Affiliation:
1. Department of Mathematics and Experimental Science, School of Education Fray Luis de Leon, Catholic University of Ávila 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
and
ℳ
(
E
6
)
{\mathcal{ {\mathcal M} }}\left({E}_{6})
be the moduli space of
E
6
{E}_{6}
-Higgs bundles over
X
X
. We consider the automorphisms
σ
+
{\sigma }_{+}
of
ℳ
(
E
6
)
{\mathcal{ {\mathcal M} }}\left({E}_{6})
defined by
σ
+
(
E
,
φ
)
=
(
E
∗
,
−
φ
t
)
{\sigma }_{+}\left(E,\varphi )=\left({E}^{\ast },-{\varphi }^{t})
, induced by the action of the outer involution of
E
6
{E}_{6}
in
ℳ
(
E
6
)
{\mathcal{ {\mathcal M} }}\left({E}_{6})
, and
σ
−
{\sigma }_{-}
defined by
σ
−
(
E
,
φ
)
=
(
E
∗
,
φ
t
)
{\sigma }_{-}\left(E,\varphi )=\left({E}^{\ast },{\varphi }^{t})
, which results from the combination of
σ
+
{\sigma }_{+}
with the involution of
ℳ
(
E
6
)
{\mathcal{ {\mathcal M} }}\left({E}_{6})
, which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of
σ
+
{\sigma }_{+}
and
σ
−
{\sigma }_{-}
, as
F
4
{F}_{4}
-Higgs bundles,
F
4
{F}_{4}
-Higgs pairs associated with the fundamental irreducible representation of
F
4
{F}_{4}
, and
PSp
(
8
,
C
)
{\rm{PSp}}\left(8,{\mathbb{C}})
-Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of
Sp
(
8
,
C
)
{\rm{Sp}}\left(8,{\mathbb{C}})
. Finally, we describe the reduced notions of semistability and polystability for these objects.
Cited by
3 articles.
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