We consider Dirac operators on odd-dimensional compact spin manifolds which are twisted by a product bundle. We show that the space of connections on the twisting bundle which yields an invertible operator has infinitely many connected components if the untwisted Dirac operator is invertible and the dimension of the twisting bundle is sufficiently large.

1.
Atiyah
,
M. F.
, “
Eigenvalues of the Dirac operator
,” in
Lecture Notes in Mathematics
(
Springer
,
Berlin
,
1985
), Vol.
1111
, pp.
251
260
.
2.
Atiyah
,
M. F.
,
Patodi
,
V. K.
, and
Singer
,
I. M.
, “
Spectral asymmetry and Riemannian geometry. III
,”
Math. Proc. Cambridge Philos. Soc.
79
,
71
99
(
1976
).
3.
Bär
,
C.
, “
Metrics with harmonic spinors
,”
Geom. Funct. Anal.
6
,
899
942
(
1996
).
4.
Booss
,
B.
and
Wojciechowski
,
K.
, “
Desuspension of splitting elliptic symbols I
,”
Ann. Global Anal. Geom.
3
,
337
383
(
1985
).
5.
Booss-Bavnbek
,
B.
and
Wojciechowski
,
K. P.
,
Elliptic Boundary Problems for Dirac Operators
(
Birkhäuser
,
1993
).
6.
Bredon
,
G. E.
,
Topology and Geometry
,
Graduate Texts in Mathematics
(
Springer
,
1993
), Vol.
139
.
7.
Davaux
,
H.
, “
An optimal inequality between scalar curvature and spectrum of the Laplacian
,”
Math. Ann.
327
,
271
292
(
2003
).
8.
Ginoux
,
N.
,
The Dirac Spectrum
,
Lecture Notes in Mathematics
(
Springer-Verlag
,
2009
), Vol.
1976
.
9.
Herzlich
,
M.
, “
Extremality of the Vafa-Witten bound on the sphere
,”
Geom. Funct. Anal.
15
,
1153
1161
(
2005
).
10.
Jardim
,
M.
and
Leão
,
R. F.
, “
On the eigenvalues of the twisted Dirac operator
,”
J. Math. Phys.
50
,
063513
(
2009
).
11.
Kronheimer
,
P.
and
Mrowka
,
T.
, “
Monopoles and three-manifolds
,” in
New Mathematical Monographs
(
Cambridge University Press
,
Cambridge
,
2007
), Vol.
10
.
12.
Lawson
,
H. B.
and
Michelsohn
,
M.-L.
,
Spin Geometry
(
Princeton University Press
,
1989
).
13.
Miatello
,
R. J.
and
Podestá
,
R. A.
, “
The spectrum of twisted Dirac operators on compact flat manifolds
,”
Trans. Am. Math. Soc.
358
,
4569
4603
(
2006
).
14.
Vafa
,
C.
and
Witten
,
E.
, “
Eigenvalue inequalities for fermions in gauge theories
,”
Commun. Math. Phys.
95
,
257
276
(
1984
).
15.
Waterstraat
,
N.
, “
A remark on the space of metrics having non-trivial harmonic spinors
,”
J. Fixed Point Theory Appl.
13
,
143
149
(
2013
); e-print arXiv:1206.0499 [math.SP].
16.
Waterstraat
,
N.
, “
A family index theorem for periodic Hamiltonian systems and bifurcation
,”
Calculus Var. Partial Differ. Equations
52
,
727
753
(
2015
); e-print arXiv:1305.5679 [math.DG].
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