A gap-labelling conjecture for the magnetic Schrödinger operator on euclidean space KK-theory, Gauge Theory and Topological Phases Lorentz center-Leiden March 2017 Moulay Tahar BENAMEUR Institut Montpellierain Alexander Grothendieck Statement of the conjecture Conjecture The range of the K -theory of the Bellissard C ∗ -algebra for the magnetic Schrödinger operator is contained in Λ(Σ, Θ, µ) where: X Λ(Σ, Θ, µ) = Pf(ΘI )ZI [µ]. 0≤|I|≤p joint work with: • Varghese Mathai (University of Adelaide). M.-T. Benameur and V. Mathai, Gap-labelling conjecture with non-zero magnetic field. [1508.01064] + ongoing research Outline of talk 1 Magnetic Schrödinger operators 2 The conjecture 3 A twisted measured index theorem for laminations 4 What is known about the conjecture. 5 Proof in the 3D case and comments about the general case. Magnetic Schrödinger operators - Consider Euclidean space Rp equipped with its usual P Riemannian metric pj=1 dxj2 . P - the magnetic field is B = 12 j,k Θjk dxj ∧ dxk = 21 dx t Θdx. Main example: constant skew-symmetric matrix Θ (p × p). -Pick a 1-form η such that dη = B. η is a magnetic potential,this defines the connection ∇ = d + iη on the trivial line bundle. Using the Riemannian metric the Hamiltonian of an electron in this field is given by H= 1 † ∇ ∇ + V = Hη + V , 2 with V smooth bounded real-valued. acting on L2 (Rp ) Magnetic Schrödinger operators Define a unitary T σ of Zp (and Rp ) on L2 (Rp ) as follows. Uγ (f )(x) = f (γ −1 x) = f (x − γ) (1) Sγ (f )(x) = exp(−iψγ (x))f (x) (2) Tγσ = Uγ ◦ Sγ . (3) where ψγ satisfies γ ∗ η − η = dψγ and ψγ (0) = 0+compatibility. One has Tγσ Hη = Hη Tγσ . The magnetic translations Tγσ satisfy Teσ = Id and Tγσ1 Tγσ2 (f )(x) = σ(x; γ1 , γ2 )Tγσ1 γ2 (f )(x), where σ(x; γ1 , γ2 ) satisfies a groupoid cocycle condition. If B is constant then σ(γ1 , γ2 ) = exp (−iψγ1 (γ2 )) is a multiplier on Zd : 1 σ(γ, e) = σ(e, γ) = 1 for all γ ∈ Zd ; 2 σ(γ1 , γ2 )σ(γ1 γ2 , γ3 ) = σ(γ1 , γ2 γ3 )σ(γ2 , γ3 ) for γj ∈ Zd . The Bellissard gap-labelling theorem The C ∗ -algebra of observables is a twisted groupoid algebra (Bellissard). We are interested in the trace-range of its K -theory. Too hard in general! Assume from now on B constant. ∃ a Cantor complete transversal Σ so that holonomy = minimal action of Zp , preserving a probability measure µ. Then the C ∗ -algebra of observables is Morita equivalent to the twisted discrete crossed product algebra A = C(Σ) oσ Zd . Twisted crossed product algebra Recall for a, b ∈ A0 = Cc (Σ × Zd ): Product: P ab(ω, γ) = γ 0 ∈Zd a(ω, γ 0 )b(γ 0−1 ω, γ − γ 0 )σ(γ − γ 0 , γ 0 ); Adjoint: a∗ (ω, γ) = a(γ −1 ω, −γ)σ(−γ, γ); Regular representation for a ∈ A0 and ψ ∈ `2 (Zd ): P πω (a)ψ(γ) = γ 0 ∈Zd a(γ 0−1 ω, γ − γ 0 )ψ(γ 0 )σ(γ − γ 0 , γ 0 ); Norm: ||a|| = supω∈Σ ||πω (a)||; The twisted crossed product C ∗ -algebra is then: A = A0 ||·|| For f C0 on spec(H), f (H) ∈ C(Σ) oσ Zd ⊗ K. P Recall the `2 trace τ µ ( fγ δγ ) = µ(fe ). Theorem (Bellissard) The spectral gap-labels for the magnetic Schrödinger operator lie in the image under the trace τ µ of K0 (C(Σ) oσ Zd ). The range of τ∗µ : K0 (C(Σ) oσ Zd ) −→ R, is the group we are interested in here. It will be our “magnetic gap-labelling group”. Bellissard’s gap-labelling conjecture (B = 0) Here gap-labels are expected to live in the frequencies group Z[µ] := µ(K 0 (Σ)). This is the Bellissard conjecture. Conjecture Under the previous assumptions, the gap-labels group is contained in the frequency module Z[µ]. τ∗µ (K0 (C(Σ) o Zd )) ⊂ µ(C(Σ, Z)) = Z[µ]. Many works in the 90’s: conjecture is true for d = 1, 2 and 3. Bellissard, Contensou, Kellendonk, Legrand, ... For general d, 3 differents proofs were given in J. Bellissard, R. Benedetti, J-M. Gambaudo; J. Kaminker, I. Putnam; M. B., H. Oyono-Oyono. Comment on the gap-labelling conjecture They used that the range of the top Chern character chp : K p (X ) → H p (X , Q) lies in H p (X , Z). This is true for low dimensions d = 1, 2, 3. For higher dimensions, this is still true when the cohomology of the quasi-crystal is torsion-free (B.-Oyono-Oyono). In work in progress with H. Oyono-Oyono, we started a new approach to integrality. Back to the magnetic gap-labelling conjecture - Goal We need a candidate for the freqencies group when B 6= 0? The magnetic gap-labelling conjecture concerns finding an optimal subgroup Λ(Σ, Θ, µ) of R such that Λ(Σ, Θ, µ) is computable and coincides with Z[µ] for B = 0. The range of the trace lies inside Λ(Σ, Θ, µ), i.e. τµ (K0 (C(Σ) oσ Zd )) ⊂ Λ(Σ, Θ, µ). √ P where σ(γ, γ 0 ) = exp(π −1 Θjk γj γk0 ) Statement of the conjecture Let Λ[dx] = Λ[dx1 , . . . , dxd ] denote the exterior algebra with generators dx1 , . . . , dxd . Given a skew-symmetric matrix Θ, recall Pf(Θ) is given when d = 2m by m 1 1 t = Pf(Θ)dx1 ∧ · · · dxd . dx Θdx m! 2 1 In general, e 2 dx t Θdx can be expressed in terms of the Pfaffians: Mathai-Quillen identity 1 e 2 dx t Θdx = X Pf(ΘI )dxI I where I runs over subsets of {1, . . . , p} with an even number of elements, and ΘI = (Θij )i,j∈I . Statement of the conjecture Recall the subgroup Z[µ] generated by µ-measures of clopens. Let I be an ordered subset of {1, . . . , d}, and let C(Σ, Z)ZI c c denote the coinvariants under the subgroup ZI of Zd . Set ZI [µ] = µ C(Σ, Z)ZI c ZI . So, for instance: Z{1,··· ,p} [µ] = Z ⊂ ZI [µ] ⊂ Z[µ] = Z∅ [µ]. Statement of the conjecture Conjecture (MGL conjecture) The range of the trace is contained in Λ(Σ, Θ, µ) where: X Λ(Σ, Θ, µ) = Pf(ΘI )ZI [µ] 0≤|I|≤d So 1 If d is even, Λ(Σ, Θ, µ) = Z[µ] + X Pf(ΘI )ZI [µ] + Pf(Θ)Z. 0<|I|<d 2 If d is odd, Λ(Σ, Θ, µ) = Z[µ] + X 0<|I|<d Pf(ΘI )ZI [µ]. Comments on the conjecture For B = 0, MGL is the Bellissard GL conjecture. For the periodic case (Σ = pt), MGL is Elliott’s theorem (’82) about the range of the trace for the rotation algebras. Integrality of Chern ⇒ MGL conjecture. So, MGL is satisfied in low dimensions d ≤ 3. Index approach ([B.-Oyono-Oyono]) unifies the previous cases and explains the conjecture More precisely... We use the twisted Baum-Connes isomorphism to reduce the problem to the computation of twisted measured indices. Indeed, twisting the Dirac operator on Rp with the connection d + iη defines the twisted BC morphism µBC : K p (X ) −→ K0 (C(Σ) oσ Zd ) by µBC (ξ) := Ind(/∂η ⊗ ξ). A twisted version of the Connes measured index theorem allows to compute the range of τµ ◦ µBC . We then have reduced the problem to a cohomological computation. The index approach The suspension X = Rp ×Zp Σ is a compact foliated space with transversal the Cantor set Σ, and with invariant transverse measure induced from µ. The monodromy groupoid is G = (Rp × Rp × Σ)/Zp . Let /∂ denote the Dirac operator on Rp and ∇ = d + iη the connection on the trivial line bundle on Rp , ∇E the lift to Rp × Σ of a connection on a vector bundle E → X . Consider the twisted Dirac operator along the leaves of the lifted foliation, DE = /∂ ⊗ ∇ ⊗ ∇E : L2 (Rp × Σ, S + ⊗ E) −→ L2 (Rp × Σ, S − ⊗ E). Then one computes that Tγ ◦ DE = DE ◦ Tγ for γ ∈ Zp .The heat kernel of D has a well defined µ-trace. The index approach For t > 0, define the Wasserman idempotent et (DE+ ) ∈ M2 (C(Σ) oσ Zp ⊗ K) as follows:   e et (DE+ ) =   −tDE− DE+ t + − e− 2 DE DE DE+ e  − + − e−tDE DE ) + DE  , DE− DE+  − + 1 − e−tDE DE − 2t DE− DE+ (1 Then the C ∗ -twisted foliated analytic index is defined as Index(DE+ ) = [et (DE+ )] − [P0 ] ∈ K0 (C(Σ) oσ Zp ), where t > 0 and P0 is the idempotent ! 0 0 P0 = 0 1 (4) The index approach The Baum-Connes map µBC : K p (X ) −→ K0 (C(Σ) oσ Zp ). is then E 7→ Index(DE+ ). It is an isomorphism (using Packer-Raeburn). Reduces the problem to computing the range of τ∗µ ◦ µBC : K p (X ) −→ R. The index theorem Theorem (B.-Mathai ’15) τµs (Index(DE+ )) = X Z Pf(ΘI ) I Σ×(0,1)p dxI ∧ ch` (FE )dµ(ϑ). Here I runs over subsets of {1, . . . , p} with an even number of elements, and ΘI is the submatrix of Θ = (Θij ) with i, j ∈ I. A McKean-Singer type argument shows that the supertrace 2 τµs (e−tD ) = τµs (Index(DE+ )) is independent of t > 0 and represents the measured twisted foliated index. The 2D case The group cohomology class of [σ] ∈ H 2 (Z2 ; R/Z) ∼ = R/Z can be identified with a real number θ, 0 ≤ θ < 1. More precisely, we take σ = e2πiθω with ω the standard symplectic form on Z2 . Theorem (2D case) One has τµ (K0 (C(Σ) oσ Z2 ) = Z[µ] + Zθ NB: Previously proved by J. Kellendonk. Index proof for 2D ∼ µBC : K 0 (X ) −→ K0 (C(Σ) oσ Z2 ) is an isomorphism, where X = Σ ×Z2 R2 . Measured twisted index theorem ⇒ Z τµ (µBC (E)) = θ rank(E) + c1 (E)dΛµ ZX = θ rank(E) + Σ ΨZ2 (c1 (ξ))dµ using the isomorphism ΨZ2 : H 2 (X , Z) → C(Σ, Z)Z2 . The result follows. Periodic potentials = Elliott’s theorem Consider the special case when the self-adjoint p-dimensional magnetic Schrödinger operator H= 1 (d + iη)† (d + iη) + V , 2 acting on L2 (Rp ) has (real-valued) periodic potential V , then much easier: Σ is just a point. The crossed product algebra is C oσ Zd = AΘ . Periodic potentials = Elliott’s theorem MGL becomes Elliott’s theorem: Theorem (purely periodic case) The range of the trace on the K-theory of AΘ is: τ (K0 (AΘ )) = X Pf(ΘI )Z. 0≤|I|≤p Proof. ∼ µΘ : K 0 (Tp ) −→ K0 (AΘ ) is an isomorphism. The twisted index theorem gives (1 leaf here!): Z X τ (µΘ (E)) = Pf(ΘI ) dxI ∧ Ch(E) I The result follows. Tp The 3D case Theorem (B.-Mathai ’15) The magnetic GL group is contained in Z[µ] + Θ12 Z12 [µ] + Θ13 Z13 [µ] + Θ23 Z23 [µ]. Theorem (B.-Mathai ’15) If for instance T3 acts minimally then Θ12 Z12 [µ] ⊂ Range(τµ ). Therefore, if each generator acts minimally, then the magnetic GL group coincides with: Z[µ] + Θ12 Z12 [µ] + Θ13 Z13 [µ] + Θ23 Z23 [µ]. Main steps of the proof for 3D Again the twisted BC isomorphism K 1 (X ) → K0 (C(Σ) oσ Z3 ) is the class of the twisted Dirac operator. Applying the gap-labelling theorem + the twisted measured index theorem, we get: Z −1 u d` u tr Range(τ∗µ ) = Z[µ]+ ∧ π ∗ B dΛµ , u ∈ K 1 (X ) . 2iπ X This index formula+compatibility of the isomorphism H ∗ (X , Z) ' H ∗ (Z3 , C(Σ, Z)), reduces the MGL conjecture to computing the range of the map ∪Θ Ψ µ H 1 (Z3 , C(Σ, Z)) −→ H 3 (Z3 , C(Σ, R) −→ C(Σ, R)Z3 −→ R. Concentrate on Θ12 ψ1 ∪ ψ2 .We have exact sequences: 0 → C hT1 ,T3 i hT2 i →H 1 (Z3 , C)→H 1 (hT1 , T3 i, C)hT2 i → 0 and 0 → C hT3 i hT1 i −→H 1 (hT1 , T3 i, C)−→ (C)hT3 i hT1 i → 0. Denote by A the image of H 1 (hT1 , T3 i, C)hT2 i in (C)hT3 i hT ,T i So A ⊂ (C)hT3 i 1 2 . hT1 i . Main steps of the proof for 3D The map ∪(ψ1 ∪ ψ2 ) : H 1 (Z3 , C) → (C)Z3 induces a well-defined morphism α : H 1 (hT1 , T3 i, C)hT2 i −→ (C)Z3 . α coincides with the composite map ∪ψ H 1 (hT1 , T3 i, C)hT2 i →1 H 2 (hT1 , T3 i, C)hT2 i → (C)Z3 . Therefore, α induces a morphism β : A → (C)Z3 . Hence, the range of ∪(ψ1 ∪ ψ2 ) on H 1 (Z3 , C) coincides with β(A). But A ⊂ H 1 (hT3 i, C)hT1 ,T2 i , so the range of µ ◦ ∪(ψ1 ∪ ψ2 ) is hT ,T i contained in the range under µ of (C)hT3 i 1 2 . QED