Holomorphic differentials have been at the crossroads of various active areas in analysis, geometry and quantum field theory – low dimensional dynamics, (higher) Teichmuller theory, moduli of Higgs bundles and their BPS spectra, stability conditions on Fukaya-type categories, etc… 

The aim of this seminar is to understand the spectral networks and spectral coordinates as constructed by Gaiotto – Moore – Neitzke in their study of quantum field theory associated to moduli of Higgs bundles. Spectral networks are trajectory structures associated to holomorphic differentials on Riemann surfaces. From these trajectory structures, we can construct spectral coordinates on the moduli of flat connections, generalizing the celebrated Fock-Goncharov coordinates. 

Our long-term goal is to build a bridge between WKB analysis of differential equations with irregular singularities and the theory of spectral networks/coordinates. 

In the case of quadratic differentials, the story is clear. Via the work of Iwaki-Nakanishi (Exact WKB analysis and cluster algebras) and Gaiotto - Moore - Neitzke (Wall-crossing, Hitchin systems, and the WKB approximation), cluster algebras serve as the bridge to connect these two areas and give the combinatorial and algebraic model  for both the monodromy data of Schrödinger type equations on Riemann surfaces and spectral networks/coordinates of SU(2)-Higgs bundles. 

However, for holomorphic differentials of higher ranks, the story on either side becomes more opaque. We hope, by connecting the two sides together, either side could shed new light to the other, i.e., spectral networks can serve as nice combinatorial models for the Stokes data of differential equations and the known WKB approximations of stokes matrices could also help us understand better the intricate structures of spectral networks in higher dimensions. 

Topics to be covered:

1. Holomorphic differentials on Riemann surfaces.

2. Cluster algebras.

3. Exact WKB analysis of Schrödinger type equations on Riemann surfaces.  

4. Voros symbols and DDP Formulas.

5. Moduli of Higgs bundles and non-abelian Hodge. 

6. Spectral networks associated to holomorphic differentials and spectral coordinates on moduli of flat connections on Riemann surfaces. 

7. BPS spectrum of the theory of Class S.

References will include but not limit to:

1. Fock V V and Goncharov A B, Moduli spaces of local systems and higher Teichmüller theory. 

2. Fomin S, Shapiro M and Thurston D, Cluster algebras and triangulated surfaces. Part I:cluster complexes.

3. Iwaki K and Nakanishi T, Exact WKB analysis and cluster algebras. 

4. Gaiotto D, Moore G W and Neitzke A, Wall-crossing, Hitchin systems, and the WKB approximation.

5. Gaiotto D, Moore G W and Neitzke A, Spectral networks.

Time:

7pm - 9pm   [June 2 - August 4, every Thursday]

6/2    6/9    6/16    6/23    6/30    7/7    7/14    7/21    7/28    8/4

Tencent Meeting：

ID：774-887-281