**Web site:** `http://www.math.umd.edu/~jmr/geomphysRIT.html`
**Meeting times:** Wednesdays, 3:15 - 5:00, MTH 1311. (We
probably won't use the whole time slot but have the room reserved for
this period.)
**Organizers:** Jonathan
Rosenberg and Hisham Sati

This will be an informal learning seminar on the general theme of Geometry and Physics. We hope that students will give most of the lectures. Topics will include various areas where modern geometry and topology interacts with ideas from theoretical physics. Examples include loop groups and loop spaces and "string topology" and "string structures", gauge theory and geometry of vector bundles and moduli spaces, etc. Since the subject is interdisciplinary, and involves a mixture of physics, geometry, Lie theory, and topology, we assume the participants have some background in one of these topics but not necessarily in all of them. We will pick out some interesting papers to read and discuss.

- Seiberg-Witten theory
- Electric-magnetic duality and Langlands duality
- Conformal field theory and moduli spaces
- Topological field theory and TQFTs (topological quantum field theories)
- Renormalization theory: topics like the renormalization group and the Hopf algebra structure of Connes, Moscovici, and Kreimer
- Geometry of loop spaces and string topology

For the fall semester, we decided to start with #1 (with a little of #2 mixed in). After that we may move to #4.

- Introduction to Yang-Mills theory (1-2 lectures, Richard Wentworth)
- Introduction to supersymmetry (1-2 lectures, Gokhan Civan, William Donnelly)
- Super-Yang-Mills theory
- Dirac's EM duality and Montonen-Olive duality (Rong Zhou)
- Seiberg-Witten theory (Ben Sibley)
- Topological field theories (if time permits)

- L. Alvarez-Gaume, S.F. Hassan, Introduction to S-Duality in N=2 Supersymmetric Gauge Theory Fortsch.Phys. 45 (1997) 159-236 (a pedagogical review of the work of Seiberg and Witten).
- A. Bilal, Duality in N=2 SUSY SU(2) Yang-Mills Theory: A pedagogical introduction to the work of Seiberg and Witten, Quantum fields and quantum space time (Cargièse, 1996), 21-43, NATO Adv. Sci. Inst. Ser. B Phys., 364, Plenum, New York, 1997.
- P. Di Vecchia, Duality in supersymmetric N=2,4 gauge theories.
- D. Freed, Five Lectures on Supersymmetry, American Mathematical Society, Providence, 1999. A very preliminary version is available here.
- D. Freed, Classical Field Theory and Supersymmetry, IAS/Park City Mathematics Series, 2001.
- P. Dirac, Quantised singularities in the electromagnetic field, Proc. Royal Soc. London A 113 (1931), 60-72; jstor:pss/95639.
- P. Deligne and D. Freed, Supersolutions, lecture notes on classical globally supersymmetric theories.
- A. Klemm, On the Geometry behind N=2 Supersymmetric Effective Actions in Four Dimensions.
- C. Montonen and D. Olive, Magnetic monopoles as gauge particles?, Phys. Lett. B 72 (1977), 117-120.
- W. Lerche, Introduction to Seiberg-Witten Theory and its Stringy Origin, Nucl. Phys. Proc. Suppl. 55B (1997), 83-117; Fortsch.Phys. 45 (1997), 293-340.
- D. Olive, Exact electromagnetic duality, Trieste talk, 1995.
- N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nuclear Phys. B 426 (1994), no. 1, 19-52. Erratum, Nuclear Phys. B 430 (1994), no. 2, 485-486; arxiv:hep-th/9407087.
- E. Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), no. 6, 769-796; arxiv:hep-th/9411102.

Date | Speaker | Title | |

9/15 | Organizational meeting | ||

9/22 | Richard Wentworth | An introduction to Yang-Mills | |

9/29 | Gokhan Civan | Introduction to supersymmetry | |

10/6start time 4:00 | William Donnelly | SUSY from a physics perspective | |

10/13start time 4:00 | Hisham Sati | Supersymmetric Yang-Mills in flat space-time | |

10/20start time 4:00 | Rong Zhou | Electric-magnetic duality | |

10/27start time 4:00 | Rong Zhou | Electric-magnetic duality (cont'd) and the Montonen-Olive conjecture | |

11/3start time 4:00 | Hisham Sati | N=2 supersymmetry and super-Yang-Mills | |

11/10back to regular time
of 3:15 for rest of semester | Hisham Sati | Extended supersymmetry and the central charge | |

11/17 | Jonathan Rosenberg | Seiberg-Witten theory, part 1 | |

11/24 | No meeting because of Thanksgiving | ||

12/1 | Jonathan Rosenberg | Seiberg-Witten theory, part 2 | |

12/8 | Ben Sibley | Witten's paper on Seiberg-Witten invariants for 4-manifolds |

The new topic for the spring semester is *topological
quantum field theories* (TQFTs). This topic is reasonably self-contained,
so you ought to be able to follow even if you didn't attend in the fall.
Basic references for this topic include:

- Lowell Abrams, Two
dimensional topological quantum field theories and Frobenius algebras,
J. Knot Theory and its Ramifications
**5**(1996), 569-587. - Michael Atiyah, "Topological quantum field theory", Publications Mathématiques de l'IHÉS 68 (1988), 175-186, available here.
- Bruce H. Bartlett, Categorical aspects of topological quantum field theories, 2005, arxiv: math/0512103.
- Dan Freed, Lectures on Topological Quantum Field Theory, 1992.
- Dan Freed, Locality and Integration in Topological Field Theory, arxiv: hep-th/9209048.
- Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, and Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups, arxiv:0905.0731.
- Anton Kapustin, Topological Field Theory, Higher Categories, and Their Applications, to appear in Proc. ICM 2010, arxiv: 1004.2307.
- Jacob Lurie, "On the Classification of Topological Field Theories", preprint 2010.
- J. Peter May, Lectures on TQFTs.
- Greg Moore, 2D Yang-Mills Theory and Topological Field Theory, Proc. ICM 1994, arxiv: hep-th/9409044.
- Albert Schwarz, The partition function of a degenerate quadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2 (1978), 247-252.
- Albert Schwarz, The partition function of a degenerate functional, Comm. Math. Phys. 67 (1979), 1-16.
- Graeme Segal, Stanford lectures on Topological Field Theories, 1999 ITP Workshop on Geometry and Physics.
- Kevin Walker, Notes on TQFTs, 2006.
- Edward Witten, "Topological quantum field theory", Comm. Math. Phys. 117 (1988), 353-386.

**Suggested order of reading:** Start with Atiyah's paper, then Freed's
notes. After that we may do Witten's CMP paper and then Segal's 1999
notes, followed by some of the more recent references.

Date | Speaker | Title |

1/26 | Organizational meeting | |

2/2
start time 4:00 | Ben Sibley | Atiyah's definition of a TQFT |

2/9 | Gokhan Civan | Basic examples following Freed's notes |

2/16 | Gokhan Civan | Basic examples following Freed's notes (cont'd): finite groups |

2/23 | Ben Sibley | Basic examples following Freed's notes (cont'd): Lie groups |

3/2 | Rong Zhou | TQFTs via the BRST symmetry formalism, I |

3/9 | Rong Zhou | TQFTs via the BRST symmetry formalism, II |

3/16
start time 4:00 | Jonathan Rosenberg | A TQFT connected to Ray-Singer torsion (after A. Schwarz) |

3/23 | Spring break, no meeting | |

3/30 | Jonathan Rosenberg | Segal's notes on TQFTs, I: Frobenius algebras |

4/6 | Steve Balady | Defining TQFTs in dimension > 2 |