BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Lisa Jeffrey (University of Toronto) DTSTART:20200917T185000Z DTEND:20200917T195000Z DTSTAMP:20260423T022929Z UID:GPRTatNU/3 DESCRIPTION:Title: Flat connections and the $SU(2)$ commutator map\nby Lisa Jeffrey (Uni versity of Toronto) as part of Geometry\, Physics\, and Representation The ory Seminar\n\n\nAbstract\nThis talk is joint work with Nan-Kuo Ho\, Paul Selick and Eugene Xia. We describe the space of conjugacy classes of repre sentations of the fundamental group of a genus 2 oriented 2-manifold into $G := SU(2)$.\n\nWe identify the cohomology ring and a cell decomposition of a\; space homotopy equivalent to the space of commuting pairs in $SU(2) $.\n\nWe compute the cohomology of the space $M:= \\mu^{-1}(-I)$\, where $ \\mu:G^4 \\to G$ is the product of commutators.\n\nWe give a new proof of the cohomology of $A:= M/G$\, both as a group and as a ring. The group str ucture is due to Atiyah and Bott in their landmark 1983 paper. The ring st ructure is due to Michael Thaddeus 1992.\n\nWe compute the cohomology of t he total space of the prequantum line bundle over $A$.\n\nWe identify the transition functions of the induced $SO(3)$ bundle $M\\to A$.\n\nTo appear in QJM (Atiyah memorial special issue). arXiv:2005.07390\n\nReferences:\n \n[1] M.F. Atiyah\, R. Bott\, The Yang-Mills equations over Riemann surfac es\, Phil. Trans. Roy. Soc. Lond. A308 (1983) 523-615.\n\n[2] T. Baird\, L . Jeffrey\, P. Selick\, The space of commuting n-tuples in $SU(2)$\, Illin ois J. Math. 55 (2011)\, no. 3\, 805–813.\n\n[3] M. Crabb\, Spaces of co mmuting elements in $SU(2)$\, Proc. Edin. Math. Soc. 54 (2011)\, no. 1\, 6 7–75.\n\n[4] N. Ho\, L. Jeffrey\, K. Nguyen\, E. Xia\, The $SU(2)$-chara cter variety of the closed surface of genus 2. Geom. Dedicata 192 (2018)\, 171–187.\n\n[5] N. Ho\, L. Jeffrey\, P. Selick\, E. Xia\, Flat connecti ons and the commutator map for $SU(2)$\, Oxford Quart. J. Math.\, to appea r (in the Atiyah memorial special issue).\n\n[6] L. Jeffrey\, A. Lindberg\ , S. Rayan\, Explicit Poincar´e duality in the cohomology ring of the $SU (2)$ character variety of a surface. Expos. Math.\, to appear.\n\n[7] M.S. Narasimhan and C.S. Seshadri\, Stable and unitary vector bundles on a com pact Riemann surface. Ann. of Math. 82 (1965) 540–567.\n\n[8] P. Newstea d\, Topological properties of some spaces of stable bundles\, Topology 6 ( 1967)\, 241–262.\n\n[9] C.T.C Wall\, Classification problems in differen tial topology. V. On certain 6-manifolds. Invent. Math. 1 (1966)\, 355–3 74\; corrigendum\, ibid.\, 2 (1966) 306.\n\n[10] M. Thaddeus\, Conformal f ield theory and the cohomology of the moduli space of stable bundles. J. D ifferential Geom. 35 (1992) 131–149.\n\n[11] E. Witten\, Two dimensional gauge theories revisited\, J. Geom. Phys. 9 (1992) 303-368.\n LOCATION:https://researchseminars.org/talk/GPRTatNU/3/ END:VEVENT END:VCALENDAR