Seminário de Teoria Quântica do Campo Topológica  

In this talk we describe some recent results relating, on the one hand, to a method of quantizing a model of gravity in two space and one time dimensions, and on the other, to an algebraic structure - quantum matrix pairs - appearing in this context, which has many similarities with quantum groups.

References:

1. J. E. Nelson and R. F. Picken, Quantum holonomies in (2+1)-dimensional gravity, gr-qc/9911005.
2. J. E. Nelson and R. F. Picken, Quantum matrix pairs, math.QA/9911015.

Introduction to the solution by Seiberg and Witten of the $N=2$ supersymmetric Yang-Mills theory (following the reference).

Reference

• W. Lerche, "Introduction to Seiberg-Witten Theory and its Stringy Origin", hep-th/9611190 .

Geometric description of singularities in Del Pezzo surfaces.

Reference:

• J.-Y. Merindol, "Les Singularites Simples Elliptiques, Leurs Deformations, les Surfaces Del Pezzo et les Transformations Quadratiques", Ann. Scient. Ec. Norm. Sup., 4 serie, 15 (1982) 17-44

Continuation of the study of aspects of the relation between

1. moduli spaces of flat $E_8$ connections on elliptic curves,
2. the deformation of complex structures on Del Pezzo surfaces and,
3. singularities on these surfaces,

are studied. The motivation comes from work on "F-theory" by Friedman, Morgan and Witten.

References

• R. Friedman, J. Morgan and E. Witten, "Vector Bundles and F theory" hep-th/9701162
• R. Friedman, J. Morgan and E. Witten, "Vector Bundles over Elliptic Fibrations", alg-geom/9709029.
• R. Friedman, J. Morgan and E. Witten, "Principal G-bundles over elliptic curves", alg-geom/9707004 .

Study equations in Lax form and understand how they give rise to a spectral curve and a line bundle on it.

References:

• Mumford, D. "Tata Lectures on Theta, vol II", Progress in Math. 43, Birkhauser 1984
• Beauville, A. "Jacobiennes des courbes spectrales et systemes hamiltoniens completement integrables", Acta Math. 164, '90