Capturing the Atiyah-Patodi-Singer index from the lattice

CAPTURING THE A TIY AH–P A TODI–SINGER INDEX FR OM THE LA TTICE SHOTO A OKI, HAJIME FUJIT A, HIDENORI FUKA Y A, MIKIO FURUT A, SHINICHIR OH MA TSUO, TETSUY A ONOGI, AND SA TOSHI Y AMAGUCHI Abstra ct. Using the Wilson Dirac op erator in lattice gauge theory with a domain-w all mass term, w e construct a discretization of the Atiy ah–P ato di–Singer index for domains with compact b oundary in a ﬂat torus. W e pro v e that, for suﬃcien tly small lattice spacings, this discretization correctly captures the con tinuum Atiy ah–P ato di–Singer index. Contents 1. In tro duction 2 2. Wilson Dirac op erators on a lattice 4 2.1. Dirac op erators in con tin uum space 4 2.2. Wilson Dirac op erators 6 3. F rom lattice to con tin uum 7 3.1. Finite element interpolator 7 3.2. Prop erties 9 4. Domain-w all Dirac op erators and the A tiy ah-P ato di-Singer index 10 4.1. Dirac op erators on manifolds with b oundaries and the APS index 10 4.2. Domain-w all fermion Dirac operators 11 4.3. The domain-wall Dirac operators and the APS index 12 4.4. Domain-w all Dirac op erators on a ﬂat torus 13 5. K -groups and un b ounded selfadjoin t op erators 13 5.1. Deﬁnitions of K p,q • -co cycles 14 5.2. Set-theoretic techniques 17 5.3. Deﬁnition of the semigroup K p,q • 20 5.4. Group structure on the semigroup K p,q • 21 5.5. Degrees 23 5.6. Riesz contin uous families 24 5.7. The inv erse element 25 5.8. Isomorphism K ∗ bounded ∼ = K ∗ Riesz 29 5.9. Gap top ology 31 6. Deﬁnition of sp ectral ﬂo w 32 7. Main theorem 34 Preprin t num ber: OU-HET-1300. 1 CAPTURING THE APS INDEX FR OM THE LA TTICE 2 7.1. Domain-w all mass term 34 7.2. Main theorem 35 7.3. Pro of of Theorem 30 37 8. Applications to the mo d-t w o APS index 39 A c knowledgmen ts 41 References 42 1. Intr oduction Lattice gauge theory oﬀers a p ow erful to ol in particle ph ysics to compute quan tum ﬁeld theories from the ﬁrst principle. By coarse-graining the space time into a discrete lattice space, the functional in tegral becomes mathematically w ell-deﬁned and n umerically calculable. Ho w ever, a naiv e discretization often fails to main tain crucial prop erties of the target con tin uum theory . T op ology is one of them. In particular, the F redholm index of Dirac op erators 1 (w e refer to it the A tiy ah- Singer(AS) index for comparison with the Atiy ah-Patodi-Singer index) has b een a main c hallenging sub ject to form ulate in lattice gauge theory . The most successful approac h was giv en b y the ov erlap Dirac op erator on ev en-dimensional square lattices [34] (and a similar form ulation by [24]), which realizes the Z 2 -grading op erator, or chiralit y op erator in ph ysics [33], with a slight mo diﬁcation to satisfy the so-called Ginsparg-Wilson relation [23]. In this approac h, the index was deﬁned b y the kernel of the ov erlap Dirac op erator, whic h is consistent with the AS index of the contin uum Dirac op erator on a torus. See [1] for a mathematical justiﬁcation and [15] for a general extension of this form ulation. In our previous work [4], w e addressed a diﬀerent approach on a lattice to extracting the AS index from massiv e Dirac op erators. W e used a mathematical relation b et ween the AS index of the massless ( Z 2 -graded) Dirac op erator and a family of massiv e Dirac op erators, whic h is well-kno wn in K -theory as the susp ension isomorphism K 0 ( pt ) ∼ = K 1 ( I , ∂ I ) . Here pt denotes a one-p oint set, and I is an in terv al with its b oundary ∂ I , consisting of the tw o end p oin ts. W e ga v e a mathematical pro of that at suﬃcien tly small lattice spacings, w e can construct a family of the massiv e Wilson Dirac op erators on a lattice, which can be iden tiﬁed as an elemen t of K 1 ( I , ∂ I ) and its spectral ﬂo w 2 giv es the same v alue of the AS index in the contin uum theory . Since the Z 2 -grading structure is lost in the massiv e Dirac operator from the b eginning, the Ginsparg-Wilson relation is not required in our form ulation and the standard Wilson Dirac operator is goo d enough. W e note 1 T o b e precise, we consider Dirac-t yp e op erators, which form a broader class than the Dirac op erators asso ciated with spin structures or spin c structures. In our work, we simply refer to Dirac-t yp e op erators as Dirac op erators. 2 The sp ectral ﬂow approach was empirically known [25] (even b efore the ov erlap fermion was kno wn) to repro duce the index but the mathematical background was not rigorously discussed. CAPTURING THE APS INDEX FR OM THE LA TTICE 3 that there are also diﬀerent mathematical formulations [31, 43] using the Wilson Dirac op erator. In this w ork, w e attempt an imp ortan t generalization of the index, the Atiy ah– P ato di–Singer (APS) index [9–11] of Dirac op erators on manifolds with boundary . The APS index is studied in ph ysics to understanding the bulk-b oundary corre- sp ondence of the fermion anomaly in the symmetry-protected topological phases [26, 28, 30, 35–37, 41, 42, 44]. Compared to the AS index on closed manifolds, the APS index is more challenging to realize on the lattice due to the following diﬃculties: • The APS b oundary condition is global and non-lo cal, making its lattice form ulation diﬃcult. In particular, it is not kno wn how to imp ose the APS b oundary condition on lattice Dirac operators. • The APS index is not top ological, i.e. , it depends on the metric and connections of the bundle near the boundary . This is in con trast to the AS index, which is a topological in v ariant. Therefore, one needs to control the metric and connection dep endence. In fact, the pro of giv en in [4] is so robust that it can also b e applied to the APS index. if we use a mathematical relation [16, 17, 19, 21] that the APS index in the con tin uum theory can b e expressed as the spectral ﬂo w of the domain-wall Dirac op erators [12, 27]. By the domain-w all fermion formulation, w e can o vercome the ab o ve t wo diﬃculties as b elo w. • W e can a void the APS b oundary condition by gluing the b oundary of the original manifold w e denote b y X + to another manifold X − sharing the same b oundary to form a closed manifold. Instead of the b oundary condition, we assign opposite signs to the mass term on X + and X − so that the nontrivial geometrical information is obtained from the X + subspace only , which is prov ed to b e the APS index on X + . This observ ation indicates that the APS index on a lattice ma y be deﬁned by the sp ectral ﬂo w of the lattice domain-wall Dirac operators [29, 39]. Once form ulated in terms of the sp ectral ﬂo w, the pro of of [4] can b e applied almost straigh tforw ardly . • W e can also a v oid the problem that the APS index is not a topological in v ariant, by just assuming that the target domain-w all Dirac op erator is inv ertible. With this assumption, w e can construct the lattice domain- w all Dirac op erator family as a mathematically well-deﬁned element of K 1 ( I , ∂ I ) . Here is a summary of this pap er. W e describ e a form ulation of K and K O groups in such a w ay that contin uum unbounded Dirac op erators and ﬁnite lattice b ounded Dirac op erators are simultaneously handled. Although such a form ulation w as essen tially kno wn in the literature [32] and discussed in our previous paper [4] as w ell, w e w ould lik e to presen t it in a selfcontained and comprehensiv e w a y treating the K and K O groups with arbitrary degree on general pairs of base spaces. Then w e consider a direct sum of contin uum and lattice domain-wall Dirac operators CAPTURING THE APS INDEX FR OM THE LA TTICE 4 and prov e in our main theorem (Theorem 30) that the combined Dirac op erator giv es a trivial element of K 1 ( I , ∂ I ) . In the pro of, our ﬁnite element interpolator b et ween the functions on the lattice and those in the con tin uum space, as well as its action on the domain-wall mass term, pla ys a k ey role to ensure that the com bined op erator is in v ertible. The equalit y betw een the t wo spectral ﬂo ws immediately follo ws (Theorem 32). As far as w e know, this is the ﬁrst mathematically rigorous form ulation of the APS index on a lattice. This form ulation is suﬃcien tly robust to allow extensions to the systems with additional symmetries. W e explicitly describ e an application to the mo d-t w o v ersion of the APS index when the lattice domain-w all Dirac op erator is real (Theorems 35 and 37). W e note that the ph ysics part of this w ork w as already published in [5] (see also [20] whic h ga ve a p erturbativ e discussion of the spectral ﬂo w of the domain-w all Dirac operators), where we presented the k ey formulas with a summary of mathematical descriptions and numerical examinations of them on tw o-dimensional lattices. In this w ork, the base manifold X = X − ∪ X + is limited to a ﬂat torus. A c- cordingly w e emplo y a square lattice as a discretization of X . The readers may ask a question if w e can extend our work to the case where X is a general curved manifold. Curren tly w e do not hav e a clear answer but there is an interesting observ ation. In our setup, the domain-wall Y can be an y curv ed submanifold of X . On such a curved domain-w all on a lattice, it w as shown that a non trivial curv ature eﬀect (gra vitational background in physics) is induced, which is consistent with the con tin uum theory [2, 3, 6–8, 14]. Embedding X in to a further higher dimensional square lattice ma y be an in teresting direction, whic h is, ho wev er, b eyond the scope of this w ork. The rest of the pap er is organized as follows. In Section 2 we construct the Wilson Dirac operator on a square lattice from a given contin uum Dirac op erator on a ﬂat torus. In Section 3 w e deﬁne a ﬁnite elemen t interpolator betw een the functions on a lattice and those in contin uum space and summarize its k ey properties pro v ed in our previous work [4]. Then in Section 4, we deﬁne the domain-wall Dirac op erator and review its relation to the APS index. In Section 5 w e presen t the formulation of K and K O groups whic h simultaneously handles b ounded and unbounded Dirac op erators. Deﬁnitions of the sp ectral ﬂo w and its mo d-tw o version are giv en in Section 6. Then in Section 7, w e state our main theorems and giv e the proof. As a non trivial example with symmetry , we also presen t an application to the mod-tw o APS index of real Dirac op erators in Section 8. 2. Wilson Dirac opera tors on a la ttice In this section, w e will construct the Wilson Dirac op erators, commonly used in lattice gauge theory , whic h we adopt as our discretization of the Dirac operators. 2.1. Dirac op erators in contin uum space. W e set up notation. Let X := T d = ( R / Z ) d b e a d -dimensional ﬂat torus. Let e 1 , . . . , e d b e the standard orthonormal CAPTURING THE APS INDEX FR OM THE LA TTICE 5 basis of R d . The tangent bundle T X is canonically isometric to the trivial bundle X × R d . The Cliﬀord algebra Cl d is generated b y { e 1 , . . . , e d } sub ject to the an ticomm utation relation { e i , e j } = e i e j + e j e i = − 2 δ ij . F or simplicity , w e ﬁrst assume that d is ev en. Let E → X b e a Cliﬀord module bundle on X ; that is, E is a Z 2 -graded Hermitian v ector bundle with a smo oth map of graded algebra bundles σ : X × Cl d → End ( E ) . W e also assume that σ ( e j ) ∗ = − σ ( e j ) . W e denote its Z 2 -grading op erator by γ . Fix a Cliﬀord connection A on E and we denote the cov arian t deriv ative in the e j direction with resp ect to A b y ∇ j . Let R X b e the injectiv e radius of X . F or arbitrary tw o p oints x, y ∈ X suc h that | x − y | < R X , there is a unique minimal geo desic from x to y . W e denote the parallel transport b y the connection A along this minimal geo desic b y U x,y : E y → E x , whic h depends smo othly on x and y . Note that U y ,x = U − 1 x,y and U x,x = id E x hold 3 . Let Γ( E ) be the space of smo oth sections of E . F or smo oth sections u, v ∈ Γ( E ) and the Hermitian metric ( · , · ) on E , w e denote their inner pro duct and the asso ciated norm b y ⟨ u, v ⟩ := Z X ( u, v ) dx, || v || L 2 := ⟨ v , v ⟩ 1 / 2 , where dx denotes the v olume elemen t of X asso ciated with the standard ﬂat metric. Let D : Γ( E ) → Γ( E ) b e the Dirac op erator deﬁned b y D u := d X j =1 σ ( e j ) ∇ j u, for u ∈ Γ( E ) , which is a ﬁrst order formally selfadjoint elliptic op erator. W e also use the L 2 1 norm || · || L 2 1 on Γ( E ) ; || v || L 2 1 :=  || v || 2 L 2 + 1 m 2 0 d X j =1 Z X |∇ j v | 2 dx  1 / 2 for v ∈ Γ( E ) , where m 0 is an arbitrary non-zero real n umber, whic h is often tak en as a typical scale of the ph ysical system w e fo cus on. W e consider the Hilb ert space L 2 ( E ) (resp. L 2 1 ( E ) ) of the completion of Γ( E ) by the norm || ∗ || L 2 (resp. || ∗ || L 2 1 ). The Dirac op erator D can b e extended as an un b ounded selfadjoin t op erator (which is denoted b y the same letter) D : L 2 ( E ) → L 2 ( E ) , 3 The set of the parallel transp orts { U x,y } is an example of the generalized link v ariables deﬁned in [4]. CAPTURING THE APS INDEX FR OM THE LA TTICE 6 with the domain L 2 1 ( E ) . 2.2. Wilson Dirac operators. Let N b e a p ositiv e integer, and set a := 1 / N . Let b X a := ( a Z / Z ) d ⊂ X b e the standard lattice: we tak e, for simplicit y , a hypercubic lattice whose size and lattice spacing are equal in every direction 4 . Set b E a := E | b X a . W e simply write b X and b E when a dep endence is not imp ortant. F or eac h lattice p oin t z ∈ b X and j ∈ { 1 , . . . , d } , w e deﬁne the link variables U z ,z + a e j : b E z + a e j → b E z , whic h is the restriction of the parallel transp ort U x,y on to the lattice b X . W e deﬁne the forward diﬀerence operator b ∇ f j : Γ( b E ) → Γ( b E ) b y  b ∇ f j u  ( z ) := U z ,z + a e j  u ( z + a e j )  − u ( z ) a for u ∈ Γ( b E ) and z ∈ b X . W e also deﬁne the backw ard diﬀerence op erator b ∇ b j : Γ( b E ) → Γ( b E ) b y  b ∇ b j u  ( z ) := u ( z ) − U z ,z − a e j  u ( z − a e j )  a for u ∈ Γ( b E ) and z ∈ b X . Note that b ∇ b j u = −  b ∇ f j u  ∗ . W e then deﬁne a co v ariant diﬀerence op erator b ∇ j : Γ( b E ) → Γ( b E ) by b ∇ j :=  b ∇ f j + b ∇ b j  / 2 . Note that b ∇ j is sk ew adjoin t. Now w e deﬁne the naive Dirac op erator b D naive : Γ( b E ) → Γ( b E ) b y b D naive u := d X j =1 σ ( e j ) b ∇ j u for u ∈ Γ( b E ) . Note that b D naive is selfadjoint. A fundamental lesson of lattice gauge theory is that the naiv e Dirac operator, though it ma y app ear to b e a natural discretization, is inadequate, since a prop erty whic h corresponds to the elliptic estimate in the contin uum theory is lost 5 . One should instead emplo y , for instance, the Wilson Dirac operator, whic h w e deﬁne b elo w. Recall that a = 1 / N is our lattice spacing and that γ is the Z 2 -grading op erator of E . 4 It would not b e diﬃcult to consider anisotropic shap e of the lattice. but we do not discuss that in this w ork. 5 In physics, the problem is known as fermion doubling since the naiv e Dirac op erator dev elops m ultiple zero p oints in the momen tum space. Due to these unph ysical doubler zeros, the naive Dirac op erator do es not recov er the elliptic estimate even in the contin uum limit a → 0 . CAPTURING THE APS INDEX FR OM THE LA TTICE 7 Deﬁnition 1. The Wilson term W : Γ( b E ) → Γ( b E ) is deﬁned b y W u := a 2 d X j =1 b ∇ f j  b ∇ f j  ∗ u for u ∈ Γ( b E ) . Deﬁnition 2. W e deﬁne the Wilson Dirac op erator b D wilson : Γ( b E ) → Γ( b E ) by b D wilson := b D naive + γ W . Note that W is selfadjoin t and so is b D wilson . There is a crucial prop ert y of the Wilson Dirac op erator which corresp onds to the elliptic estimate in the con tinuum theory . In [4][Theorem 4.7], it w as sho wn that there exist t wo p ositive a -indep enden t constan ts a 2 > 0 and C > 0 such that the follo wing inequality uniformly holds for any ﬁnite lattice spacing satisfying 0 < a ≤ a 2 and arbitrary ϕ ∈ Γ( b E ) . d X j =1 || b ∇ f j ϕ || 2 ≤ 2 || b D wilson a ϕ || 2 + C || ϕ || 2 , where we hav e put the subscript a to the Wilson Dirac op erator to remind that it dep ends on the lattice spacing a . Finally let us introduce the lattice v ersions of L 2 and L 2 1 norms ; || v a || L 2 :=  a d X z ∈ b X | v a ( z ) | 2  1 / 2 , || v a || L 2 1 :=   || v a || 2 L 2 + a d m 2 0 X z ∈ b X d X i =1 | ( b ∇ f j v a )( z ) | 2   1 / 2 for v a ∈ Γ( b E ) . Since Γ( b E ) is ﬁnite dimensional for ﬁxed a , the completions with resp ect to the norms || · || L 2 and || · || L 2 1 coincide with Γ( b E ) itself. But in the con tin uum limit a → 0 , the t w o con verge to the diﬀerent con tin uum coun terparts. 3. Fr om la ttice to continuum In this section, w e recall a “ﬁnite element metho d” interpolating the lattice theory with the con tin uum theory , which was discussed in our previous pap er [4]. 3.1. Finite elemen t interpolator. W e ﬁrst deﬁne an op erator ι a : Γ( b E ) → Γ( E ) , whic h interpolates functions on the lattice and those on the con tin uum torus. Using the translational symmetry of the metric, the diﬀerence of the co ordinates like x − z is w ell-deﬁned. CAPTURING THE APS INDEX FR OM THE LA TTICE 8 W e ﬁrst deﬁne a p erio dic function ρ (1) a : R → R with p erio d 1 (see Figure 2) b y ρ (1) a ( t ) := 1 a max { 0 , 1 − t/a, 1 − (1 − t ) /a } , which induce functions ¯ ρ a : S 1 → R and ρ a : X → R , ρ a ( x ) := d Y i =1 ¯ ρ a ( x i ) ( x = ( x 1 , . . . , x d ) ∈ X ) . W e will use this ρ a ( x ) as a cut-oﬀ function, which hav e the following prop erties. 0 1/2 a 1/ a -1 a 1 t ρ a (1) ( t ) Figure 1. The function ρ (1) a ( t ) . The function ¯ ρ a ( t ) i s its restriction to t ∈ [0 , 1] where the t w o end p oints are iden tiﬁed. (i) F or arbitrary x ∈ X , we hav e a d X z ∈ b X ρ a ( x − z ) = 1 . and for arbitrary z ∈ b X , w e ha v e Z x ∈ X ρ a ( x − z ) dx = 1 , where dx is the v olume form on X . (ii) F or B = { 0 } ∪ {± e k | 1 ≤ k ≤ d } , w e hav e a d X e ∈ B Z x ∈ X ρ a ( x ) ρ a ( x − ae ) dx = 1 . Using the ab o v e cut-oﬀ function, w e deﬁne the map, the ﬁnite element in terp ola- tor, ι a : Γ( b E ) → Γ( E ) b y , ( ι a ϕ )( x ) := a d X z ∈ b X ρ a ( x − z ) U x,z ϕ ( z ) ( ϕ ∈ Γ( b E ) , x ∈ X ) CAPTURING THE APS INDEX FR OM THE LA TTICE 9 and its adjoin t ι ∗ a : Γ( E ) → Γ( b E ) b y ( ι ∗ a ψ )( z ) := Z x ∈ X ρ a ( z − x ) U − 1 x,z ψ ( x ) dx ( ψ ∈ Γ( E ) , z ∈ b X ) . F or later conv enience, w e also deﬁne ι a | ϕ ( x ) | := a d X z ∈ b X ρ a ( x − z ) | ϕ ( z ) | for ϕ ∈ Γ( b E ) and x ∈ X . Note that ρ a ( z − x ) is nonzero only when x is inside a unit hypercub e of the lattice where z is one of the v ertices. 3.2. Prop erties. W e next review the k ey prop erties of the ﬁnite element interpola- tor ι a . One can see that neither ι a ι ∗ a nor ι ∗ a ι a is the identit y map. This phenomenon is due to the nonlo cal nature of these op erators. Note, how ev er, that ρ a ( x − z ) is nonzero only when | x i − z i | < a for all i . In the contin uum limit a → 0 , they b eha ve lik e the identit y , which is guaran teed by the follo wing properties pro v ed in [4]. (i) ι a and ι ∗ a ha v e the same ﬁnite op erator norm || ι a || = || ι ∗ a || . (ii) [Prop osition 4.3 of [4]] F or k = 0 , 1 , the op erator norm of ι a : L 2 k ( b E ) → L 2 k ( E ) is uniformly b ounded with resp ect to a . Lik ewise, the op erator norm of ι ∗ a : L 2 k ( E ) → L 2 k ( b E ) is uniformly b ounded with resp ect to a . Here w e set L 2 0 ( · ) = L 2 ( · ) . (iii) [Prop osition 4.4 of [4]] There exists C > 0 which is indep enden t of a suc h that for an y a and ϕ ∈ L 2 ( b E ) || ι ∗ a ι a ϕ − ϕ || 2 L 2 ≤ C a || ϕ || 2 L 2 1 . (iv) [Prop osition 4.5 of [4]] F or arbitrary ψ ∈ Γ( E ) ⊂ L 2 ( E ) , in the limit a → 0 , ι a ι ∗ a ψ conv erges to ψ in the strong L 2 -sense: ι a ι ∗ a ψ − → ψ . In particular, for arbitrary ψ , ψ ′ ∈ Γ( E ) ⊂ L 2 ( E ) , w e hav e ⟨ ι ∗ a ψ ′ , ι ∗ a ψ ⟩ → ⟨ ψ ′ , ψ ⟩ in the a → 0 limit. (v) [Prop osition 4.6 of [4]] F or arbitrary ψ ∈ Γ( E ) ⊂ L 2 ( E ) , in the limit a → 0 , w e ha ve a strong L 2 -con v ergence ι a ( b D wilson ) ∗ ι ∗ a ψ − → D ∗ ψ ( L 2 ) . Namely , for arbitrary ψ , ψ ′ ∈ Γ( E ) , w e hav e ⟨ ι ∗ a ψ ′ , ( b D wilson ) ∗ ι ∗ a ψ ⟩ → ⟨ ψ ′ , D ∗ ψ ⟩ in the a → 0 limit. CAPTURING THE APS INDEX FR OM THE LA TTICE 10 4. Domain-w all Dira c opera tors and the A tiy ah-P a todi-Singer index In this section, we will review the domain-wall Dirac operator and its relation to the Atiy ah-P ato di-Singer index [18]. 4.1. Dirac op erators on manifolds with b oundaries and the APS index. W e deﬁne c , ϵ , and Γ by c =  0 1 − 1 0  , ϵ =  0 1 1 0  , and Γ =  1 0 0 − 1  . They satisfy that c 2 = − 1 , ϵ 2 = Γ 2 = 1 , Γ = cϵ , and they all an ti-comm ute. Let X + b e an oriented even-dimensional Riemannian manifold with a boundary Y . W e assume that Y has a collar neigh b ourho o d isometric to the standard pro duct I × Y with an in terv al I to which the co ordinate is assigned b y u . Note that we do not assume the connectedness of Y . Let S = S + ⊕ S − b e a Z 2 - graded Hermitian v ector bundle on X with the Z 2 -grading op erator γ suc h that γ | S ± = ± 1 . W e denote the space of the smo oth sections 6 of S on X b y C ∞ ( X ; S ) . Let D : C ∞ ( X ; S ) → C ∞ ( X ; S ) b e a ﬁrst-order, formally selfadjoint, elliptic partial diﬀeren tial operator. W e assume that D is o dd in the sense that it anti-comm utes with γ . W e assume that S and D are standard in the following sense: there exist a Hermitian vector bundle E on Y and a bundle isomorphism from S | I × Y to C 2 ⊗ E as Z 2 -graded Hermitian v ector bundles suc h that, under this isomorphism, D tak es the form D = c ⊗ ∂ u + ϵ ⊗ D Y =  0 ∂ u + D Y − ∂ u + D Y 0  , where the grading of C 2 ⊗ E is given b y Γ ⊗ id and D Y : C ∞ ( Y ; E ) → C ∞ ( Y ; E ) is a formally selfadjoin t, elliptic partial diﬀerential op erator. In this pap er, w e will concentrate on the case when D Y has no zero eigenv alues, and assume this condition. Let C ∞ ( X + ; S ± : P D Y ) :=  f ∈ C ∞ ( X + ; S ± ) | P D Y ( f | Y ) = 0  , where P D Y : L 2 ( Y ; E ) → L 2 ( Y ; E ) denotes the spectral pro jection on to the span of the eigensections of D Y with p ositive eigenv alues. This is kno wn as the APS boundary condition. W e deﬁne the APS index [18] of D b y Ind APS ( D ) := dim  Ker D + ∩ C ∞ ( X + ; S + | X + : P D Y )  − dim  Ker D − ∩ C ∞ ( X + ; S − | X + : P D Y )  , where D ± = D P ± with the pro jection operator P ± = (id ± Γ) / 2 . 6 In this section, w e take a notation which sp eciﬁes the base manifold. CAPTURING THE APS INDEX FR OM THE LA TTICE 11 4.2. Domain-w all fermion Dirac op erators. Let X b e a closed orien ted ev en dimensional Riemannian manifold. Let S b e a Z 2 -graded Hermitian v ector bundle on X with the Z 2 -grading op erator γ . Let D : C ∞ ( X ; S ) → C ∞ ( X ; S ) b e an o dd, ﬁrst-order, formally selfadjoin t, elliptic partial diﬀerential op erator. Let Y ⊂ X b e a separating submanifold that decomposes X in to the union of t w o compact manifolds X + and X − whic h share the common b oundary Y . Let κ : X → [ − 1 , 1] b e an L ∞ -function such that κ ≡ ± 1 on X ± \ Y . W e deﬁne the domain-w all fermion Dirac op erator by D − mκγ , where the mass parameter m takes a positive v alue. W e also introduce a one-parameter family κ t with t ∈ ( −∞ , + ∞ ) , suc h that κ t = ( κ ( t ≥ 1) − 1 ( t ≤ − 1) where we assume that κ t is smo oth with resp ect to t in the range I := [ − 1 , 1] . The asso ciated one-parameter family of the massiv e Dirac op erator is given by D − mκ t γ . Later assuming D − mκ t γ is in v ertible at t = ± 1 , w e consider the sp ectral ﬂo w sf [ D − mκ t γ ] whic h is an element of the K 1 ( I , ∂ I ) group, where ∂ I is the b oundary of I whic h consists of tw o end points {− 1 , 1 } . The deﬁnitions of the group K 1 ( I , ∂ I ) and the asso ciated spectral ﬂo w will b e giv en in the later sections. In physics, it is known that the domain-wall fermion Dirac op erator has the so-called “edge states”, whic h are lo w eigenstates whose amplitude is exp onentially lo calized at the domain-wall satisfying a diﬀeren t “b oundary condition” from the APS b oundary condition introduced in the previous section. In order to see this, let us consider R × Y instead of a closed manifold X , and a Hermitian vector bundle E on Y . W e denote the co ordinate of R b y u . Let ¯ S = C 2 ⊗ E b e a Z 2 -graded Hermitian vector bundle on R × Y with the Z 2 -grading op erator γ = Γ ⊗ id E . Let κ 0 : R → R b e a sign function suc h that κ 0 ( ± u ) = ± 1 for u > 0 . Let ¯ D : C ∞ ( R × Y ; ¯ S ) → C ∞ ( R × Y ; ¯ S ) b e an odd, ﬁrst-order, formally selfadjoin t, elliptic partial diﬀeren tial op erator. The corresp onding domain-w all fermion Dirac op erator, whic h is essentially selfadjoint on L 2 ( R × Y ; C 2 ⊗ E ) , tak es the form ¯ D − mκ 0 γ = c ⊗ ∂ u + ϵ ⊗ D Y − mκ 0 Γ ⊗ id E =  − mκ 0 id E ∂ u + D Y − ∂ u + D Y + mκ 0 id E  , where D Y : C ∞ ( Y ; E ) → C ∞ ( Y ; E ) is a formally selfadjoin t, elliptic partial diﬀeren- tial op erator. The edge-states are deﬁned as vectors in the in tersection of the k ernel CAPTURING THE APS INDEX FR OM THE LA TTICE 12 of c ⊗ ∂ u − mκ 0 Γ ⊗ id E and the eigenspace of ϵ ⊗ D Y . Since c ⊗ ∂ u − mκ 0 Γ ⊗ id E and ϵ ⊗ D Y an ti-comm ute, eac h edge-state has the form v − ⊗ ψ λ D Y exp( − m | u | ) , for an eigensection ψ λ D Y of D Y with the eigen v alue λ D Y and v − = (1 / √ 2 , − 1 / √ 2 ) T ( T denotes the transp ose). The L 2 -condition for edge-state implies ϵv − = − v − . It is also imp ortan t to note that the condition ϵv − = − v − is diﬀerent from the APS b oundary condition. 4.3. The domain-wall Dirac op erators and the APS index. In this subsection, w e discuss a nontrivial relation b etw een the domain-w all fermion Dirac operator and the APS index pro v ed in [18]. W e consider the same X = X + ∪ X − , the Hermitian vector bundle S and the domain-w all fermion Dirac operator D − mκγ as in the previous subsection. Here as in the subsection 4.1, let us assume a collar neighbourho o d near the domain-w all: w e ass ume the product structure I ′ × Y , where the interv al I ′ is a double of I so that ( I ′ × Y ) ∩ X ± = I × Y . W e also assume that D Y has no zero eigenv alue. In [18], it w as pro ved that (1) Ind APS ( D | X + ) = − η ( D − mκγ ) − η ( D + mγ ) 2 , where η denotes the Atiy ah-Patodi-Singer η -inv arian t [9]. The pro of of the abov e equalit y was giv en b y considering a certain em b edding of (( −∞ , 0) × Y ) ∩ X + in to R × X where the coordinate for R is denoted b y t . W e pull back the bundle S on X to R × X , which will b e denoted b y the same sym b ol. W e in tro duce a selfadjoin t op erator b D m : L 2 ( R × X ; S ⊕ S ) → L 2 ( R × X ; S ⊕ S ) deﬁned by b D m := c ⊗ ∂ t + ϵ ⊗ ( D − mκ t γ ) =  0 D − mκ t γ + ∂ t D − mκ t γ − ∂ t 0  . Note that b D m is an odd operator with resp ect to the grading of S ⊕ S = C 2 ⊗ S giv en by Γ ⊗ id . In fact, the both sides of (1) corresp ond to diﬀeren t ev aluations of the same index of b D m . One ev aluation uses the lo calization of the zero eigensections whic h are lo calized in the neigh b ourho o d of the submanifold in X determined b y ˜ m ( · , t ) = 0 , whic h is diﬀeomorphic to (( −∞ , 0) × Y ) ∩ X + , when the mass parameter m is suﬃciently large. Then the pro duct form ula of the index [22] as well as [9, Prop osition 3.11] indicates that the index equals to the APS index on X + : Ind ( b D m ) = Ind APS ( D | X + ) . Another ev aluation employs the APS index theorem, where [9, Proposition 3.11] indicates that Ind ( b D m ) equals to the APS index on I × X with t wo b oundaries of the interv al I . Note that the constant term in the asymptotic expansion of the CAPTURING THE APS INDEX FR OM THE LA TTICE 13 heat k ernel v anishes on suc h an o dd-dimensional manifold as I × X . Therefore, the only b oundary η -inv arian ts con tributes to the index, and w e hav e Ind ( b D m ) = − η ( D − mκγ ) − η ( D + mγ ) 2 . In this work, we use third ev aluation using the sp ectral ﬂow. F ollowing the standard argument, we can sho w that dimKer ( b D m ˆ P ± ) with ˆ P ± = (1 ± Γ ⊗ id ) / 2 corresp onds to the n umber of crossing zero eigenstates of D − mκ t γ from neg- ativ e/p ositive to p ositive/negativ e, resp ectively . Th us w e ha v e the follo wing mo diﬁcation of the theorem in [18]. Theorem 3. Ind APS ( D | X + ) = sf [ D − mκ t γ ] holds, wher e sf denotes the sp e ctr al ﬂow deﬁne d later in Se ction 6. 4.4. Domain-w all Dirac op erators on a ﬂat torus. In this w ork, we consider a d -dimensional square lattice with perio dic or antiperio dic boundary conditions. Therefore, the manifold X w e consider in the con tin uum limit is limited to a ﬂat torus only . The domain-wall b et ween X + and X − can b e any d − 1 -dimensional curved submanifold of X . With this curved domain-wall, a spin connection and non trivial curv ature (whic h corresp onds to a gravitational bac kground) is induced as sho wn in [2, 3, 6, 7]. In our main theorem in which we compare the sp ectral ﬂo w of the lattice and contin uum domain-w all fermion Dirac op erators, w e do not require the collar-lik e structure near the domain-w all b etw een X + and X − . Therefore, it is not guaran teed that sf [ D − mκ t γ ] as w ell as its lattice version satisﬁes the standard form of the APS index theorem, whic h is a sum of the curv ature term and the η in v ariant of a b oundary Dirac op erator. Still we would lik e to call it the APS index in this article for simplicit y . 5. K -gr oups and unbounded self adjoint opera tors In this pap er, we compare the sp ectral ﬂo w of contin uum Dirac op erators with that of lattice Dirac op erators as elements of K -theory . F or this purp ose, it is desirable to hav e a deﬁnition of K -groups and K O -groups that allows one to treat un b ounded and b ounded selfadjoin t operators on an equal fo oting. Accordingly , in this section, w e presen t a self-con tained deﬁnition of K -groups and K O -groups of arbitrary degree using unbounded selfadjoint op erators, and pro v e that it is naturally isomorphic to the standard deﬁnition formulated in terms of b ounded selfadjoin t op erators. Although there are several p ossible choices of top ology on the space of un b ounded selfadjoin t op erators, w e adopt the Riesz top ology . Related w ork in this direction includes that of Lesch [32]. A detailed treatment of these topics will app ear in a forthcoming pap er. CAPTURING THE APS INDEX FR OM THE LA TTICE 14 W e note that neither the suspension isomorphism nor Bott p erio dicity is pro v ed in this section. In later sections, we use the susp ension isomorphism, e.g. K 0 ( pt ) ∼ = K 1 ([0 , 1] , { 0 , 1 } ) , via the isomorphism established here b y identifying it with the corresp onding suspension isomorphism in the standard K -theory or K O -theory . F or notational simplicit y , we explain only the deﬁnition of K -groups; the mo diﬁ- cation to K O -groups is straigh tforward. W e will construct tw o ab elian semigroups, for each p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . , K p,q bounded ( X , A ) and K p,q Riesz ( X , A ) for a compact Hausdorﬀ space X and a closed subset A ⊂ X . W e will show in Section 5.4 that both semigroups are in fact groups. K p,q bounded ( X , A ) , deﬁned using b ounded selfadjoint op erators, is naturally identiﬁed with the standard K n ( X , A ) for n = p − q . K p,q Riesz ( X , A ) is deﬁned using un b ounded selfadjoin t op erators equipp ed with the Riesz top ology . In Theorem 23, we will pro v e that K p,q bounded ( X , A ) and K p,q Riesz ( X , A ) are canonically isomorphic. 5.1. Deﬁnitions of K p,q • -co cycles. W e b egin by recalling the deﬁnition of the Riesz top ology . Let H b e a separable Hilb ert space o v er C . Let B ( H ) denote the space of b ounded op erators on H equipp ed with the norm top ology ∥·∥ op . Let B sa ( H ) ⊂ B ( H ) denote the space of b ounded selfadjoin t op erators on H . Let C sa ( H ) denote the space of densely deﬁned un b ounded selfadjoin t op erators on H . In this pap er, all un b ounded operators are assumed to b e densely deﬁned, and w e use the term “un b ounded op erator" to mean a possibly unbounded op erator. Moreo v er, the term “Hilb ert space" is understo o d to include ﬁnite-dimensional v ector spaces equipp ed with a Hermitian inner pro duct; the same conv ention applies to Hilb ert bundles. W e deﬁne the Riesz transform T Riesz : C sa ( H ) → B sa ( H ) via functional calculus asso ciated with the homeomorphism T Riesz : R → ( − 1 , 1) , λ 7→ λ √ 1 + λ 2 . The Riesz top ology on C sa ( H ) is deﬁned as the pullback of the norm topology on B sa ( H ) via T Riesz . W e note that h ∈ B sa ( H ) is in Im ( T Riesz ) if and only if ∥ h ∥ op ≤ 1 and Ker( h ± id H ) = { 0 } . Let p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . . The Cliﬀord algebra Cl q +1 ,p is the ∗ -algebra ov er R generated by { ϵ 0 , ϵ 1 , . . . , ϵ q , e 1 , e 2 , . . . , e p } CAPTURING THE APS INDEX FR OM THE LA TTICE 15 sub ject to the an ticomm utation relations { ϵ k , ϵ k ′ } = 2 δ k,k ′ , { e l , e l ′ } = − 2 δ l,l ′ , { ϵ k , e l } = 0 , ϵ ∗ k = ϵ k , e ∗ l = − e l for k , k ′ = 0 , 1 , . . . , q and l , l ′ = 1 , 2 , . . . , p . Note our conv en tion for p and q : q starts from − 1 , and the order of p and q in the Cliﬀord algebra Cl q +1 ,p is rev ersed. W e adopt the conv ention that, when q = − 1 , there are no generators ϵ k . W e denote b y Hom ∗ ( C 0 , C 1 ) the set of ∗ -homomorphisms b et w een ∗ -algebras C 0 and C 1 . Deﬁnition 4. W e deﬁne B p,q sa ( H ) :=  ( c, h ) ∈ Hom ∗  Cl q +1 ,p , B ( H )  × B sa ( H )   ♡  C p,q sa ( H ) :=  ( c, h ) ∈ Hom ∗  Cl q +1 ,p , B ( H )  × C sa ( H )   ♡  , where the condition ♡ means that { c ( ϵ k ) , h } = 0 , { c ( e l ) , h } = 0 for each k = 0 , 1 , . . . , q and l = 1 , 2 , . . . , p . Let X b e a compact Hausdorﬀ space. Let H → X b e a Hilb ert bundle o v er X with ﬁbre a separable Hilb ert space o v er C . W e assume throughout this pap er that the structure group of a Hilb ert bundle is equipp ed with the norm top ology . W e denote by H x the ﬁbre of H at x ∈ X . W e w ould like to consider contin uity for bundle maps betw een Hilb ert bundles; ho w ever, since we work in an un b ounded setting, w e deﬁne it carefully as follo ws. Let x 0 ∈ X . Let U b e an op en neigh b orho o d of x 0 and ϕ : U × H x 0 ∼ = H| U a local trivialisation of H o v er U . F or each x ∈ U , w e let ϕ ∗ x : B ( H x ) → B ( H x 0 ) b e the bijection induced b y ϕ . W e ﬁrst consider families of b ounded op erators. Let ( c, h ) =  { c x } x ∈ X , { h x } x ∈ X  b e a family of ( c x , h x ) ∈ B p,q sa ( H x ) parametrized b y x ∈ X . W e deﬁne a map F ϕ : U →  B ( H x 0 )  ( q +1)+ p +1 b y x 7→  ϕ ∗ x ( c x ( ϵ 0 )) , . . . , ϕ ∗ x ( c x ( ϵ q )) , ϕ ∗ x ( c x ( e 1 )) , . . . , ϕ ∗ x ( c x ( e p )) , ϕ ∗ x ( h x )  . Deﬁnition 5. Let ( c, h ) b e a family as ab o v e. (i) Let x 0 ∈ X . W e say that ( c, h ) is con tin uous at x 0 if there exist an op en neigh b orho o d U of x 0 and a lo cal trivialisation ϕ of H o v er U suc h that the map F ϕ : U →  B ( H x 0 )  ( q +1)+ p +1 deﬁned as ab o ve is con tinuous at x 0 with respect to the norm top ology on B ( H x 0 ) . Note that the con tin uity of F ϕ at x 0 do es not dep end on the c hoice of U and ϕ . CAPTURING THE APS INDEX FR OM THE LA TTICE 16 (ii) W e sa y that ( c, h ) is con tin uous if ( c, h ) is con tinuous at each x 0 ∈ X . W e next consider families of unbounded operators. Let ( c, h ) =  { c x } x ∈ X , { h x } x ∈ X  b e a family of ( c x , h x ) ∈ C p,q sa ( H x ) parametrized by x ∈ X . Recall that h x is un- b ounded. W e deﬁne a map F Riesz ϕ : U →  B ( H x 0 )  ( q +1)+ p +1 b y x 7→  ϕ ∗ x ( c x ( ϵ 0 )) , . . . , ϕ ∗ x ( c x ( ϵ q )) , ϕ ∗ x ( c x ( e 1 )) , . . . , ϕ ∗ x ( c x ( e p )) , ϕ ∗ x ( T Riesz ( h x ))  . Deﬁnition 6. Let ( c, h ) b e an unbounded family as ab o ve. (i) Let x 0 ∈ X . W e say that ( c, h ) is Riesz-contin uous at x 0 if there exist an op en neighborho o d U of x 0 and a lo cal trivialisation ϕ of H o v er U suc h that the map F Riesz ϕ : U →  B ( H x 0 )  ( q +1)+ p +1 deﬁned as ab o v e is con tin uous at x 0 with respect to the norm top ology on B ( H x 0 ) . Note that the con tin uit y of F Riesz ϕ at x 0 do es not depend on the c hoice of U and ϕ b ecause we equip the structure group of a Hilbert bundle with the norm top ology . (ii) W e sa y that ( c, h ) is Riesz-con tinuous if ( c, h ) is Riesz-con tinuous at each x 0 ∈ X . Ha ving established these preliminaries, w e no w introduce the follo wing spaces, whic h will b e used to deﬁne K -co cycles. Deﬁnition 7. Let X b e a compact Hausdorﬀ space and H → X a Hilbert bundle o v er X . (i) W e denote by B p,q sa ( H ) the set of all b ounded families ( c, h ) =  { c x } x ∈ X , { h x } x ∈ X  with ( c x , h x ) ∈ B p,q sa ( H x ) for eac h x ∈ X that are con tinuous in the sense deﬁned ab o v e. (ii) W e denote by C p,q sa ( H ) the set of all unbounded families ( c, h ) =  { c x } x ∈ X , { h x } x ∈ X  with ( c x , h x ) ∈ C p,q sa ( H x ) for each x ∈ X that are Riesz-contin uous in th e sense deﬁned ab o ve. W e deﬁne K -co cycles as follows. Deﬁnition 8. Let p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . . Let X b e a compact Hausdorﬀ space and A ⊂ X a closed subset. W e use the symbol • to stand for either “ Riesz ” or “ b ounded ”. W e sa y that a triple α = ( H , c, h ) is a K p,q • -co cycle or simply K -co cycle if the following four properties are satisﬁed: (i) H is a Hilb ert bundle o v er X with ﬁbre a separable Hilb ert space. (ii) If • = b ounded , then ( c, h ) ∈ B p,q sa ( H ) , and if • = Riesz , then ( c, h ) ∈ C p,q sa ( H ) . (iii) h x is F redholm for eac h x ∈ X . (iv) Ker h a = { 0 } for eac h a ∈ A . CAPTURING THE APS INDEX FR OM THE LA TTICE 17 W e write α ∈ K p,q • ( X , A ) if α is a K p,q • -co cycle. R emark 9 . Recall our con ven tion that the term “un b ounded op erator” is used to mean a p ossibly un b ounded operator. Accordingly , a family h = { h x } ma y consist of b ounded operators at some p oin ts and unbounded op erators at others, but this causes no diﬃculty for the notion of contin uit y deﬁned ab ov e. Recall also our con v ention that the term “Hilb ert space” is understo o d to include ﬁnite-dimensional v ector spaces equipp ed with a Hermitian inner pro duct. Consequently , the ﬁbres of a Hilb ert bundle ma y b e ﬁnite-dimensional or inﬁnite-dimensional, p ossibly v arying from one connected comp onen t to another. In the deﬁnition of the K -group, only the behaviour of the sp ectrum near zero is essen tial, and the F redholm condition precisely ensures that the sp ectrum near zero consists of isolated eigen v alues with ﬁnite multiplicit y . By con trast, the distinction b et ween b ounded and un b ounded op erators concerns the b eha viour of the sp ectrum at inﬁnit y , far aw ay from zero, and is therefore completely inessen tial from the viewp oin t of K -theory . R emark 10 . W e note that the collection of all K p,q • -co cycles does not form a set; th us, the expression α ∈ K p,q • ( X , A ) is just a notation, and K p,q • ( X , A ) is not a set but just a formal expression. W e will discuss set-theoretic issues in Section 5.2. W e regard a K -co cycle α = ( H , c, h ) as a Hilbert bundle equipp ed with additional structure. The notions of isomorphism, pullback, and direct sum for K -co cycles are deﬁned as follows. Deﬁnition 11. Let p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . . (i) Let X b e a compact Hausdorﬀ space and A ⊂ X a closed subset. Let α = ( H , c, h ) , α ′ = ( H ′ , c ′ , h ′ ) ∈ K p,q • ( X , A ) . An isomorphism ϕ : α → α ′ is a Hilb ert bundle isomorphism ϕ : H → H ′ suc h that c = ϕ ∗ c ′ and h = ϕ ∗ h ′ . If there exists such an isomorphism, we denote α ∼ = α ′ . (ii) Let X , Y b e compact Hausdorﬀ spaces and A ⊂ X and B ⊂ Y closed subsets. Let f : X → Y b e a contin uous map such that f ( A ) ⊂ B . F or α = ( H , c, h ) ∈ K p,q • ( Y , B ) , the pullback f ∗ α is deﬁned to b e the triple ( f ∗ H , f ∗ c, f ∗ h ) . Note that f ∗ α ∈ K p,q • ( X , A ) . (iii) Let X b e a compact Hausdorﬀ space and A, A ′ ⊂ X closed subsets. Let α = ( H , c, h ) ∈ K p,q • ( X , A ) and α ′ = ( H ′ , ′ c ′ , h ′ ) ∈ K p,q • ( X , A ′ ) . The direct sum α ⊕ α ′ is deﬁned to b e ( H ⊕ H ′ , c ⊕ c ′ , h ⊕ h ′ ) . Note that α ⊕ α ′ ∈ K p,q • ( X , A ∩ A ′ ) . 5.2. Set-theoretic techniques. If the collection of K p,q • -co cycles w ere a set, one could in tro duce an equiv alence relation on it and deﬁne K p,q • -groups as the corresp onding quotien t. Since this is not the case, w e pro ceed as follows. This is a standard and frequen tly used tec hnique. In thi s pap er, since w e restrict ourselv es to Hilb ert bundles whose ﬁbres are separable, w e do not use the notion of univ erses. Deﬁnition 12. Let X b e a compact Hausdorﬀ space and A ⊂ X a closed subset. CAPTURING THE APS INDEX FR OM THE LA TTICE 18 (i) Let S b e a set. W e denote b y Hilb ert ( S ) the set of all Hilb ert bundles ov er X with ﬁbre a separable Hilb ert space whose total spaces are subsets of S . (ii) Let S b e a set. W e denote by K p,q • ,S ( X , A ) a set { ( H , c, h ) ∈ K p,q • ( X , A ) | H ∈ Hilb ert( S ) } . (iii) W e denote by ∼ = S the relation on the set K p,q • ,S ( X , A ) obtained b y restricting the isomorphism ∼ = of K p,q • -co cycles. Note that ∼ = S is an equiv alence relation. F or α S ∈ K p,q • ,S ( X , A ) , we denote its equiv alence class by ( α S mo d ∼ = S ) . W e denote by | S | the cardinality of a set S and by |H| the cardinality of the total space of a Hilb ert bundle H . W e denote by ℓ 2 ( Z ) the standard separable Hilb ert space whose standard basis is parametrized b y Z . Lemma 13. L et X b e a c omp act Hausdorﬀ sp ac e and A, A ′ ⊂ X b e close d subsets. L et S and T b e sets such that | S | , | T | ≥ | X × ℓ 2 ( Z ) | . (i) F or any α ∈ K p,q • ( X , A ) , ther e exists α S ∈ K p,q • ,S ( X , A ) such that α S ∼ = α . Mor e over, the e quivalenc e class ( α S mo d ∼ = S ) ∈  K p,q • ,S ( X , A ) / ∼ = S  dep ends only on α and S . (ii) Ther e exists a unique map ⊕ S :  K p,q • ,S ( X , A ) / ∼ = S  ×  K p,q • ,S ( X , A ′ ) / ∼ = S  →  K p,q • ,S ( X , A ∩ A ′ ) / ∼ = S  such that, for any α S ∈ K p,q • ,S ( X , A ) , β S ∈ K p,q • ,S ( X , A ′ ) , and γ S ∈ K p,q • ,S ( X , A ∩ A ′ ) , we have ( α S mo d ∼ = S ) ⊕ S ( β S mo d ∼ = S ) = ( γ S mo d ∼ = S ) if and only if α S ⊕ β S ∼ = γ S as K p,q • -c o cycles. (iii) Ther e exists a unique bije ction Φ T ,S :  K p,q • ,S ( X , A ) / ∼ = S  →  K p,q • ,T ( X , A ) / ∼ = T  such that, for any α S ∈ K p,q • ,S ( X , A ) and α T ∈ K p,q • ,T ( X , A ) , we have Φ T ,S  ( α S mo d ∼ = S )  = ( α T mo d ∼ = T ) if and only if α S ∼ = α T as K p,q • -c o cycles. (iv) ⊕ S and ⊕ T ar e c omp atible with Φ T ,S , that is, we have Φ T ,S  ( α S mo d ∼ = S ) ⊕ S ( β S mo d ∼ = S )  =  Φ T ,S  ( α S mo d ∼ = S )  ⊕ T  Φ T ,S  ( β S mo d ∼ = S )  for any α S ∈ K p,q • ,S ( X , A ) and β S ∈ K p,q • ,S ( X , A ′ ) . Pr o of. Let α ∈ K p,q • ( X , A ) . Since | S | ≥ | X × ℓ 2 ( Z ) | = |H| , there exists an injection from the total space of H in to S . This implies the ﬁrst assertion. The remaining assertions follow immediately from the ﬁrst one. □ CAPTURING THE APS INDEX FR OM THE LA TTICE 19 The existence of the bijection Φ T ,S in the third assertion of Lemma 13 suggests, at an informal lev el, that the set  K p,q • ,S ( X , A ) / ∼ = S  do es not dep end on the c hoice of S pro vided that | S | ≥ | X × ℓ 2 ( Z ) | . T o mak e this observ ation precise, w e ﬁx such a set S for eac h X as follo ws. Deﬁnition 14. Let X b e a compact Hausdorﬀ space and A, A ′ ⊂ X closed subsets. Set S X := X × ℓ 2 ( Z ) . (i) W e deﬁne  K p,q • ( X , A ) / ∼ =  :=  K p,q • ,S X ( X , A ) / ∼ = S X  . Note that  K p,q • ( X , A ) / ∼ =  is just a formal expression. (ii) W e simply write ⊕ for ⊕ S X , that is, ⊕ := ⊕ S X :  K p,q • ( X , A ) / ∼ =  ×  K p,q • ( X , A ′ ) / ∼ =  →  K p,q • ( X , A ∩ A ′ ) / ∼ =  . Deﬁnition 15. Let X , Y b e compact Hausdorﬀ spaces and A ⊂ X and B ⊂ Y closed subsets. Let f : X → Y b e a contin uous map such that f ( A ) ⊂ B . W e deﬁne a map f ∗ :  K p,q • ( Y , B ) / ∼ =  →  K p,q • ( X , A ) / ∼ =  b y the pullback deﬁned in Deﬁnition 11, which is w ell deﬁned by Lemma 13. This map f ∗ is characterized by the property that f ∗ ( β mo d ∼ = S Y ) = ( α mo d ∼ = S X ) if and only if f ∗ β ∼ = α for an y α ∈ K p,q • ( X , A ) and β ∈ K p,q • ( Y , B ) . F or α ∈ K p,q • ( X , A ) , by the ﬁrst assertion of Lemma 13, the equiv alence class α S mo d ∼ = S dep ends only on α ; thus, b y a slight abuse of language, we also write ( α mo d ∼ = ) := ( α S mo d ∼ = S ) ∈  K p,q • ,S ( X , A ) / ∼ = S  , although, strictly speaking, ( α mo d ∼ = ) is just a formal expression. Then, w e can rephrase the deﬁnitions ab o v e simply as follo ws: • W e deﬁne ( α mo d ∼ = ) ⊕ ( α ′ mo d ∼ = ) := ( α ⊕ α ′ mo d ∼ = ) ∈  K p,q • ( X , A ∩ A ′ ) / ∼ =  for any α ∈ K p,q • ( X , A ) and α ′ ∈ K p,q • ( X , A ′ ) . • W e deﬁne f ∗ ( β mo d ∼ = ) :=  ( f ∗ β ) mo d ∼ =  ∈  K p,q • ( X , A ) / ∼ =  for any β ∈ K p,q • ( Y , B ) . CAPTURING THE APS INDEX FR OM THE LA TTICE 20 5.3. Deﬁnition of the semigroup K p,q • . With these preparations in place, we are ﬁnally in a p osition to deﬁne the semigroup K p,q • . Let X b e a compact Hausdorﬀ space and A ⊂ X a closed subset. Let i 0 : X → X × [0 , 1] and i 1 : X → X × [0 , 1] b e giv en by i t ( x ) = ( x, t ) for t = 0 , 1 . W e use the symbol • to stand for either “ Riesz ” or “ b ounded ”. Let p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . . W e b egin b y deﬁning a relation ∼ on the set  K p,q • ( X , A ) / ∼ =  . F or α 0 , α 1 ∈ K p,q • ( X , A ) , we say that ( α 0 mo d ∼ = ) ∼ ( α 1 mo d ∼ = ) if and only if there exist β 0 , β 1 ∈ K p,q • ( X , X ) and e α ∈ K p,q • ( X × [0 , 1] , A × [0 , 1]) suc h that ( α 0 mo d ∼ = ) ⊕ ( β 0 mo d ∼ = ) = i ∗ 0 ( e α mo d ∼ = ) ( α 1 mo d ∼ = ) ⊕ ( β 1 mo d ∼ = ) = i ∗ 1 ( e α mo d ∼ = ) . It can be sho wn by standard argumen ts that ∼ is an equiv alence relation. Using ∼ , we deﬁne the semigroup K p,q • as follows. Deﬁnition 16. W e deﬁne K p,q • ( X , A ) :=  K p,q • ( X , A ) / ∼ =  / ∼ , and we write [ α ] := ( α mo d ∼ = ) / ∼ for α ∈ K p,q • ( X , A ) . By deﬁnition, ev ery element of K p,q • ( X , A ) is of the form [ α ] for some α ∈ K p,q • ( X , A ) . Moreo v er, for α 0 , α 1 ∈ K p,q • ( X , A ) , we hav e [ α 0 ] = [ α 1 ] ∈ K p,q • ( X , A ) if and only if there exist β 0 , β 1 ∈ K p,q • ( X , X ) and e α ∈ K p,q • ( X × [0 , 1] , A × [0 , 1]) suc h that α 0 ⊕ β 0 ∼ = i ∗ 0 e α and α 1 ⊕ β 1 ∼ = i ∗ 1 e α . By abuse of language, we call α t ⊕ β t a stabilization of α t for t = 0 , 1 , and we sa y that i ∗ 0 e α and i ∗ 1 e α are homotopic. With this con v en tion, for α 0 , α 1 ∈ K p,q • ( X , A ) , w e hav e [ α 0 ] = [ α 1 ] if and only if they are homotopic after stabilization, in a w a y analogous to the standard construction. The direct sum ⊕ on  K p,q • ( X , A ) / ∼ =  is clearly compatible with the equiv alence relation ∼ , and hence induces a comm utative semigroup structure + on K p,q • ( X , A ) b y [ α ] + [ α ′ ] := [ α ⊕ α ′ ] for any α, α ′ ∈ K p,q • ( X , A ) . The identit y is describ ed as follo ws. Let O X b e the trivial pro duct Hilb ert bundle X × { 0 } . F or ( H , c, h ) ∈ K p,q • ( X , A ) , if H = O X , then c and h are uniquely determined. W e hav e [ α ] + [ O X , c, h ] = [ α ] ∈ K p,q • ( X , A ) for any α ∈ K p,q • ( X , A ) . W e write 0 for [ O X , c, h ] . CAPTURING THE APS INDEX FR OM THE LA TTICE 21 5.4. Group structure on the semigroup K p,q • . Let X b e a compact Hausdorﬀ space and A ⊂ X a closed subset. In this subsection, we sho w that the ab elian semigroup K p,q • ( X , A ) deﬁned so far is in fact a group. Let p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . . Recall that the Cliﬀord algebra Cl q +1 ,p is the ∗ -algebra o ver R generated b y { ϵ 0 , . . . , ϵ q , e 1 , . . . , e p } sub ject to th e relations { ϵ k , ϵ k ′ } = 2 δ k,k ′ , { e l , e l ′ } = − 2 δ l,l ′ , { ϵ k , e l } = 0 , ϵ ∗ k = ϵ k , e ∗ l = − e l . W e deﬁne a graded inv olution Γ : Cl q +1 ,p → Cl q +1 ,p b y Γ( ϵ k ) = − ϵ k , Γ( e l ) = − e l for k = 0 , . . . , q and l = 1 , . . . , p . F or a ∗ -algebra B and a ∗ -homomorphism c : Cl q +1 ,p → B , w e deﬁne − c := c ◦ Γ ; that is, − c is a ∗ -homomorphism c haracterized b y ( − c )( ϵ k ) := −  c ( ϵ k )  , ( − c )( e l ) := −  c ( e l )  for k = 0 , . . . , q and l = 1 , . . . , p . Note that − c is not the p oint wise negativ e of c ; rather, the sign dep ends on the grading. W e deﬁne − α := ( H , − c, − h ) ∈ K p,q • ( X , A ) for α = ( H , c, h ) ∈ K p,q • ( X , A ) . The following prop osition is not only a basic prop ert y of K p,q • ( X , A ) but also serv es as a protot yp e for Prop osition 21, the k ey argumen t of this pap er. A ccordingly , w e include a somewhat detailed explanation. Suc h a complicated argumen t is required b ecause the seemingly trivial con tinuit y of the addition op erator C sa × B sa → C sa , ( A, B ) 7→ A + B b ecomes unexpectedly delicate when the space of un b ounded selfadjoin t op erators C sa is equipp ed with the Riesz top ology , owing to the nonlinearity of the Riesz transform. By con trast, arguments are m uc h simpler if one works with the gap top ology . Ho wev er, w orking with the gap top ology would b e somewhat excessiv e for our purp oses, and therefore we choose in this pap er to w ork with the more familiar Riesz top ology . Prop osition 17. W e have [ α ] + [ − α ] = 0 in K p,q • ( X , A ) for any α ∈ K p,q • ( X , A ) . In p articular, K p,q • ( X , A ) is a gr oup. Pr o of. Let α = ( H , c, h ) ∈ K p,q • ( X , A ) . W e denote b y π : X × [0 , 1] → X the pro jection on to X . W e deﬁne the triple e α = ( e H , e c, e h ) on X × [0 , 1] by setting e H := π ∗ H ⊗ R R 2 , e c ( x,t ) =  c x 0 0 − c x  , e h ( x,t ) =  h x t t − h x  for ( x, t ) ∈ X × [0 , 1] . W e ﬁrst sho w that e α ∈ K p,q • ( X × [0 , 1] , A × [0 , 1]) ; that is, in either case • = b ounded or Riesz , we v erify that e c and e h dep end contin uously CAPTURING THE APS INDEX FR OM THE LA TTICE 22 on ( x, t ) ∈ X × [0 , 1] , that e h ( x,t ) is F redholm for all ( x, t ) ∈ X × [0 , 1] and that Ker e h ( x,t ) = { 0 } for eac h ( x, t ) ∈ A × [0 , 1] . The con tinuit y of e c follo ws directly from that of c . W e sho w the con tin uity of e h as follows: • = b ounded case: The con tin uit y of e h follows directly from that of h . • = Riesz case: Recall that the Riesz transform is deﬁned via T Riesz ( λ ) = λ/ √ 1 + λ 2 . W e ha ve T Riesz ( e h ( x,t ) ) = e h ( x,t ) q 1 + e h 2 ( x,t ) = 1 p 1 + t 2 + h 2 x  h x t t − h x  for each ( x, t ) ∈ X × [0 , 1] . Then, w e ha v e, as λ → ±∞ , 1 √ 1 + t 2 + λ 2  λ t t − λ  − →  ± 1 0 0 ∓ 1  uniformly with respect to t ∈ [0 , 1] . Therefore, there exists a contin uous map ϕ : [ − 1 , 1] × [0 , 1] → Mat 2 ( R ) such that 1 √ 1 + t 2 + λ 2  λ t t − λ  = ϕ  T Riesz ( λ ) , t  for each ( λ, t ) ∈ R × [0 , 1] . Consequently , w e ha ve T Riesz ( e h ( x,t ) ) = ϕ  T Riesz ( h x ) , t  for eac h ( x, t ) ∈ X × [0 , 1] . Th us, e h dep ends contin uously on ( x, t ) ∈ X × [0 , 1] . W e next sho w that e h ( x,t ) is F redholm for ( x, t ) ∈ X × [0 , 1] and that Ker e h ( x,t ) = { 0 } for ( x, t ) ∈ A × [0 , 1] . Fix ( x, t ) ∈ X × [0 , 1] . F or t  = 0 , w e ha ve e h 2 ( x,t ) =  h 2 x + t 2 0 0 h 2 x + t 2  > 0 . Hence, e h ( x,t ) is inv ertible. In particular, e h ( x,t ) is F redholm. F or t = 0 , we hav e e h ( x, 0) = h x ⊕ ( − h x ) so the F redholm prop ert y of e h ( x, 0) and Ker e h ( x, 0) = { 0 } for x ∈ A follow. No w w e deﬁne α 0 := α ⊕ ( − α ) ∈ K p,q • ( X , A ) , α 1 := 0 ∈ K p,q • ( X , A ) , β 0 := 0 ∈ K p,q • ( X , X ) , β 1 := ( H ⊗ R R 2 , c 1 , h 1 ) ∈ K p,q • ( X , X ) , where ( c 1 ) x :=  c x 0 0 c x  , ( h 1 ) x :=  h x 1 1 − h x  CAPTURING THE APS INDEX FR OM THE LA TTICE 23 for each x ∈ X . Then, w e hav e α 0 ⊕ β 0 ∼ = i ∗ 0 e α and α 1 ⊕ β 1 ∼ = i ∗ 1 e α. Consequen tly , w e hav e [ α ⊕ ( − α )] = [0] . Th us, we hav e pro v ed that [ α ] + [ − α ] = 0 . □ 5.5. Degrees. In this subsection, w e explain how to assign degrees to K -groups, thereb y completing their deﬁnition. Prop osition 18. L et X b e a c omp act Hausdorﬀ sp ac e and A ⊂ X a close d subset. Ther e is a natur al isomorphism K p,q • ( X , A ) → K p +1 ,q +1 • ( X , A ) for e ach p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . . Pr o of. W e deﬁne ϵ R 2 , ϵ ′ R 2 , e R 2 ∈ Mat 2 ( R ) by ϵ R 2 :=  1 0 0 1  , ϵ ′ R 2 :=  0 1 1 0  , e R 2 :=  0 − 1 1 0  . Then, ϵ 2 R 2 = ( ϵ ′ R 2 ) 2 = 1 and e 2 R 2 = − 1 . W e ﬁrst deﬁne a homomorphism δ : K p,q • ( X , A ) → K p +1 ,q +1 • ( X , A ) . as follows. F or ( H , c, h ) ∈ K p,q • ( X , A ) , set δ ([( H , c, h )]) =: [( H ′ , c ′ , h ′ )] , where H ′ := R 2 ⊗ R H , c ′ ( ϵ k ) := ϵ R 2 ⊗ c ( ϵ k ) , c ′ ( ϵ q +1 ) := ϵ ′ R 2 ⊗ id c ′ ( e l ) := ϵ R 2 ⊗ c ( e l ) , c ′ ( e p +1 ) := e R 2 ⊗ id , h ′ := ϵ R 2 ⊗ h for k = 0 , 1 , . . . , q and l = 1 , 2 , . . . , p . W e next deﬁne a homomorphism ρ : K p +1 ,q +1 • ( X , A ) → K p,q • ( X , A ) . as follo ws. F or ( H ′ , c ′ , h ′ ) ∈ K p +1 ,q +1 • ( X , A ) , set ρ ([( H ′ , c ′ , h ′ )]) =: [( H , c, h )] , where H := Ker  c ′ ( ϵ q +1 ) c ′ ( e p +1 ) − id ⊗ id  ⊂ H ′ , c ( ϵ k ) := c ′ ( ϵ k ) | H , c ( e l ) := c ′ ( e l ) | H , h := h ′ | H for k = 0 , 1 , . . . , q and l = 1 , 2 , . . . , p . Then, one can chec k that δ and ρ are m utually in v erse. □ This prop osition gives rise to a direct system . . . ∼ = → K p,q • ( X , A ) ∼ = → K p +1 ,q +1 • ( X , A ) ∼ = → K p +2 ,q +2 • ( X , A ) ∼ = → . . . and, using this system, w e ﬁnally deﬁne the K -groups. CAPTURING THE APS INDEX FR OM THE LA TTICE 24 Deﬁnition 19. Let X b e a compact Hausdorﬀ space and A ⊂ X a closed subset. W e deﬁne the abelian group K n • ( X , A ) as the direct limit K n • ( X , A ) := lim − →  · · · → K p,q • ( X , A ) → K p +1 ,q +1 • ( X , A ) → · · ·  p − q = n for n ∈ Z . 5.6. Riesz contin uous families. In this subsection, we presen t a criterion for Riesz con tinuit y of families of unbounded selfadjoin t operators, whic h will be used later. W e deﬁne the Riesz-contin uity of a family of un b ounded selfadjoin t operators as follo ws. See Deﬁnition 6. Let H b e a Hilb ert bundle o ver a compact Hausdorﬀ space X . Let x 0 ∈ X , and let U b e an op en neigh b ourho o d of x 0 and ϕ : U × H x 0 ∼ = H| U a lo cal trivialisation of H ov er U . F or eac h x ∈ U , w e let ϕ ∗ x : C sa ( H x ) → C sa ( H x 0 ) b e the bijection induced by ϕ . Let h = { h x } x ∈ X b e a family of h x ∈ C sa ( H x ) parametrized by x ∈ X . W e deﬁne a map F Riesz ϕ : U → B ( H x 0 ) by x 7→ T Riesz ( ϕ ∗ x h x ) . W e say that h is Riesz-con tin uous at x 0 if F Riesz ϕ is con tin uous at x 0 with resp ect to the norm top ology on B ( H x 0 ) , and that h is Riesz-con tinuous if h is Riesz-con tinuous at each x 0 ∈ X . The following prop osition was sho wn implicitly b y Lesch [32, PROPOSITION 2.2] using the theory of op erator-monotonic increasing functions, the key p oint b eing the operator monotonicit y of the square root t 7→ √ t . W e give an alternative argumen t based on the theory of op er ator-Lipschitz functions. The k ey observ ation here is that the Riesz transform is op erator-Lipsc hitz. F or the theory of operator-Lipschitz functions, w e refer the reader to the surv ey pap er [38] and the references therein; for the reader’s con v enience, we brieﬂy collect here the results that are needed in the sequel. A function f : R → R is called op erator-Lipsc hitz if there exists a constan t C > 0 suc h that ∥ f ( A ) − f ( B ) ∥ op ≤ C ∥ A − B ∥ op for any (p ossibly unbounded) selfadjoint op erator A and B on a Hilb ert space for which A − B is b ounded. By [38, Theorem 1.4.4], a function f : R → R is op erator-Lipsc hitz if f b elongs to the Besov space B 1 ∞ , 1 ( R ) . The theory of Beso v spaces is also neatly summarized in [38, Section 2.1]. No w T Riesz ∈ B 1 ∞ , 1 ( R ) . Th us, T Riesz is op erator-Lipsc hitz. Prop osition 20. L et X b e a c omp act Hausdorﬀ sp ac e. L et H b e a Hilb ert bund le over X . L et h = { h x } x ∈ X b e a family of unb ounde d selfadjoint op er ators h x ∈ C sa ( H x ) p ar ametrize d by x ∈ X . W e assume that, for e ach x 0 ∈ X , ther e exist an op en neighb ourho o d U of x 0 and ϕ : U × H x 0 → H | U such that ϕ = id at x 0 , that CAPTURING THE APS INDEX FR OM THE LA TTICE 25 the op er ator ϕ ∗ x h x − h x 0 extends to a b ounde d op er ator m x ∈ B ( H x 0 ) for e ach x ∈ U , and that the op er ator-value d map U ∋ x 7→ m x ∈ B ( H x 0 ) is c ontinuous with r esp e ct to the norm top olo gy on B ( H x 0 ) . Then, the family { h x } is c ontinuous with r esp e ct to the Riesz top olo gy. Pr o of. Fix x 0 ∈ X and an op en neighbourho o d U of x 0 as in the assumption of the prop osition. Since T Riesz is op erator-Lipsc hitz, there exists a constan t C > 0 such that (2) ∥ T Riesz ( ϕ ∗ x h x ) − T Riesz ( h x 0 ) ∥ op ≤ C ∥ ϕ ∗ x h x − h x 0 ∥ op = C ∥ m x ∥ op for any x ∈ U . Since ϕ = id at x 0 , we ha v e m x 0 = 0 . Moreov er, x 7→ m x is norm- con tin uous at x 0 b y assumption. Thus, b y the abov e inequality , x 7→ T Riesz ( ϕ ∗ x h x ) is norm-contin uous at x 0 . Consequen tly , the family { h x } is Riesz-con tin uous. □ 5.7. The inv erse element. In this subsection, w e explain a method for sho wing that [ α ⊕ ( − α )] = 0 in the K -group for a K -co cycle α . W e ﬁrst ﬁx the notation. F or a (p ossibly unbounded) selfadjoin t operator h on a Hilb ert space H , w e write its sp ectral decomp osition as h = Z R λ dE h ( λ ) , where E h denotes the sp ectral measure asso ciated with h . W e denote b y σ ( h ) the sp ectrum of h . W e also deﬁne H h< − λ 0 := Im E h  { λ ∈ R | λ < − λ 0 }  , H h>λ 0 := Im E h  { λ ∈ R | λ > λ 0 }  , H | h | <λ 0 := Im E h  { λ ∈ R | | λ | < λ 0 }  , H | h | >λ 0 := Im E h  { λ ∈ R | | λ | > λ 0 }  for λ 0 > 0 . W e adopt the same notation for families of selfadjoint op erators on Hilb ert bundles. Let H b e a Hilb ert bundle o ver a compact Hausdorﬀ space X and h b e a Riesz- con tin uous family of un b ounded selfadjoin t operators. Fix x 0 ∈ X . F or an y Λ > 0 , there exists λ 0 ∈ (0 , Λ) such that ± λ 0 / ∈ σ ( h x 0 ) . Fix such a λ 0 . Then, there exists an op en neigh b ourho o d U 0 of x 0 suc h that ± λ 0 / ∈ σ ( h x ) for an y x ∈ U 0 . Th us, we obtain an orthogonal decomp osition H| U 0 =  H| U 0  h< − λ 0 ⊕  H| U 0  | h | <λ 0 ⊕  H| U 0  h>λ 0 b y the Riesz con tinuit y of h . The follo wing prop osition contains the key argument of this section; its protot yp e app ears as Prop osition 17. Prop osition 21. L et X b e a c omp act Hausdorﬀ sp ac e and A ⊂ X a close d subset. L et p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . . L et α = ( H , c, h ) , α ′ = ( H ′ , c ′ , h ′ ) ∈ CAPTURING THE APS INDEX FR OM THE LA TTICE 26 K p,q • ( X , A ) . Supp ose that ther e exist λ 0 > 0 and a family f = { f x } x ∈ X of b ounde d op er ators f x : H x → H ′ x p ar ametrize d by x ∈ X that satisfy the fol lowing c onditions: (i) F or e ach x ∈ X , the sp e ctr a ( σ ( h x ) ∩ [ − λ 0 , λ 0 ]) and ( σ ( h ′ x ) ∩ [ − λ 0 , λ 0 ]) c onsist only of isolate d eigenvalues with ﬁnite multiplicity. (ii) The family x 7→ f x is c ontinuous with r esp e ct to the norm top olo gy. (iii) F or e ach x ∈ X , we have f x ◦ c x ( g ) = c ′ x ( g ) ◦ f x for g ∈ Cl q +1 ,p . (iv) F or e ach x ∈ X , we have f x ◦ h x = h ′ x ◦ f x . (v) F or e ach x ∈ X , the c omp osition  E h ′ x ([ − λ 0 , λ 0 ])  ◦ f x ◦ ( E h x ([ − λ 0 , λ 0 ])) : Im E h x ([ − λ 0 , λ 0 ]) → Im E h ′ x ([ − λ 0 , λ 0 ]) is a unitary isomorphism. Then, we have [ α ′ ⊕ ( − α )] = 0 ∈ K p,q • ( X , A ) . In p articular, [ α ′ ] = [ α ] ∈ K p,q • ( X , A ) . Pr o of. Let α = ( H , c, h ) , α ′ = ( H ′ , c ′ , h ′ ) ∈ K p,q • ( X , A ) satisfy the ab ov e assump- tions. W e will sho w that α ′ ⊕ ( − α ) ∼ 0 . Let ξ λ 0 : R → R be an even contin uous function (see Figure 2) deﬁned as ξ λ 0 =      (1 / 3) λ 0 if | λ | ≤ (1 / 3) λ 0 (2 / 3) λ 0 − | λ | if (1 / 3) λ 0 < | λ | < (2 / 3) λ 0 0 if | λ | ≥ (2 / 3) λ 0 for λ ∈ R , and set ( ψ h,h ′ ,λ 0 ) x := ξ λ 0 ( h ′ x ) ◦ f x ◦ ξ λ 0 ( h x ) : H x → H ′ x for eac h x ∈ X . Then, ( ψ h,h ′ ,λ 0 ) x is a ﬁnite-rank op erator and ∥ ( ψ h,h ′ ,λ 0 ) x ∥ op ≤ 1 / 3 . 0 1/3 2/3 1 -1 -2/3 -1/3 1/3 2/3 1 λ / λ 0 ξ λ 0 ( λ )/ λ 0 Figure 2. The function ξ λ 0 ( λ ) normalized b y λ 0 . CAPTURING THE APS INDEX FR OM THE LA TTICE 27 W e deﬁne the triple e α = ( e H , e c, e h ) on X × [0 , 1] by setting e H := π ∗ H ′ ⊕ π ∗ H , e c ( x,t ) =  c ′ x 0 0 − c x  , e h ( x,t ) =  h ′ x t ( ψ h,h ′ ,λ 0 ) x t ( ψ h,h ′ ,λ 0 ) ∗ x − h x  for ( x, t ) ∈ X × [0 , 1] , where π : X × [0 , 1] → X is the pro jection on to X . W e ﬁrst sho w that e α ∈ K p,q • ( X × [0 , 1] , A × [0 , 1]) ; that is, in either case • = b ounded or Riesz , w e v erify that e c and e h dep end con tin uously on ( x, t ) ∈ X × [0 , 1] and that e h ( x,t ) is F redholm at eac h ( x, t ) ∈ X × [0 , 1] . The con tinuit y of e c follo ws directly from that of c . W e sho w the con tin uity of e h as follows: • = b ounded case: The contin uit y of e h follo ws from those of h and t ( ψ h,h ′ ,λ 0 ) . • = Riesz case: Fix ( x 0 , t 0 ) ∈ X × [0 , 1] . Let λ 1 b e a constan t that satisﬁes 2 3 λ 0 < λ 1 < λ 0 , ± λ 1 / ∈ σ ( h x 0 ) , and ± λ 1 / ∈ σ ( h ′ x 0 ) F or t = 0 , the Riesz-contin uit y of the family ( x, 0) 7→ e h ( x, 0) follo ws directly from that of x 7→ h ′ x and x 7→ − h x . Therefore, we ha ve an orthogonal decomp osition H| U 0 × [0 , 1] =  H| U 0 × [0 , 1]  e h ( x, 0) < − λ 1 ⊕  H| U 0 × [0 , 1]  | e h ( x, 0) | <λ 1 ⊕  H| U 0 × [0 , 1]  e h ( x, 0) >λ 1 . By the c hoice of ξ λ 0 and λ 1 , for t > 0 , the diﬀ erence e h ( x,t ) − e h ( x, 0) pre- serv es the ab o v e decomposition. Hence, the Riesz transform T Riesz ( e h ( x,t ) ) preserv es the decomp osition. Since its restriction on the second summand is a ﬁnite-rank op erator, it is con tin uous at ( x 0 , t 0 ) . Moreo v er, since the diﬀerence e h ( x,t ) − e h ( x, 0) acts trivially on the ﬁrst and the third summands, the con tin uity of the restriction follows from those of T Riesz ( h ′ ) and T Riesz ( h ) . Th us, e h is Riesz-con tin uous at ( x 0 , t 0 ) . W e next show that e h ( x 0 ,t 0 ) is F redholm for eac h ( x 0 , t 0 ) ∈ X × [0 , 1] . F or t 0 = 0 , the F redholm prop erty of e h ( x,t ) follo ws directly from that of h ′ x and h x . F or t 0  = 0 , w e consider the decomposition H ( x 0 ,t 0 ) =  H ( x 0 ,t 0 )  e h ( x 0 ,t 0 ) < − λ 1 ⊕  H ( x 0 ,t 0 )  | e h ( x 0 ,t 0 ) | <λ 1 ⊕  H ( x 0 ,t 0 )  e h ( x 0 ,t 0 ) >λ 1 . It suﬃces to v erify the F redholm prop ert y on each summand. On the ﬁrst and third summands, w e ha v e e h ( x 0 ,t 0 ) = h ′ x 0 ⊕ ( − h x 0 ) , whi c h is F redholm. The restriction to the second summand is a ﬁnite-rank op erator, hence is F redholm. W e next show that Ker e h ( x 0 ,t 0 ) = { 0 } for eac h ( x 0 , t 0 ) ∈ A × [0 , 1] . F or t 0 = 0 , w e ha ve e h x 0 , 0 = h ′ x 0 ⊕ ( − h x 0 ) , and, hence, Ker e h ( x 0 , 0) = { 0 } . Suppose that t 0  = 0 . CAPTURING THE APS INDEX FR OM THE LA TTICE 28 Let v ′ ⊕ v ∈ Ker e h ( x 0 ,t 0 ) . Then, we ha ve h ′ x 0 v ′ + t 0 ( ψ h,h ′ ,λ 0 ) x 0 v = 0 , t 0 ( ψ h,h ′ ,λ 0 ) ∗ x 0 v ′ − h x 0 v = 0 . Hence, we hav e ( h ′ x 0 ) 2 v ′ + t 0 h ′ x 0 ( ψ h,h ′ ,λ 0 ) x 0 v = 0 , t 0 ( ψ h,h ′ ,λ 0 ) x 0 ( ψ h,h ′ ,λ 0 ) ∗ x 0 v ′ − ( ψ h,h ′ ,λ 0 ) x 0 h x 0 v = 0 . Since ( ψ h,h ′ ,λ 0 ) x 0 ◦ h x 0 = h ′ x 0 ◦ ( ψ h,h ′ ,λ 0 ) x 0 , the second equation implies t 0 h ′ x 0 ( ψ h,h ′ ,λ 0 ) x 0 v = t 0 ( ψ h,h ′ ,λ 0 ) x 0 h x 0 v = t 2 0 ( ψ h,h ′ ,λ 0 ) x 0 ( ψ h,h ′ ,λ 0 ) ∗ x 0 v ′ . Hence, we hav e ( h ′ x 0 ) 2 v ′ + t 2 0 ( ψ h,h ′ ,λ 0 ) x 0 ( ψ h,h ′ ,λ 0 ) ∗ x 0 v ′ = 0 . Th us, w e ha v e h ′ x 0 v ′ = 0 and ( ψ h,h ′ ,λ 0 ) ∗ x 0 v ′ = 0 . Then, h ′ x 0 v ′ = 0 implies v ′ ∈ E h ′ x 0 ( { 0 } ) . On the other hand, since ( ψ h,h ′ ,λ 0 ) ∗ x 0 = ξ λ 0 ( h ) ◦ f ∗ x 0 ◦ ξ λ 0 ( h ′ ) and f ∗ is an isomorphism, ( ψ h,h ′ ,λ 0 ) ∗ x 0 v ′ = 0 implies v ′ ∈ E h ′ x 0  { λ ∈ R | ξ λ 0 = 0 }  . By the deﬁnition of ξ λ 0 , w e ha v e { 0 } ∩ { λ ∈ R | ξ λ 0 = 0 } = ∅ . Thus, v ′ ∈ E h ′ x 0 ( ∅ ) = { 0 } . In the same wa y , we hav e v = 0 . Consequen tly , we ha ve pro v ed that Ker e h ( x 0 ,t 0 ) = { 0 } . No w w e deﬁne α 0 := α ′ ⊕ ( − α ) ∈ K p,q • ( X , A ) , α 1 := 0 ∈ K p,q • ( X , A ) β 0 := 0 ∈ K p,q • ( X , X ) , β 1 := ( H ′ ⊕ H , c 1 , h 1 ) ∈ K p,q • ( X , X ) , where ( c 1 ) x :=  c ′ x 0 0 − c x  , ( h 1 ) x :=  h ′ x 1 1 − h x  for eac h x ∈ X . Then, we hav e α 0 ⊕ β 0 ∼ = i ∗ 0 e α and α 1 ⊕ β 1 ∼ = i ∗ 1 e α . Consequen tly , w e ha ve [ α ′ ⊕ ( − α )] = [0] . Thus, w e ha ve prov ed that [ α ] + [ − α ] = 0 . □ By combining Prop ositions 20 and Prop osition 21, w e obtain a criterion for t w o K -co cycles deﬁned b y un b ounded selfadjoint op erators to represent the same elemen t in the K -group. This criterion will b e used in the pro of of the main theorem (Theorem 30). Theorem 22. L et X b e a c omp act Hausdorﬀ sp ac e and A ⊂ X a close d subset. L et p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . . L et α = ( H , c, h ) , α ′ = ( H ′ , c ′ , h ′ ) ∈ K p,q Riesz ( X , A ) . L et f = { f x } x ∈ X b e a family of b ounde d op er ators f x : H x → H ′ x p ar ametrize d by x ∈ X . W e assume the fol lowing c onditions: CAPTURING THE APS INDEX FR OM THE LA TTICE 29 (i) F or e ach x 0 ∈ X , ther e exist an op en neighb ourho o d U of x 0 and ϕ : U × H x 0 → H| U such that ϕ = id at x 0 , that the op er ator ϕ ∗ x h x − h x 0 extends to a b ounde d op er ator m x ∈ B ( H x 0 ) for e ach x ∈ U . Mor e over, the op er ator- value d map U ∋ x 7→ m x ∈ B ( H x 0 ) is c ontinuous with r esp e ct to the norm top olo gy on B ( H x 0 ) . (ii) W e imp ose the same assumption on α ′ . (iii) F or e ach x ∈ X , we have f x ◦ c x ( g ) = c ′ x ( g ) ◦ f x for g ∈ Cl q +1 ,p . (iv) W e have Ker  h ′ x f x f ∗ x − h x  = 0 for e ach x ∈ X . Then, we have [ α ′ ⊕ ( − α )] = 0 ∈ K p,q • ( X , A ) . In p articular, [ α ′ ] = [ α ] ∈ K p,q • ( X , A ) . Pr o of. Let e H := ( H ⊕ H ′ ) × [0 , 1] . F or each ( x, t ) ∈ X × [0 , 1] , we set e c ( x,t ) := c ′ x ⊕ c x and e h ( x,t ) :=  h ′ x tf x tf ∗ x − h x  , and let e c = { e c ( x,t ) } ( x,t ) ∈ X × [0 , 1] and e h = { e h ( x,t ) } ( x,t ) ∈ X × [0 , 1] . By Prop osition 20, the family h is Riesz-contin uous. Hence, e α := ( e H , e c, e h ) ∈ K p,q Riesz ( X × [0 , 1] , A × [0 , 1]) . Then, using e α , w e can argue exactly as in the pro of of Proposition 21 to conclude that [ α ′ ⊕ ( − α )] = 0 ∈ K p,q • ( X , A ) . □ 5.8. Isomorphism K ∗ bounded ∼ = K ∗ Riesz . Let X b e a compact Hausdorﬀ space and A ⊂ X a closed subset. Let p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . . In this subsection, w e ﬁnally establish the isomorphism K p,q bounded ( X , A ) ∼ = K p,q Riesz ( X , A ) . W e note that, if f : R → R is a bounded con tin uous function and h = { h x } x ∈ X is a norm-con tin uous family of bounded selfadjoin t op erators h x ∈ B sa ( H x ) on a Hilb ert bundle H o v er X , then the family f ( h ) := { f ( h x ) } is also norm-contin uous. W e deﬁne a homomorphism τ bounded Riesz : K p,q bounded ( X , A ) → K p,q Riesz ( X , A ) b y simply sending [ α ] 7→ [ α ] for α ∈ K p,q bounded ( X , A ) , which is clearly well deﬁned. T o deﬁne the in v erse, w e in tro duce an auxiliary function T R → [-1/2 , 1/2] : R → R b y setting T R → [-1/2 , 1/2] ( λ ) :=      − 1 2 if λ ≤ − 1 2 λ if | λ | < 1 2 1 2 if λ ≥ 1 2 for λ ∈ R . This function is b ounded and contin uous. W e then deﬁne a homomor- phism τ Riesz bounded : K p,q Riesz ( X , A ) → K p,q bounded ( X , A ) CAPTURING THE APS INDEX FR OM THE LA TTICE 30 b y setting [( H ′ , c ′ , h ′ )] 7→ [( H ′ , c ′ , T R → [-1/2 , 1/2] ( h ′ ))] for ( H ′ , c ′ , h ′ ) ∈ K p,q Riesz ( X , A ) . The well-deﬁnedness of this homomorphism follows from the existence of a con tin uous function T [-1 , 1] → [-1/2 , 1/2] : [ − 1 , 1] → [ − 1 / 2 , 1 / 2] suc h that T R → [-1/2 , 1/2] = T [-1 , 1] → [-1/2 , 1/2] ◦ T Riesz . -1 -1/2 0 1/2 1 -1 -1/2 1/2 1 λ T ( λ ) Figure 3. The function T R → [ − 1 / 2 , 1 / 2] ( λ ) . The homomorphisms τ bounded Riesz and τ Riesz bounded are compatible with the direct sys- tems { K p,q • ( X , A ) → K p +1 ,q +1 • ( X , A ) } p − q = n . Th us, for each n ∈ Z , they induce homomorphisms τ bounded Riesz : K n bounded ( X , A ) → K n Riesz ( X , A ) τ Riesz bounded : K n Riesz ( X , A ) → K n bounded ( X , A ) , whic h w e denote by the same notation. Theorem 23. L et X b e a c omp act Hausdorﬀ sp ac e and A ⊂ X a close d subset. F or e ach n ∈ Z , the homomorphisms τ bounded Riesz and τ Riesz bounded b etwe en K n Riesz ( X , A ) and K n bounded ( X , A ) ar e mutual ly inverse. In p articular, we have a natur al isomorphism K n Riesz ( X , A ) ∼ = K n bounded ( X , A ) for al l n ∈ Z . The pro of relies on Prop osition 21, which provides the k ey argumen t in this pap er. Pr o of. Let n ∈ Z , and ﬁx p = 0 , 1 , 2 , . . . and q = − 1 , 0 , 1 , . . . suc h that p − q = n . W e ﬁrst pro ve that τ Riesz bounded ◦ τ bounded Riesz = id . Let α = ( H , c, h ) ∈ K p,q bounded ( X , A ) , and set α ′ := ( H , c, T R → [-1/2 , 1/2] ( h )) . Since X is compact and h is a con tin uous family , there exists λ 0 ∈ (0 , 1 / 2) such that ( σ ( h x ) ∩ [ − λ 0 , λ 0 ]) consists only of isolated CAPTURING THE APS INDEX FR OM THE LA TTICE 31 eigen v alues with ﬁnite multiplicit y for eac h x ∈ X . Fix suc h a λ 0 and let f := id . Then, by Prop osition 21, we obtain [ α ] = [ α ′ ] . W e next pro v e that τ bounded Riesz ◦ τ Riesz bounded = id . Let β := ( H ′ , c ′ , h ′ ) ∈ K p,q Riesz ( X , A ) , and set β ′ := ( H ′ , c ′ , T R → [-1/2 , 1/2] ( h ′ )) . Fix again λ 0 ∈ (0 , 1 / 2) such that ( σ ( h ′ x ) ∩ [ − λ 0 , λ 0 ]) consists only of isolated eigenv alues with ﬁnite m ultiplicit y for eac h x ∈ X , and let f := id . Then, b y Prop osition 21, w e obtain [ β ] = [ β ′ ] . □ This completes our construction of K -groups. 5.9. Gap top ology. In this ﬁnal subsection, although somewhat tangen tial to the main line of argumen t, w e state a prop osition that pla ys an essential role in deﬁning K -groups using the gap top ology . The use of the gap top ology will b e necessary when we generalize our main theorem (Theorem 30) to families. In Prop osition 21, the assumption that ϕ = id at x 0 w as imposed in order to ensure that m x 0 = 0 . How ever, when extending the argument to families, this assumption m ust b e remo ved. When working wi th the Riesz top ology , this seems diﬃcult b ecause the Riesz transform is highly nonlinear. W e begin b y recalling the deﬁnition of the gap top ology . Let H b e a separable Hilb ert space o v er C . Recall that C sa ( H ) and U ( H ) , B sa ( H ) ⊂ B ( H ) denote the spaces of un b ounded selfadjoin t op erators, unitary op erators, bounded selfadjoin t op erators, and b ounded operators on H resp ectiv ely . W e deﬁne the Cayley trans- form T Cayley : C sa ( H ) → U ( H ) via functional calculus associated with an injectiv e con tin uous map T Cayley : R → U (1) \ { 1 } , λ → λ − i λ + i . The gap topology on C sa ( H ) is deﬁned as the pullbac k of the norm top ology on U ( H ) ⊂ B ( H ) via T Cayley . W e note that Im T Cayley = { U ∈ U ( H ) | Ker ( U − id ) = { 0 }} . Moreo v er, we observe that T Cayley ( A ) − T Cayley ( B ) = A − i A + i − B − i B + i =  1 − 2 i A + i  −  1 − 2 i B + i  = − 2 i  1 A + i − 1 B + i  for A, B ∈ C sa ( H ) . Hence, the gap top ology is the w eakest top ology suc h that the maps A 7→ ( A ± i ) − 1 are con tinuous. W e also remark that the gap top ology is strictly weak er than the Riesz top ology . The follo wing prop osition should b e regarded as the counterpart, for the gap top ology , of inequalit y (2) , whic h has pla y ed a k ey role in the pro of of Riesz con tin uity and follows from the theory of op erator-Lipschitz functions. Prop osition 24. W e have ∥ T Cayley ( A + K ) − T Cayley ( B + L ) ∥ op ≤ 2 ∥ K − L ∥ op +(1+ ∥ L ∥ op ) 2 ∥ T Cayley ( A ) − T Cayley ( B ) ∥ op CAPTURING THE APS INDEX FR OM THE LA TTICE 32 for A, B ∈ C sa ( H ) and K , L ∈ B sa ( H ) . Pr o of. W e hav e T Cayley ( A + K ) − T Cayley ( B + L ) = 2 i  ( A + K + i ) − 1 − ( B + L + i ) − 1  = 2 i  ( A + K + i ) − 1 − ( A + L + i ) − 1 + ( A + L + i ) − 1 − ( B + L + i ) − 1  = 2 i  ( A + K + i ) − 1 − ( A + L + i ) − 1  + 2 i  ( A + L + i ) − 1 − ( B + L + i ) − 1  . By the second resolven t identit y , w e ha v e 2 i  ( A + K + i ) − 1 − ( A + L + i ) − 1  = 2 i ( A + K + i ) − 1 (( A + L + i ) − ( A + K + i )) ( A + L + i ) − 1 = 2 i ( A + K + i ) − 1 ( L − K )( A + L + i ) − 1 . Hence, we hav e ∥ 2 i  ( A + K + i ) − 1 − ( A + L + i ) − 1  ∥ op = 2 ∥ ( A + K + i ) − 1 ( L − K )( A + L + i ) − 1 ∥ op ≤ 2 ∥ L − K ∥ op . Noting that b oth ( A + L + i ) − 1 ( A + i ) and ( B + i )( B + L + i ) − 1 extend to b ounded op erators, w e ha v e 2 i  ( A + L + i ) − 1 − ( B + L + i ) − 1  = ( A + L + i ) − 1 ( A + i ) · 2 i  ( A + i ) − 1 − ( B + i ) − 1  · ( B + i )( B + L + i ) − 1 = ( A + L + i ) − 1 ( A + i ) · ( T Cayley ( B ) − T Cayley ( A )) · ( B + i )( B + L + i ) − 1 . Th us, w e obtain ∥ 2 i (( A + L + i ) − 1 − ( B + L + i ) − 1 ) ∥ op = ∥ ( A + L + i ) − 1 ( A + i ) ∥ op ∥ T Cayley ( B ) − T Cayley ( A ) ∥ op ∥ ( B + i )( B + L + i ) − 1 ∥ op ≤ (1 + ∥ L ∥ op ) · ∥ T Cayley ( B ) − T Cayley ( A ) ∥ op · (1 + ∥ L ∥ op ) = (1 + ∥ L ∥ op ) 2 ∥ T Cayley ( B ) − T Cayley ( A ) ∥ op . This completes the pro of. □ 6. Definition of spectral flow In our main theorems in Section 7, K 1 ( I , ∂ I ) of an in terv al I and its t wo endp oin ts ∂ I , pla ys the essen tial role. The elemen ts of K 1 ( I , ∂ I ) are classiﬁed b y the sp ectral ﬂo w deﬁned b elow. In Section 8, w e also discuss the mo d-tw o version of the APS index of Dirac op erators when it is real and sk ewsymmetric. The corresp onding mo d-t wo sp ectral ﬂo w of lattice Dirac op erators, whic h is real, Z 2 -graded and selfadjoin t, classiﬁes elements of K O 0 ( I , ∂ I ) . By abuse of notation, w e also call an elemen t of K 1 ( I , ∂ I ) itself the sp ectral ﬂow, as well as an elemen t of K O 0 ( I , ∂ I ) the mo d-t w o spectral ﬂo w. CAPTURING THE APS INDEX FR OM THE LA TTICE 33 Let I = [ − 1 , 1] b e an interv al. Let H → I b e a Hilb ert bundle o ver I with ﬁbre a separable Hilb ert space o ver C . Supp ose { h t } t ∈ I is a Riesz-con tin uous one-parameter family of un b ounded selfadjoint F redholm op erators on H with Ker h t = { 0 } for t ∈ ∂ I = {− 1 , 1 } . F or eac h t , there exists Λ t suc h that h t has no sp ectrum in the range { λ | − Λ t ≤ λ ≤ Λ t } except for ﬁnite eigenv alues with ﬁnite m ultiplicities. Let us in tro duce a ﬁnite n umber of p oints t 0 = − 1 < t 1 < · · · t n = +1 in I for whic h we assign the v alues λ 1 , . . . , λ n suc h that (i) λ 1 = λ n = 0 . (ii) F or an y t ∈ I in the range t k − 1 ≤ t ≤ t k , λ k is in the range − Λ t < λ k < Λ t and is not an eigen v alue of h t . F or the k -th set ( t k , λ k ) for 0 < k < n , we assign sgn k and d k as follows. F or λ k  = λ k +1 , we set sgn k = λ k − λ k +1 | λ k − λ k +1 | , and d k b y sum of dimensions of the eigenspace with the eigen v alues in the range b et ween λ k and λ k +1 . When λ k = λ k +1 , we assign that sgn k = 0 and d k = 0 . Deﬁnition 25 (Sp ectral ﬂow) . Let H b e a complex Hilb ert bundle ov er I and { h t } t ∈ I b e a Riesz-contin uous family of un b ounded selfadjoin t F redholm operators on H . The sp ectral ﬂow of { h t } t ∈ I is deﬁned b y sf [ { h t } t ∈ I ] = X 0 0 suc h that for an y y ∈ Y and x ∈ X if ρ a ( x − y )  = 0 then | x − y | < l a for any a . Let N la ( Y ) b e the ( l a - ) neigh b ourho o d of the domain-w all Y giv en b y N la ( Y ) = { x ∈ X | ∃ y ∈ Y , | x − y | < la } . Noting that the maximal v alue of ¯ κ t is tw o, and the volume of N la ( Y ) is o ( a ) , we hav e an inequality Z X | ¯ κ t | p dx ≤ C ′ 2 l a. with a p dep enden t constan t C ′ 2 . Then the next prop osition follo ws. □ CAPTURING THE APS INDEX FR OM THE LA TTICE 35 Prop osition 28. F or ϕ ∈ L 2 ( b X a , b E a ) , the fol lowing ine quality holds. || κ t ι a ϕ − ι a ( b κ t ϕ ) || L 2 ≤ || ¯ κ t ι a ( | ϕ | ) || L 2 . Pr o of. F or x ∈ X , the explicit computation shows ( κ t ι a ϕ − ι a ( b κ t ϕ ))( x ) = X z ∈ b X ρ a ( x − z )( κ t ( x ) − b κ t ( z )) U x,z ϕ ( z ) . Therefore, we hav e the following inequalities | κ t ι a ϕ − ι a ( b κ t ϕ ) | ( x ) ≤ X z ∈ b X ρ a ( x − z ) | κ t ( x ) − b κ t ( z ) || U x,z ϕ ( z ) | ≤ ¯ κ t ( x ) X z ∈ b X ρ a ( x − z ) | ϕ ( z ) | = ¯ κ t ι a ( | ϕ | )( x ) , whic h leads to the prop osition abov e. □ Prop osition 29. Ther e exists a c onstant C 3 such that || ¯ κ t ι a ( | ϕ | ) || L 2 ≤ C 3 || ¯ κ t || L d · || ι a ( | ϕ | ) || L 2 1 holds for any ϕ ∈ L 2 ( b X a , b E a ) , wher e d is the dimension of X . Pr o of. A ccording to the Hölder inequality and Sob olev inequality there exists a constan t C 3 suc h that || ¯ κ t ι a ( | ϕ | ) || L 2 ≤ || ¯ κ t || L d · || ι a ( | ϕ | ) || L p ≤ || ¯ κ t || L d · C 3 || ι a ( | ϕ | ) || L 2 1 , holds where p satisﬁes 1 / 2 = 1 /d + 1 /p . □ F rom the ab ov e Prop ositions. 27,28 and 29, we hav e || κ t ι a ϕ − ι a ( b κ t ϕ ) || L 2 ≤ || ¯ κ t ι a ( | ϕ | ) || L 2 ≤ C 3 || ¯ κ t || L d · || ι a ( | ϕ | ) || L 2 1 ≤ C 2 C 3 a 1 /d || ι a ( | ϕ | ) || L 2 1 ≤ C 2 C 3 a 1 /d C 1 || ϕ || L 2 1 , with a p ositiv e constan t C 1 , which prov es the Prop osition 26. 7.2. Main theorem. Let I = [ − 1 , 1] b e a line segment parametrized by t whose t w o end p oints are denoted b y ∂ I = {− 1 , 1 } . W e compare t wo one-parameter families of the contin uum and lattice domain-wall Dirac operators: { D − mκ t γ } t ∈ I , { b D wilson − m b κ t γ } t ∈ I , with κ t and b κ t deﬁned in the previous subsection. W e assume that at t = 1 D − mκ 1 γ is inv ertible. D − mκ − 1 γ is also in v ertible whic h trivially follows from ( D − mκ − 1 γ ) 2 = D 2 + m 2 . F rom a general argumen t, w e can sho w that D − mκ t γ is a F redholm operator at any v alue of t . CAPTURING THE APS INDEX FR OM THE LA TTICE 36 A ccording to Prop osition 22, the un b ounded con tinuum op erator D − mκ t γ is Riesz-con tin uous with resp ect to t , since they at diﬀeren t t diﬀer only b y the mass term mκ t γ , which is a compact b ounded op erator. Therefore, the family { D − mκ t γ } t ∈ I can b e regarded as an elemen t of K 1 ( I , ∂ I ) , which is classiﬁed by the sp ectral ﬂow sf [ D − mκ t γ ] ∈ Z . Then the question is if the lattice Dirac op erator family { b D wilson − m b κ t γ } t ∈ I can b e iden tiﬁed as a w ell-deﬁned elemen t of K 1 ( I , ∂ I ) or not. Let us deﬁne a contin uum-lattice com bined domain-w all fermion Dirac operator D cmb a ( m, t, s ) : Γ( E ) ⊕ Γ( b E a ) → Γ( E ) ⊕ Γ( b E a ) by D cmb a ( m, t, s ) :=  D − mκ t γ s ι a s ι ∗ a − ( b D wilson − m b κ t γ )  =  D 0 0 − b D wilson  − m  κ t γ 0 0 − b κ t γ  + s  0 ι a ι ∗ a 0  with an additional parameter s ∈ [0 , 1] . Theorem 30. Fix a staple-shap e d p ar ameter r e gion P in the t - s plane dr awn in Figur e 4 starting fr om ( s, t ) = ( − 1 , 0) , via ( − 1 , 1) , (1 , 1) then to (1 , 0) . Ther e exists a c onstant a 1 and m 1 such that for arbitr ary lattic e sp acing a = 1 / N satisfying 0 < a ≤ a 1 , and arbitr ary mass p ar ameter m > m 1 D cmb a ( m, t, s ) is invertible at any p oint ( s, t ) on P . Figure 4. The staple-shap ed parameter region in the t - s plane where w e pro ve that the lattice-contin uum com bined Dirac op erator D cmb ( m, t, s ) is in v ertible. Then the corollary of Theorem 30 b elo w and our main theorem follo w. CAPTURING THE APS INDEX FR OM THE LA TTICE 37 Corollary 31. The sp e ctr al ﬂow sf [ b D wilson − m b κ t γ ] is wel l-deﬁne d as an element of K 1 ( I , ∂ I ) , wher e I = [ − 1 , 1] and ∂ I = {− 1 , 1 } . Theorem 32 (APS Index of lattice Dirac operator) . F or any lattic e sp acing a = 1 / N < a 1 , the fol lowing holds. sf [ D − mκ t γ ] = sf [ b D wilson − m b κ t γ ] ∈ K 1 ( I , ∂ I ) ∼ = Z , wher e I = [ − 1 , 1] and ∂ I = {− 1 , 1 } . Pr o of. The claim immediately follo ws from Prop osition 22. □ When Y has a collar neighbourho o d as in Section 4.3, the sp ectral ﬂo w sf [ D − mκ t γ ] equals to the APS index Ind APS ( D | X + ) . Therefore, we regard sf [ b D wilson − m b κ t γ ] as a lattice formulation of the APS index. Since b D wilson − m b κ t γ is a ﬁnite-sized matrix, the follo wing equalit y immediately follo ws b y using the fact that η ( b D wilson + mγ ) = 0 pro v ed in Prop osition C.1 of [4]. Corollary 33. sf [ b D wilson − m b κ t γ ] = − 1 2 η ( b D wilson − m b κγ ) . 7.3. Pro of of Theorem 30. Supp ose that Theorem 30 do es not hold. Then there should exist a series lab eled b y i = 1 , 2 , . . . comp osed b y • a i = 1 / N i → 0 , • ( t i , s i ) ∈ P (the staple-shaped region in Figure 4), • ( ψ i , ϕ i ) ∈ L 2 ( E ) ⊕ L 2 ( b E a i ) whic h satisfy for an y i that (i) || ψ i || 2 L 2 + || ϕ i || 2 L 2 = 1 and (ii) (4) D cmb a i ( m, t i , s i )  ψ i ϕ i  = 0 . T aking subsequences, w e can assume without loss of generalit y that ( t i , s i ) con v erges to a point ( t ∞ , s ∞ )  = (0 , 0) : t i → t ∞ and s i → s ∞ . Let us decomp ose (4) in to the tw o equations. ( D − mκ t i γ ) ψ i + s i ι a i ϕ i = 0 , s i ι ∗ a i ψ i − ( b D wilson a i − m b κ t i γ ) ϕ i = 0 . F rom the ﬁrst equation we ha ve the following uniform bound, || ψ i || 2 L 2 1 ≤ C ( || D ψ i || 2 L 2 + || ψ i || 2 L 2 ) = C ( || mκ t i γ ψ i − s i ι a i ϕ i || 2 L 2 + || ψ i || 2 L 2 ) ≤ C (( m 2 || κ t i || 2 + 1) || ψ i || 2 L 2 + s 2 i || ι a i ϕ i || 2 L 2 ) ≤ C ′ CAPTURING THE APS INDEX FR OM THE LA TTICE 38 with p ositiv e constants C and C ′ . F or the last inequality we hav e used the prop ert y (iii) in Section 3.2. F rom the second equation, w e ha v e || ϕ i || 2 L 2 1 ≤ C ( || b D wilson a i ϕ i || 2 L 2 + || ϕ i || 2 L 2 ) ≤ C ( s 2 i || ι ∗ a i ψ i || 2 L 2 + ( m 2 || b κ a i || 2 + 1) || ϕ i || 2 L 2 ) < C ′ , with p ositiv e constants C and C ′ . Here w e hav e used the prop erty (iv) in Section 3.2. Th us, ψ i and ι a i ϕ i are uniformly L 2 1 b ounded in L 2 ( E ) . T aking subsequences, w e can assume without loss of generalit y that ι a i ϕ i w eakly con v erges in L 2 1 ( E ) to a v ector ψ ′ ∞ and ψ i w eakly conv erges in L 2 1 ( E ) to another v ector ψ ∞ . Moreo v er, these are strong con v ergence in L 2 according to the Rellich theorem. F rom the prop erty (iii) in Section 3.2, w e can also show th at || ι a i ϕ i || 2 L 2 − || ϕ i || 2 L 2 ≤ C a i || ϕ i || 2 L 2 1 with a constan t C so that the s eries ϕ i strongly conv erges to a lattice v ector ϕ ∞ in L 2 . Then we can conclude that || ψ ∞ || 2 L 2 + || ϕ ∞ || 2 L 2 = 1 . Lemma 34. With the assumptions made ab ove for the pr o of by c ontr adiction, the fol lowing e quation holds.  D − mκ t ∞ γ s ∞ s ∞ − ( D − mκ t ∞ γ )   ψ ∞ ψ ′ ∞  = 0 . Pr o of. F or an y Ψ ∈ C ∞ ( E ) , the weak limit of the inner product ⟨ D ψ i , Ψ ⟩ L 2 b ecomes ⟨ D ψ i , Ψ ⟩ L 2 = ⟨ ψ i , D ∗ Ψ ⟩ L 2 → ⟨ ψ ∞ , D ∗ Ψ ⟩ L 2 = ⟨ D ψ ∞ , Ψ ⟩ L 2 . Similarly , w e hav e ⟨ mκ t i γ ψ i − s i ( ι a i ϕ i ) , Ψ ⟩ L 2 → ⟨ mκ t ∞ γ ψ ∞ − s ∞ ψ ′ ∞ , Ψ ⟩ L 2 . F rom the upp er comp onen t of (4), w e ha v e ⟨ ( D − mκ t ∞ γ ) ψ ∞ + s ∞ ψ ′ ∞ , Ψ ⟩ L 2 = 0 . By applying ι a i to the lo wer comp onent of (4) , w e obtain an equation in L 2 ( E ) , − ι a i ( b D wilson a i − m b κ t i γ ) ϕ i + s i ι a i ι ∗ a i ψ i = 0 . F or Ψ ∈ C ∞ ( E ) , w e hav e ⟨ ι a i ( b D wilson a i ϕ i ) , Ψ ⟩ L 2 = ⟨ ϕ i , b D wilson ∗ a i ( ι ∗ a i Ψ) ⟩ L 2 , and ⟨ ι a i ( b D wilson a i ι ∗ a i ι a i ) ϕ i , Ψ ⟩ L 2 = ⟨ ι ∗ a i ι a i ϕ i , b D wilson ∗ a i ( ι ∗ a i Ψ) ⟩ L 2 = ⟨ ι a i ϕ i , ι a i b D wilson ∗ a i ( ι ∗ a i Ψ) ⟩ L 2 → ⟨ ψ ′ ∞ , D Ψ ⟩ L 2 , where w e ha v e used the prop erty (v) in Section 3.2. Comparing the ab o ve t wo, w e obtain |⟨ ι a i ( b D wilson a i ϕ i ) , Ψ ⟩ L 2 − ⟨ ι a i ( b D wilson a i ι ∗ a i ι a i ) ϕ i , Ψ ⟩ L 2 | ≤ || ( ι ∗ a i ι a i − id) ϕ i || L 2 · || b D wilson ∗ a i ( ι ∗ a i Ψ) || L 2 → 0 , CAPTURING THE APS INDEX FR OM THE LA TTICE 39 where w e ha v e used the prop ert y (iii) in Section 3.2 and that || b D wilson ∗ a i ( ι ∗ a i Ψ) || L 2 is b ounded b y C || Ψ || L 2 1 with some constan t C . Therefore, w e ha v e for an y Ψ ∈ C ∞ ( E ) , ⟨ ι a i ( b D wilson a i ϕ i ) , Ψ ⟩ L 2 → ⟨ D ψ ′ ∞ , Ψ ⟩ L 2 . Similarly , for any Ψ ∈ C ∞ ( E ) , the following w eak con v ergence is obtained. ⟨ s i ι a i ι ∗ a i ψ i , Ψ ⟩ L 2 = ⟨ ψ i , s i ι a i ι ∗ a i Ψ ⟩ L 2 → ⟨ s ∞ ψ ∞ , Ψ ⟩ L 2 . Finally from the b ound b elo w for an y Ψ ∈ C ∞ ( E ) , w e hav e |⟨ m ι a i ( b κ t i γ ϕ i ) , Ψ ⟩ L 2 − ⟨ mκ t i γ ι a i ϕ i , Ψ) ⟩| ≤ m || ( ι a i b κ t i − κ t i ι a i ) γ ϕ i || L 2 · || Ψ || L 2 ≤ mC a 1 /d i || γ ϕ i || L 2 · || Ψ || L 2 , where Prop osition 26 is used in the second inequalit y , and we ha ve the following w eak con vergence of the series ⟨ m ι a i ( b κ t i γ ϕ i ) , Ψ ⟩ L 2 → ⟨ mκ t ∞ γ ψ ′ ∞ , Ψ ⟩ L 2 and ⟨ s ∞ ψ ∞ − ( D − mκ t ∞ γ ) ψ ′ ∞ , Ψ ⟩ L 2 = 0 , holds. □ When Lemma 34 holds, b y applying the square of the op erator, w e ha v e  D − mκ t ∞ γ s ∞ s ∞ − ( D − mκ t ∞ γ )  2  ψ ∞ ψ ′ ∞  =  ( D − mκ t ∞ γ ) 2 + s 2 ∞ 0 0 ( D − mκ t ∞ γ ) 2 + s 2 ∞   ψ ∞ ψ ′ ∞  = 0 . Since ( D − mκ ± 1 γ ) 2 is in vertible from the assumption, ( D − mκ t ∞ γ ) 2 + s 2 ∞ is ev erywhere inv ertible on the staple-shap ed region ( t ∞ , s ∞ ) ∈ P . Therefore, ψ ∞ and ψ ′ ∞ m ust b e zero, whic h contradicts with the condition || ψ ∞ || 2 L 2 + || ϕ ∞ || 2 L 2 = 1 . This prov es Theorem 30. Theorem 32 immediately follo ws from Theorem 30. □ 8. Applica tions to the mod-two APS index So far, we ha v e discussed the standard Z -v alued index of Dirac op erators in even dimensions and the corresp onding sp ectral ﬂow of the massive Dirac op erators, whic h giv e elemen ts of K 1 ( I , ∂ I ) . Since our formulation of K n or K O n for arbitrary degree n in Section 5 is so general that we can deal with unbounded contin uum Dirac op erators and b ounded lattice Dirac op erators with the corresp onding symmetries. In particular, it is imp ortan t in ph ysics to formulate the Z 2 -v alued index of real sk ewsymmetric Dirac op erators in order to describ e non-lo cal t yp es of anomaly [40]. In the original deﬁnition, the Z 2 -v alued index giv es an elemen t of the group K O − 1 ( pt ) , where the degree n = − 1 indicates absence of the Z 2 -grading or chiralit y CAPTURING THE APS INDEX FR OM THE LA TTICE 40 op erator. The corresponding mo d-t wo v ersion of the spectral ﬂo w giv es an elemen t of the group K O 0 ( I , ∂ I ) 7 . In this section, we form ulate a lattice version of the mo d-t wo APS index using the mo d-t wo sp ectral ﬂo w. Let X := T d = ( R / Z ) d b e a d -dimensional ﬂat torus, where d can b e an y positive in teger. In this section, it is con venien t to introduce the basis ϵ 1 , . . . , ϵ d of the Cliﬀord algebra Cl d , satisfying the anticomm utation relation { ϵ i , ϵ j } = ϵ i ϵ j + ϵ j ϵ i = 2 δ ij . Let E → X b e a Cliﬀord mo dule bundle on X with a smo oth bundle map σ : X × Cl d → End ( E ) . W e assume that E is of the form E R ⊗ R C for an Euclidean v ector bundle E R and σ ( ϵ i ) is a real symmetric operator on E , i.e. , a symmetric op erator on E R for all i . Let Y ⊂ X b e a separating submanifold that decomp oses X in to the union of t w o compact manifolds X + and X − whic h share the common b oundary Y . W e in tro duce the same one-parameter family of the domain-w all function κ t as in (3) , where t is in the range I = [ − 1 , 1] and the t w o end p oints are denoted b y ∂ I . Fix a Cliﬀord connection A on E preserving E R , whic h determines the co v ariant deriv ative ∇ j in the e j direction for eac h j = 1 , · · · d , and the parallel transp ort as in Section 2. W e denote the space of smo oth sections of E b y Γ( E ) . Let D : Γ( E ) → Γ( E ) be a real and sk ewsymmetric ﬁrst order elliptic op erator deﬁned b y D u := d X j =1 σ ( ϵ j ) ∇ j u, for u ∈ Γ( E ) . A distinct prop erty of the fermion system in odd dimensions is that the massiv e Dirac op erators D − mκ t id , with a positive parameter m > 0 , is neither selfadjoin t nor skew adjoin t 8 in general. In order to represent the K O 0 ( I , ∂ I ) elements, we deﬁne the follo wing one-parameter family of the real symmetric op erators { H t } t ∈ I on Γ( E ) ⊕ Γ( E ) where H t :=  0 D − mκ t id D ∗ − mκ t id 0  , with an assumption that H t =1 is in vertible ( H t = − 1 is also trivially in v ertible). This op erator is Riesz-con tinuous with resp ect to t . When X − = {∅} and κ t = t id , it is not diﬃcult to conﬁrm that the mo d- t w o spectral ﬂo w: sf 2 [ { H t } t ∈ I ] , which is the n umber of zero-crossing pairs of the eigen v alues of H t along t ∈ [ − 1 , 1] , agrees with the dimension of Ker D mo dulo 7 A similar isomorphism K O − 2 (pt) ∼ = K O − 1 ( I , ∂ I ) ∼ = Z 2 also w orks. 8 Recall that in even dimensions, we can use the Z 2 -grading op erator γ to make γ ( D − mκ t id ) selfadjoin t. CAPTURING THE APS INDEX FR OM THE LA TTICE 41 t w o. In [17], it was prov ed that the mo d-tw o spectral ﬂo w of { H t } t ∈ I equals to the mo d-t wo APS index of D on X + . No w let us deﬁne the lattice Wilson Dirac op erator b D wilson : Γ( b E ) → Γ( b E ) using the same notation as in Section 2.2, b D wilson := d X i =1 σ ( ϵ j ) b ∇ j + a 2 d X j =1 b ∇ f j  b ∇ f j  ∗ , whic h is real. Note here that the ﬁrst term is skew adjoin t but the second Wilson term is selfadjoin t. W e also deﬁne the lattice v ersion of the real symmetric op erator b H t : Γ( b E a ) ⊕ Γ( b E a ) → Γ( b E a ) ⊕ Γ( b E a ) by b H t := 0 b D wilson − m b κ t id b D wilson ∗ − m b κ t id 0 ! . Let us deﬁne a contin uum-lattice com bined domain-w all fermion Dirac operator H cmb a ( m, t, s ) : Γ( E ) ⊕ Γ( E ) ⊕ Γ( b E a ) ⊕ Γ( b E a ) → Γ( E ) ⊕ Γ( E ) ⊕ Γ( b E a ) ⊕ Γ( b E a ) by H cmb a ( m, t, s ) :=  H t id 2 × 2 ⊗ s ι a id 2 × 2 ⊗ s ι ∗ a − b H t  with an additional parameter s ∈ [0 , 1] . Theorem 35. On the p ath P in Figur e 4, ther e exists a c onstant a 1 and m 1 such that for arbitr ary lattic e sp acing a = 1 / N satisfying 0 < a ≤ a 1 , and arbitr ary mass p ar ameter m > m 1 H cmb a ( m, t, s ) is invertible at any p oint ( s, t ) on P . Corollary 36. The mo d-two sp e ctr al ﬂow sf 2 [ { b H t } t ∈ I ] is wel l-deﬁne d as an element of K O 0 ( I , ∂ I ) . Theorem 37 (Mo d-tw o APS Index of lattice Dirac op erator) . F or any lattic e sp acing a = 1 / N < a 1 , the fol lowing holds. sf 2 [ { H t } t ∈ I ] = sf 2 [ { b H t } t ∈ I ] ∈ K O 0 ( I , ∂ I ) . Pr o of. The pro of go es in the essentially same wa y as that of Theorems 30 and 32. □ When Y has a collar neighbourho o d as in Section 4.3, the sp ectral ﬂow sf 2 [ { H t } t ∈ I ] equals to the mo d-t w o APS index Ind APS mod - 2 ( D | X + ) , which w as prov ed in [17]. Therefore, sf 2 [ { b H t } t ∈ I ] can b e regarded as its lattice form ulation. 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(SA) Interdisciplinar y Theoretical and Ma thema tical Sciences Pr ogram (iTHEMS), RIKEN, W ako, Jap an Email addr ess : shotoaoki@g.ecc.u-tokyo.ac.jp (HF uj) F acul ty of Science, Jap an Women’s University, Mejirod ai, Bunkyo-ku, Tokyo 112-8681, Jap an Email addr ess : fujitah@fc.jwu.ac.jp (HF uk, TO, and SY) Dep ar tment of Physics, Osaka University, Osaka, Jap an Email addr ess : hfukaya@het.phys.sci.osaka-u.ac.jp Email addr ess : onogi@phys.sci.osaka-u.ac.jp Email addr ess : yamaguch@het.phys.sci.osaka-u.ac.jp (MF) Gradua te School of Ma thema tical Sciences, The University of Tokyo, Tokyo, Jap an Email addr ess : furuta@ms.u-tokyo.ac.jp (SM) Gradua te School of Ma thema tics, Nago y a University, Nago y a, Jap an Email addr ess : shinichiroh@math.nagoya-u.ac.jp

Capturing the Atiyah-Patodi-Singer index from the lattice

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