arXiv subjects

Graeme Wilkin

Publications and source records attributed to Graeme Wilkin.

24 records · Page 2Linked to original sources

arXiv2010

Morse theory of the moment map for representations of quivers

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-Narasimhan-Jordan-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.

math.DG↗

arXiv2010

Cohomology of SL(2,C) character varieties of surface groups and the action of the Torelli group

We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat PSL(2,C) connections. The non-trivial part consists of the even alternating products of degree two Prym representations, so that the kernel of the action is precisely the Prym-Torelli group. We compute the Betti numbers of the ordinary cohomology of the moduli space of flat SL(2,C) connections. Using results of Cappell-Lee-Miller we show that the Prym-Torelli group, which acts trivially on equivariant cohomology, acts non-trivially on ordinary cohomology.

math.GT↗

arXiv2010

Morse theory for the space of Higgs G-bundles

Fix a $C^\infty$ principal $G$--bundle $E^0_G$ on a compact connected Riemann surface $X$, where $G$ is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang--Mills--Higgs functional on the cotangent bundle of the space of all smooth connections on $E^0_G$. We prove that this flow preserves the subset of Higgs $G$--bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs $G$--bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.

math.DG↗

arXiv2009

Homotopy groups of moduli spaces of stable quiver representations

The purpose of this paper is to describe a method for computing homotopy groups of the space of $α$-stable representations of a quiver with fixed dimension vector and stability parameter $α$. The main result is that the homotopy groups of this space are trivial up to a certain dimension, which depends on the quiver, the choice of dimension vector, and the choice of parameter. As a corollary we also compute low-dimensional homotopy groups of the moduli space of $α$-stable representations of the quiver with fixed dimension vector, and apply the theory to the space of non-degenerate polygons in three-dimensional Euclidean space.

math.SG↗

arXiv2009

Morse Theory and Hyperkahler Kirwan Surjectivity for Higgs Bundles

This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott's original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperkähler Kirwan map is surjective for the non-fixed determinant case.

math.SG↗

arXiv2008

Morse Theory for the Space of Higgs Bundles

Here we prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions $(A'', ϕ)$ converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the Harder-Narasimhan-Seshadri filtration of $(A'', ϕ)$. In particular, the results of this paper show that the failure of hyperkähler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a failure of the existence of a Morse theory.

math.DG↗