Brian Collier

Asymptotics of Higgs bundles in the Hitchin component

In this paper we pursue a more geometric approach to compactification of the Hitchin component. Our main motivation is Wolfs harmonic map interpretation of Thurstons compactification of Teichmuller space with measured foliations. Using Hitchins parameterization of the Hitchin component by holomorphic differentials, we study asymptotics of certain rays of representations. More precisely, along these rays we solve the Hitchin equations asymptotically and use the solution to study the asymptotics of the parallel transport operator of the associated flat connection. The asymptotics of the corresponding family of equivariant harmonic maps to the symmetric space SL(n,R)/SO(n,R) proves a conjecture of Katzarkov, Noll, Pandit and Simpson [17] on the Hitchin WKB problem in our setting.

Parallel transport on principal bundles over stacks

Abstract In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldorf from manifolds to stacks. In the process of proving our main result we simplify Schreiber and Waldorf’s original definition of a transport functor for principal bundles with connections over manifolds and provide a more direct proof of the correspondence between principal bundles with connections and transport functors.

Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars

Abstract For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups SO ( p , q ) with 2 p q . These groups lie outside formerly know classes of groups associated with exotic components.