Dirac-Schrödinger operators and the irrational torus
aa r X i v : . [ m a t h . K T ] A ug DIRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS
HEATH EMERSON, NATHANIEL BUTLER, AND TYLER SCHULZA
BSTRACT . Dirac-Schr¨odinger operators on the real line coupled with a natural family ofrepresentations of the irrational rotation algebra are used to build a 2-dimensional spectraltriple over the Schwartz algebra of the irrational torus. These spectral cycles (‘Heisenbergcycles’) fit easily into Connes’ framework of Noncommutative Geometry: they are regularand give rise to meromorphic zeta functions, which we compute geometrically via Mehler’sformula for the harmonic oscillator. We compute the Connes’-Moscovici Index formula forthe Heisenberg cycles and determine their Chern characters, which are mixed-degree co-cochains with the standard trace in degree zero and Connes’ curvature cocycle times therotation parameter, in degree 2. We apply the resulting index computations to prove a resultabout the b-twist renormalization morphisms arising in a previous paper of the first authorand A. Duwenig.
1. I
NTRODUCTION
The irrational rotation algebra A ℏ : = C ( T ) ⋊ ℏ Z , the crossed product of C ( T ) = C ( R / Z ) by the action of Z by translation by ℏ mod Z on T , is one of the key motivating exam-ples in Noncommutative Geometry. Early results of Connes and Rieffel classified finitelygenerated projective modules over A ℏ , or over its natural Schwartz subalgebra A ∞ ℏ , by ananalogue of the first Chern number of a line bundle over T , defined c ( e ) : = π i · τ ( e [ δ ( e ) , δ ( e )]) ,where δ , δ are the derivations of A ℏ generating the natural R -action. In fact these num-bers are integers , a fact related to the Quantum Hall effect in solid state physics.The reason for the integrality lies in the following. The densely defined operators ∂∂ x , ∂∂ y on L ( T ) assemble to the operator¯ ∂ : = " ∂∂ x − i ∂∂ y ∂∂ x + i ∂∂ y .on L ( T ) ⊕ L ( T ) , and the representation of C ( T ) on L ( T ) by multiplication operatorscan be adjusted by introducing phase factors to give a representation λ ℏ : A ℏ → B (cid:0) L ( T ) (cid:1) which makes the triple (cid:0) L ( T ) ⊕ L ( T ) , λ ℏ , ¯ ∂ (cid:1) a 2-summable spectral triple over A ∞ ℏ whose Chern character may be computed using the Local Index Formula of Connes andMoscovici to be the class of the cyclic cocycle(1.1) τ ( a , a , a ) = τ (cid:0) a δ ( a ) δ ( a ) − a δ ( a ) δ ( a ) (cid:1) , a , a , a ∈ A ℏ . Date : August 25, 2020.
Key words and phrases.
K-theory, K-homology, Noncommutative Geometry.This research was supported by an NSERC Discovery grant and the NSERC USRA program.
The integrality of the Chern numbers c ( e ) follows from the Connes-Moscovici IndexTheorem which implies that for any idempotent e ∈ A ∞ ℏ , c ( e ) = h [ e ] , [ ¯ ∂ ] i ∈ Z ,where the right hand side is the pairing between K-theory and K-homology.In this note we study the noncommutative geometry of A ℏ from a slightly differentangle, originating in the relationship between A ℏ with the Heisenberg group.For any ℏ ∈ R , A ℏ has a natural representation π ~ on L ( R ) by letting periodic functionson R act by multiplication operators and the integers by translation by multiplies of ℏ . Theoperators x and d / dx , realizing the canonical anti-commutation relations, assemble to form D = (cid:20) x − d / dxx + d / dx (cid:21) , whose closure is self-adjoint. The operator D commutes modbounded operators with group translations and smooth bounded functions with boundedderivatives, and D is roughly the harmonic oscillator on R , which has discrete spectrumgrowing at a linear rate. It follows that the triple(1.2) (cid:18) L ( R ) ⊕ L ( R ) , π ℏ ⊕ π ℏ , D = (cid:20) x − d / dxx + d / dx (cid:21)(cid:19) defines a 2-dimensional even spectral cycle and class [ D ℏ ] ∈ KK ( A ℏ , C ) . We call (1.2)the Heisenberg cycle . Since A ℏ only depends on the congruence class of ℏ mod Z but theHeisenberg ℏ -cycle depends on ℏ as a real number, this process defines a Z -parameterizedfamily of spectral cycles on A ℏ for any ℏ ∈ R .It also seems a point of interest that the same cycle is defined over the non-separableC*-algebra C u ( R ) ⋊ ℏ Z , where C u ( R ) is the C*-algebra of bounded, uniformly continuousfunctions on R , and pulls back to [ D ℏ ] under the natural inclusion A ℏ ⊂ C u ( R ) ⋊ Z , but wefocus primarily on A ℏ here.There is quite a lot of topological information contained in the Heisenberg classes. If ℏ is an integer we get a cycle for C ( T × b Z ) . When ℏ = [ D ] ∈ KK ( C ( T × b Z ) , C ) equals theK-homology class of a point in T × b Z . If ℏ = T × b Z twisted by the Poincar´e bundle: the class [ D ] isthus equivalent to the spin dual of the Fourier-Mukai transform.In general, we show that if ℏ is fixed, then the classes { [ D ℏ + b ] } b ∈ Z are permuted transi-tively by the action of the cyclic group of invertibles in KK ( A ℏ , A ℏ ) generated by a certain twist T (see [6]). The twist is obtained by Poincar´e dualizing a certain finitely generatedprojective module over A ℏ ⊗ A ℏ and a spectral cycle for the twist is constructed in [6]represented by bundle of Dirac-Schr¨odinger operators over T .Our main goal in this note is to compute the Chern character of the Heisenberg cy-cles using the Connes-Moscovic Local Index Theorem [5]. The key problem in applyingthe Local Index Theorem to the Heisenberg cycles is understanding the zeta functions Tr( aH − s ) where H is the harmonic oscillator and a ∈ A ℏ , the main interest being a ∈ C ( T ) a periodic function on R .We describe a closed linear subspace D of C u ( R ) of functions having the asymptoticmean property: that the limits lim x →± ∞ x Z x f ( t ) dt exist. Taking the average of the two limits one obtains a linear functional µ : D ⊂ C u ( R ) → C , µ ( f ) : = µ − ( f ) + µ + ( f ) µ ± ( f ) : = lim x → ∞ x Z ± x f ( t ) dt . IRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS 3
The class contains all periodic functions, and the main result of interest is that if f ∈ C ( T ) isperiodic of periodic ρ then Tr( f H − s ) extends meromorphically to C with a single, simplepole at s =
1, and residue given by
Res Tr( f ) = ρ Z ρ f ( t ) dt ,the usual mean of f over the circle.Based on these calculations, we compute that the (local) Chern character of [ D ℏ ] in thesense of Connes and Moscovici is represented by the mixed-degree cyclic co-chain τ + ℏ · τ ,for A ∞ ℏ , where τ is the trace on A ℏ and τ given by (1.1). In particular, the K-homologyK-theory pairing of [ D ℏ ] with a projection e ∈ A ∞ ℏ is specified in local terms by the indexformula h [ e ] , [ D ℏ ] i = τ ( e ) − ℏ · c ( e ) ,where c is Connes’ first Chern number. If e = p ℏ is the Rieffel projection, we obtain theformula h [ p ℏ ] , [ D ℏ ] i = − ⌊ ℏ ⌋ where ⌊ ℏ ⌋ is the greatest integer < ℏ . The discontinuities of the index at the integers aredue, at least in a practical sense, to the construction of the Rieffel projection which requiresa representative 0 < ℏ < analytic problemthis solves; it would seem almost certainly related to some of the noncommutative ellipticproblems considered by Connes [3] using difference operators on the real line. We havenot checked this.We conclude by noting that an interesting feature of the these arguments is that if onecombines the Chern character calculation above with the integrality of first Chern numbers,the fact that τ ∗ (K ( A ℏ )) = Z + Zℏ where τ is the trace on A ℏ , follows immediately. In particular, one does not need to computethe K-theory to solve this ‘gap labelling’ problem; it suffices to extract the cohomologicalinformation supplied by the Local Index Theorem for two distinct spectral triples.2. S PECTRAL CYCLES FROM THE CANONICAL ANTI - COMMUTATION RELATIONS
The Heisenberg group H = { x z y | x , y , z ∈ R } has Lie algebra h the 3-by-3strictly upper triangular matrices under matrix commutator. Let X , Y be the elements X = , Y = ,of h . Then [ X , Y ] = Z : = ,while Z is central in h . It follows that if π is any irreducible representation of H , π ( Z ) = π ([ X , Y ]) = [ π ( X ) , π ( Y )] is a multiple of the identity operator: [ π ( X ) , π ( Y )] = ℏ , HEATH EMERSON, NATHANIEL BUTLER, AND TYLER SCHULZ for some ℏ ∈ R , a ‘Planck constant.’The name Heisenberg group originates in these relations, which have the same form asthe canonical commutation relations in quantum mechanics, where x and ddx model positionand momentum operators.From the above remarks, we obtain a classification of irreducible representations of H .Either ℏ =
0, in which case π ( Z ) = π ( X ) and π ( Y ) commute, which implies therepresentation is 1-dimensional, and is completely determined by the pair of real numbers ( π ( X ) , π ( Y )) , or ℏ =
0, in which case one can show that the representation is isomorphicto the following interesting representation π ℏ of h by unbounded operators on L ( R ) . Let π ℏ ( X ) = x , and π ℏ ( Y ) = ℏ ddx .Then [ x , ℏ ddx ] = ℏ , so the required identity is satisfied to give a representation.Application of functional calculus to the operators x and ddx produces the operators u = e π ix , v ℏ : = e − ℏ ddx ,where u is multiplication by the periodic function e π ix and ( v ℏ ) ξ ( x ) = ξ ( x − ℏ ) .We have uv λ = e − π i ℏ v λ u .If ℏ ∈ R \ Q then the irrational rotation algebra is the C*-algebra A ℏ : = C ( T ) ⋊ ℏ Z ,where Z acts on the circle T : = R / Z with generator the automorphism induced by trans-lation by ℏ mod Z . If U ∈ C ( T ) ⋊ h Z is the generator U ( t ) = e π it of C ( T ) and V thegenerator of the Z action in the crossed-product, then a quick computation shows that UV = e − π i ℏ VU ∈ A ℏ ,and it follows that we obtain, for each ℏ , rational or not, a representation π ℏ : A ℏ → B ( L ( R )) of A ℏ on L ( R ) .We are going to fit these representations into a family of spectral cycles for KK ( A ℏ , C ) ,using the properties of the harmonic oscillatorH : = − d dx + x ,a second-order elliptic operator on R , whose domain we will take initially to be the Schwartzspace S ( R ) .Let A = x + ddx , initial domain the Schwartz space S ( R ) , and A ∗ = x − ddx . The relations(2.1) AA ∗ = H + A ∗ A = H − [ A , A ∗ ] = [ H , A ] = − A , [ H , A ∗ ] = A ∗ .hold as operators on S . (See [10].)Now set ψ : = π − · e − x ∈ L ( R ) . In quantum mechanics, ψ is called the ground state ,and the states inductively defined by ψ k : = ( k ) − · A ∗ ψ k − the ‘excited states’. Observe IRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS 5 that due to HA ∗ = A ∗ H + A ∗ , from (2.1), we see by induction that ψ k is a unit-lengtheigenvector of H with eigenvalue 2 k + H ψ k = ( k ) − · HA ∗ ψ k − = ( k ) − · ( A ∗ H + A ∗ ) ξ k − = ( k ) − · (( k − ) · A ∗ ψ k − + A ∗ ψ k − ) = ( k + ) · ψ k .It follows from [ H , A ] = − A that A ψ k = √ k · ψ k − , A ∗ ψ k = √ k + · ψ k + . Remark 2.1.
The eigenvectors of H are given by ξ k = H k ( x ) e − x where H k is the k th Hermite polynomial . This follows from induction using the recurrence H k ( x ) = ( k ) − · (cid:0) xH k − ( x ) − H ′ k − ( x ) (cid:1) to define the polynomials.The vectors { ψ k } form an orthonormal basis for L ( R ) by the Stone-Weierstrass Theo-rem, and each ψ k is in the Schwartz class S ( R ) .With respect to this basis, H is diagonal with eigenvalues the odd integers 1, 3, 5, . . . : H = · · · · · · · · ·· · · · · · · · · · · · .In particular, H has a canonical extension to a self-adjoint operator on L ( R ), and f ( H ) is a compact operator for all f ∈ C ( R ) , and a bounded operator for all f ∈ C b ( R ) .If f ∈ L ( R ) , let ( ˆ f ( n )) denote the sequence of its Fourier coefficients with respect tothe eigenbasis for L ( R ) for H discussed above. Lemma 2.2.
If f ∈ L ( R ) , then f ∈ S if and only if ( ˆ f ( n )) is a rapidly decreasing sequenceof integers: | ˆ f ( n ) | = O ( n − k ) for any k.Proof. If f ∈ S then H f is in S , as is clear from the definition of H . Similarly, H k f ∈ S for all k . Since d H k f ( n ) = ( n + ) k · ˆ f ( n ) ,and since this is an L -sequence, and hence bounded, we get, for each k a constant C suchthat ( n + ) k · | ˆ f ( n ) | ≤ C k and so | ˆ f ( n ) | = O ( n − k ) follows.Conversely, suppose that f ∈ L ( R ) and that ( ˆ f ( n )) is a rapidly decreasing sequence.Then f is, by definition, in the domains of A , A ∗ , H , and all positive powers of theseoperators. Since A + A ∗ = x and A − A ∗ = ddx , it follows that x k f and d k fdx k are in L ( R ) forall k . Hence f ∈ S . (cid:3) HEATH EMERSON, NATHANIEL BUTLER, AND TYLER SCHULZ
Let D be the unbounded operator D = (cid:20) A ∗ A (cid:21) on L ( R ) ⊕ L ( R ) , defined initially on Schwartz functions; it admits a canonical extensionto a densely defined self-adjoint operator. We have D = (cid:20) H − H + (cid:21) and hence 1 + D = (cid:20) H H + (cid:21) which is diagonal with respect to the basis describedabove, and invertible.The analogue of C ∞ ( T ) for A ℏ is the Schwartz subalgebra A ∞ ℏ : = . ∑ n , m a nm U n V m ∈ A ℏ , ∑ | a nm | · ( + m + n ) k < ∞ , ∀ k of rapidly decaying sequences in U , V . It is well-known to be dense and holomorphicallyclosed. Lemma 2.3.
Let ℏ ∈ R . Then the commutator (cid:20)(cid:20) π ℏ ( a ) π ℏ ( a ) (cid:21) , D (cid:21) is bounded for a ∈ A ∞ ℏ .Proof. The operator ddx commutes with unitaries on L ( R ) induced by translations of R ,while x commutes mod bounded operators with translations and commutes with multipli-cation operators by arbitrary bounded functions on R . Finally, if f ∈ C ∞ ( T ) , regarded asa periodic function on R , then f ′ is bounded on R . It follows that [ f , ddx ] = f ′ is bounded.This implies that [ π ℏ ( a ) , D ] is bounded for a ∈ C ∞ ( T )[ Z ] . Note that, suppressing the rep-resentations π ℏ in the notation, [ U n , D ] = (cid:20) − nU n nU n (cid:21) , [ V m , D ] = (cid:20) mV n mV n (cid:21) and it follows that [ a , D ] is bounded for a ∈ A ℏ in the Schwartz algebra. (cid:3) Definition 2.4.
The
Heisenberg cycle is the even, 2-dimensional unbounded Fredholmmodule (cid:18) L ( R ) ⊕ L ( R ) , π ℏ ⊕ π ℏ , D = (cid:20) A ∗ A (cid:21)(cid:19) ,for )KK ( A ℏ , C ) , defining a spectral triple over the Schwartz subalgebra A ∞ ℏ of the rotationalgebra A ℏ : = C ( T ) ⋊ ℏ Z .We denote by [ D ℏ ] the class in KK ( C ( T ) ⋊ ℏ Z , C ) of the corresponding Fredholmmodule, given by L ( R ) ⊕ L ( R ) , π ℏ ⊕ π ℏ , F : = χ ( D ) = " A ∗ ( H + ) − AH − . IRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS 7
The spectrum of D = (cid:20) H − H + (cid:21) grows linearly so | D | − lies in the Dixmier class L ( ∞ ) and the spectral triple is 2-summable. Using the Dixmier trace we obtain a trace τ ( a ) : = Tr ω ( π ℏ ( a ) · | D | − ) on A ℏ which we will study in the next section. Remark 2.5. If f ∈ C ∞ b ( R ) is a smooth bounded function with bounded first derivative,acting by a multiplication operator on L ( R ) , then the commutator [ f , D ] = (cid:20) − f ′ f ′ (cid:21) isbounded. Let C u ( R ) : = { f ∈ C b ( R ) | f is uniformly continuous } Then the translation action of Z induced by ℏ ∈ R gives an automorphic action of Z on C u ( R ) . Note that if C ∞ u ( R ) denotes the bounded functions on R all of whose derivatives arebounded, then C ∞ u ( R ) is dense in C u ( R ) . and if π ℏ denotes the representation of C u ( R ) ⋊ ℏ Z induced by letting functions act by multiplication operators, the integers by translation bymultiples of ℏ , then (cid:18) L ( R ) ⊕ L ( R ) , π ℏ ⊕ π ℏ , D = (cid:20) A ∗ A (cid:21)(cid:19) is a 2-dimensional spectral cycle for C u ( R ) ⋊ ℏ Z .The inclusion A ℏ = C ( T ) ⋊ ℏ Z → C u ( R ) ⋊ ℏ Z pulls this back to the Heisenberg cycle.Since the spectrum of the harmonic oscillator H is the odd integers, for any boundedoperator W , W H − s is trace class whenever Re( s ) >
1, and
Tr(
W H − s ) is analytic in the cor-responding half-plane. In the next section we will show that the zeta functions Tr( π h ( a ) · H − s ) , a = ∑ n ∈ Z f n [ n ] ∈ C u ( R )[ Z ] , are entire away from the identity element of Z , and soare primarily interesting for functions a = f ∈ C u ( R ) .We will show that the Dixmier trace Tr ω ( f H − ) of f ∈ C u ( R ) in a suitable class, agreeswith a geometrically defined asymptotic mean of f , which for periodic functions agreeswith the usual mean of a periodic function, and that for periodic functions, the zeta func-tions Tr( π h ( a ) · H − s ) extend to meromorphic functions on C for a ∈ A ∞ ℏ .We start by noting that such zeta functions are entire for compactly supported functions W = f ∈ C c ( R ) . Lemma 2.6.
The operator f H − s is trace class for any f ∈ C c ( R ) and any s ∈ C .Proof. With A = x + ddx , a direct calculation with the basis { ψ n } shows that AH − and H − A , A ∗ H − and H − A ∗ extend to bounded operators. Since A + A ∗ = x , it followsthat p ( x ) H − k and H − k p ( x ) are bounded for any polynomial of degree k . We deduce that H − k ( + x ) k is bounded and invertible, whence that ( + x ) − k H k is bounded. It followsthat ( + x ) − k H − s is trace class if Re( s ) > − k . Now if f ∈ C c ( R ) , given s , factor f as f ( x ) H − s = g ( x )( + x ) − k H − s for appropriate k big enough relative to Re( s ) , where g ( x ) = f ( x )( + x ) k . Since f iscompactly supported g is bounded, and ( + x ) − k H − s is trace-class for Re( s ) > − k ,whence so is f H − s . (cid:3) HEATH EMERSON, NATHANIEL BUTLER, AND TYLER SCHULZ
3. T
HE HARMONIC OSCILLATOR AND THE ASYMPTOTIC MEAN PROPERTY
The heat equation for the harmonic oscillator seeks a smooth kernel k t = k t ( x , y ) suchthat ( ∂∂ t + H ) · φ t = φ t ( x ) = Z R k t ( x , y ) φ ( y ) dy ,for φ ∈ S ( R ) , and t ≥
0, together with the initial conditionlim t → φ t = φ .Proposing the ansatz k t ( x , y ) = exp (cid:16) a t x + b t xy + a t y + c t (cid:17) and setting this equal to 0 and solving for coefficients gives the ordinary differential equa-tions ˙ a t = a t − = b t , ˙ c t = a t .Solving these gives a t = − coth ( t + C ) , b t = csch ( t + C ) , c t = −
12 log sinh ( t + C ) + D .Using the initial conditions we get C = D = log ( π ) − . See [1].We obtain the following, called Mehler’s formula [9].
Lemma 3.1.
The heat kernel for the harmonic oscillator is given by (3.1) k t ( x , y ) = ( π sinh 2 t ) − · exp (cid:18) − ( x + y ) coth 2 t + xy csch2 t (cid:19) Remark 3.2.
Mehler’s formula [9] was stated originally in the form(3.2) ∞ ∑ n = u n n n ! H n ( x ) H n ( y ) = √ − u exp (cid:18) − u ( x + y ) − uxy − u (cid:19) .where the H n are the Hermite polynomials (see Remark 2.1) and − < ρ <
1. The normal-ized eigenvectors of H are given by ψ n ( x ) = H n ( x ) p n n ! √ π e − x .Using this orthonormal basis for the calculation, we get that e − tH is an integral operatorwith kernel(3.3) k t ( x , y ) = ∞ ∑ n = e − t ( n + ) ψ n ( x ) ψ n ( y ) = e − t √ π ∞ ∑ n = e − tn n n ! exp ( − x + y ) H n ( x ) H n ( y )= e − t √ π · exp ( − x + y ) · ∞ ∑ n = e − tn n n ! H n ( x ) H n ( y ) , = e − t √ π · exp ( − x + y ) · √ − e − t exp (cid:18) − e − t ( x + y ) − e − t xy − e − t (cid:19) .using (3.2).By further manipulations we get IRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS 9 (3.4) = √ π sinh 2 t · exp (cid:18) − ( x + y ) · ( e − t − e − t + ) + xy · ( e − t − e − t ) (cid:19) = √ π sinh 2 t · exp (cid:18) − ( x + y ) · ( coth 2 t ) + xy csch ( t ) (cid:19) ,yielding (3.1).Another useful variant is(3.5) k t ( x , y ) = √ π sinh 2 t exp (cid:18) − tanh t · ( x + y ) − coth t · ( x − y ) (cid:19) Definition 3.3.
Let f ∈ C u ( R ) . We say f has the asymptotic mean property if there exist µ ± such that lim x →± ∞ x R x f ( u ) du = µ ± .We call the average µ ( f ) : = µ − + µ + the asymptotic mean of f . Proposition 3.4.
Let f ∈ C u ( R ) . a) If f is periodic, of period ρ , then f has the asymptotic mean property andµ + ( f ) = µ − ( f ) = ρ Z ρ f ( u ) du , Hence the asymptotic mean of f is the ordinary mean of f over [ ρ ] . b) If lim x →± ∞ f ( x ) = : f ( ± ∞ ) exist, then f has the asymptotic mean property andµ ± ( f ) = f ( ± ∞ ) , µ ( f ) = f (+ ∞ )+ f ( − ∞ ) . c) The collection D ⊂ C u ( R ) of functions with the asymptotic mean property is aclosed linear subspace of C u ( R ) , invariant under translations and dilations of R .The asymptotic mean defines a continuous linear functional µ : D → C .Proof. For x ≥ n ρ ≤ x < ( n + ) ρ . Then n = x ρ − ε where 0 ≤ ε < ρ and(3.6) 1 x Z x f ( t ) dt = x Z n ρ f ( t ) dt + x Z xn ρ f ( t ) dt = nx Z ρ f ( t ) dt + x Z xn ρ f ( t ) dt = ρ Z ρ f ( t ) dt + x (cid:18) Z xn ρ f ( t ) dt − ε Z ρ f ( t ) dt (cid:19) and the error term is O ( / x ) as x → ∞ .The other statements are obvious. (cid:3) In particular any f ∈ C ( R ) has the asymptotic mean property and µ ( f ) = µ ± ( f ) = Tr( f H − s ) is the following integral formula, derived from Mehler’s formula. Lemma 3.5.
If f ∈ C u ( R ) and Re ( s ) > then (3.7) Γ ( s ) · Tr( f H − s ) = √ π · Z Z R t s − csch t · f ( x √ coth t ) · e − x dxdt + ρ where ρ is an entire function. Proof.
Let f ∈ C u ( R ) . Using Mehler’s formula:(3.8) Γ ( s ) · Tr( f H − s ) = Z ∞ t s − Z R f ( x ) k t ( x , x ) dxdt = Z ∞ Z R t s − √ π sinh 2 t · f ( x ) · exp (cid:0) − tanh t · x (cid:1) dxdt .The function Z ∞ Z R t s − √ π sinh 2 t · f ( x ) · exp (cid:0) − tanh t · x (cid:1) dxdt extends to an entire function on C . Making the change of variables x x √ tanh ( t ) to theremaining part and a little algebra gives(3.9) 12 √ π · Z Z R t s − csch t · f ( x √ coth t ) · e − x dxdt .as required. (cid:3) Theorem 3.6.
For f ∈ C u ( R ) , the limit lim s → + ( s − ) · Tr( f H − s ) exists if and only if the limit lim α → ∞ √ π Z Z R f (cid:16) xt α (cid:17) e − x dxdtexists, and the two limits are equal.Proof. By the previous Lemma, the first limit is given by(3.10) lim α → ∞ √ π Z t α csch ( t α ) Z R f (cid:16) x p coth ( t α ) (cid:17) e − x dxdt The function t α csch ( t α ) is analytic, with power-series of the form 1 + ∑ n ≥ a n t α n . Further-more, the function g α ( t ) = √ π Z R f (cid:16) x p coth ( t α ) (cid:17) e − x dx is continuous and bounded by k f k , so a n t α n g α ( t ) → t ∈ [
0, 1 ) and n ≥
1. Itfollows that we can then write Equation (3.10) as the limitlim α → ∞ √ π Z Z R f (cid:16) x p coth ( t α ) (cid:17) e − x dxdt It is then straight-forward to check that this limit agrees with the second limit in the state-ment of the theorem. (cid:3) (cid:3)
Corollary 3.7.
For f ∈ C u ( R ) , if f has the asymptotic mean property with asymptoticmean µ ( f ) then Tr ω ( f H − ) = µ ( f ) where Tr ω is the Dixmier trace. IRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS 11
Proof.
If we apply integration-by-parts to the expression in Theorem 3.7, we obtain1 √ π Z Z R (cid:18) Z x f (cid:16) ut α (cid:17) du (cid:19) xe − x dxdt = √ π Z Z R (cid:18) t α Z x / t α f ( u ) du (cid:19) xe − x dxdt Multiplying and dividing by x , this becomes1 √ π Z Z R (cid:18) t α x Z x / t α f ( u ) du (cid:19) x e − x dxdt For either x > x <
0, the function t α x Z x / t α f ( u ) du converges pointwise to the appro-priate limit as α → ∞ . Since Z x e − x dx = √ π (cid:3) (cid:3) Let D ⊆ C u ( R ) be the set of uniformly continuous bounded functions with the asymp-totic mean property. Define a map δ : D → C u ( R ) by δ ( f )( x ) = Z x f ( t ) dt − µ − ( f ) x · χ ( − ∞ ,0 ] − µ + ( f ) x χ [ ∞ ) ,where µ ± ( f ) = lim x →± ∞ x R x f ( t ) dt . Corollary 3.8.
If f is periodic, then f ∈ dom ( δ n ) for all n =
1, 2, . . . , andµ ± ( δ n ( f )) = µ ± ( δ n ( f )) = n =
1, 2, . . .
Proof.
Observe first that if f is periodic, then δ ( f ) is periodic (of the same period), say ρ since if x ≥ δ ( f )( x + ρ ) = Z x + ρ f ( t ) dt − µ + ( f )( x + ρ ) = (cid:18) Z ρ f ( t ) dt − L f , + ρ (cid:19) + (cid:18) Z x f ( t ) dt − µ f , + x (cid:19) Since µ + ( f ) = ρ R ρ f ( t ) dt , it follows that I ( f )( x + ρ ) = Z x f ( t ) dt − µ + ( f ) x = δ ( f )( x ) By Proposition 3.4 f has the asymptotic mean property. Since δ ( f ) is periodic, it doesas well. Iterating this argument, we see that periodic implies the smooth asymptotic meanproperty. To see µ ± ( δ ( f )) =
0, write f as a Fourier series for x ≥ f ( x ) = ∑ n ∈ Z a n e π inx / T then δ ( f )( x ) is equal to δ ( f )( x ) = ∑ n ∈ Z , n = a n T π in e π inx / T The anti-derivative of δ ( f ) is bounded, so the positive part of the result holds; the negativeis similar. (cid:3) Theorem 3.9.
If f ∈ C u ( R ) is periodic of periodic ρ then Tr( f H − s ) extends meromorphi-cally to C with a single simple pole at s = and residue there given by Res s = Tr( f H − s ) = ρ Z ρ f ( t ) dt . Proof. (Of Theorem 3.9).By the Lemma, Γ ( s ) · Tr( f H − s ) = √ π Z ∞ t s − csch t Z R f ( x √ coth t ) e − x dxdt Using ddx ( δ ( f )) = f − µ ± ( f ) as x → ± ∞ and applying integration by parts to the abovegives(3.11) 1 √ π Z ∞ t s − csch t · µ + ( f ) Z ∞ x e − x dxdt + √ π Z ∞ t s − csch t · µ − ( f ) Z − ∞ x e − x dxdt + √ π Z ∞ t s − csch t √ tanh t Z R δ ( f )( x p coth ( t )) · xe − x dxdt The first two terms sum to M ( csch t ) · µ ( f ) , where M ( csch t ) is the Mellin transform ofcsch t . It is known that M ( csch t ) = ( − − s ) Γ ( s ) ζ ( s ) , which is meromorphic on thewhole complex plane; dividing this by Γ ( s ) , we have a single simple pole at s =
1. Theresidue there is given by the Dixmier trace, by by Corollary 3.7 is µ ( f ) , which equals one-half the ordinary mean of f by Proposition 3.4.The last term in Equation 3.11 (which we will write as Ψ ( s ) ) is convergent for ℜ ( s ) > /
2. Since f is periodic, µ ± ( δ f ) = ddx (cid:0) δ f (cid:1) = δ f ,and performing the corresponding integration by parts we obtain Ψ ( s ) = √ π Z ∞ t s − csch t tanh t Z R δ ( f )( x √ coth t )( x − ) e − x dxdt Repeating this process, we in fact obtain the following expression for Ψ ( s ) vaild for any n ≥ Ψ ( s ) = ( − ) n √ π Z ∞ t s − csch t tanh n / t Z R δ n ( f )( x √ coth t ) d n dx n e − x dxdt For each n , this expression is convergent for ℜ ( s ) > − n /
2. Therefore Ψ ( s ) is an entireanalytic function. (cid:3) Lemma 3.10.
Let U α be the unitary induced by translation on the real line by α = . Thenif f ∈ C u ( R ) , the function Tr( fU α H − s ) extends analytically to an entire function.Proof. We have Γ ( s ) · Tr( AH − s ) = Z ∞ t s − Tr( Ae − tH ) dt for any A . We put A = fU α .The operator fU α e − tH is a compact integral operator with kernel k ′ t ( x , y ) = f ( x ) k t ( x − α , y ) . IRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS 13
Hence, by Mehler’s formula(3.12) Γ ( s ) · Tr( fU α H − s ) = Z ∞ Z R t s − k t ( x − α , x ) dxdt = Z ∞ Z R t s − ( π sinh 2 t ) − f ( x ) exp (cid:18) − ( x − α ) t − α t (cid:19) dxdt As t →
0, coth t → ∞ and hence for α = t . Hence this formula defines the required analytic extension. (cid:3) Corollary 3.11.
If a ∈ A ∞ ℏ then the function Tr( π ℏ ( a ) · H − s ) , Re( s ) > extends meromorphically to C , with a single simple pole at s = , and if a ∈ A ∞ ℏ then Res s = Tr( π ℏ ( a ) · H − s ) = τ ( a ) where τ : A ℏ → C is the standard trace.
4. T HE L OCAL I NDEX F ORMULA FOR THE H EISENBERG CYCLES
Our objective in this note is to use the Local Index Theorem of [5] in dimension 2 tocompute the pairing of the class [ D ℏ ] with K ( A ℏ ) . We will use the development of theLocal Index Formula by N. Higson in [8] and our results on the harmonic oscillator residuetrace of the previous section.In the index formula for the Heisenberg cycles over A ℏ which we are going to describe,there is a 0-dimensional part of the formula, and a 2-dimensional part. The 0-dimensionalpart, as we will show, is given by the the standard trace on A ℏ .The 2-dimensional part is a multiple of Connes’ ‘curvature’ cocycle, which we reviewfirst.In the paper [2], Alain Connes described an invariant of a finitely generated projectivemodule over A ℏ , generalizing the first Chern number of a complex vector bundle over T .Connes’ construction was the following. Let A be any C*-algebra endowed with anaction of R by automorphisms with ( s , t ) acting by α s ◦ β t .Let δ i : A → A be the densely defined derivations δ ( a ) : = lim t → α t ( a ) − at , δ ( a ) : = lim t → β t ( a ) − at , a ∈ A ∞ ,where A ∞ = ∩ n , m dom ( δ n ) ∩ dom ( δ m ) , the *-subalgebra of elements such that ( s , t ) α s ( β t ( a )) is smooth.In addition, let τ : A → C be an R -invariant tracial state. Definition 4.1.
In the above notation, Connes’ invariant of a f.g.p. module eA ∞ , where e is a projection in A ∞ , is given by c ( e ) : = π i τ ( e [ δ ( e ) , δ ( e )]) .We call θ ( e ) : = e [ δ ( e ) , δ ( e )] the curvature of e and c ( e ) the first Chern number of e .The curvature of e is an A ∞ -module endomorphism of eA ∞ . Theorem 4.2. (Connes, [4] ) ) The numberc ( e ) : = π i τ ( e [ δ ( e ) , δ ( e )]) only depends on the equivalence class of e in K ( A ∞ ) .Moreover, c ( E ⊕ E ′ ) = c ( E )+ c ( E ′ ) and c thus determines a group homomorphism K ( A ) → R . The last statement follows from the fact that A ∞ ⊂ A is holomorphically closed.Let ℏ ∈ R and A ℏ = C ( T ) ⋊ h Z the corresponding rotation algebra, with u ∈ A ℏ thegenerator of the Z -action. Then the R -action with α t ( f ) = f ( x − t ) , α t ( u ) = u ; β t ( u n ) = e π int u n , β t ( f ) = f , gives rise to the derivations δ ( ∑ n f n [ n ]) = ∑ n f ′ n [ n ] , δ ( ∑ n f n [ n ]) = ∑ n π in · f n [ n ] .The following calculation from [4] is reproduced below for the benefit of the reader. Lemma 4.3.
Let p ℏ ∈ A ℏ = C ( T ) ⋊ ℏ Z be the Rieffel projection Thenc ( p ℏ ) = + Proof.
For brevity, for f ∈ C ( T ) understood as a 2 π -periodic function on R , let f ℏ ( x ) : = f ( x − ℏ ) denote the action.The Rieffel projection is given by p ℏ = f + gu + g − ℏ u ∗ where f and g are suitably chosen functions. For a ∈ (
0, 1 ) , and ε > f equalszero on [ a ] and on [ a + ℏ + ε , 2 π ] , and f = [ a + ε , a + ℏ ] . We choose f so that f ( x ) + f ( x + ℏ ) =
1. We set g = p f − f on [ a + ℏ , a + ℏ + ε ] and is zero otherwise. Wethen have to compute c ( e ) = π i τ ([ δ ( p ℏ ) , δ ( p ℏ )] . We first compute(4.1) 12 π i [ δ ( p ℏ ) , δ ( p ℏ )] = u ∗ ( g f ′ − g f ′ ℏ ) + (cid:16) ( gg ′ ) − ℏ − gg ′ (cid:17) + ( g f ′ − g f ′ ℏ ) u Multiplying this on the left by p ℏ produces a terrific mess, but we are only interested in itstrace, so the only part which is relevant is(4.2) g ( f ′ − f ′ ℏ ) + f (cid:16) ( gg ′ ) − ℏ − gg ′ (cid:17) + (cid:16) g ( f ′ − f ′ ℏ ) (cid:17) − ℏ which we want to integrate over T .Set w = g , v = f − f ℏ . The integral is given by(4.3) Z wv ′ + f (cid:16) w ′− ℏ − w ′ (cid:17) + ( wv ′ ) − ℏ The middle term is Z f w ′− ℏ − Z f w ′ = Z f ℏ w ′ − Z f w ′ = − Z vw ′ .Hence we are reduced to computing Z wv ′ − vw ′ + ( wv ′ ) − h = Z wv ′ − vw ′ = Z wv ′ , IRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS 15 by integration by parts. Next, since f ℏ = − f on supp( g ) , we can replace f ′ − f ′ ℏ by − f ′ and get(4.4) − Z f ′ ( f − f ) = − Z f ′ f + Z f ′ f = − Z ( f ) ′ + Z ( f ) ′ = − = g . This completes thecalculation. (cid:3) The first Chern class of any projection in A ∞ ℏ is an integer, a fact related to the QuantumHall Effect (see [3]). This is due to agreement of the curvature of a projection with theindex pairing of the projection with the quantized Dolbeault spectral cycle discussed at theend of the section.This is a consequence of the Local Index Formula of Connes and Moscovici, which weare going to work out in the case of the Heisenberg cycles, but first state in low dimensions. Theorem 4.4. (Connes-Moscovici, [5] .) Let ( H , π , D ) be an even, -dimensional spectralFredholm module for a unital C*-algebra A, regular and with the meromorphic continua-tion property over A ∞ ⊂ A.Let [ D ] ∈ KK ( A , C ) be the class of the triple. Let ∆ : = D , and let ε be the gradingoperator on H.Define functionals • ψ : A ∞ → C , ψ ( a ) : = Res s = Γ ( s ) · Tr( ε a ( ∆ + proj ker D ) − s ) , and • ψ : A ∞ ⊗ A ∞ ⊗ A ∞ → C , , ψ ( a , a , a ) : =
12 Res s = ( ε a [ D , a ][ D , a ] ∆ − s ) . Then, if e ∈ A ∞ is a projection, then h [ e ] , [ D ] i = ψ ( e ) − ψ ( e −
12 , e , e ) , where h [ e ] , [ D ] i ∈ Z is the pairing between the K ( A ) -class [ e ] and the KK ( A , C ) class [ D ] . The Local Index Formula in the case of the Heisenberg cycles is as follows.
Theorem 4.5.
Let ℏ ∈ R and A ℏ : = C ( T ) ⋊ ℏ Z the corresponding rotation algebra.Let [ D ℏ ] be the class of the Heisenberg cycle (Definition 2.4) (cid:18) L ( R ) ⊕ L ( R ) , π ℏ , D : = (cid:20) x − d / dxx + d / dx (cid:21)(cid:19) . Then the associated functionals are given by • ψ : A → C , ψ ( a ) : = τ ( a ) , and • ψ : A ⊗ A ⊗ A → C , , ψ ( a , a , a ) = ℏ π i · τ (cid:0) a δ ( a ) δ ( a ) − a δ ( a ) δ ( a ) (cid:1) . where δ and δ are the standard derivations of A ℏ , τ the standard trace. Corollary 4.6.
Let e ∈ A ∞ ℏ be a projection, [ e ] ∈ K ( A ℏ ) its class. Then h [ e ] , [ D ℏ ] i = τ ( e ) − ℏ · c ( e ) , where c ( e ) is its first Chern number.In particular, h [ p ℏ ] , [ D h ] i = − ⌊ ℏ ⌋ where ⌊ ℏ ⌋ is the greatest integer < ℏ . We first take care of the regularity hypothesis.
Lemma 4.7.
The spectral triple of Theorem 4.5 over A ∞ ℏ is regular.Proof. In order to show that spectral cycle is regular, it suffices to show that the followingbasic estimate holds (c.f. Theorem 4.26 of [8]): For any X in the algebra generated by theWeyl algebra and C ∞ u ( R ) ⋊ Z of order ≤ q , there exists some ε > k ξ k + k ξ k q ≥ ε k X ξ k , ∀ ξ ∈ S where S is the Schwartz space of R and k ξ k q = k ξ k + k H q / ξ k , for H the harmonicoscillator.Every element of the above algebra can be expressed as a linear combination of termsof the form X = x n ( d m / dx m ) a for a in C ∞ u ( R ) ⋊ Z ; it suffices to check the estimate onoperators of this form. This operator has order ≤ n + m , and we find that k X ξ k ≤ x n D m a ξ , x n D m a ξ ≤ D m x n D m a ξ , a ξ ≤ H ( n + m ) a ξ , a ξ ≤ H n + m a ξ , H n + m a ξ = k a ξ k n + m − k a ξ k ≤ k a ξ k n + m Taking square roots, we obtain k X ξ k ≤ k a ξ k n + m . Since a is a bounded map of the Sobolevspace W n + m to itself, we have k a ξ k n + m ≤ k a k n + m k ξ k n + m , and the result follows for ε = k a k − n + m . Since the smallest eigenvalue of H is 1, we have k ξ k q ≤ k ξ k r for q ≤ r , and sothe estimate holds on X for all q ≥ n + m , as desired. (cid:3) Proof. (Of Theorem 4.5).We refer to Theorem 4.4. The Hilbert space for the Heisenberg triple is the directsum of two copies of L ( R ) , and D = (cid:20) x − d / dxx + d / dx (cid:21) . The kernel of D is thesame as the kernel of x + d / dx , and is spanned by the ground state ψ ( x ) = π − e − x , and ∆ = (cid:20) H − H + (cid:21) where H is the harmonic oscillator. Hence ∆ + proj ker D = (cid:20) H − + pr ker D H + (cid:21) .The first copy of the Hilbert space L ( R ) is even in the grading, the second is odd. Hencewe need to compute the residue of the difference Γ ( s ) · Tr( π ℏ ( a ) · ( H − + pr ker D ) − s ) − Γ ( s ) · Tr( π ℏ ( a ) · ( H + ) − s ) .for a ∈ A ℏ = C ( T ) ⋊ ℏ Z .The operator H − + pr ker D is diagonal with respect to the basis ψ , ψ , . . . of eigenvec-tors of H , with spectrum 1, 2, 4, 6, 8, . . . . Let f ∈ C ( T ) , thought of as a Z -periodic functionon R . Let k ∈ Z , and set a = f [ k ] . By Mehler’s formula IRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS 17 (4.5) k t ( x , y ) = ∞ ∑ n = e − nt φ n ( x ) φ n ( y ) = √ π sinh 2 t exp (cid:18) − tanh t · ( x + y ) − coth t · ( x − y ) (cid:19) We now compute:
Tr( π ℏ ( a ) · ( H − + pr ker D ) − s ) = Γ ( s ) h f [ k ] φ , φ i + Z ∞ t s − Z R f ( x ) ∞ ∑ n = e − nt φ n ( x − k ) φ n ( x ) ! dxdt = Γ ( s ) h f [ k ] φ , φ i + √ π Z ∞ t s − Z R f ( x ) " √ − e − t × exp (cid:18) − tanh ( t ) ( x − k ) − coth ( t ) k (cid:19) − exp (cid:18) − x + ( x − k ) (cid:19) dxdt Similarly Γ ( s ) · Tr( π ℏ ( a ) · ( H + ) − s ) = − s Γ ( s ) h f [ k ] φ , φ i + Z ∞ t s − e − t Z R f ( x ) ∞ ∑ n = e − nt φ n ( x − k ) φ n ( x ) ! dxdt = − s Γ ( s ) h f [ k ] φ , φ i + √ π Z ∞ t s − e − t Z R f ( x ) " √ − e − t × exp (cid:18) − tanh ( t ) ( x − k ) − coth ( t ) k (cid:19) − exp (cid:18) − x + ( x − k ) (cid:19) dxdt Taking the difference of these two, we obtain the meromorphic function Ψ ( f [ k ]) s = ( − − s ) · Γ ( s ) h f [ k ] φ , φ i + √ π Z ∞ t s − (cid:0) − e − t (cid:1) Z R f ( x ) " √ − e − t × exp (cid:18) − tanh ( t ) ( x − k ) − coth ( t ) k (cid:19) − exp (cid:18) − x + ( x − k ) (cid:19) dxdt This formula is valid for all
Re( s ) >
0, potentially with a simple pole at s =
0. Taking theresidue at 0, we obtain:Res s = Ψ ( f [ k ]) s = lim s → √ π Z ∞ st s − (cid:0) − e − t (cid:1) Z R f ( x ) " √ − e − t exp − tanh ( t ) × ( x − k ) − coth ( t ) k ! − exp (cid:18) − x + ( x − k ) (cid:19) dxdt = lim α → ∞ √ π Z ∞ (cid:16) − e − t α (cid:17) Z R f ( x ) " √ − e − t α exp − tanh ( t α ) × ( x − k ) − coth ( t n ) k ! − exp (cid:18) − x + ( x − k ) (cid:19) dxdt = lim α → ∞ √ π Z Z R f ( x ) p tanh ( t α ) exp − tanh ( t α ) × ( x − k ) − coth ( t α ) k ! dxdt The last equality follows from dominated convergence applied to the integrals from 1 to ∞ ,and the fact that 1 − e − t α → t ∈ [
0, 1 ] .For k =
0, the coth ( t n ) term dominates in the limit, and so the above expression tends to 0as n → ∞ . When k =
0, we arrive at the following expression:Res s = Ψ ( f ) s = lim n → ∞ √ π Z Z R f (cid:16) x p coth ( t n ) (cid:17) e − x dxdt = Res Tr( f ) Next, we compute Ψ . Expand [ D , a ][ D , a ] as a block matrix (cid:2) D , a (cid:3) (cid:2) D , a (cid:3) = (cid:20) (cid:2) x − d / dx , a (cid:3) (cid:2) x + d / dx , a (cid:3) (cid:2) x + d / dx , a (cid:3) (cid:2) x − d / dx , a (cid:3) (cid:21) givingRes s = Ψ ( a , a , a ) s = Res Tr (cid:0) ε a (cid:2) D , a (cid:3) (cid:2) D , a (cid:3) ∆ − (cid:1) = Res Tr (cid:0) a (cid:2) x , a (cid:3) (cid:2) d / dx , a (cid:3)(cid:1) − Res Tr (cid:0) a (cid:2) d / dx , a (cid:3) (cid:2) x , a (cid:3)(cid:1) IRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS 19
Now [ x , a ] = − π i · δ ( a ) and [ d / dx , a ] = δ ( a ) . Hence Res Tr (cid:0) a (cid:2) x , a (cid:3) (cid:2) d / dx , a (cid:3)(cid:1) − Res Tr (cid:0) a (cid:2) d / dx , a (cid:3)(cid:1) = π i (cid:0) a δ ( a ) δ ( a ) − a δ ( a ) δ ( a ) (cid:1) .This completes the proof. (cid:3) We close with a couple of applications.The Noncommutative Geometry of A ℏ has been traditionally studied in connection withthe quantized Dirac- Dolbeault spectral cycle over A ℏ defined in the following way.The Hilbert space is L ( T ) ⊕ L ( T ) , evenly graded. The operator is¯ ∂ : = " ∂∂ x − i ∂∂ y ∂∂ x + i ∂∂ y .The representation is the direct sum of two copies of the representation λ : A ℏ → B (cid:0) L ( T ) (cid:1) ,which is specified by the covariant pair ( λ ( f ) ξ ) ( x , y ) = f ( x ) ξ ( x , y ) , ( π ( n ) ξ ) ( x , y ) = e π iny ξ ( x − n ℏ , y ) .One can check that the Dolbeault cycle is 2-dimensional, and is regular with the mero-morphic extension property over A ∞ ℏ . Proposition 4.8.
Let [ ¯ ∂ ] ∈ KK ( A ℏ , C ) be the class of the Dolbeault cycle. If e ∈ A ∞ ℏ is aprojection, then h [ ¯ ∂ ] , [ e ] i = c ( e ) , where c ( e ) is the first Chern number of e. For the Dolbeault operator, one has¯ ∂ = " ∂ ∂ x + ∂ ∂ y ∂ ∂ x + ∂ ∂ y ,and the kernel of ¯ ∂ is 2-dimensional: spanned by a copy of the constant functions in thefirst L ( T ) , and by a copy of the constant functions in the second copy as well. Thissymmetry implies that the 0th functional Ψ associated to this spectral cycle is zero.In particular, this implies the integrality of the first Chern number, for any projection e ∈ A ℏ . Corollary 4.9.
Suppose ℏ ∈ R is nonzero. Then if τ : A ℏ → C is the trace, τ ∗ : K ( A ℏ ) → R the induced group homomorphism, then then range of τ ∗ (K ( A ℏ )) is the subgroup Z + ℏZ ⊂ R . Proof. If e ∈ A ∞ ℏ is a projection, then application of our results above gives that τ ( e ) + ℏ · c ( e ) is an integer. On the other hand, c ( e ) is an integer. This implies τ ( e ) = m + n ℏ for a pair of integers m , n . Finally, A ∞ ℏ is dense and holomorphically closed in A ℏ , so anyprojection in A ℏ is represented by a projection in A ∞ ℏ . (cid:3) The interest in this argument is that it computes the range of the trace using a combina-tion of two spectral cycles, but does not rely on computation of K ∗ ( A ℏ ) . It is conceivablethat such a method could be applied in connection with other ‘Gap-Labelling’ problems.
5. T
RANSVERSE FOLIATIONS
There is a well-known procedure for producing finitely generated projective (f.g.p.)modules over A ℏ , using Morita equivalence. Fixing one of the standard linear loops in T determines a Morita equivalence of A ℏ with the C*-algebra B ℏ = C ( T ) ⋊ ℏ R of theKronecker foliation F ℏ into lines of slope ℏ . On the other hand, any linear loop in T istransverse to F ℏ . These linear loops, parameterized by pairs of relatively prime integers,determine therefore Morita equivalences between A ℏ and what turn out to be other rotationalgebras, and in particular, unital algebras. Hence they determine f.g.p. modules over A ℏ ;these parameterize (the positive part of) the K-theory.In [6] we showed that there is an analogue of this procedure using non-compact transver-sals: if ℏ ′ = ℏ then the Kronecker foliations F ℏ and F ℏ ′ are transverse. Their product fo-liates T × T and its restriction to the diagonal gives an equivalent ´etale groupoid. Thisreasoning produces for every ℏ ′ = ℏ a f.g.p. module L ℏ , ℏ ′ over A ℏ ⊗ A ℏ ′ and corresponding K -theory class [ L ℏ , ℏ ′ ] ∈ KK ( C , A ℏ , ⊗ A ℏ ′ ) . In particular, fixing ~ ′ = ~ + b for any integer b = A ℏ ⊗ A ℏ . We denote it L b .Let PD : KK ( C , A ℏ ⊗ A ℏ ) → KK ( A ℏ , A h ) ,be Connes’ Poincar´e duality map [3]. In [6] it is proved that(5.1) PD ([ L b ]) = τ b ,where τ b is the Kasparov morphism defined in terms of Dirac-Schr¨odinger operators asfollows. We take the standard right Hilbert A ℏ -module L ( R ) ⊗ A ℏ . We let Z act on theleft by the following formula, where we designate a dense set of elements of our Hilbertmodule in the form ∑ n ∈ Z ξ n · [ n ] , with ξ n ∈ L ( R ) ⊗ C ( T ) : k · ∑ n ∈ Z ξ n · [ n ] ! : = ∑ n ∈ Z k ( ξ n ) · [ k + n ] .where k ( ξ )([ x ] , t ) = ξ ([ x − k ℏ ] , t − k ) , ξ ∈ L ( R ) ⊗ C ( T ) .Let f ∈ C ( T ) act by f · ∑ n ∈ Z ξ n · [ n ] ! : = ∑ n ∈ Z f b · ξ n · [ n ] .where f b ( t , [ x ]) = f ([ x + tb ]) .These two assignments determine a covariant pair and representation π ℏ : A ℏ → B ( L ( R ) ⊗ A ℏ ) . Definition 5.1.
The
Heisenberg twist τ b ∈ KK ( A ℏ , A h ) is the class of the spectral cycle (cid:0) L ( R ) ⊗ A ℏ ⊕ L ( R ) ⊗ A ℏ , π h ⊕ π h , D ⊗ A ℏ (cid:1) ,with D : = (cid:20) x − d / dxx + d / dx (cid:21) .The equality (5.1) gives rise to an explicit geometric cycle representing the unit inConnes’ duality [6]. The relationship between the Heisenberg twist and the Heisenbergcycles is implied by the following Lemma. Lemma 5.2.
Let ℏ , µ ∈ R . IRAC-SCHR ¨ODINGER OPERATORS AND THE IRRATIONAL TORUS 21 a) The group multiplication m : T × T → T intertwines the diagonal Z -action on T × T by group addition of ( ℏ , µ ) and group addition on T of ℏ + µ, and so determinesa *-homomorphism χ : A ℏ + µ → A ℏ ⊗ A µ . In this notation: χ ∗ ([ D ℏ ] ⊗ C [ D µ ]) = [ D ℏ + µ ] .b) If µ = b ∈ Z , then τ b = χ ∗ ([ D b ] ⊗ A ℏ ) where τ b is the Heisenberg twist. c) τ b ⊗ A ℏ [ D ℏ ] = [ D ℏ + b ] for any ℏ ∈ R , b ∈ Z .Proof. For c) we have τ b ⊗ A ℏ [ D ℏ ]) = χ ∗ ([ D b ] ⊗ A ℏ ) ⊗ A ℏ [ D ℏ ] = χ ∗ ([ D b ] ⊗ C [ D h ]) = [ D b + ℏ ] using first part b) and then part a).b) follows from an inspection at the level of cycles: they differ only in the representa-tions, which are clearly homotopic.We now prove a).Consider χ ∗ ([ D ℏ ] ⊗ C [ D µ ]) , a class in KK( A θ + η , C ) . By the standard method of com-puting external products, it is represented by the following spectral cycle. The Hilbertspace is module is L ( R , C ) ⊗ L ( R , C ) and operator D ⊗ + ⊗ D . With u , v ∈ A ℏ thestandard unitary generators, u = z , v = [ ] , the representation is given by: ( u · φ )( x , y ) = e π i ( x + y ) φ ( x , y ) , ( v · φ )( x , y ) = φ ( x − θ , y − η ) We apply a homotopy to the representation, with t ∈ [
0, 1 / ] : ( π t ( v ) · φ )( x , y ) = φ ( x − ( − t ) θ − t η , y − t θ − ( − t ) η ) The resulting cycle is ( L ( R , C ) ⊗ L ( R , C ) , ∆ , D ⊗ + ⊗ D ) , where ∆ is the “diagonal”representation of A θ + η : ( u φ )( x , y ) = e π i ( x + y ) φ ( x , y ) , ( v φ )( x , y ) = φ (cid:18) x − θ + η y − θ + η (cid:19) Next consider the class [ D ℏ + µ ] . The the unit in 1 C ∈ KK( C , C ) can be represented bythe cycle ( L ( R , C ) , 1, D ) . Taking the intersection product of this with [ D θ + η ] yields acycle which is equivalent to [ D θ + η ] , but more closely resembles the cycle described in theprevious paragraph: the Hilbert space is L ( R , C ) ⊗ L ( R , C ) , the operator is D ⊗ + ⊗ D , and the representation is given by: ( u · φ )( x , y ) = e π ix φ ( x , y ) , ( v · φ )( x , y ) = φ ( x − θ − η , y ) From here we take a homotopy by rotating this representation around R to lie along thediagonal (i.e. so that ( a · φ )( x , y ) depends only on x + y ), and the result follows.By b) the equation τ b ⊗ A ℏ [ D ℏ ] = [ D ℏ + b ] follows immediately. (cid:3) Corollary 5.3.
The Heisenberg twist acts by the identity on K ( A ℏ ) .With respect to the ordered basis { [ ] , [ p ℏ ] } for K ( A ℏ ) , where p ℏ is the Rieffel projec-tion, the morphism τ b acts by matrix multiplication by (cid:20) b (cid:21) . Proof.
The first statement follows from [6].Consider ( τ b ) ∗ ([ p h ]) ∈ K ( A ℏ ) . Write ( τ b ) ∗ ([ p h ]) = x [ ] + y [ p ℏ ] .Pairing both sides with [ D h ] and using the Index Theorem Corollary ?? twice gives x = ( τ b ) ∗ ([ p h ]) ⊗ A ℏ [ D ℏ ] = [ p ℏ ] ⊗ A ℏ ( τ b ) ∗ ([ D h ]) = [ p ℏ ] ⊗ A ℏ [ D h + b ] = b .Pairing the same equation with [ D ℏ + ] and computing give that y = (cid:3) It follows from similar simple arguments that in the classical case ℏ ∈ Z , the Heisenbergclasses [ D b ] ∈ KK ( C ( T ) , C ) = K ( T ) are given by [ D b ] = [ pt ] + b · [ ¯ ∂ ] ∈ K ( T ) .For b =
1, we have noted that [ D ] = [ ¯ ∂ · P ] . This corresponds to [ ¯ ∂ · P ] = [ pt ] + [ ¯ ∂ ] , whichof course follows from the Riemann-Roch formula. We have h [ D n ] , [ E ] i = dim E + n · c ( E ) ,for any complex vector bundle E over T . Therefore the classes [ D n ] taken together deter-mine both the the dimension and first Chern number, the two basic invariants of a complexvector bundle over T . R EFERENCES[1] N. Berline, E. Getzler, M. Vergne,
Heat kernels and Dirac operators , Grundlehren der mathematicischenWissenshcaften (1992), Springer-Verlag Hedelberg NewYork.[2] A. Connes:
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The residue index theorem of Connes and Moscovici . Surveys in noncommutative geometry,Clay Math. Proc., 6, Amer. Math. Soc., Providence, RI, (2006), 71126.[9] F.G. Mehler: ” ¨Uber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Func-tionen hherer Ordnung”, Journal fr die Reine und Angewandte Mathematik 66. 161– 176.[10] . J. Roe:
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