Flat connections and the $SU(2)$ commutator map

Abstract: This talk is joint work with Nan-Kuo Ho, Paul Selick and Eugene Xia. We describe the space of conjugacy classes of representations of the fundamental group of a genus 2 oriented 2-manifold into $G := SU(2)$.

We identify the cohomology ring and a cell decomposition of a; space homotopy equivalent to the space of commuting pairs in $SU(2)$.

We compute the cohomology of the space $M:= \mu^{-1}(-I)$, where $\mu:G^4 \to G$ is the product of commutators.

We give a new proof of the cohomology of $A:= M/G$, both as a group and as a ring. The group structure is due to Atiyah and Bott in their landmark 1983 paper. The ring structure is due to Michael Thaddeus 1992.

We compute the cohomology of the total space of the prequantum line bundle over $A$.

We identify the transition functions of the induced $SO(3)$ bundle $M\to A$.

To appear in QJM (Atiyah memorial special issue). arXiv:2005.07390

References:

[1] M.F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A308 (1983) 523-615.

[2] T. Baird, L. Jeffrey, P. Selick, The space of commuting n-tuples in $SU(2)$, Illinois J. Math. 55 (2011), no. 3, 805–813.

[3] M. Crabb, Spaces of commuting elements in $SU(2)$, Proc. Edin. Math. Soc. 54 (2011), no. 1, 67–75.

[4] N. Ho, L. Jeffrey, K. Nguyen, E. Xia, The $SU(2)$-character variety of the closed surface of genus 2. Geom. Dedicata 192 (2018), 171–187.

[5] N. Ho, L. Jeffrey, P. Selick, E. Xia, Flat connections and the commutator map for $SU(2)$, Oxford Quart. J. Math., to appear (in the Atiyah memorial special issue).

[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.

[7] M.S. Narasimhan and C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface. Ann. of Math. 82 (1965) 540–567.

[8] P. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967), 241–262.

[9] C.T.C Wall, Classification problems in differential topology. V. On certain 6-manifolds. Invent. Math. 1 (1966), 355–374; corrigendum, ibid., 2 (1966) 306.

[10] M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles. J. Differential Geom. 35 (1992) 131–149.

[11] E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303-368.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

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