Branes and Representations

Wednesdays 12:40–2pm, 939 Evans, CCN 54506

Topic

We will explore Gukov and Witten's proposal of brane quantization as a new and improved geometric quantization, following their construction of the irreducible unitary representations of SL(2,R). Their proposal offers possible new insight to the solution of a classical unsolved problem, a geometric construction of the unitary non-tempered representations of a real semi-simple Lie group. This will be an excuse for us to review the basics of the structure and representation theory of real semi-simple Lie groups, where many of the statements are beautiful, yet sophisticated refinements of analogues for compact Lie groups. Familiarity with the representation theory of the latter is strongly desirable, although the first 6-7 lectures should be accessible just from knowledge of SU(2) and the basics of hyper-Kaehler manifolds.

References will be updated.

Time and energy permitting, we will review the application of the same new brany ideas to Chern-Simons theory. (This seems unlikely now)

Tentative schedule of lectures

Date
 Topic
 Speaker
 References
1/28
 Introduction and outline of the seminar
 Teleman
 8, 9
2/4
 Branes and Gukov-Witten quantization 1
 Ryan
 1, 2 Notes
2/11
 Branes and Gukov-Witten quantization 2
 Ryan
 1, 2
2/18
 Hyper-Kähler structures on co-adjoint orbits Teleman
 10
2/25
 Representations of SL(2, R) via brane quantization
 Alex Takeda
 1, 2. Notes
2/27 Review of Representation theory of compact Lie groups Ben Gammage
3/4 Representations of SL(2, R) via brane quantization 2 Alex Takeda
3/6
 Structure of real semi-simple Lie groups QC
 3, 4
3/11
 Harish-Chandra module, infinitesimal characters, admissible representations Kiran Luecke
 3
3/18
 Discrete series representations. Character formula. Teleman
 3, 4, 5
3/25
 Spring Break

4/1
 Parabolic induction and Langlands classification George Melvin
 3, 4, 6
4/8
 Geometry and Physics lectures -- no meeting see announcement
4/15 ???

4/22 ???

4/29 Co-adjoint orbits and their geometric quantization Teleman


References

(Berkeley Library and sometimes source links)

  1. Gukov and Witten, Branes and Quantization
  2. Gukov, Quantization via Mirror Symmetry 
  3. Adams, Vogan: Representation theory of Lie groups, IAS/Park City Mathematics series Vol. 8. Specifically,
    Representations of semi-simple Lie groups, by Knapp and Trapa
    The method of coadjoint orbits for real semi-simple Lie groups, by Vogan
  4. Howe, A century of Lie theory
  5. Vergne, On Rossmann's character formula for Discrete Series
  6. Knapp, Zuckerman: Classification of irreducible tempered representations of semisimple groups I, II. Ann. Math. 116, 1992
  7. Vogan: Unitary representations and complex analysis. In Representation theory and Complex Analysis, CIME Summer School 2004
  8. Bill Casselman, Representations of SL(2, R)
  9. Vogan, Representations of SL(2, R)
  10. Kronheimer, A Hyper-Kähler structure on the co-adjoint orbits of a simple Lie group. J. London Math. Soc 42 (1990)