Table Of ContentThe kernel of Dirac operators on S3 and R3
0
0
0 L´aszl´o Erd˝os ∗ Jan Philip Solovej†
2
n School of Mathematics Department of Mathematics
a
J
GeorgiaTech University of Copenhagen
4
2
Atlanta GA-30332, USA Universitetsparken 5,
1
v DK-2100 Copenhagen, Denmark
6
3
0 January 21, 2000
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0
0
0
/
h Abstract
p
-
h In this paper we describean intrinsically geometric way of produc-
t
a ing magnetic fields on S3 and R3 for which the corresponding Dirac
m
operators have a non-trivial kernel. In many cases we are able to
:
v
compute the dimension of the kernel. In particular we can give exam-
i
X
ples where the kernel has any given dimension. This generalizes the
r
a
examples of Loss and Yau [LY].
AMS 1991 Subject Classification: 53A50, 57R15, 58G10, 81Q05, 81Q10
Running title: Kernel of Dirac operators
∗L. Erdo˝s was supported by the N.S.F. grant DMS-9970323
†J.P. Solovej was supported in parts by the EU TMR-grant FMRX-CT 96-0001 by
MaPhySto – Centre for Mathematical Physics and Stochastics, funded by a grant from
TheDanishNationalResearchFoundationandbyagrantfromtheDanishNaturalScience
Research Council
1
1 Introduction
In [LY] Loss and Yau proved the existence of a magnetic field B = ∇×A :
R3 → R3 with the property that |B|2 < ∞ and such that the Dirac
R3
operator R
σ ·(−i∇−A) (1)
has a nonvanishing kernel in L2(R3;C2).
The significance of this result was its implications to the stability of mat-
ter (electrons and nuclei) coupled to classical electromagnetic fields. This
model was studied in the series of papers [FLL, LY, LL]. The existence of a
square integrable zero mode for the Dirac operator corresponding to a square
integrable magnetic field implies that matter cannot be stable unless there
is an upper bound on the fine structure constant.
Loss and Yau gave a very explicit construction of a magnetic field B =
∇×A andof a corresponding zero mode, i.e. a solution to [σ·(−i∇−A)]ψ =
0. They also discussed a general way of constructing a vector potential A
for a given ψ so that ψ be in the kernel of the Dirac operator (1). However
their methods gave only one element of the kernel for each magnetic field
constructed. Moreover, the proofs are very computational and they some-
what left the origin of these zero modes unexplored. Further examples of
zero modes were given later in [E] and [AMN1], based upon ideas from the
Loss and Yau construction.
In this paper we discuss a more geometric way of constructing Dirac
operatorswithanontrivialkernelonR3. Moreprecisely, wedescribeafamily
of magnetic fields on the 3-sphere S3 for which we can give a characterization
ofthespectrum andinparticular forsomeofthese fieldswe canalsocalculate
the dimension of the kernel. It is a well known fact (see [H] and Theorem 23
2
below) that the dimension of the kernel of the Dirac operator is a conformal
invariant. Since R3 is conformally invariant to the 3-sphere with a point
removed we can use the construction on S3 to learn about the kernel of Dirac
operators on R3.
On a general Riemannian manifold one can define the Dirac operator if
one has a Spin structure, a corresponding spinor bundle, and an appropriate
covariant derivative (a Spin connection). The kernel of this (nonmagnetic)
Dirac operator has been studied in [B]. If one is interested in Dirac operators
withmagneticfields onemust consider insteadSpinc structures, Spinc spinor
bundles and a Spinc connections. The magnetic field is then related to the
curvature of the connection (see Definition 3). On R3 these structures reduce
to the well known objects. The Spinc spinors are simply maps from R3 → C2
and the Dirac operators are of the form (1).
On2-dimensionalmanifoldsandingeneraloneven dimensional manifolds
the Atiyah-Singer Index Theorem often gives nontrivial information on the
index of the Dirac operator. In certain cases one knows from vanishing
theorems that the index is equal to the dimension of the kernel. One example
of such a result is the Aharonov-Casher Theorem (see Theorem 37) which
holds for Dirac operators on R2 and S2. Characteristic for the index theorem
is of course that the index is expressed in terms of topological quantities,
whereas in general the dimension of the kernel is not a topological invariant.
For odd-dimensional manifolds it is not easy to get information about the
dimension of the kernel from index theorems. Given a Dirac operator (with
magnetic field) on S3 we do not know in general how to say anything about
its kernel.
Inthis paper we explain how, for certainmagnetic fields onS3 one may, in
a sense, separate variables and reduce the problem to a problem on S2, where
3
one can use the Aharonov-Casher Theorem, and a problem on S1 which can
be solved explicitly.
Our construction can be used on other manifolds than S3.
In Sect. 2 we discuss Spinc structures and define the Dirac operator and
magnetic fields. In Sect. 4 we discuss how the Dirac operator changes under
conformaltransformations. InSect.5wedescribehowtoliftSpinc structures
from 2 to 3-dimensional manifolds. One can of course not always do this. It
requires that there is a map of the type known as a Riemannian submersion
fromthe3tothe2-dimensionalmanifold. Ifsuchamapexists wegivealower
bound on the kernel of the 3-dimensional Dirac operator using the index
Theorem for the 2-dimensional Dirac operator (see Theorem 31 in Sect. 6).
Finally in Sect. 8 we give the more detailed results for S3. In this case we
use the Hopf map as the Riemannian submersion from S3 to S2. The Hopf
map however has much stronger properties than just being a Riemannian
submersion. These properties allow us to separate variables for the Dirac
operator on S3. Our main results for S3 can be found in Theorem 35 and the
remarks following it.
In particular, we show that one can construct Dirac operators on R3 and
S3 having kernels of any given dimension. Examples of Dirac operators on
R3 with degenerate kernels were recently given independently in [AMN2] for
a subclass of the magnetic fields considered here. The exact degeneracy was
however not proved there.
Our results were announced in [ES].
2 Spinc bundles
4
1. DEFINITION (Spinc spinor bundle). LetM bea3-dimensionalRie-
mannian manifold. A Spinc spinor bundle Ψ over M is a 2-dimensional com-
plex vector bundle over M with inner product and an isometry σ : T∗M →
Ψ(2), where
Ψ(2) := {A ∈ End(Ψ) : A = A∗,TrA = 0}.
The inner product on Ψ(2) is given by (A,B) := 1Tr[AB]. A spinor bundle
2
over a 2-dimensional Riemannian manifold is defined in the same way except
that then σ is only an injective partial isometry. The map σ is called the
Clifford multiplication of the spinor bundle Ψ.
Note that if A,B ∈ Ψ2 then {A,B} := AB +BA = Tr[AB]I = 2(A,B)I
and therefore
{σ(α),σ(β)} = 2(α,β)I, for all α,β ∈ T∗M (2)
where (·,·) in the last equation denotes the metric (inner product) on T∗M.
We have here used the convention that the Clifford multiplication is Hermi-
tian rather than anti-Hermitian, which is the more common in the mathe-
matics literature.
2. DEFINITION ( Spin spinor bundle). A Spinc spinor bundle Ψ over
a 2 or 3-dimensional manifold M is said to be a Spin spinor bundle if there
exists an antilinear bundle isometry C : Ψ → Ψ such that (η,Cη) = 0 and
C2η = −η for all η ∈ Ψ. In the physics literature the map C is often referred
to as charge conjugation. An equivalent way to say that a Spinc bundle is
actually a Spin bundle is to say that the determinant line bundle Ψ ∧ Ψ is
trivial.
3. REMARK. We shall use mostly Spinc spinor bundles in our results. For
brevity, we shall refer to them simply as spinor bundles. Spin spinor bundles
will be mentioned only in some additional remarks.
5
The following proposition shows that in 3-dimensions the spinor bundle
gives a natural orientation of M.
4. PROPOSITION. Let Ψ be a spinor bundle over a 3-dimensional man-
ifold M with Clifford multiplication σ. If e1,e2,e3 is an orthonormal basis in
T∗M then iσ(e1)σ(e2)σ(e3) = ±I.
p
Proof. Let a = iσ(e1)σ(e2)σ(e3). It follows immediately from (2) that a is
Hermitian and commutes with σ(ej), j = 1,2,3. Hence a is a real scalar.
Againfrom(2)itisclearthatσ(ej)2 = I,forj = 1,2,3andhencea2 = I.
5. DEFINITION (Positive orientation). We say that e1,e2,e3 is a pos-
itively oriented basis if iσ(e1)σ(e2)σ(e3) = −I.
6. REMARK (Spinors over non-orientable manifolds). Proposition 4 shows
that a spinor bundle with C2 fibers can exist only on orientable manifolds.
However, onemayalsodefinespinorbundlesovernon-orientable3-dimensional
manifolds but in this case it should be a C4 bundle and the Clifford multi-
plication map should be an injective partial isometry satisfying (2).
7. PROPOSITION (Basis for Ψ gives basis for T M). Let ξ be a lo-
∗ ±
cal orthonormal basis of spinor fields. Define the vectors e ,e such that
1 2
α(e )+iα(e ) = (ξ ,σ(α)ξ )
1 2 − +
holds for all one-forms α. Then e ,e are orthonormal.
1 2
In the case when M is 3-dimensional also define e such that
3
α(e ) = (ξ ,σ(α)ξ )
3 + +
for all α. Then e ,e ,e is a positively oriented orthonormal basis.
1 2 3
6
Proof. Wetreatthe3-dimensionalcase, the2-dimensionalcaseissimilar. We
prove instead that there is an orthonormal basis of forms e1,e2,e3 for which
e , e , e is the dual basis. This follows if we show that σ(e1), σ(e2), σ(e3)
1 2 3
are orthonormal in Ψ(2). Of course we define e1, e2, e3 by ej(e ) = δ , for
i ij
i,j = 1,.... We then see from the definitions of e , e , e that the matrices
1 2 3
of σ(e1), σ(e2), σ(e3), in the basis ξ are the standard Pauli matrices (note
±
that (ξ ,σ(α)ξ ) = −(ξ ,σ(α)ξ ) since σ(α) is traceless)
+ + − −
0 1 0 −i 1 0
, , .
1 0 i 0 0 −1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
The orthonormality and positivity follows from simple matrix calculations.
8. DEFINITION (Spinc Connection). A connection ∇ ona spinor bun-
dle Ψ is said to be a Spinc connection if for all tangent vectors X ∈ T M we
∗
have
(i.) X(ξ,η) = (∇ ξ,η)+(ξ,∇ η) for all sections ξ,η in Ψ.
X X
(ii.) [∇ ,σ(α)] = σ(∇ α) for all one-forms α on M. Here ∇ α refers to
X X X
the Levi-Civita connection acting on one-forms.
9. PROPOSITION (Local expression for ∇ ). Let ξ be a local or-
X ±
thonormal basis of spinor fields on a 3-dimensional manifold and let e ,e ,e
1 2 3
be the orthonormal basis defined in Prop. 7. Then for all vectors X
(ξ ,∇ ξ ) (ξ ,∇ ξ )
+ X + + X −
(ξ ,∇ ξ ) (ξ ,∇ ξ )
− X + − X −
7
(e ,∇ e ) −(e ,∇ e )−i(e ,∇ e )
1 X 2 3 X 2 3 X 1
= i1 −iα(X)I,
2
−(e ,∇ e )+i(e ,∇ e ) −(e ,∇ e )
3 X 2 3 X 1 1 X 2
where α is the real local one-form given by
α(X) = i1(ξ ,∇ ξ )+i1(ξ ,∇ ξ ).
2 + X + 2 − X −
The same formulas are true in the 2-dimensional case if we simply replace
e by 0 everywhere.
3
Proof. Let e1,e2,e3 be the dual basis to e ,e ,e . Using that X(ω(e )) =
1 2 3 j
∇ ω(e )+ω(∇ e ) for any one-form ω we find, with ω = ej, that
X j X j
(e ,∇ e )+i(e ,∇ e ) = ej(∇ e )+iej(∇ e )
j X 1 j X 2 X 1 X 2
= X(ξ ,σ(ej)ξ )−(ξ ,σ(∇ ej)ξ )
− + − X +
= (∇ ξ ,σ(ej)ξ )+(ξ ,σ(ej)∇ ξ ).
X − + − X +
Since σ(e3)ξ = ±ξ (see the proof of Prop. 7) we find
± ±
(e ,∇ e )+i(e ,∇ e ) = (∇ ξ ,ξ )−(ξ ,∇ ξ ) = −2(ξ ,∇ ξ )
3 X 1 3 X 2 X − + − X + − X +
where we have also used that 0 = X(ξ ,ξ ) = (∇ ξ ,ξ ) + (ξ ,∇ ξ ).
− + X − + − X +
This also gives
(ξ ,∇ ξ ) = −(ξ ,∇ ξ ) = 1(e ,∇ e )−i1(e ,∇ e ).
+ X − − X + 2 3 X 1 2 3 X 2
Using σ(e1)ξ = ξ we obtain
± ∓
(ξ ,∇ ξ )−(ξ ,∇ ξ ) = (∇ ξ ,ξ )+(ξ ,∇ ξ )
+ X + − X − X − − + X +
= (e ,∇ e )+i(e ,∇ e ) = i(e ,∇ e )
1 X 1 1 X 2 1 X 2
where we have used that (e ,∇ e ) = 1X(e ,e ) = 0. It only remains to
1 X 1 2 1 1
show that α is real. This follows from 0 = X(ξ ,ξ ) = 2Re(ξ ,∇ ξ ).
± ± ± X ±
8
In the 2-dimensional case we conclude from (∇ ej,ej) = 0 that ∇ e1 =
X X
(∇ e1,e2)e2 and ∇ e2 = (∇ e2,e1)e1. Since σ(e1)σ(e2)ξ = iξ we have
X X X + +
σ(e1)σ(e2)∇ ξ = ∇ (σ(e1)σ(e2)ξ )−σ(∇ e1)σ(e2)ξ −σ(e1)σ(∇ e2)ξ
X + X + X + X +
= i∇ ξ − (∇ e1,e2)+(∇ e2,e1) ξ = i∇ ξ ,
X + X X + X +
(cid:0) (cid:1)
where we have used that (∇ e1,e2) + (∇ e2,e1) = X(e1,e2) = 0. There-
X X
fore ∇ ξ is an eigenvector with eigenvalue i of the antihermitean operator
X +
σ(e1)σ(e2). Since σ(e1)σ(e2)ξ = −iξ we conclude that (ξ ,∇ ξ ) = 0
− − − X +
in the 2-dimensional case. The other statements in the 2-dimensional case
follow as for the 3-dimensional case.
10. DEFINITION (Curvature tensor and magnetic form on Ψ). As
usual the curvature tensor is defined by
R (X,Y)ξ = ∇ ∇ ξ −∇ ∇ ξ −∇ ξ,
Ψ X Y Y X [X,Y]
where ξ is a spinor section and X,Y are vector fields on M. The Magnetic
2-form β on M is defined via the trace of R as
Ψ
β(X,Y) = i1Tr[R (X,Y)]. (3)
2 Ψ
11. PROPOSITION. The operator iR (X,Y) is Hermitian on Ψ and β
Ψ
is a real closed two-form. Locally β = dα, where α is the one-form defined
in Proposition 9.
Proof. In fact it follows from the definition of a Spinc connection that
X(Y ((ξ,η))) = (∇ ∇ ξ,η)+(ξ,∇ ∇ η)+(∇ ξ,∇ η)+(∇ ξ,∇ η).
X Y X Y X Y Y X
9
Hence
(∇ ∇ ξ −∇ ∇ ξ,η)+(ξ,∇ ∇ η −∇ ∇ η)
X Y Y X X Y Y X
= X(Y ((ξ,η)))−Y (X ((ξ,η)))
= [X,Y]((ξ,η)) = (∇ ξ,η)+(ξ,∇ η)
[X,Y] [X,Y]
and thus (R (X,Y)ξ,η)+(ξ,R (X,Y)η) = 0.
Ψ Ψ
That β is real is a consequence of iR being Hermitian. That it is a
Ψ
2-form, i.e, that it is antisymmetric is immediate from the definition. That
β = dα will be shown in the proof of Theorem 12, hence it immediately
follows that β is closed.
12. THEOREM (Spinor curvature in terms of Riem. curvature). Let
R denote the Riemann curvature tensor of M and let {e } be any orthonor-
i
mal basis of vector fields and {ei} the dual basis of one-forms. For all vectors
X and Y we then have the identity
R (X,Y) = 1 (e ,R(X,Y)e )σ(ei)σ(ej)−iβ(X,Y)I
Ψ 4 i j
ij
X
Proof. It is enough to check this identity at a point p ∈ M. It is also clear
that the identity is independent of the choice of orthonormal basis of vectors.
We may choose an orthonormal spinor basis ξ such that ∇ ξ = 0 at p for
± X ±
all vectors X. The corresponding vectors e defined in Prop. 7 gives a local
i
geodesic basis at p, i.e., ∇ e = 0. Moreover, the one-form α from Prop. 9
ej i
vanishes at the point p. Using the local expression for the Spinc connection
given in Prop. 9 we find
R (X,Y)ξ = i (±(e ,R(X,Y)e )ξ −((e ,R(X,Y)e )∓i(e ,R(X,Y)e ))ξ )
Ψ ± 2 1 2 ± 3 2 3 1 ∓
−i(∇ α(Y)−∇ α(X))ξ
X Y ±
10
