Annals of Mathematics, 150 (1999), 1177–1179
Addendum to
Classification of irreducible holonomies
of torsion-free affine connections
By Sergei Merkulov and Lorenz Schwachhöfer*
The real form Spin(6, H) ⊂ End(R32 ) of Spin(12, C) ⊂ End(C32 ) is absolutely irreducible and thus satisfies the algebraic identities (40) and (41).
Therefore, it also occurs as an exotic holonomy and the associated supermanifold Mg admits a SUSY-invariant polynomial. This real form has been
erraneously omitted in our paper.
Also, the two real four-dimensional exotic holonomies, whose occurrences
were unknown at the time of writing, have been shown to exist very recently
by R. Bryant [B].
With these corrections, Table 3 and the table in Theorem C should read
as follows.
∗ The
original article appeared in 150 (1999), 77–149.
1178
¨
SERGEI MERKULOV AND LORENZ SCHWACHHOFER
Table 3: List of exotic holonomies
group G
representation V
TR · Spin(5, 5)
R16
R16
C16 ' R32
TR · Spin(1, 9)
TC · Spin(10, C)
R27
R27
TR · E16
TR · E46
TC · EC
6
C27 ' R54
TR · SL(2, R)
¯ 3 R2 ' R4
C∗ · SL(2, C)
R∗ · Sp(2, R)
C∗ · Sp(2, C)
¯ 3 C2 ' R8
SL(2, C)
¯ 3 C2 ' R8
C
R∗ · SO(2) · SL(2, R)
C∗ · SU(2)
R4
4
' R8
R2 ⊗ R 2 ' R4
C2 ' R4
C2 ' R4
C2 ' R4
Hλ · SU(2)
Hλ · SU(1, 1)
SL(2, R) · SO(p, q)
R2 ⊗ Rp+q ' R2(p+q)
Hn ' R4n
C2 ⊗ Cn ' R4n
Sp(1) · SO(n, H)
SL(2, C) · SO(n, C)
>3
n>2
n>3
p+q
R56
R56
E57
E77
EC
7
R112 ' C56
Sp(3, R)
R14 ⊂ Λ3 R6
R28 ' C14 ⊂ Λ3 C6
Sp(3, C)
SL(6, R)
R20 ' Λ3 R6
R20
R20
R40 ' Λ3 C6
SU(1, 5)
SU(3, 3)
SL(6, C)
R32
R32
R32
C32 ' R64
Spin(2, 10)
Spin(6, 6)
Spin(6, H)
Spin(12, C)
Notation:
restrictions/remarks
TF denotes any connected Lie subgroup of
Hλ =
©
e(2πi+λ)t | t ∈ R
ª
⊆ C∗ , λ > 0.
F∗ ,
ADDENDUM: CLASSIFICATION OF IRREDUCIBLE HOLONOMIES
1179
Table from Theorem C
Group G
Representation space
Group G
Representation space
Sp(n, R)
R
C2n
E57
R56
R56
C56
R32
R32
R32
C32
2n
Sp(n, C)
SL(2, R)
R4 ' ¯ 3 R2
C4 ' ¯ 3 C2
SL(2, C)
SL(2, R) · SO(p, q)
SL(2, C) · SO(n, C)
Sp(1)SO(n, H)
SL(6, R)
SU(1, 5)
SU(3, 3)
SL(6, C)
E77
R
2(p+q)
, p+q
>3
C2n , n > 3
Hn ' R4n , n > 2
R20 ' Λ3 R6
R20
R20
C20 ' Λ3 C6
E7C
Spin(2, 10)
Spin(6, 6)
Spin(6, H)
Spin(12, C)
Sp(3, R)
Sp(3, C)
R14 ⊂ Λ3 R6
C14 ⊂ Λ3 C6
Glasgow University, Glasgow, UK
E-mail address: sm@maths.gla.ac.uk
Mathematisches Institut, Universität Leipzig, Leipzig, Germany
E-mail address: schwachh@mathematik.uni-leipzig.de
References
[B] R. Bryant, Recent advances in the theory of holonomy, Séminaire Bourbaki, 51ème année,
1998-99, n◦ 861; xxx mathematics e-print archive: math.DG/9910059; Astérisque, to appear.