SLIDE 1 An index theorem for end-periodic operators ∗
Tomasz Mrowka1 Daniel Ruberman2 Nikolai Saveliev3
1Department of Mathematics
Massachusetts Institute of Technology
2Department of Mathematics
Brandeis University
3Department of Mathematics
University of Miami
University of Minnesota, November 2011
∗http://arxiv.org/pdf/1105.0260, http://arxiv.org/pdf/0905.4319
SLIDE 2 Introduction
Background: Index of elliptic operators on closed manifolds. Setup: X a closed Riemannian manifold; E → X and F → X vector bundles; D : C∞(E) → C∞(F) an elliptic differential operator. Examples to keep in mind: E = Λeven(X), F = Λodd(X); operator D = d + d∗. X 4k-dimensional oriented manifold; D = signature
X 4k-dimensional spin manifold; E, F spinor bundles S±; D+ = chiral Dirac operator.
SLIDE 3 Usually complete C∞(E) to Sobolev space L2
k(E).
Key fact: on closed manifold ellipticity ⇒ Fredholm property: D has closed range, dim(ker(D)) and dim(coker(D)) finite. Definition: ind(D) = dim(ker(D)) − dim(coker(D)). The Atiyah-Singer Index theorem (1963): ind(D) =
AS(D) with AS(D) a characteristic class associated to D and X. Basic example: Dirac operator D+ on spin manifold AS = A, a polynomial in Pontrjagin classes.
SLIDE 4 What if X has boundary or is not compact? Atiyah-Patodi-Singer (1975) consider case X compact, ∂X = Y. Add collar to boundary to get M = X ∪Y R+ × Y. On the end, D+ ∼ = d
dt + DY.
APS show: D+ : L2
1(M; S+) → L2(M; S−) is Fredholm if
ker(DY) = {0}. Index formula involves a new term, η(DY). ind(D+
M) =
2η(DY).
SLIDE 5 The η-invariant is a spectral invariant: it is the value at s = 0 of the meromorphic extension of the function
λ ∈ Spec(DY). Intuitively, η measures spectral asymmetry of DY η(DY) =# of positive eigenvalues of DY − # of negative eigenvalues of DY. General case of non-compact X: too hard!
SLIDE 6
Periodic end manifolds and operators
M has a periodic end (cf. Taubes) if M = Z ∪ W ∪ W ∪ W ∪ · · · where Z and W are compact manifolds with ∂Z = Y and ∂W = −Y ∪ Y:
Z Y Y Y W W
Model for the end of M is X = · · · ∪ W ∪ W ∪ W ∪ · · · , the infinite cyclic cover of X = W/ ∼. Pick a smooth function f : X → S1 that classifies this covering.
SLIDE 7 Special case: X = S1 × Y, with X = R × Y. Lift a metric on X to X; extend over Z to define a metric on M. End-periodic bundles and operators have natural definitions in this setting. The operator on the end should be lifted from X. Taubes investigated the general problem of when such
- perators are Fredholm on weighted L2 spaces. A necessary
condition is that ind(DX) = 0. For Dirac operator (on spin manifolds) we have Theorem(R-S 2006) Suppose that ind(D+
X ) = 0. Then the
M : L2 1 → L2 is Fredholm for a generic metric on X.
SLIDE 8 Index theorem for Dirac operators
For simplicity, I’ll state the index theorem for the Dirac operator: Theorem(M-R-S 2011) Suppose that ind(D+
X ) = 0, and pick a
generic metric on X. Then ind(D+
M) =
ω −
f ∗(dθ) ∧ ω − 1 2η(X), where dω = A and η(X) is the end-periodic eta-invariant described below. Remark If supp f ∗(dθ) lies in a neighborhood (−ǫ, ǫ) × Y then ind(D+
M) =
2η(X).
SLIDE 9 The end-periodic η-invariant
The adjoint of D+ is the negative Dirac operator D−. Both extend to holomorphic families D±
z = D± − ln z · f ∗(dθ)
Then η(X, g) = 1 πi ∞
Tr
z e−tD−
z D+ z
dz z dt For X = S1 × Y, and g a product metric, this is η(Y, gY).
SLIDE 10
End-periodic spectral flow
The index on M depends on the choice of metric g on X. Index change calculated by end-periodic spectral flow Points z ∈ C∗ where D±
z have non-zero kernel are called
spectral points.
1 Figure: Spectral points
SLIDE 11 In 1-parameter family gt, the L2–index is constant as long as D±
z remains Fredholm.
Equivalently (Taubes) no spectral points z ∈ S1 appear. Spectral curves For zt spectral points for D+(gt): S =
(zt, t) ∈ C × [0, 1] Spectral flow formula: For gt a generic path, ind(D+(g1)) − ind(D+(g0)) is the intersection number SF(D+(g0), D+(g1)) = (S ·
SLIDE 12
C t S
Figure: Spectral flow
SLIDE 13
Application to 4-manifolds
Investigation originated in study invariants of 4-manifolds. Seiberg-Witten theory assigns to a 4-manifold X and Spinc structure s, a number SW(X, s), by counting irreducible solutions (up to equivalence) to the Seiberg-Witten equations. Variables: Spinc connection A; spinor ψ ∈ C∞(S+) D+
A (g)ψ = 0
F +
A + r 2q(ψ) = µ
where g is a metric on X, and µ ∈ Ω2
+(X; iR).
SLIDE 14 Equations depend on metric on X and 2-form µ. Generic perturbation µ makes moduli space smooth,
- riented 0-manifold with no reducibles (ψ = 0).
Signed count of irreducible (ψ = 0) solutions to µ-perturbed Seiberg-Witten equations ⇒ SW(X, g, µ) ∈ Z. Independent from g and µ if b+
2 X > 1.
For applications to problems in topology–want to understand this when X has the homology of S1 × S3.
SLIDE 15
Because b+
2 X = 0, SW(X, g, µ) metric/perturbation dependent.
Theorem (M-R-S 2009) For generic metric/perturbations (g0, µ0) and (g1, µ1), the change in SW invariants is given by end-periodic spectral flow; SW(X, g1, µ1)−SW(X, g0, µ0) = SF(D+(g0)+iβ0, D+(g1)+iβ1) where µi = d+βi. This leads to a well-defined invariant SW(X) by adding a periodic-index term to SW(X, g, µ)
SLIDE 16
Definition: Consider the quantity λSW(X, g, β) = SW(X, g, β) − ind(D+
β (M, g)) − 1
8sign(Z) associated to X, and any periodic-end spin manifold M with end modeled on X. Theorem (M-R-S 2009) λSW(X, g, β) is independent of choice of Z, metric, and perturbation, and gives a C∞ invariant of X. λSW(X, g, β), reduced modulo 2, is the classical Rohlin invariant of X.
SLIDE 17 Brief sketch of index theorem proof
Heat equation method: on closed manifold ind(D+) = lim
t→∞
- tr(e−tD−D+) − tr(e−tD+D−)
- Traces are defined by integrating heat kernels K(t; x, x).
Characteristic classes appear when we consider limt→0. On non-compact manifold, the operators are not trace-class because integral doesn’t converge. Work instead with normalized traces Tr♭(D−D+) = lim
N→∞
tr(K(t; x, x)) − (N + 1)
tr( K(t; x, x))
MN = Z∪ N copies of W, K = heat kernel on M, K = heat kernel on
- X. The η invariant appears at the end the calculation.
