Cohomological gauge theory, quiver matrix models and Donaldson–Thomas theory

Abstract

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.

Elliptic non-Abelian Donaldson-Thomas invariants of ℂ3

“…We underline that in the QM case the higher-rank result does not factorise in Abelian contributions, due to the presence of non-trivial twisted sectors under the orbifold. We instead confirm that the factorisation holds in the matrix model limit, as conjectured in [26] and verified in [44,53]. The relevant formula for the matrix model case is (3.23), that we checked with our techniques up to 8 th order in the instanton expansion.…”

On Framed Quivers, BPS Invariants and Defects

“…At large radius the problem of studying Donaldson-Thomas invariants on a Calabi-Yau manifold X can be approached via a cohomological gauge theory [26,12]. This is a particular version of a topological quantum field theory obtained by the topological twist of six dimensional Yang-Mills theory.…”

“…In both of these chambers one can make some progress by using field and string theory concepts. In particular all of these problems can be approached from the point of view of a certain six dimensional topological Yang-Mills theory [26,12], where the relevant moduli spaces arise as moduli spaces of generalized instanton configurations. In this survey we will highlight those concepts which originate from quivers and their representations, in accordance with the theme of this volume.…”

“…In both chambers the relevant BPS invariants can be understood from the point of view of a supersymmetric quiver quantum mechanics which localizes onto the moduli spaces of BPS configurations. This quiver quantum mechanics is a physical counterpart of studying the intersection theory of the generalized instanton moduli spaces via equivariant localization [12,13,15,14,16].…”

BPS Hall algebra of scattering Hall states

“…String theory proposes an efficient and rather universal description of BPS states in terms of D-branes (see, for example, [55]), whose IR dynamics [16] gives rise to gauge theories admitting quiver description. There are multiple examples (see, for instance, [14,1,11,2] and references therein) in the literature implementing a quiver description to derive right counting of BPS states of the corresponding initial gauge or gravity theory.…”