Introduction Background: Index of elliptic operators on closed - PowerPoint PPT Presentation

An index theorem for end-periodic operators ∗ Tomasz Mrowka 1 Daniel Ruberman 2 Nikolai Saveliev 3 1 Department of Mathematics Massachusetts Institute of Technology 2 Department of Mathematics Brandeis University 3 Department of Mathematics University of Miami University of Minnesota, November 2011 ∗ http://arxiv.org/pdf/1105.0260, http://arxiv.org/pdf/0905.4319

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 = Λ even ( X ) , F = Λ odd ( X ) ; operator D = d + d ∗ . X 4 k -dimensional oriented manifold; D = signature operator. X 4 k -dimensional spin manifold; E , F spinor bundles S ± ; D + = chiral Dirac operator.

Usually complete C ∞ ( E ) to Sobolev space L 2 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 ) X 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.

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 + D Y . APS show: D + : L 2 1 ( M ; S + ) → L 2 ( M ; S − ) is Fredholm if ker ( D Y ) = { 0 } . Index formula involves a new term, η ( D Y ) . � A − 1 � ind ( D + M ) = 2 η ( D Y ) . M

The η -invariant is a spectral invariant: it is the value at s = 0 of the meromorphic extension of the function � sign λ · | λ | − s , λ ∈ Spec ( D Y ) . Intuitively, η measures spectral asymmetry of D Y η ( D Y ) =# of positive eigenvalues of D Y − # of negative eigenvalues of D Y . General case of non-compact X : too hard!

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 W W Y Y Y Model for the end of M is � X = · · · ∪ W ∪ W ∪ W ∪ · · · , the infinite cyclic cover of X = W / ∼ . Pick a smooth function f : X → S 1 that classifies this covering.

Special case: X = S 1 × 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 operators are Fredholm on weighted L 2 spaces. A necessary condition is that ind ( D X ) = 0. For Dirac operator (on spin manifolds) we have Theorem (R-S 2006) Suppose that ind ( D + X ) = 0. Then the 1 → L 2 is Fredholm for a generic metric on X . operator D + M : L 2

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 � � � f ∗ ( d θ ) ∧ ω − 1 ind ( D + � M ) = A + ω − 2 η ( X ) , M Y X where d ω = � A and η ( X ) is the end-periodic eta-invariant described below. Remark If supp f ∗ ( d θ ) lies in a neighborhood ( − ǫ, ǫ ) × Y then � A − 1 � ind ( D + M ) = 2 η ( X ) . M

The end-periodic η -invariant The adjoint of D + is the negative Dirac operator D − . Both extend to holomorphic families z = D ± − ln z · f ∗ ( d θ ) D ± Then � ∞ � � � dz η ( X , g ) = 1 z D + df · D + z e − tD − Tr z dt z π i 0 | z | = 1 For X = S 1 × Y , and g a product metric, this is η ( Y , g Y ) .

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

In 1-parameter family g t , the L 2 –index is constant as long as D ± z remains Fredholm. Equivalently (Taubes) no spectral points z ∈ S 1 appear. Spectral curves For z t spectral points for D + ( g t ) : � S = ( z t , t ) ∈ C × [ 0 , 1 ] t ∈ [ 0 , 1 ] Spectral flow formula: For g t a generic path, ind ( D + ( g 1 )) − ind ( D + ( g 0 )) is the intersection number � � S 1 × [ 0 , 1 ] SF ( D + ( g 0 ) , D + ( g 1 )) = ( S · .

S t C Figure: Spectral flow

Application to 4-manifolds Investigation originated in study invariants of 4-manifolds. Seiberg-Witten theory assigns to a 4-manifold X and Spin c structure s , a number SW ( X , s ) , by counting irreducible solutions (up to equivalence) to the Seiberg-Witten equations. Variables: Spin c connection A ; spinor ψ ∈ C ∞ ( S + ) D + A ( g ) ψ = 0 F + A + r 2 q ( ψ ) = µ where g is a metric on X , and µ ∈ Ω 2 + ( X ; i R ) .

Equations depend on metric on X and 2-form µ . Generic perturbation µ makes moduli space smooth, oriented 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 S 1 × S 3 .

Because b + 2 X = 0, SW ( X , g , µ ) metric/perturbation dependent. Theorem (M-R-S 2009) For generic metric/perturbations ( g 0 , µ 0 ) and ( g 1 , µ 1 ) , the change in SW invariants is given by end-periodic spectral flow; SW ( X , g 1 , µ 1 ) − SW ( X , g 0 , µ 0 ) = SF ( D + ( g 0 )+ i β 0 , D + ( g 1 )+ i β 1 ) where µ i = d + β i . This leads to a well-defined invariant SW ( X ) by adding a periodic-index term to SW ( X , g , µ )

Definition: Consider the quantity β ( M , g )) − 1 λ SW ( X , g , β ) = SW ( X , g , β ) − ind ( D + 8sign ( Z ) associated to X , and any periodic-end spin manifold M with end modeled on � X . Theorem (M-R-S 2009) λ SW ( X , g , β ) is independent of choice of Z , metric, and perturbation, and gives a C ∞ invariant of X . λ SW ( X , g , β ) , reduced modulo 2, is the classical Rohlin invariant of X .

Brief sketch of index theorem proof Heat equation method: on closed manifold � � tr ( e − tD − D + ) − tr ( e − tD + D − ) ind ( D + ) = lim t →∞ Traces are defined by integrating heat kernels K ( t ; x , x ) . Characteristic classes appear when we consider lim t → 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 tr ( � tr ( K ( t ; x , x )) − ( N + 1 ) K ( t ; x , x )) . N →∞ M N W M N = Z ∪ N copies of W , K = heat kernel on M , � K = heat kernel on � X . The η invariant appears at the end the calculation.