Morse theory of the moment map for representations of quivers

Abstract

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-NarasimhanJordan-Hölder filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts.

Self-dual quiver moduli and orientifold Donaldson-Thomas invariants

“…Instead, a σ,θ d can be interpreted as the Poincaré series of the moduli stack [R σ,θ d /G σ d ]. For similar interpretations in the case of G-bundles over curves and ordinary quiver representations see [1], [25] and [16] respectively.…”

“…Let Q be a finite type quiver with involution and let θ be a σ-generic stability. The orientifold DT series A σ,θ admits a factorization of the form (16). Explicitly,…”

“…(3) The form of equations (15) and (16) reflect the difference between ordinary and σ-Jordan-Hölder filtrations: in the latter there is only one self-dual factor, leading to the linear structure of equation (16). (4) The factorization of A σ by equations (15) and (16) encodes all ordinary and orientifold DT invariants and can be regarded as a way of extracting the pure orientifold invariants from A σ .…”

“…In finite type examples the wall-crossing formulas can be reformulated as quantum dilogarithm identities holding representations of quantum tori. We use these identities to define orientifold DT invariants of finite type quivers; see equation (16). These invariants satisfy an orientifold modification of the primitive wall-crossing formula proposed in the physics literature [7].…”

The Topological Field Theory of Data: a program towards a novel strategy for data mining through data language

“…This assumes particular significance, because the induced Morse stratification, [21], is essentially the same as the Harder-Narasimhan [22] stratification of algebraic geometry, thus one can construct the analog of local co-ordinates around the Morse strata and represent the negative normal bundle to the critical sets. What relates Morse with homology is the property that the number ν i of critical points of index i of a given function f M is equal to the number of i cells in the simplicial complex structure obtained climbing f M , that bears on b i .…”