Topological Quantum Field Theory Seminar  

I will explain how geometric descriptions of genera determine geometric descriptions of the associated cohomology theories and then give some examples. Then I will try to say something about the case of elliptic genera. For these the geometric description is still not rigorous.

References (I have copies of the non-web references, in case any one is interested):

1. Haven't looked at this paper but it has a cool title: Dijkgraaf, R.; Moore, G.; Verlinde, E.; Verlinde, H., Elliptic genera of symmetric products and second quantized strings. Comm. Math. Phys. 185 (1997), no. 1, 197--209. hep-th/9608096
2. Witten, Ed., Elliptic genera and quantum field theory. Comm. Math. Phys. 109 (1987), no. 4, 525--536. Postscript from KEK library
3. Hopkins, Michael J. Characters and elliptic cohomology. Advances in homotopy theory (Cortona, 1988), 87--104, London Math. Soc. Lecture Note Ser., 139, Cambridge Univ. Press, Cambridge-New York, 1989
4. M. J. Hopkins, M. Ando, and N. P. Strickland, "Elliptic spectra, the Witten genus, and the theorem of the cube", dvi file
5. Segal, G. "Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others)". Séminaire Bourbaki, Vol. 1987/88. Astérisque No. 161-162, (1988), Exp. No. 695, 4, 187--201 (1989).

Bibliografia:

A. Na exposição seguirei muito de perto:

1. Cartan Elie, La theorie des groupes finis et continus et la geometrie differentielle. Gauthiers-Villars, 1937.
2. Cartan Elie, La methode du repere mobile, la theorie des groupes continus et les espaces generalises. Hermann, 1935.

B. Outras referências mais actuais e avançadas (que eu não vou abordar):

1. Griffiths P., On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. Journal 41 (1974), 775-814.
2. Griffiths P., Harris J., Algebraic geometry and local differential geometry. Ann. Sci. Ecole Norm. Sup. 12 (1979), 355-452.
3. Akivis M. A., Goldberg V. V., Projective differential geometry of submanifolds. North-Holland, 1993.
4. Akivis M. A., Goldberg V. V., Conformal differential geometry and its generalizations. John Wiley and Sons, Inc., 1996.

C. Aplicações (que eu não vou abordar):

1. Razumov A. V., Frenet Frames and Toda Systems, math.DG/9901023
2. Fels, M., Olver, P. J., Moving coframes I. A practical algorithm. Acta Appl. Math. 51 (1998) 161-213.
3. Fels, M., Olver, P. J., Moving coframes II. Regularization and theoretical foundations. Acta Appl. Math. 55 (1999) 127-208.

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