Affiliation:
1. Department of Mathematics and Experimental Science, University College of Education Fray Luis de Leon, Catholic University of Avila, C/ Tirso de Molina , 44 , 47010 Valladolid , Spain
Abstract
Abstract
Let
X
X
be a compact Riemann surface of genus
g
≥
2
g\ge 2
,
G
G
be a semisimple complex Lie group and
ρ
:
G
→
GL
(
V
)
\rho :G\to {\rm{GL}}\left(V)
be a complex representation of
G
G
. Given a principal
G
G
-bundle
E
E
over
X
X
, a vector bundle
E
(
V
)
E\left(V)
whose typical fiber is a copy of
V
V
is induced. A
(
G
,
ρ
)
\left(G,\rho )
-Higgs pair is a pair
(
E
,
φ
)
\left(E,\varphi )
, where
E
E
is a principal
G
G
-bundle over
X
X
and
φ
\varphi
is a holomorphic global section of
E
(
V
)
⊗
L
E\left(V)\otimes L
,
L
L
being a fixed line bundle over
X
X
. In this work, Higgs pairs of this type are considered for
G
=
Spin
(
8
,
C
)
G={\rm{Spin}}\left(8,{\mathbb{C}})
and the three irreducible eight-dimensional complex representations which
Spin
(
8
,
C
)
{\rm{Spin}}\left(8,{\mathbb{C}})
admits. In particular, the reduced notions of stability, semistability, and polystability for these specific Higgs pairs are given, and it is proved that the corresponding moduli spaces are isomorphic, and a precise expression for the stable and not simple Higgs pairs associated with one of the three announced representations of
Spin
(
8
,
C
)
{\rm{Spin}}\left(8,{\mathbb{C}})
is described.