Moduli Spaces of Parabolic Connections and Parabolic Bundles and Geometric Langlands by M-H Saito

Explore the intricacies of moduli spaces and their connections to Geometric Langlands theory in this comprehensive lecture. Delve into A-parabolic connections, λ-connections, and their properties, including residues, local exponents, and the Fuchs relation. Examine parabolic and quasiparabolic bundles, stability conditions, and the construction of moduli spaces for stable parabolic connections and Higgs bundles. Investigate deformation theory, symplectic structures, and the relationship between parabolic connections and bundles. Analyze specific cases such as the Painlevé VI equation and results from Arinkin and Lysenko. Gain insights into the interplay between mathematics and theoretical physics in this advanced exploration of Quantum Fields, Geometry, and Representation Theory.

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Moduli spaces of parabolic connections and parabolic bundles and Geometric Langland's
1. Moduli spaces of A-parabolic connections 1.1. Settings.
1.2. λ-connections. λ E C.
λ ≠0: linear connection
1.3 Residues and local exponents
1.4. Fuchs relation
1.6. Genericity for local exponents.
1.7. Parabolic connections
1.8. Quasiparabolic bundles.
1.9. Parabolic stability on quasiparbolic bundles.
1.10. a-stable v-parabolic connections.
1.11. Moduli spaces of a- stable parabolic connections and a-stable parabolic Higgs bundles.
1.12. Existence of algebraic moduli space of a-stable v-parabolic con- connections.
1.13. As in the similar way,
1.14. The Moduli space of connections, Painleve VI case.
1.15. The Moduli space of parabolic Higgs bundles.
2. DEFORMATION THEORY AND SYMPLECTIC STRUCTURE
3. Moduli Spaces of Parabolic Bundles
4. The image of v-parabolic connections For simplicity, we propose the following:
Theorem 4.1.
4.1. The coarse moduli for C1 = Pl,
4.2. C = P1 and t = 1. ... .t;.
5. A RESULT OF ARINKS AND LYSENKO
Theorem 5.1 The functor
Q&A