Interpretations of Stability for Twisted Quiver Representations on the Projective Line
Abstract
The Kobayashi-Hitchin correspondence shows that the moduli space of stable Higgs bundles MX(r, d)
corresponds directly with solutions to the Hitchin equations, which are self-dual, dimensionally-reduced Yang Mills equations written on a smooth Hermitian bundle E of rank r ≥ 1 and degree d on a smooth compact
Riemann surface X of genus g ≥ 2 [5]. We may expand this correspondence to all g ≥ 0 when we consider
twisted versions of the Hitchin equations. As surveyed by Rayan [14], the moduli space MX(r, d) can be
equipped with a natural U(1) action and the fixed points of this action can be encoded in a “twisted”
representation of an A-type quiver,
•
(r1,d1)
•
(r2,d2)
· · · •
(rn,dn)
,
φ1 φ2 φn−1
where Pn
i=1 ri = r,
Pn
i=1 di = d and φi
is a bundle map from a rank ri
, degree di
, bundle to a rank ri+1 and
degree di+1 bundle tensored by a fixed holomorphic line bundle L. Moreover, in the special case when X is the
projective line, the Birkhoff-Grothendieck theorem says that vector bundles in the above quiver decompose
into a direct sum of line bundles. Expanding each node accordingly, this allows for many interesting types
of quivers, such as argyle quivers as explored by Rayan and Sundbo [15].
This thesis aims to introduce the reader to stable quiver representations in a twisted category of bundles
on X. We begin by reviewing the standard theory of linear quiver representations as well as the theory of
holomorphic vector bundles on algebraic curves. After this background material, we introduce the notion of
a stable vector bundle defined in terms of the Mumford slope condition [9] and then extend this definition
more generally to stable twisted quiver representations in the category of bundles on X. From these twisted
representations we introduce several associated induced ordinary quiver representations. Finally, we present
necessary conditions for stability as linear programming problems when X = P
1
for quiver representations
of type (2,1) and type (2,2) and discuss how these necessary stability conditions are manifested in the
aforementioned induced ordinary quiver representations