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Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory. (English) Zbl 1192.81309
Summary: We study the relation between Donaldson-Thomas theory of Calabi-Yau threefolds and a six-dimensional topological Yang-Mills theory. Our main example is the topological \(U(N)\) gauge theory on flat space in its Coulomb branch. To evaluate its partition function we use equivariant localization techniques on its noncommutative deformation. As a result the gauge theory localizes on noncommutative instantons which can be classified in terms of \(N\)-coloured three-dimensional Young diagrams. We give to these noncommutative instantons a geometrical description in terms of certain stable framed coherent sheaves on projective space by using a higher-dimensional generalization of the ADHM formalism. From this formalism we construct a topological matrix quantum mechanics which computes an index of BPS states and provides an alternative approach to the six-dimensional gauge theory.

MSC:
81T45 Topological field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
83E30 String and superstring theories in gravitational theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
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