Hitchin Systems, Nonabelian Hodge and Wall Crossing

(org by M. Alim, A. Balasubramanian, F. Beck, A. Brochier, J. Teschner)

This is a webpage for the ZMP Seminar (Summer’ 17) on Hitchin Systems, Non Abelian Hodge and Wall Crossing

The Seminar is divided into two halves. The first half (Apr13-May 11) will be an introduction to various mathematical foundations regarding Hitchin Systems. In the second half (Jun 01 - Jul 06), we will study the problem of Wall crossing in four dimensional N=2 theories. The solution to this problem in specific theories turns out to intimately involve the Hitchin System. Some of the foundational material that is discussed in the first half will be useful for the second half.

Schedule

Note : On Apr 13 and Jun 01, there will be two talks as part of the ZMP Seminar

DatesPotential TopicReferencesSpeakersNotes/Summary
Apr 13-IIntro to Hitchin (Self-duality Eqns, Hyperkahler structure)[1,6a,2]Áron Szabó Speaker's Notes
Apr 13-IIHarmonic Metric and Nonabelian Hodge[6a,4] and
Sec 2 of [7]		Arpan SahaSpeaker's Notes
Apr 27More NAH, Twistor Space Construction [8,10] for NAH ;
[6c], Sec 3 of [11] for Twistor Space ; 		Florian Beck (NAH) , Troy Figiel (Twistor) Speaker's Notes (for Twistors, updated on May 11)
May 11Twistor Space Construction (contd)[6c], Sec 3 of [11], [5] Troy Figiel(see updated link above)
Jun 01-IThe Hitchin Integrable System [6b, 3], Chapter 1 of [18] Adrien BrochierSpeaker's notes
Jun 01-IIIntroduction to wall crossing in N=2 theories Murad Alim
Jun 22Special Geometry of Integrable SystemsFlorian Beck
Cancelled(G20)BPS Spectrum and the Hitchin System Aswin Balasubramanian -

References

This is a collection of some basic references for the seminar.

First Half

Introductory Lectures/Notes

  1. What is a Higgs Bundle ? (AMS Notices) Bradlow, Garcia-Prada, Gothen.
  2. Moduli of Higgs Bundles , A. Neitzke’s notes
  3. JT's notes
  4. Higgs bundles and local systems on Riemann surfaces, R. Wentworth

Papers

  1. Hyperkahler Metrics and Supersymmetry , Hitchin-Karlhede-Lindstrom-Rocek
  2. Hitchin’s Papers :
    1. Self-duality equations on a Riemann Surface
    2. Stable bundles and Integrable Systems
    3. Bourbaki Talk : Hyperkahler Manifolds (pdf)
  3. Opers vs Non-Abelian Hodge , Dumitrescu et al
  4. Non-abelian Hodge Theory (pdf), C. Simpson
  5. Higgs Bundles and Local Systems, C. Simpson
  6. Non-abelian Hodge Theory, K. Corlette
  7. Four-dimensional wall-crossing via three-dimensional field theory, Gaiotto-Moore-Neitzke

Second Half

Introductory Lectures/Notes

  1. N=2 Supersymmetric dynamics for pedestrians , Y. Tachikawa
  2. Introduction to Seiberg-Witten Integrable Systems, R. Donagi
  3. Hitchin Systems in N=2 Field Theory, A. Neitzke

Papers

Apart from the GMN paper ([11] above), we will also use :
  1. Seiberg-Witten’s papers :
    1. Electric Magnetic Duality, Monopole Condensation and confinement in N=2 SYM
    2. Monopoles, Duality and chiral symmetry breaking in N=2 SQCD
    3. Gauge dynamics and compactification to three dimensions
  2. Special Kahler Manifolds, D. Freed
  3. Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, M. Kontsevich and Y. Soibelman
  4. More On Gauge Theory and Geometric Langlands, E. Witten