Abstract
The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod the action of the Weyl group, or equivalently to the conjugacy classes of commuting pairs of elements in G. Since the component group for a non-simply connected group is given by some finite dimensional subgroup in the centralizer of an n-tuple, we use diagram automorphisms of the extended Dynkin diagram to prove properties of centralizers of pairs of elements in G.
Highlights
Classifying the moduli space of gauge equivalence classes of flat connections on a principal G−bundle over a compact Riemann surface Σg of genus g is of interest from various perspectives
Atiyah-Bott [2] proved that this moduli space is equivalent to the finite dimensional representation space {Hom(π1(Σg), G)}/G by constructing a symplectic structure on the moduli space by symplectic reduction from the infinite-dimensional sympletic manifold of all connections
If A is a flat connection on a K−bundle over T 3 the holonomy of A is defined by the conjugacy classes of commuting triples in K
Summary
Classifying the moduli space of gauge equivalence classes of flat connections on a principal G−bundle over a compact Riemann surface Σg of genus g is of interest from various perspectives. Kac-Smilga [9] proved that computing the number of quantum vacuum states over T 3 is equivalent to classifying commuting triples in a simple, compact, connected Lie group G. Given G is a compact, connected, semisimple Lie group, they proved that principal G-bundles ζ with flat connections over a maximal two torus T 2 are classified up to restricted gauge equivalence by classifying commuting pairs of elements in the connected covering Gof G that commute up to the center. The choice of a lift xis unique up to an element in Ker(π) ∼= π1(G) which is identified as a subgroup of the center of the connected covering Extending this for a c-pair: for k ∈ Ker(π), [x, y] = [kx, ky] = c because k ∈ Ker(π) commutes with every element in Gand is invariant under the choice of x, y.
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