Donaldson-Thomas invariants and cluster transformations*

[N] A. Neitzke, *Hitchin systems in N=2 field theory*

[N2] A. Neitzke, *What is a BPS state?*

[SW] N. Seiberg, E. Witten, *Monopole Condensation, and Confinement in N = 2 Supersymmetric Yang-Mills Theory*

[STWZ] V. Shende, D. Treumann, H. Williams, and E. Zaslow, *Cluster varieties from Legendrian knots*

[W] Witten, *Dynamics of Quantum FIeld Theory* (lectures from IAS QFT, vol. 2)