Global topology of the Hitchin system - GEOM

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Arithmetic harmonic analysis on ˆˇM B for any finite group G, [Frobenius 1896], . . . , . . . , TQFT [Freed–Quinn 1993] ❀ a1, b1, . . . , ag , bg ∈ G| |G| [ai, bi] = z = 2g−1 χ(z) χ(1) 2g−1 when ζn ∈ F ∗ q, i.e n|q − 1, we get E(MB; q) Katz = |M d B(Fq)| = (q−1) χ∈Irr(GLn(Fq)) χ∈Irr(G) |GLn(Fq)| 2g−2 χ(1) 2g−2 · χ(ζd n · I ) χ(1) Irr(GLn(Fq)) described combinatorially by [Green, 1955] ❀ formula for E(MB; q) [Hausel–Villegas, 2008] when n|q − 1 E( ˇMB; q) Katz = | ˇM d B(Fq)| = χ∈Irr(SLn(Fq)) |SLn(Fq)| 2g−2 χ(1) 2g−2 · χ(ζd n · I ) χ(1) Irr(SLn(Fq)) more difficult; only need value of χ(ζ d n · I ) ❀ Clifford theory ❀ calculation of E( ˇMB; q) by [Mereb, 2010] 32 / 39
Character table of GL2(Fq) 33 / 39
Page 1 and 2: Global topology of the Hitchin syst
Page 3 and 4: Mirror Symmetry phenomenon first ar
Page 5 and 6: Strominger-Yau-Zaslow X CY 3-fold Y
Page 7 and 8: Hitchin system Hamiltonian system:
Page 9 and 10: The moduli space of vector bundles
Page 11 and 12: Cohomology of ˆ Ň The cohomologie
Page 13 and 14: Hitchin map for SLn and PGLn for SL
Page 15 and 16: Proper Hitchin map (E, φ) ∈ T
Page 17 and 18: PGLn Hitchin system over the same H
Page 19 and 20: Generic fibres of the Hitchin map a
Page 21 and 22: Symmetries of the GLn and PGLn Hitc
Page 23 and 24: Duality of the Hitchin fibres short
Page 25 and 26: E-polynomials (Deligne 1972) constr
Page 27 and 28: Topological mirror symmetry conject
Page 29 and 30: Example PGL2 γ ∈ Γ = Pic 0 (C)[
Page 31: Arithmetic technique to calculate E
Page 35 and 36: Betti - Topological Mirror Test for
Page 37 and 38: P = W conjecture recall Hitchin map
Page 39: Heuristics from TMS to FL let Γ be

Global topology of the Hitchin system - GEOM

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