c Pleiades Publishing, Ltd., 2017. ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2017, Vol. 298, pp. 256–293.  Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 298, pp. 276–314.

Spin Geometry of Dirac and Noncommutative Geometry of Connes A. G. Sergeev a Received December 11, 2016

Abstract—The review is devoted to the interpretation of the Dirac spin geometry in terms of noncommutative geometry. In particular, we give an idea of the proof of the theorem stating that the classical Dirac geometry is a noncommutative spin geometry in the sense of Connes, as well as an idea of the proof of the converse theorem stating that any noncommutative spin geometry over the algebra of smooth functions on a smooth manifold is the Dirac spin geometry. DOI: 10.1134/S0081543817060177

One of the main goals of noncommutative geometry is to interpret the basic notions of analysis, topology, and geometry in terms of Banach algebras. In our previous publications on this topic we paid attention to the classical objects of analysis such as the integral and differential (cf. [9]) and constructed noncommutative analogs of some classical spaces of function theory such as the Sobolev space of half-differentiable functions and the group of quasisymmetric homeomorphisms of the circle (cf. [7, 6]). In this review we proceed to translate the basic notions of spin geometry into the language of Banach algebras. Our main goal is to present the ideas underlying the correspondence between the classical Dirac geometry and its noncommutative counterpart. For this reason we mainly focus on the principal items while omitting some technical proofs. In such cases we refer to other publications in which these proofs can be found. In the first part we give a short review of spin geometry (a more detailed exposition of this subject can be found in Lawson and Michelsohn’s book [5]). The main topics of this part are Clifford algebras, spin groups and their representations, spin structures on the bundles, spin connections, and the Dirac operator. The second part is devoted to noncommutative geometry and its interrelations with the Dirac spin geometry. It starts with the central notion of spectral triple and with the definition of noncommutative spin geometry in terms of this notion. In Section 12 we formulate and give an idea of the proof of the main Theorem 2 stating that the Dirac spin geometry is a noncommutative spin geometry in the sense of the given definition. In Section 13 we formulate and give an idea of the proof of Theorem 3, which can be considered as the converse of Theorem 2. It states that any noncommutative spin geometry over the algebra of smooth functions on a smooth manifold is the Dirac spin geometry. In the Appendix, for the reader’s convenience, we have collected the basic definitions related to the Hochschild homology and cohomology. As the basic references on noncommutative geometry, we have used the books [1–4] and the lecture notes [8] of the course delivered to the students of the Scientific and Educational Center of the Steklov Mathematical Institute in 2014–2015. a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia.

E-mail address: [email protected]

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I. SPIN GEOMETRY 1. CLIFFORD ALGEBRAS 1.1. Definition. Let V be the n-dimensional Euclidean space provided with an orthonormal basis {ei }ni=1 . Definition 1. The Clifford algebra Cl(n) is an associative algebra Cl(V ) over the field R generated by the elements 1, e1 , . . . , en satisfying the relations e2i = −1,

ei ej + ej ei = 0 for i = j.

This definition immediately implies that V ⊂ Cl(V ) and uv + vu = −2(u, v)

for all u, v ∈ V.

As a real vector space, Cl(V ) has dimension 2n and can be provided with the basis given by 1 and the elements of the form eI := ei1 ei2 . . . eik , where I = {i1 , i2 , . . . , ik } is a strictly increasing set of |I| := k indices with 1 ≤ i1 < . . . < ik ≤ n. In particular, any element x ∈ Cl(V ) can be written as x=



xI eI ,

I

where we add the subset I = 0 to the collection {I} of index sets and put e0 := 1. Using this representation, we can introduce a natural inner product on Cl(V ) given by the formula  xI y I (x, y) := I

(which does not depend on the choice of the orthonormal basis {ei }ni=1 ). Denote by Clk (V ) the subspace of Cl(V ) consisting of degree k elements, which are linear combinations of the basis elements eI with |I| = k (assuming that I = 0 for k = 0). Moreover, introduce the subspaces of Cl(V ) of the form Clev (V ) :=



Clk (V ),

Clod (V ) :=

k even



Clk (V ).

k odd

Then Clev (V ) is a subalgebra in Cl(V ) and Cl(V ) = Clev (V ) ⊕ Clod (V ). 1.2. Universal property. The notion of the Clifford algebra Cl(V ) does not in fact depend on the choice of the orthonormal basis {ei }ni=1 due to the following universal property, which can be taken for a definition of this algebra. Proposition 1. The Clifford algebra Cl(V ) is a unique associative R-algebra with unit which contains the Euclidean space V and has the following property: for any associative R-algebra A with unit 1A and for an arbitrary linear map f : V → A satisfying the condition f (v)f (v) = −|v|2 · 1A , PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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there exists a unique extension of f to an algebra homomorphism f: Cl(V ) → A such that the following diagram is commutative: V

f

A

i

Cl(V )

f

Proof. To prove the formulated universal property, we will use the following equivalent definition of the Clifford algebra. Denote by T (V ) =

∞ 

V ⊗k

k=0

the tensor algebra of V and consider the ideal J (V ) of this algebra generated by the elements of the form v ⊗ v + |v|2 · 1. The Clifford algebra Cl(V ) coincides with the quotient of the tensor algebra T (V ) by this ideal: Cl(V ) ∼ = T (V )/J (V ). Now we return to the proof of the universal property of Cl(V ). Any linear map f : V → A extends uniquely to an algebra homomorphism f: T (V ) → A. By assumption this homomorphism vanishes on the ideal J (V ), so f can be pushed down to an algebra homomorphism f: Cl(V ) → A.  Here are examples of Clifford algebras of low dimensions: (1) Cl(R) = C with e1 = i; (2) Cl(R2 ) = H with e1 = i, e2 = j, and e1 e2 = k; (3) Cl(R4 ) = Mat2 (H) is the space of quaternionic 2 × 2 matrices. 1.3. Multiplicative group. Denote by Cl× (V ) the group of invertible elements of the Clifford algebra Cl(V ). It is a Lie group, which contains V \ {0} since the inverse v −1 of any element v ∈ V \ {0} is given by the formula v v −1 = − 2 . |v| The Lie algebra of Cl× (V ) is the algebra cl(V ) identified as a set with Cl(V ) and provided with the Lie bracket of the form [x, y] := xy − yx. The multiplicative group Cl× (V ) acts on the algebra Cl(V ) by the adjoint representation w → Adw x := wxw−1 ,

w ∈ Cl× (V ).

The differential of this action is a Lie algebra homomorphism ad : cl(V ) → Der Cl(V ) from cl(V ) into the algebra Der Cl(V ) of derivations of Cl(V ) given by the formula ady x := [y, x]

for y ∈ cl(V ),

x ∈ Cl(V ).

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For any u ∈ V \ {0} the map − Adu v = v − 2

(u, v) u, |u|2

v ∈ V,

is the reflection with respect to the hyperplane u⊥ orthogonal to u. In order to get rid of the minus sign on the left-hand side of the latter equality, instead of the adjoint representation Ad it is convenient to use the action of the group Cl× (V ) on the algebra Cl(V ) given by the twisted adjoint representation of the form w → πw (x) := χ(w)xw−1 ,

x ∈ Cl(V ),

w ∈ Cl× (V ),

where χ is the grading map given on the homogeneous elements of degree k from Cl× (V ) by the formula χ(w) := (−1)deg w w = (−1)k w. Then the map πu : V → V determined by the element u ∈ V \ {0} will coincide with the reflection with respect to the hyperplane u⊥ . Taking this into account, consider a subgroup of Cl× (V ) consisting of the elements x ∈ Cl× (V ) with the property πx (V ) = V . As pointed out above, all elements v ∈ V \ {0} have this property, so it is worthwhile to introduce the following Definition 2. The Clifford group Γ(V ) ≡ Γ(n) is the subgroup of Cl× (V ) generated by the elements v ∈ V \ {0}. Any element of Γ(V ) generates a nondegenerate linear transformation of V , so we have a homomorphism π : Γ(V ) → GL(V ). This homomorphism takes values in the orthogonal group O(V ). Indeed, since any element x ∈ Γ(V ) is represented as the product x = v1 . . . vk of some elements v1 , . . . , vk ∈ V \ {0}, the corresponding transformation πx is the composition of reflections associated with the elements vi , i.e., it belongs to O(V ). Moreover, the homomorphism π : Γ(V ) → O(V ) is an epimorphism, since any orthogonal transformation is the composition of reflections. The homomorphism π : Γ(V ) → O(V ) can be included into the exact sequence of group homomorphisms of the form → O(V ) → 1. 1 → R× → Γ(V ) − π

We also set SΓ(V ) := Γ(V ) ∩ Clev (V ). 1.4. Complexified Clifford algebra. The definition of Clifford algebras given above immediately extends to the case of vector spaces V over the field C of complex numbers provided with a nondegenerate symmetric bilinear form (determined uniquely up to multiplication by a nonzero complex number). We also introduce the complexified Clifford algebra Cl(V ) of the n-dimensional real vector space V by setting Cl(V ) := Cl(V ) ⊗R C. This algebra is isomorphic to the Clifford algebra Cl(V C ) of the complexified vector space V C := V ⊗R C provided with a nondegenerate complex quadratic form. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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2. SPIN GROUPS 2.1. The group Pin. Definition 3. The group Pin(V ) is defined as the subgroup of Γ(V ) generated by the unit vectors from V , i.e., by the vectors v ∈ V with |v| = 1. As in the case of the Clifford group, we have a homomorphism π : Pin(V ) → O(V ), which can be included into the exact sequence of group homomorphisms π

→ O(V ) → 1. 1 → Z2 → Pin(V ) − 2.2. The group Spin. Definition 4. The group Spin(V ) is the connected component of unity in the group Pin(V ). Otherwise, it can be defined as Spin(V ) = Pin(V ) ∩ Clev (V ). As in the case of the group Pin(V ), there is an exact sequence of group homomorphisms π

→ SO(V ) → 1. 1 → Z2 → Spin(V ) − For n > 2 the group Spin(n) is a simply connected covering of the group SO(V ). Here are examples of Spin groups of low dimensions: (1) Spin(2) = U(1) = SO(2); (2) Spin(3) = SU(2); (3) Spin(4) = SU(2) × SU(2). 3. RELATION TO THE EXTERIOR ALGEBRA 3.1. Definition. The exterior algebra Λ(V ) =

n 

Λk (V )

k=0

of the n-dimensional real vector space V can be defined as the quotient of the tensor algebra T (V ) by the ideal I(V ) generated by the elements of the form u ⊗ v + v ⊗ u. Otherwise, Λ(V ) can be defined as the associative algebra with unit that contains V in which the multiplication ∧ satisfies the relation u∧v+v∧u=0

for all u, v ∈ V.

The exterior algebra, just as the Clifford one, has a universal property based on the above relation. For a fixed orthonormal basis {ei }ni=1 in V , introduce the subspace Λk (V ) consisting of degree k elements generated by 1 and the elements of the form εI := ei1 ∧ . . . ∧ eik , where I = {i1 , . . . , ik } is a strictly increasing subset of indices from the set {1, . . . , n}: 1 ≤ i1 < this definition by setting I = 0 and e0 = 1 for k = 0. The dimension . . . < ik ≤ n. We complement  n k of Λ (V ) is equal to k , which implies that the dimension of Λ(V ) is equal to 2n ; i.e., it coincides with the dimension of the Clifford algebra Cl(n). Hence we can expect that for the n-dimensional Euclidean space V there should exist a close relation between Λ(V ) and Cl(V ) (such a relation will be established below in Subsection 3.3). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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3.2. Derivations and transposition. Let us introduce inner and exterior derivations of the algebra Λ(V ). We begin with the inner product of a form and an element of the dual space ξ ∈ V  by setting ι(ξ) · 1 = 0 and ι(ξ)(v1 ∧ . . . ∧ vk ) =

k 

ξ(vj )v1 ∧ . . . ∧ vj ∧ . . . ∧ vk ,

j=1

where v1 , . . . , vk ∈ V and the hat over vj means that this element should be omitted. Then for any ξ, η ∈ V  we will have ι(ξ)ι(η) + ι(η)ι(ξ) = 0. Using now the universal property of Λ(V  ), we can extend the constructed map ι : V  → End Λ(V ) to the whole algebra Λ(V  ) as a map ι : Λ(V  ) → End Λ(V ). On the other hand, on Λ(V ) we have the operation of exterior product, which generates a homomorphism  : Λ(V ) → End Λ(V ). The introduced operations satisfy the following commutation relations: [(u), (v)] = (u)(v) + (v)(u) = 0, [ι(ξ), ι(η)] = ι(ξ)ι(η) + ι(η)ι(ξ) = 0, [ι(ξ), (v)] = ι(ξ)(v) + (v)ι(ξ) = ξ(v) for all u, v ∈ V and ξ, η ∈ V  . The tensor algebra T (V ) has a special involutive antiautomorphism equal to the identity on V . This automorphism is called the transposition and is uniquely determined by the formula (v1 ⊗ . . . ⊗ vk )T := vk ⊗ . . . ⊗ v1 . Since this operation preserves the ideal I(V ), it can be pushed down to the exterior algebra Λ(V ). Moreover, η T = (−1)k(k−1)/2 η on the forms η ∈ Λk (V ). In the complex case we denote by V C = V ⊗R C the complexification of the space V . On V C there is an operation of complex conjugation v → v¯, which extends naturally to the whole algebra Λ(V C ). Define an involution on Λ(V C ) by setting η )T . η → η ∗ := (¯ This map determines a conjugate linear antiautomorphism of the algebra Λ(V C ). 3.3. Relation between the exterior algebra and the Clifford algebra. Consider the map s : V → End Λ(V ) given by the formula s(v) = (v) + ι(v  ),

v ∈ V,

where v  ∈ V  is the covector dual to v defined as v  : u ∈ V → (v, u). The map s extends to a homomorphism of the whole tensor algebra s : T (V ) → End Λ(V ). The commutation relations for the inner and exterior derivations imply that s(u)s(v) + s(v)s(u) = −2(u, v) · 1. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Hence, the map s vanishes on the ideal J (V ); so it can be pushed down to a homomorphism s : Cl(V ) → End Λ(V ). Consider next the map σ : Cl(V ) → Λ(V ), called the symbol map, given by the formula σ(x) := s(x) · 1, where 1 is regarded as an element of Λ0 (V ). The map Alt : Λ(V ) → Cl(V ) converse to σ is called the alternation map and is given by the formula Alt(v1 ∧ . . . ∧ vk ) =

1  (−1)sgn p vp(1) . . . vp(k) , k! p∈Sk

where the summation is taken over all permutations p ∈ Sk of the set {1, . . . , k} and sgn p denotes the parity of p. 3.4. Involution and volume element. The transposition map introduced above on the tensor algebra T (V ) preserves the ideal J (V ) and so can be pushed down to an antiautomorphism of the Clifford algebra Cl(V ) with the following property: (v1 . . . vk )T = vk . . . v1 . Note that the alternation map Alt intertwines the transpositions in Λ(V ) and Cl(V ). In the complex case the involution on the Clifford algebra Cl(V ) can be defined by the formula x)T , x → x∗ := (¯ where x → x ¯ is the complex conjugation on Cl(V ). Again the alternation map Alt intertwines the involutions in Λ(V ) and Cl(V ). With the help of the transposition we can introduce another useful map N : x → xT x which is defined on the elements x from the Clifford group Γ(V ). Since any element x ∈ Γ(V ) can be represented in the form x = v1 . . . vk , the map N takes values in R× . This map determines a group homomorphism N : Γ(V ) → R× ,

x → xT x,

which is called the norm and has the property N(λx) = λ2 N(x) for λ ∈ R× . The alternation of the volume element ω ∈ Λn (V ) of the n-dimensional vector space V yields an element ω ∈ Cl(V ) (denoted by the same letter ω), which is called the volume element of the Clifford algebra or its chiral element. The square of ω is a scalar, which can easily be checked by choosing an orthonormal basis {ei }ni=1 of the Euclidean space V and setting ω = e1 . . . en . Then ω 2 = (−1)n(n+1)/2 . Moreover, for odd n the element ω is central, while for even n we have xω = ωχ(x) for all x ∈ Cl(n). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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4. THE GROUP Spinc 4.1. Definition. Consider the complexified Clifford algebra Cl(V ) = Cl(V C ), where V C is the complexification of the space V , and provide it with a Hermitian inner product defined by the equality z, w := (¯ z , w) on elements z, w ∈ V C and extended to the whole algebra Cl(V C ). Recall that this algebra has the z )T on elements z ∈ Cl(V ). involution defined by the equality z ∗ = (¯ Let Γ(V C ) be the Clifford group of V C consisting of the elements z ∈ Cl× (V C ) that satisfy the condition πz (V C ) = V C , so that πz ∈ O(V C ). Definition 5. Introduce the following groups:   Pinc (V ) = z ∈ Γ(V C ) : z ∗ z = 1 , Γc (V ) = z ∈ Γ(V C ) : z ∗ z ∈ R+ , Spinc (V ) = Pinc (V ) ∩ SΓ(V C ). It can be shown that an element z ∈ Γ(V C ) belongs to the subgroup Γc (V ) if and only if the map πz ∈ O(V C ) preserves the real subspace V . In other words, Γc (V ) coincides with the inverse image (with respect to the map π) of the subgroup O(V ) ⊂ O(V C ). The exact sequence for the group Γ(V C ) transforms into the exact sequence of group homomorphisms 1 → C× → Γc (V ) → O(V ) → 1, which induces the exact sequences 1 → U(1) → Pinc (V ) → O(V ) → 1,

1 → U(1) → Spinc (V ) → SO(V ) → 1,

since C× ∩ Pinc (V ) = C× ∩ Spinc (V ) = U(1). These sequences imply that the groups Pinc (V ) and Spinc (V ) are the central extensions of the groups O(V ) and SO(V ), respectively, by U(1). 4.2. Exact sequences for the group Spinc (V ). Otherwise, the group Spinc (V ) can be defined as a subgroup of Γc (V ) of the form  Spinc (V ) = z = xeiθ : x ∈ Spin(V ), θ ∈ R . In other words, the group Spinc (V ) is the quotient of the group Spin(V ) × U(1) by the equivalence relation (x, eiθ ) ∼ (−x, −eiθ ). This statement can be written in the form of the exact sequence δ

→ U(1) → 1, 1 → Spin(V ) → Spinc (V ) − where δ : xeiθ → e2iθ . In other words, Spinc (V ) = Spin(V ) ×Z2 U(1). The norm homomorphism N : Γc (V ) → C× ,

z → z T z,

defined on the elements of the Clifford group Γ(V C ), takes values in U(1) when restricted to the elements of Pinc (V ). The combination of this homomorphism with the exact sequences written above yields (π,N)

1 → Z2 → Γc (V ) −−−→ O(V ) × C× → 1, (π,N)

1 → Z2 → Pinc (V ) −−−→ O(V ) × U(1) → 1, PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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and (π,N)

1 → Z2 → Spinc (V ) −−−→ SO(V ) × U(1) → 1, where the homomorphism π is given by the map z → πz (v) = χ(z)vz −1 . The Lie algebra of the group Spinc (V ) coincides with spinc (V ) = cl2 (V ) ⊕ iR, where cl2 (V ) is the quadratic component (coinciding as a set with Cl2 (V )) of the Clifford algebra Cl(V ) provided with the commutator as the Lie bracket. Here are examples of Spinc groups of low dimensions: (1) Cl(R) = C ⊕ C, and the group Spinc (R) coincides with the group U(1) embedded into C ⊕ C by the diagonal map; (2) Cl(R2 ) = Mat2 (C), and the group Spinc (R2 ) coincides with the group U(1) × U(1) consisting of diagonal unitary matrices in Mat2 (C). 5. SPIN REPRESENTATION 5.1. Clifford modules. Definition 6. The Clifford representation is a homomorphism ρ : Cl(V ) → End S from the Clifford algebra Cl(V ) into the algebra of linear operators acting in a complex vector space S, which is called the Clifford module over Cl(V ) or the space of spinors for the algebra Cl(V ). We will assume that the space S is equipped with a Hermitian inner product. Otherwise, the Clifford module can be defined as a linear map ρ : V → End S such that ρ(u)ρ(v) + ρ(v)ρ(u) + 2(u, v) · 1 = 0

for all

u, v ∈ V.

Then by the universal property such a map extends to a representation ρ : Cl(V ) → End S. The action of the representation ρ on the space S is often denoted by ρ(x)s := x · s

for x ∈ Cl(V ),

s∈S

and called the Clifford multiplication. 5.2. Semispinor spaces. For the n-dimensional Euclidean vector space V provided with an orthonormal basis {ei }ni=1 , the Clifford algebra Cl(n) has the volume element ω = e1 . . . en . In the complex case, along with ω, we can introduce the complex volume element ω c given by the formula ω c := i[(n+1)/2] ω, where [·] denotes the integer part of a number. For odd n the elements ω and ω c belong to the center of the Clifford algebra. Moreover, ω 2 = 1 for n ≡ 3, 4 mod 4,

(ω c )2 = 1 for all n.

Consider first the real volume element and suppose that n ≡ 3, 4 mod 4, so that ω 2 = 1. Then the elements π± := (1 ± ω)/2 are the mutually orthogonal idempotents and π+ + π− = 1. Hence the Clifford algebra Cl(V ) can be decomposed as Cl(V ) = Cl+ (V ) ⊕ Cl− (V ),

where Cl± (V ) := π± Cl(V ).

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In the same way any Clifford module S over Cl(V ) can be represented in the form S = S+ ⊕ S−, where S ± are the eigenspaces of the operator ρ(ω) with eigenvalues ±1. The subspaces S ± are called the semispinor spaces. In the complex case for any odd n we have an analogous representation of the complexified Clifford algebra Cl(V ) = Cl+ (V ) ⊕ Cl− (V ),

where Cl± (V ) := (1 ± ω c )Cl(V ).

5.3. Description of irreducible Clifford modules. Proposition 2. Let ρ : Cl(n) → End S be an irreducible representation of the Clifford algebra and n = 4m + 3. Then ρ(ω) = ±Id. Both possibilities are realized, and the corresponding Clifford representations are not equivalent to each other. An analogous assertion holds for the complexified Clifford algebra Cl(n) for any odd n. Proof. Represent the space of spinors S as the direct sum S = S + ⊕ S − of (±1)-eigenspaces of the operator ρ(ω). These subspaces are invariant under the Clifford multiplication because the element ρ(ω) is central. Since the representation S is irreducible, we have either S = S + or S = S − , which proves the first assertion of the proposition. These irreducible representations are not equivalent to each other because if ρ(ω) = ±Id, i.e., ρ is a multiple of Id, then the same property holds for any equivalent representation ρ1 and ρ1 (ω) will be a multiple of Id with the same proportionality coefficient as ρ(ω). In order to check that both possibilities are realized, it is sufficient to consider the action of the Clifford algebra Cl(V ) on Cl± (V ) by left multiplication. The proof in the complex case is analogous to the above.  Before turning to the study of irreducible representations of the Clifford algebra in the case n = 4m, let us prove the following lemma, which relates the Clifford algebra Cl(n − 1) with the even part of the Clifford algebra Cl(n). Lemma 1. For any n > 1 we have an algebra isomorphism Cl(n − 1) → Clev (n). Proof. Choose an orthonormal basis e1 , . . . , en−1 , en in the space Rn and denote by Rn−1 the subspace generated by the first n − 1 vectors of this basis. Consider the map f : Rn−1 → Clev (n) given on the basis elements by the formula i = 1, . . . , n − 1.

n−1 . We Compute the value of (f (x))2 on an arbitrary element x = n−1 i=1 xi ei from the space R have       2 xi en ei xj en ej = xi xj en ei en ej = xi xj ei ej = xx = −x2 · 1. (f (x)) = f (ei ) = en ei ,

i

j

i,j

i,j

Hence by the universal property the constructed map extends to an algebra homomorphism f : Cl(n − 1) → Clev (n). Considering it on the basis elements, one easily sees that this map is an isomorphism. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Proposition 3. Let ρ : Cl(n) → End S be an irreducible representation of the Clifford algebra and n = 4m. Consider the decomposition of the space S into the direct sum of semispinor subspaces S = S+ ⊕ S−. Then each of the subspaces S ± is invariant under the even part Clev (n) of the Clifford algebra Cl(n). These subspaces correspond to two different representations of the algebra Cl(n − 1) ∼ = Clev (n). An analogous assertion holds for the complexified Clifford algebra Cl(n) for any even n. Proof. The subspaces S ± are invariant under Clev (n) since for even n the volume element ω commutes with any element from Clev (n). Under the action of the isomorphism Cl(n − 1) ∼ = Clev (n) from Lemma 1, the volume element ω  of the algebra Cl(n − 1) is sent to the volume element ω ∈ Clev (n) of the algebra Cl(n). Hence, ρ(ω  ) = Id on S + and ρ(ω  ) = −Id on S − . By the previous proposition these representations are not equivalent to each other. The proof in the complex case is similar to the above.  5.4. Spin representation. Definition 7. The real spin representation of the group Spin(n) is a group homomorphism Δn : Spin(n) → GL(S) obtained by restricting an irreducible representation ρ : Cl(n) → End S of the Clifford algebra Cl(n) to the group Spin(n) ⊂ Clev (n) ⊂ Cl(n). Proposition 4. For n = 4m + 3 the representation Δn is irreducible and, being the restriction of an irreducible representation of the algebra Cl(n), does not depend on the choice of this representation. For n = 4m the representation Δ4m can be decomposed into the direct sum − Δ4m = Δ+ 4m ⊕ Δ4m

of two inequivalent irreducible representations of the group Spin(n). Proof. If n = 4m + 3, then the grading automorphism χ interchanges the terms Cl+ (n) and − Cl (n) in the decomposition Cl(n) = Cl+ (n) ⊕ Cl− (n), since χ(ω) = −ω. Hence, the elements of the subalgebra Clev (n), which are invariant under χ by definition, should have the form (x, χ(x)) ∈ Cl+ (n) ⊕ Cl− (n); i.e., the subalgebra Clev (n) is embedded diagonally into Cl+ (n) ⊕ Cl− (n). Since the two irreducible representations of the algebra Cl(n) differ from each other by the homomorphism χ, they become equivalent when restricted to Clev (n). This proves the first assertion of the proposition. If n = 4m, then the restriction of a representation of the algebra Cl(n) to the even part Clev (n) decomposes into the direct sum of two inequivalent irreducible representations. Each of these representations restricts to the group Spin(n) as an irreducible one because the group Spin(n) contains the basis of the algebra Clev (n) considered as a vector space.  Definition 8. The complex spin representation of the group Spin(n) is a group homomorphism Δcn : Spin(n) → GL(S, C) obtained by restricting an irreducible representation ρ : Cl(n) → End S of the complexified Clifford algebra Cl(n) to the group Spin(n) ⊂ Clev (n) ⊂ Cl(n). Proposition 5. For odd n the representation Δcn is irreducible and, being the restriction of an irreducible representation of the algebra Cl(n), does not depend on the choice of this representation. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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For even n = 2m the representation Δc2m can be decomposed into the direct sum Δc2m = (Δc2m )+ ⊕ (Δc2m )− of inequivalent irreducible representations of the group Spin(n). The proof is analogous to that of the previous proposition. Here are examples of spin representations of low dimensions: (1) the representation Δ3 has dimension 2, which yields a homomorphism (depending on the choice of an orthonormal basis) Spin(3) → SU(2), which is an isomorphism by the dimension-counting argument; i.e., Spin(3) ∼ = SU(2); ± (2) both representations Δ4 are two-dimensional; as in the previous case, the representation − Δ+ 4 ⊕ Δ4 yields an isomorphism Spin(4) ∼ = SU(2) × SU(2). 5.5. Spin representation in the complex case. Let V be a 2m-dimensional Euclidean vector space identified with the m-dimensional complex vector space. We will assume that this space is K¨ahler, i.e., it has a Hermitian inner product ·, · compatible with the Euclidean one. Denote by V ∗ the Hermitian dual space of V . Its complexification VC∗ = V ∗ ⊗R C can be represented in the form VC∗ = V 1,0 ⊕ V 0,1 . Hence for any vector v ∈ V the dual covector v ∗ can be represented as v ∗ = v 1,0 + v 0,1 . We will construct a canonical representation of the Clifford algebra Cl(n), n = 2m, in the space Scan := Λ0,∗ V ∗ =

m 

Λ0,q (V ∗ ).

q=0

The representation ρcan : V → End Scan in this space is defined by the formula ρcan (v)η = v 0,1 ∧ η − v 0,1  η,

v ∈ V,

η ∈ Λ0,q (V ∗ ),

where  stands for the inner product. It can be checked that ρcan (v) ◦ ρcan (v)η = −v2 η, so by the universal property the map ρcan : V → End Scan extends to a representation of the Clifford algebra ρcan : Cl(n) → End Scan . The semispinor spaces coincide in this case with + = Λ0,ev (V ∗ ), Scan

− Scan = Λ0,od (V ∗ ).

6. SPIN STRUCTURES 6.1. Spin structures on vector bundles. Let p : E → M be a real vector bundle of rank n over a smooth compact oriented Riemannian manifold M . We will assume that this bundle is Riemannian, i.e., in each fiber Ex , x ∈ M , a positive definite inner product (·, ·) is given that depends smoothly on the point x ∈ M . We will also assume that E is oriented, i.e., in each fiber Ex an orientation is given that depends smoothly on the point x ∈ M . In contrast to the Riemannian structure, which exists on any smooth vector bundle, the orientability of E takes place only under the following topological condition. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Proposition 6. A bundle p : E → M is orientable if and only if its first Stiefel–Whitney class vanishes: w1 (E) = 0. For a proof see [5, Ch. II, § 1]. Assume now that p : E → M is an orientable Riemannian vector bundle of rank n and PSO (E) is the bundle of oriented orthonormal bases (below called frames for brevity) in the fibers of E. Suppose that n ≥ 2, so that the homomorphism π : Spin(n) → SO(n) is a double covering. Definition 9. The spin structure on the vector bundle p : E → M is given by a principal Spin(n)-bundle PSpin (E) together with the double covering bundle map Π : PSpin (E) → PSO (E), which is Spin-equivariant in the sense that Π(sg) = Π(s)π(g)

s ∈ PSpin (E),

for all

g ∈ Spin(n).

Here, the action of Spin(n) on PSO (E) is given by the homomorphism π : Spin(n) → SO(n). This definition can be rewritten in the form of the commutative diagram PSpin (E)

Π

PSO (E)

M in which the restriction of Π to the fibers coincides with the homomorphism π. Proposition 7. A spin structure on a vector bundle p : E → M exists if and only if the second Stiefel–Whitney class vanishes: w2 (E) = 0. Under this condition, different spin structures on E are indexed by the elements of the cohomology group H 1 (M, Z2 ). For a proof see [5, Ch. II, § 1]. Definition 10. An oriented Riemannian manifold M is called a spin manifold if its tangent bundle T M admits a spin structure. Here are examples of spin manifolds: (1) a complex manifold M is a spin manifold if and only if its first Chern class is even, i.e., c1 (M ) ≡ 0 mod 2; this fact follows from the relation w2 (M ) ≡ c1 (M ) mod 2; (2) the complex projective space CPn is a spin manifold if and only if n is odd; (3) let Σg be a compact Riemann surface of genus g; then it has 22g different spin structures. 6.2. Spinor and Clifford bundles. Recall a general construction of the associated bundle. Let π : P → M be a principal G-bundle over a manifold M and F be another smooth bundle on which the group G acts smoothly from the right, i.e., there is a homomorphism ρ : G → Diff(F ). Then we can construct a new bundle P ×ρ F with fiber F and structure group G. To this end consider the left action of G on the product P × F given by the formula g(s, f ) := (gs, f ρ(g −1 )),

where

s ∈ P,

f ∈ F,

g ∈ G.

The quotient of P × F by this action is denoted by P ×ρ F , and its elements are given by the orbits [s, f ] of the elements (s, f ) ∈ P × F under the above action of G. The projection πρ : P ×ρ F → M is defined by the formula πρ ([s, f ]) := π(s). If ρ : G → GL(V ) is a linear representation of the group G in a vector space V , then the associated bundle P ×ρ F is a vector bundle over M . PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Example 1. Let M be an oriented Riemannian manifold of dimension n and P = PSO (M ) be the principal SO(n)-bundle of frames on M . Denote by ρ the standard action of the group SO(n) on the space Rn . Then T M = PSO (M ) ×ρ Rn . In a more general way, if E → M is an oriented Riemannian vector bundle over M , then E = PSO (E) ×ρ Rn . We turn now to the Clifford algebras. Every orthogonal transform of the space V = Rn generates an automorphism of the Clifford algebra Cl(n). Indeed, such a transform maps the tensor algebra T (V ) into itself and preserves the ideal J (V ), which determines the Clifford algebra, so it can be pushed down to the Clifford algebra Cl(n). Thus, we have a representation cl : SO(n) → Aut(Cl(n)). Definition 11. Let E → M be an oriented Riemannian vector bundle. Its Clifford bundle is the bundle of the form Cl(E) := PSO (E) ×cl Cl(n). In other words, Cl(E) is the bundle of Clifford algebras constructed from the bundle E of Euclidean spaces. Therefore, all constructions that have been introduced for the Clifford algebras also extend to the Clifford bundles. In particular, we can introduce a natural inner product on Cl(V ). Definition 12. Let E be an oriented Riemannian vector bundle provided with a spin structure Π : PSpin (E) → PSO (E) and Δn : Spin(n) → GL(S) be the real spin representation. The real spinor bundle over E is the bundle of the form S(E) := PSpin (E) ×Δn S. If Δcn : Spin(n) → GL(S c , C) is the complex spin representation, then the bundle S c (E) := PSpin (E) ×Δcn S c is called the complex spinor bundle. We will assume in the sequel that the bundles S and S c are provided with Hermitian structures. Having defined the spinor bundles, we can introduce semispinor bundles. To this end consider the section ω c of the bundle Cl(E) of rank n = 2m that is given at the point x ∈ M by the formula ω c = im e1 . . . e2m in terms of an oriented orthonormal basis {ej } of the space Ex . Then (ω c )2 = 1 and we can define c (E) as the (±1)-eigenbundles of the operator of Clifford multiplication the semispinor bundles S± c by ω . So c (E) = PSpin (E) ×(Δc2m )± S c . S±

For n = 4m an analogous construction also exists in the real case. Namely, let ω be a section of the bundle Cl(E) given by the formula ω = e1 . . . e4m in terms of an oriented orthonormal basis {ej } of the space Ex . Then ω 2 = 1 and we have the decomposition S(E) = S+ (E) ⊕ S− (E) into the direct sum of (±1)-eigenbundles of the operator of Clifford multiplication by ω. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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7. SPINOR CONNECTIONS 7.1. Connections in principal bundles. Let π : P → M be a principal G-bundle over a smooth manifold M of dimension n. Denote by V the distribution in the tangent bundle T P formed by the spaces Vp tangent to the fiber of P at p ∈ P : Vp = {v ∈ Pp : π∗ (v) = 0}. The subspace Vp is called the vertical subspace at the point p. Note that for any principal bundle π : P → M there is a homomorphism of the Lie algebra g of the Lie group G into the Lie algebra Vect(P ) of smooth vector fields on P . This homomorphism is determined by the left action of the group G on P ; namely, with an element ξ of the Lie algebra g, we associate the vector field X given at the point p ∈ P by the formula Xp = p∗ (ξ) :=

 d exp{tξ} · p t=0 . dt

The vector fields X obtained in this way are vertical; moreover, for any v ∈ Vp there exists a vector field X of the above type such that Xp = v. In other words, the assignment g  ξ → Xp ∈ Vp allows us to identify the vertical subspace Vp with the Lie algebra g. Definition 13. The connection in the principal bundle π : P → M is a smooth distribution H : P  p → Hp of subspaces Hp ⊂ Tp P , called horizontal, that has the following properties: (1) the tangent map π∗ : Hp → Tπ(p) M is an isomorphism of vector spaces for every p ∈ P ; (2) the distribution H is G-invariant; i.e., g∗ Hp = Hpg for all p ∈ P and g ∈ G, where by g∗ we denote the map tangent to the right action of g on P . In other words, the tangent space Tp P at any point p ∈ P can be decomposed into the direct sum Tp P = Vp ⊕ Hp of vertical and horizontal subspaces; i.e., the connection is a G-invariant mean of choosing a distribution H complementary to the vertical distribution V . At every point p ∈ P the horizontal subspace Hp determines the projection Tp P → Vp parallel to Hp . Using the above isomorphism Vp ∼ = g, we can construct a linear map ωp : Tp P → g. This defines a 1-form ω on P with values in the Lie algebra g. The 1-form ω is called the connection form on P and has the following properties: (1) ω is vertical; i.e., it vanishes on horizontal vectors; (2) for any element ξ ∈ g the following equality holds: ω(Xp ) = ξ

for all

p ∈ P,

where Xp = p∗ (ξ); (3) ω is equivariant in the sense that g∗ (ω) = Adg ω. From any smooth 1-form ω satisfying these properties, one can reconstruct the connection H by setting Hp := Ker ωp . Definition 14. The curvature of a connection H in the principal bundle π : P → M is a smooth 2-form Ω on P with values in the Lie algebra g that is defined by the equality Ω = dω + [ω, ω]. This form is horizontal; i.e., it vanishes on any pair of vectors if at least one of them is vertical. And this form is equivariant; i.e., it transforms under the action of elements g ∈ G by the formula g∗ Ω = Adg Ω. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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7.2. Riemannian connections. Let PSO (E) be the principal SO(n)-bundle of frames (oriented orthonormal bases) of an oriented Riemannian vector bundle E of rank n over an oriented Riemannian manifold M . The Lie algebra so(n) of the Lie group SO(n) coincides with the space of real skew-symmetric n × n matrices, so the connection form ω on this bundle can be considered as an n × n matrix ω = (ωij ) with entries ωij being 1-forms satisfying the relation ωij + ωji = 0. The curvature Ω in this case will be given by a matrix Ω = (Ωij ) whose entries are 2-forms of the type n  ωik ∧ ωkj , Ωij = dωij + k=1

while the map Ad acts by the formula Adg ω = gωg−1 . Definition 15. The covariant derivative on E is a linear map ∇ : Γ(E) → Γ(T ∗ M ⊗ E) satisfying the Leibniz rule ∇(f s) = df ⊗ s + f ∇s

for all f ∈ C ∞ (M ),

s ∈ Γ(E).

If X is a smooth tangent vector field on M , then the convolution with X generates a linear map ∇X : Γ(E) → Γ(E), called the covariant derivative along X. Let ω be a connection form on PSO (E) and e = {e1 , . . . , en } be a local orthonormal basis of sections of the bundle E in a neighborhood U of a point x0 ∈ M . Then e determines a local section e : U → PSO (E) of the bundle PSO (E) over U . The dual tangent map e∗ : T ∗ (PSO (E)) → T ∗ U allows one to transport the connection form ω = (ωij ) to U by setting ω  := e∗ ω. With this remark taken into account, we have the following Proposition 8. Let ω be a connection form on the bundle PSO (E). Then it uniquely determines a covariant derivative ∇ on E given in terms of a local orthonormal basis of sections of E by the formula n  ω ij ⊗ ej , where ω  = e∗ ω. (7.1) ∇ei = j=1

This covariant derivative is compatible with the Riemannian structure of E in the sense that Xs, s  = ∇X s, s  + s, ∇X s  for any smooth tangent vector field X and any smooth sections s, s ∈ Γ(E), where ·, · denotes the inner product on E. The constructed covariant derivative is called Riemannian. Conversely, any covariant derivative ∇ on E that is compatible with the Riemannian structure determines a unique connection with the connection form ω on PSO (E) defined by the formula (7.1). For a proof see [5, Ch. II, § 4]. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Let ∇ be a covariant derivative in the bundle E. Consider the composition  ∇

∇

→ Γ(T ∗ M ⊗ E) − → Γ(Λ2 (T ∗ M ) ⊗ E), Γ(E) −  is a natural extension of the covariant derivative ∇ to the sections of the bundle T ∗ M ⊗ E where ∇ of the form η ⊗ s with η a 1-form on M and s a section of E. This extension is defined by the formula  ⊗ s) := dη ⊗ s − η ∧ ∇s. ∇(η  ◦ ∇ is called the Riemannian curvature. The composition R := ∇ Proposition 9. With the notation of Proposition 8, let Ω = (Ωij ) be the curvature of the connection form ω. Then in terms of a local orthonormal basis of sections of the bundle E the Riemannian curvature is given by the formula Rei =

n 

 ij ⊗ ej , Ω

where

 = (Ω  ij ) = e∗ Ω. Ω

j=1

For any smooth tangent vector fields X and Y on M we have   RX,Y s = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] s. The map RX,Y : Γ(E) → Γ(E), called the curvature transform, has the following symmetry property: RX,Y s, s  + s, RX,Y s  = 0. For a proof see [5, Ch. II, § 4]. 7.3. Connections in Clifford and spinor bundles. The construction of connections in Clifford bundles is based on the following argument. Let π : P = PG → M be a principal G-bundle over a manifold M . Suppose a faithful representation ρ : G → SO(n) of the group G in the space Rn is given. Denote by E = Eρ the Riemannian vector bundle over M associated with P . In other words, E = Eρ = P ×ρ Rn . Then, given a connection H on the bundle P , we can construct an induced canonical connection Hρ on the principal bundle P (E) = P (Eρ ) = P ×ρ SO(n), where the group G acts on SO(n) from the right via the homomorphism ρ. In order to construct the desired connection Hρ on P (E), we extend the connection H defined on the bundle P trivially to the direct product P × SO(n) and then push the obtained connection to P ×ρ SO(n) using the invariance of the connection H. This defines the connection Hρ on the bundle P (Eρ ). Note that there is a canonical G-equivariant map i : P → P (E) given by the formula P  p → [p, Id], where [p, h] denotes the class of the pair (p, h) ∈ P × SO(n) in the quotient P ×ρ SO(n). This map is an embedding due to the faithfulness of the representation ρ. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Proposition 10. Let ω be the connection form of a connection H on the bundle PG with curvature Ω. Denote by ωρ the form of the induced connection Hρ on the bundle P (Eρ ) with curvature Ωρ . Then ωρ |P = ρ∗ ω,

Ωρ |P = ρ∗ Ω,

where ρ∗ : g → so(n) is the Lie algebra homomorphism tangent to the representation ρ : G → SO(n). For a proof see [5, Ch. II, § 4]. We apply the described construction to Clifford bundles. Let E → M be an oriented Riemannian bundle of rank n equipped with a Riemannian connection. Recall that the Clifford bundle Cl(E) is the bundle associated with the principal SO(n)-bundle PSO (E) via the homomorphism cl : SO(n) → Aut(Cl(n)), i.e., Cl(E) = PSO (E) ×cl Cl(n). Hence, in accordance with the described construction, the Riemannian connection on E generates a connection on the Clifford bundle Cl(E) in a canonical way. The covariant derivative corresponding to this connection has the following characteristic property. Proposition 11. The covariant derivative ∇ of the constructed connection on the Clifford bundle Cl(E) acts as a differentiation of sections of the Clifford bundle; i.e., ∇(σ · τ ) = (∇σ) · τ + σ · (∇τ ) for any sections σ, τ ∈ Γ(Cl(E)). Note that under the canonical identification of Cl(E) with the bundle Λ(E) the covariant derivative ∇ will correspond to the differentiation of Λ(E) that preserves the subbundles Λk (E). This implies, in particular, that ∇ also preserves the subbundles Clev (E) and Clod (E), and the volume element ω = e1 . . . en is parallel with respect to ∇, i.e., ∇ω = 0. So for n ≡ 3, 4 mod 4 the subbundles Cl± (E) are also preserved by the covariant derivative ∇. The proof of the next proposition can be found in [5, Ch. II, § 4]. Proposition 12. For any pair of tangent vector fields X, Y in a neighborhood of a point x ∈ M the curvature transform RX,Y : Cl(Ex ) → Cl(Ex ) is a differentiation, i.e., RX,Y (σ · τ ) = RX,Y (σ) · τ + σ · RX,Y (τ ) for any sections σ, τ ∈ Cl(Ex ). This transform also preserves the subspaces Clev (Ex ), Clod (Ex ), and Cl± (Ex ). Assume now that the bundle E is equipped with a spin structure, i.e., there is a bundle epimorphism Π : PSpin (E) → PSO (E). Consider the associated spinor bundle S(E) = PSpin (E) ×Δn S constructed using the spin representation Δn : Spin(n) → S. Then the connection on the bundle PSO (E) can be pulled up with the help of the epimorphism Π to a connection on the bundle PSpin (E). The obtained connection in turn generates a connection on the spinor bundle S(E) in a canonical way. It satisfies analogs of Propositions 11 and 12. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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In the case when E = T M , we set PSO (M ) := PSO (T M ) and Cl(M ) := Cl(T M ). If we have a connection ∇ on the bundle PSO (M ), then we can define the following tensor field T : with any pair of vector fields X, Y in a neighborhood of a point x ∈ M , we associate the quantity TX,Y := ∇X Y − ∇Y X − [X, Y ], called the torsion tensor of the connection ∇. The tensor T determines a 2-form on the manifold M with values in the tangent bundle T M . Theorem 1. Let M be a Riemannian manifold and PSO (M ) be the bundle of frames on T M . Then there exists a unique connection on PSO (M ) with vanishing torsion tensor. Such a connection is called the canonical Riemannian connection or the Levi-Civita connection on M . It induces the canonical connection on the Clifford bundle Cl(M ). In the case when M admits a spin structure, this connection also induces the canonical Riemannian connection on the bundle PSpin (M ) and, hence, on any spinor bundle associated with PSpin (M ). The curvature tensor R of the canonical Riemannian connection on M has the following symmetry properties: RX,Y Z + RY,Z X + RZ,X Y = 0,

RX,Y Z, U  = RZ,U X, Y 

for any tangent vector fields X, Y , Z, and U in a neighborhood of a point x ∈ M . 8. THE DIRAC OPERATOR 8.1. Main definitions. Let M be an oriented Riemannian manifold. Using the Riemannian metric, we can, in particular, identify the tangent space Tx M at any point x ∈ M with the cotangent space Tx∗ M . Denote by Cl(M ) the Clifford bundle over M . Suppose a bundle S of Clifford modules over M is given so that every fiber Sx is a module over the Clifford algebra Cl(Tx M ). We will also assume that S is Riemannian and provided with a Riemannian connection ∇. Denote by ρ : Γ(Cl(M )) ⊗ S → S the Clifford multiplication map on S. Definition 16. The Dirac operator on S generated by the connection ∇ is a linear differential operator D : Γ(S) → Γ(S) of the form ∇

j⊗1

ρ

→ Γ(T ∗ M ⊗ S) −−→ Γ(Cl(M ) ⊗ S) − → Γ(S), Γ(S) − where the embedding j comes from the identification of Λ(T ∗ M ) = Λ(T M ) with Cl(T M ). In terms of a local orthonormal basis {e1 , . . . , en } of the bundle T M , the Dirac operator D is given by the formula n  ej · ∇ej s, Ds = j=1

where s ∈ Γ(S). The operator D 2 is called the Dirac Laplacian. Lemma 2. Let f ∈ C ∞ (M ) be a smooth function on M treated as the operator of multiplication by sections from Γ(S). Then [D, f ] = ρ(df ). Proof. Since the multiplication by f commutes with the Clifford multiplication, we have [D, f ]s = ρ∇(f s) − f ρ(∇s) = ρ(df ⊗ s) = ρ(df )s for all s ∈ Γ(S).



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Recall that the principal symbol of a differential operator D : Γ(E) → Γ(E) given in local coordinates x in a neighborhood of a point x0 ∈ M by the formula D=



aα (x)

|α|≤m

∂ |α| ∂xα

is the map that assigns the linear map σξ (D) : Ex → Ex of the form  aα (x)ξ α σξ (D) = im |α|=m

to every x ∈ M and every covector ξ = ξj dxj ∈ Tx∗ M \ {0}. The operator D is called elliptic if the map σξ (D) : Ex → Ex is nondegenerate for any x ∈ M and ξ ∈ Tx∗ M \ {0}. Lemma 3. For every ξ ∈ Tx∗ M \ {0} the principal symbols of the Dirac operator and Dirac Laplacian are equal to and

σξ (D) = iξ

σξ (D 2 ) = ξ2 ,

where both operators act on S via Clifford multiplication. In particular, both operators D and D 2 are elliptic. Proof. Fix a point x0 ∈ M and a local frame {e1 , . . . , en } of the bundle T M in a neighborhood of x0 . Choose local coordinates in a neighborhood of x0 so that x0 = 0 and ej = ∂/∂xj at x0 . Then for any local trivialization of the bundle S in a neighborhood of x0 the covariant derivative ∇ej in a neighborhood of x0 will be written in the form ∂ + (zeroth-order terms), ∂xj

∇ej =

while the Dirac operator will have the form D=

 j

ej

∂ + (zeroth-order terms). ∂xj



So, for any covector ξ = ξj dxj ∈ Tx∗ M \ {0} in a neighborhood of x0 , we have the following relations:  ej ξj = iξ, σξ (D 2 ) = (σξ (D))2 = −ξ · ξ = ξ2 .  σξ (D) = i j

8.2. General properties. Now we impose some natural additional conditions on the bundle of Clifford modules S. First, we will suppose that the bundle S is equipped with a Riemannian structure that is compatible with the Clifford multiplication in the sense that the Clifford multiplication by unit vectors from T M is an orthogonal transform of S, so that for any x ∈ M the relation e · s1 , e · s2  = s1 , s2 

(8.1)

holds for any unit vector e ∈ Tx M and arbitrary sections s1 , s2 ∈ Γ(S) at x. Second, we will assume that the bundle S is provided with a Clifford connection ∇ satisfying the Leibniz rule ∇(σ · s) = (∇σ) · s + σ · (∇s), where σ ∈ Γ(Cl(M )), s ∈ Γ(S), and by ∇σ we mean the action of the canonical Riemannian connection on the Clifford bundle. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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The bundles S of Clifford modules that have the above properties will be called Dirac bundles. Introduce the inner product on S by the formula  (s1 , s2 ) := s1 , s2  vol, M

where s1 , s2 ∈ Γ(S) and vol is the volume form on M . Proposition 13. The Dirac operator D on a Dirac bundle S is formally self-adjoint; i.e., (Ds1 , s2 ) = (s1 , Ds2 ) for any smooth sections s1 , s2 ∈ C ∞ (S) with compact supports on M . Proof. Fix a point x0 ∈ M and a local frame {e1 , . . . , en } of the bundle T M in a neighborhood of x0 so that ∇ei ej = 0 at x0 . (One can construct such a frame by choosing it first at the point x0 and then extending it to a neighborhood of x0 with the help of parallel transport along geodesics issued from x0 .) Then the computation at the point x0 yields Ds1 , s2  =

 j

=−

ej ∇ej s1 , s2  = −

 ∇ej s1 , ej · s2  j

 ej s1 , ej · s2  + s1 , Ds2 . ej s1 , ej · s2  − s1 , (∇ej ej )s2  − s1 , ej ∇ej s2  = −

 j

j

Introduce the vector field X defined by the equality X, Y  = −s1 , Y s2  for an arbitrary tangent vector field Y . In terms of this vector field we can rewrite the first term in the last formula in the above chain of equalities in the form     ej s1 , ej · s2  = − ej X, ej  = ∇ej X, ej  = : div X. ej X, ej  − X, ∇ej ej  = − j

j

j

j

Thus, we have shown that Ds1 , s2  = div X + s1 , Ds2 . Since s1 and s2 are compactly supported, the first term on the right-hand side vanishes after the integration over M and we obtain the required assertion.  Remark 1. In the case of a manifold M with boundary ∂M , the above argument yields the following Stokes formula:  ν · s1 , s2  vol, (Ds1 , s2 ) − (s1 , Ds2 ) = ∂M

where ν is the outward normal to ∂M . Remark 2. It follows from the general theory of elliptic operators that any weak solution of the equation Ds = 0 is, in fact, C ∞ -smooth. If the manifold M is compact, then this theory also implies that the space of solutions of the equation Ds = 0 is finite-dimensional. Denote by L2 (S) the space of L2 -sections of the bundle S obtained by taking the closure of the 2 space Γ∞ 0 (S) of smooth sections of S with compact supports with respect to the L -norm introduced PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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above. The Dirac operator D is a symmetric operator on Γ∞ 0 (S) and so admits the closure with 2 respect to the norm of L (S). The operator thus obtained is an unbounded self-adjoint operator in L2 (S). Recall that the Dirac operator D has the principal symbol iξ, and the principal symbol of its square D 2 is equal to ξ2 , which coincides with the principal symbol of the Laplace–Beltrami operator on M . If we introduce the spinor Laplacian ΔS as  S ◦ ∇S ), ΔS := − Trg (∇ where ∇S is a spinor connection on S, then this operator will be related to the Dirac Laplacian by the formula 1 D 2 = ΔS + scalg , 4 where scalg is the scalar curvature of (M, g). This formula, called the Lichnerowicz formula (cf. [5, Ch. II, § 8]), is a particular case of the general Weitzenb¨ ock formula. 8.3. Classical Dirac operator. Let M = Rn and S = Rn × S0 where S0 is some fixed Clifford module over Cl(n). In this case the Dirac operator D is a differential operator with constant coefficients of the form n  ∂ γj , D= ∂xj j=1

which acts on S0 -valued functions defined on Rn . Here, γj are the Dirac matrices, i.e., linear maps γj : S0 → S0 satisfying the relations γj γk + γk γj = −2δjk

for all j, k = 1, . . . , n. It follows from these relations that D 2 = Δ · Id, where Δ = − nj=1 ∂ 2 /∂x2j is the Laplacian on Rn . Here are examples of Dirac operators in low dimensions. 1. For n = 1 the Clifford algebra Cl(1) is C and the operator D is i ∂/∂x1 . 2. For n = 2 the Clifford algebra Cl(2) is H = C ⊕ C = Clev (2) ⊕ Clod (2) and the operator D interchanges Clev (2) and Clod (2). Introduce real coordinates on H by writing the quaternions q ∈ H in the form q = x0 1 + x1 e1 + x2 e2 + x3 e3 . If we identify Clev (2) and Clod (2) with C by using the maps u + ve2 e1 ↔ u + iv ↔ ue1 + ve2 , then the operator D = e1 ∂/∂x1 + e2 ∂/∂x2 will be given by the matrix   0 −∂/∂z D= , ∂/∂ z¯ 0 where ∂/∂ z¯ = ∂/∂x1 + i ∂/∂x2 . In other words, the restriction of this operator to the space Clev (2) coincides with the Cauchy–Riemann operator. 3. For n = 3 the Clifford algebra Cl(3) is H ⊕ H and S0 = H. This algebra has two representations in H that act as follows. We identify the space R3 with the space Im H of imaginary quaternions by introducing the standard basis in Im H given by the imaginary units i, j, and k. Then the action of these representations in H will be given by the multiplication by basis quaternions from the left or from the right, respectively. By choosing the left action, we see that the Dirac operator coincides with the operator of the form ∂ ∂ ∂ +j +k D=i ∂x1 ∂x2 ∂x3 acting on the H-valued functions on the space R3 = Im H. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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4. For n = 4 the Clifford algebra Cl(4) is Mat2 (H) and S0 = H ⊕ H = Clev (4) ⊕ Clod (4). As in the case n = 2, the operator D interchanges Clev (4) with Clod (4). We identify the space R4 with H by choosing the standard basis in H formed by 1, i, j, and k. We also introduce a quaternionic analog of the Cauchy–Riemann operator acting on the functions H → H by the formula ∂ ∂ ∂ ∂ ∂ = +i +j +k , ∂ q¯ ∂x0 ∂x1 ∂x2 ∂x3 or in terms of the Pauli matrices     1 0 i 0 , σ1 = , σ0 = 0 1 0 −i

σ2 =

  0 −1 , 1 0

 σ3 =

0 i i 0



by the formula ∂ ∂ ∂ ∂ ∂ = σ0 + σ1 + σ2 + σ3 . ∂ q¯ ∂x0 ∂x1 ∂x2 ∂x3 Then the Dirac operator D acting on the functions with values in S0 = H ⊕ H will be given by a matrix of the form   0 −∂/∂q D= . ∂/∂ q¯ 0 9. Spinc -STRUCTURE 9.1. Spinc -structures on principal bundles. Let M be a compact oriented Riemannian manifold of dimension n and PSO → M be the principal SO(n)-bundle of frames (oriented orthonormal bases) on M . Then the Spinc -structure on PSO → M is the pullback of this bundle to a principal Spinc (n)-bundle over M . Let us give a more formal Definition 17. The Spinc -structure on the principal bundle PSO → M is a principal Spinc (n)bundle PSpinc → M together with a Spinc (n)-equivariant bundle epimorphism PSO

PSpinc M

where Spinc (n) acts on the bundle PSO → M via the homomorphism π : Spinc (n) → SO(n). With the bundle PSpinc → M , we associate a principal U(1)-bundle PU(1) → M together with a Spinc (n)-equivariant bundle epimorphism so that the diagram PU(1)

PSpinc M

where Spinc (n) acts on the bundle PU(1) → M via the homomorphism δ : Spinc (n) → U(1), is commutative. The complex line bundle L → M associated with the principal bundle PU(1) → M is called the characteristic bundle, and its Chern class c1 (L) is called the characteristic class of the Spinc -structure under consideration. In terms of the introduced bundle the Spinc -structure can also be defined in the following equivalent way. Definition 18. Let PSO → M be the principal SO(n)-bundle of frames on M . Then the Spinc -structure on PSO → M is given by a principal Spinc (n)-bundle PSpinc → M and a principal PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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U(1)-bundle PU(1) → M together with a Spinc (n)-equivariant bundle epimorphism PSO × PU(1)

PSpinc M

where Spinc (n) acts on the bundle PSO × PU(1) via the homomorphism (π, δ). The above definition can be extended to arbitrary oriented Riemannian vector bundles E → M of rank n over a compact oriented Riemannian manifold M that are associated with the principal bundle PSO → M , i.e., E = PSO ×SO(n) Rn . Definition 19. The Spinc -structure on a bundle E → M is an extension of its structure group from SO(n) to Spinc (n). In other words, the bundle E → M admits a Spinc -structure if it is the bundle associated with a principal Spinc (n)-bundle PSpinc → M , i.e., there exists a Spinc (n)equivariant bundle epimorphism making the diagram PSpinc ×Spinc (n) Rn

E M

commutative, where the group Spinc (n) acts on Rn via the homomorphism π : Spinc (n) → SO(n). If, in particular, we take the tangent bundle T M of an n-dimensional Riemannian manifold M as E, then a Spinc -structure on T M is called the Spinc -structure on the manifold M . Proposition 14. A principal SO(n)-bundle PSO → M admits a Spinc -structure if and only if its second Stiefel–Whitney class w2 (PSO ) is the reduction modulo 2 of some integral class c ∈ H 2 (M, Z), i.e., w2 (PSO ) ≡ c mod 2. For a proof see [5, Theorem D.2]. An analogous assertion is true for the oriented Riemannian vector bundles E → M over M . Recall that the bundle PSO → M admits a Spin-structure if and only if its second Stiefel– Whitney class vanishes: w2 (PSO ) = 0. This implies the following Example 2. Any principal bundle PSO → M provided with a spin structure has a canonical Spinc -structure. In this case the principal Spinc -bundle PSpinc → M is defined as PSpinc = PSpin ×δ U(1), where U(1) denotes the trivial U(1)-bundle over M and the group Spinc (n) acts on the bundle on the right-hand side as Spinc (n) = Spin(n) ×Z2 U(1). Example 3. Any complex vector bundle E → M has a canonical Spinc -structure. Indeed, in this case w2 (E) ≡ c1 (E) mod 2, so the existence of a desired Spinc -structure follows from Proposition 14 (more precisely, from its analog for vector bundles). 9.2. Spinc -structures on complex manifolds. Let M be a Spinc -manifold of dimension n. The complex spinor bundle over M is a complex vector bundle S of the form S = PSpinc ×Δcn S0 , where S0 is a Clifford module and Δcn : Spinc (n) → GL(S0 , C) is the spin representation. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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If n is even, then there are two irreducible representations of the complexified Clifford algebra, which become equivalent when restricted to Spinc (n). If a Spinc -manifold M is complex, then it has a canonical Spinc -structure Scan = Λ0,∗ (M ). The Clifford multiplication ρ : Cl(M ) → End Scan acts in this case as follows. With an arbitrary tangent vector v, we associate the linear map ρ(v) : Λ0,∗ (M ) → Λ0,∗ (M ) given by the formula ρ(v)η = v 0,1 ∧ η − v 0,1  η, where v ∗ = v 1,0 + v 0,1 is the covector Hermitian conjugate to v. Then ρ(v)(ρ(v)η) = −v2 η, so the map ρ extends by the universal property to a representation of the whole Clifford algebra Cl(M ). Other Spinc -structures on a manifold M equipped with the canonical Spinc -structure can be constructed by multiplying tensorially the canonical spinor bundle Scan by some complex line bundle L → M , i.e., by setting S(L) := Scan ⊗ L

and

PSpinc (L) := PSpin ×δ PU(1) (L),

where PU(1) (L) is the principal U(1)-bundle associated with L, and the action of Spinc (n) on the bundle on the right-hand side is given by the homomorphism (π, δ) : Spinc (n) → SO(n) × U(1). So we have an action of the group H 2 (M, Z), which parameterizes the equivalence classes of complex line bundles over M , on the space of Spinc -structures. The quotient by this action, i.e., the space of different Spinc -structures on M , is identified with the group H 1 (M, Z2 ). 9.3. Relation to the spin structure. The definition of Spinc -connections on the bundles equipped with a Spinc -structure is analogous to the definition of Spin-connections on spinor bundles. Such connections can be described as follows. Proposition 15. Let M be a Spinc -manifold which is simultaneously a spin manifold. Then for any Spinc -structure on M corresponding to a complex line bundle L → M and for an arbitrary U(1)-connection on the associated principal bundle PU(1) → M, one can construct a canonical connection on PSpinc → M which is the pullback of the connection on PSO × PU(1) given by the tensor product of the canonical Riemannian connection on PSO and the chosen U(1)-connection on PU(1) . For a proof see [5, Proposition D.11]. In fact, the spin requirement imposed on M is superfluous. The above construction extends to the case of general Spinc -manifolds M if one replaces the U(1)-connection on L by the so-called virtual connection on the virtual bundle L1/2 (cf. [5]). In conclusion consider the problem of describing the spin structure in terms of the Spinc structure. Let M be a Spinc -manifold and S be the corresponding spinor bundle equipped with a Hermitian metric. Proposition 16 (see [2]). A manifold M is a spin manifold if and only if there exists an antilinear isometry C : S → S with the following properties: (1) C(sf ) = (Cs)f¯ for s ∈ Γ(S) and f ∈ C ∞ (M ); (2) C(σs) = χ(¯ σ )(Cs) for s ∈ Γ(S) and σ ∈ Γ(Cl(M )); (3) Cs, Cs  = s , s for s, s ∈ Γ(S). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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II. NONCOMMUTATIVE SPIN GEOMETRY 10. SPECTRAL TRIPLES Definition 20. The spectral triple for an algebra A is a triple (A, H, D) consisting of the algebra A, a Hilbert space H in which a representation π of A by bounded linear operators is given, and a self-adjoint operator D in H with compact resolvent that satisfies the following property: the commutator [D, a] of D with any element a ∈ A (more precisely, with the representation operator π(a) determined by this element) is a bounded linear operator in H. Definition 21. The real spectral triple for an algebra A with involution is a spectral triple together with an antiunitary operator C in H for which the map b → Cb∗ C −1 defines an action of the opposite algebra Ao on H that commutes with the action of A, i.e., [a, Cb∗ C −1 ] = 0

for all

a, b ∈ A.

(10.1)

Recall that the algebra Ao opposite to A is defined as the algebra Ao = {ao : a ∈ A} provided with the multiplication law ao bo := (ba)o for a, b ∈ A. An operator C : H → H is said to be antiunitary if it defines an antilinear bijection H → H that has the following antisymmetricity property: (Cξ, Cη) = (η, ξ)

for all

ξ, η ∈ H

and C 2 = ±1.

If π is an action of the algebra A on H, then the action π o of the opposite algebra Ao on H will be given by the formula π o (b) := Cπ(b∗ )C −1 . Condition (10.1) means that the representations π and π o commute with each other. 11. DEFINITION OF THE NONCOMMUTATIVE SPIN GEOMETRY 11.1. Dimension. • There exists a nonnegative integer n, called the dimension of the geometry, for which D −1 ∈ −n has finite / Ln+ Ln+ (H) but D −1 ∈ 0 (H). This implies, in particular, that the operator |D| and nonzero Dixmier trace (the definition and properties of the Dixmier trace can be found in [8, 9]). In the case when the algebra A and the Hilbert space H are finite-dimensional, the dimension of the geometry is assumed to be zero. Note that the dimension of the geometry is uniquely determined by the above condition. Indeed, if an operator T belongs to Lp (H) with p ≤ n, then |T |n has a trace in the usual sense, which implies that its Dixmier trace vanishes. In particular, if D −1 ∈ Lr+ (H) with r < n, then D −1 ∈ Lp (H) with p = (n + r)/2 and, hence, D −1 ∈ Ln+ 0 (H), contrary to our assumption. 11.2. Regularity. D := A ∪ [D, A] • The spectral triple (A, H, D) should be regular. This means that the algebra A generated by A and the operators of the form [D, a] with a ∈ A should satisfy the following condition: D = A ∪ [D, A] ⊂ Dom∞ δ, where δ(T ) := [|D|, T ]. A PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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D belongs to the smooth The formulated condition means, in other words, that the algebra A ∞ definition domain Dom δ of the differentiation operator δ. In particular, the operators of the form [D, a] belong to the definition domains Dom δk of all natural powers δk of the operator δ. Consider this condition in more detail. For a regular spectral triple (A, H, D) one can introduce the Sobolev scale of Hilbert spaces Hs := Dom|D|s , with the norms

s ∈ R,

 2 ξ2s := ξ2 + |D|s ξ  .

For s > t there is a continuous embedding Hs ⊂ Ht , and the intersection H

∞

:=



H = s

s∈R

∞ 

Hk = Dom∞ |D|

k=0

is a Frechet space with the metric given by the family of seminorms · k , k ∈ N. Denote by OprD the space of operators of order r, i.e., linear operators T : H∞ → H∞ such that for every s ∈ R there exists a positive constant Cs for which T ξs−r ≤ Cs ξs . In other words, the operator T extends to a bounded operator Hs → Hs−r . D ⊂ Op0 and the operators b − |D|b|D|−1 belong Then the regularity condition implies that A D −1 D . to OpD for all b ∈ A 11.3. Finiteness. • The algebra A is a pre-C ∗ -algebra, and the space of smooth vectors H

∞

=

∞ 

Dom D k

k=0

is a finitely generated projective A-module. We start from the first condition and recall that a subalgebra A in a C ∗ -algebra B is a preif it is complete in some locally convex topology which is finer than the topology of B and if it is closed under the holomorphic functional calculus. More precisely, since A ⊂ Dom∞ δ, we can introduce the Frechet topology on A determined by the seminorms a → δk (a), k ∈ N. If A is complete in this topology, then  An , A=

C ∗ -algebra

n

where An is

the Banach algebra obtained by the completion of the algebra A with respect to the norm a → nk=0 δk (a). In this case the algebra A will satisfy the first condition. So it is natural to incorporate this property into the finiteness condition in the general case as well. Consider next the second condition in the formulation of the finiteness property. According to this condition, we can find a number m ∈ N and an idempotent e ∈ Matm (A) for which there is an isomorphism H∞ → mAe of the left A-module H∞ and the A-module mAe associated with the idempotent e. Replacing the idempotent e with a projector p from the algebra Matm (A) by the Kaplansky formula (cf. [2, 8]), we find that H∞ = mAp. The endomorphism algebra EndA H∞ in this case is identified with the algebra p Matm (A) p = pAm ⊗A mAp. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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The signs in relations (11.1) depending on j

j mod 8 0 2 4 6

ε1 + − − +

ε2 + + + +

ε3 + − + −

j mod 8 1 3 5 7

ε1 + − − +

ε2 − + − +

11.4. Reality. A key role in the formulation of this condition is played by the notion of real spectral data for the algebra A with involution τ , which depends on the parity of the number j ≡ n mod 8. Definition 22. The real spectral data of index j ∈ Z8 for the algebra A with involution τ consist of the following collection of data (A, H, D, C, χ) for even j and (A, H, D, C) for odd j: (1) (A, H, D) is a spectral triple for the algebra A; (2) C is an antilinear isometry on H that is compatible with the involution τ , which means that CaC −1 = τ (a) for all a ∈ A; (3) for even j, χ is a grading operator on H that anticommutes with D; (4) the operators D, C, and χ satisfy the commutation relations C 2 = ε1 Id,

CD = ε2 DC,

(11.1)

Cχ = ε3 χC,

where ε1 , ε2 , and ε3 are the signs ± depending on j as indicated in the table. The choice of signs in the table is dictated by the representation theory of real Clifford algebras Clp,q associated with nondegenerate quadratic forms of signature (p, q) (a detailed discussion of Clifford algebras Clp,q and real spectral triples can be found in [2]). Recall that we have representations π of the algebra A and π o of the opposite algebra Ao in the Hilbert space H. Consider the following representation π ⊗ π o of the algebra A ⊗ Ao , which is the tensor product of the algebras A and Ao : π ⊗ π o : a ⊗ bo → aCb∗ C −1 , where ∗ is the involution in the algebra A. We can introduce an involution in A ⊗ Ao defined by the formula τ (a ⊗ bo ) := b∗ ⊗ (a∗ )o . The element of A ⊗ Ao on the right-hand side corresponds to the operator in H acting by the formula b∗ CaC −1 = CaC −1 b∗ = C(aC −1 b∗ C)C −1 .

(11.2)

Since the conjugation operator satisfies the condition C 2 = ±1, we have C −1 b∗ C = Cb∗ C −1 and formula (11.2) can be rewritten as the relation b∗ CaC −1 = C(aCb∗ C −1 )C −1 , which means that the conjugation operator C is compatible with the involution τ . Hence, in this situation we can apply Definition 22 of real spectral data for the algebra A ⊗ Ao . • The conjugation operator C satisfies the commutation relations (11.1), so that (A, H, D, C, χ) form a collection of real spectral data of index j ≡ n mod 8 for the algebra A ⊗ Ao with involution τ . The conjugation operator C will also be called the real structure for the spectral triple (A, H, D). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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11.5. First order. • The representation π o of the algebra Ao commutes not only with the representation π of the algebra A but also with all operators of the form [D, a] with a ∈ A; i.e.,   for all a, b ∈ A. [D, a], Cb∗ C −1 = 0 This definition is symmetric in A and Ao since the Jacobi identity implies that       [D, a], Cb∗ C −1 + a, [D, Cb∗ C −1 ] = D, [a, Cb∗ C −1 ] = 0, so

  a, [D, Cb∗ C −1 ] = 0.

11.6. Orientation. Using the first-order condition, we can construct a representation of Hochschild chains on A with values in the algebra A ⊗ Ao (the definitions of the Hochschild homology and cohomology are given in the Appendix). First of all, note that this algebra is an A-bimodule with respect to the bimodule structure determined by the relation a (a ⊗ bo )a := a aa ⊗ bo . The representation mentioned above is given on the homogeneous Hochschild k-chains from Ck (A, A ⊗ Ao ) by the formula   πD (a ⊗ bo ) ⊗ a1 ⊗ . . . ⊗ ak := aCb∗ C −1 [D, a1 ] . . . [D, ak ]. Now we can formulate the condition determining the volume form. • There exists a Hochschild cycle c ∈ Zn (A, A ⊗ Ao ) such that πD (c) = χ in the case of even dimension n of the geometry. In the odd-dimensional case this condition reduces to the relation πD (c) = 1. 11.7. Poincar´ e duality. This condition is a reformulation of the classical Poincar´e duality in terms of K-theory. Namely, using the index map for the operator D, one can construct (cf. [2]) additive pairings on the K-groups: Ki (A) × Ki (A) → Z

with

i = 0, 1.

• The Poincar´e duality means that these additive pairings on the groups K0 (A) and K1 (A) are nondegenerate. 11.8. Definition of the noncommutative spin geometry. Definition 23. The noncommutative spin geometry is a spectral triple (A, H, D) satisfying the above seven conditions. 12. THE DIRAC GEOMETRY AS A NONCOMMUTATIVE SPIN GEOMETRY Let M be a compact oriented Riemannian manifold equipped with a Spinc -structure. In other words, there is a spinor Riemannian bundle S together with an antilinear isometry C : S → S. Definition 24. The Dirac geometry on M is the quintuple G = (A, H, D, C, χ) where (A, H, D) is a spectral triple and (1) A = C ∞ (M ); (2) H = L2 (M, S) is the Hilbert space of spinors which is obtained as the completion of the space of smooth sections Γ∞ (M, S) with respect to the norm defined by the inner PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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285

 s, t vol,

(s, t) := M

where ·, · is the Euclidean inner product on S and vol is the volume form on M ; (3) D is the operator on H obtained as the closure of the Dirac operator D = ρ ◦ ∇ on Γ∞ (M, S) given by the composition of the Clifford multiplication ρ and the spin connection ∇ ≡ ∇S ; (4) C is the conjugation operator determining the spin structure on M ; (5) χ = ρ(ω) is the grading operator if the dimension of M is even, and χ = 1 if the dimension of M is odd. We will show that the Dirac geometry on M is a noncommutative spin geometry in the sense of Definition 23, i.e., it satisfies the seven conditions listed in Section 11. Since the algebra A = C ∞ (M ) is commutative, some of these conditions are simplified. For instance, the opposite algebra Ao in this case coincides with the original algebra A, and the representation π o of this algebra coincides with the representation π of the algebra A. The relation [a, Cb∗ C −1 ] = 0 transforms into the condition [a, b] = 0 of commutativity of the algebra A. The Hochschild cycle c ∈ Zn (A, A ⊗ Ao ) that determines the orientation can be considered in this case as an element of Zn (A). Indeed, the above representation πD by the bounded linear operators acting in H reduces in this case to the representation of the group Ck (A) of Hochschild chains in L(H) given by the formula πD (a0 ⊗ . . . ⊗ ak ) = a0 [D, a1 ] . . . [D, ak ]. The kernel of this map contains the subcomplex Dk (A) generated by the chains of the form a0 ⊗ . . . ⊗ 1 ⊗ . . . ⊗ ak in which at least one of the elements ai is equal to 1. Pushing down to the quotient Ωk (A) = Ck (A)/Dk (A), we can consider πD as a homomorphism of A-modules πD : Ωk (A) → L(H). The main result of this section is the following Theorem 2. The Dirac geometry is a noncommutative spin geometry. In other words, the Dirac geometry satisfies the conditions listed in Section 11. In this section we present the most important points in the proof of this result while referring for details to the book [2]. 12.1. Dimension. The dimension of the geometry G coincides with the dimension of the manifold M . Indeed, the square of the Dirac operator D 2 has the principal symbol σ2 (D 2 )(x, ξ) = ξ2 , which coincides with the principal symbol of the Laplace–Beltrami operator Δ on M . So  −n/2   · Id = σ−n Δ−n/2 · Id, σ−n (|D|−n ) = ξ2 where Id is the identity operator on S. This implies that the operator |D|−n is a measurable operator of Dixmier class (for details on measurable operators of Dixmier class and the properties of the Wodzicki residue, see [2, 8, 9]). Indeed, the Wodzicki residue in this case is equal to   Res(f |D|−n ) = rank S · Res f Δ−n/2 PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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for f ∈ C ∞ (M ). Accordingly, the noncommutative integral is written in the form  n(2π)n with cn = [n/2] . f |D|−n = cn Tr+ (f |D|−n ) 2 Ωn In particular,



|D|−n = cn Tr+ (|D|−n ) = 1;

i.e., the operator |D|−n belongs to L1+ (H) but does not belong to the space L1+ 0 (H). 12.2. Regularity.

Since [D, f ] = ρ(df ) by Lemma 2, we have [D, f ] = ρ(df ) = df ;

i.e., the operator [D, f ] is bounded in H for every f ∈ C ∞ (M ). D To prove the regularity of the spectral triple (A, H, D), we should also check that the algebra A ∞ generated by A and [D, A] belongs to the smooth domain Dom δ, where δ(T ) := [|D|, T ] (the proof of this fact is given in [2]). 12.3. Finiteness. The algebra A = C ∞ (M ) is closed under the holomorphic functional calculus, since a function f ∈ C ∞ (M ) is invertible in this algebra if and only if it has no zeros and in this case the inverse function 1/f also belongs to C ∞ (M ). The smooth domain of the operator D coincides with H∞ = C ∞ (M, S). The latter space is a finitely generated projective module over C ∞ (M ) by the Serre–Swan theorem (cf. [2, 8]). 12.4. Reality. The proof of the fact that the quintuple (A, H, D, C, χ) is a collection of real spectral data for the algebra A = C ∞ (M ) can be found in [2]. 12.5. First order. This condition in the case under consideration takes the form   [D, f ], g = [df, g] = 0 for f, g ∈ C ∞ (M ) and is obviously satisfied for the commutative algebra A = C ∞ (M ). 12.6. Orientation. The Hochschild n-cycle c ∈ Zn (A) coincides in the case under study with the volume form vol of the oriented Riemannian manifold M . 12.7. Poincar´ e duality. This condition is satisfied because for the algebra A = C ∞ (M ) it reduces to the usual Poincar´e duality between the de Rham homology and cohomology of the manifold M . 13. NONCOMMUTATIVE SPIN GEOMETRY OVER THE ALGEBRA A = C ∞ (M ) A noncommutative spin geometry G = (A, H, D, C, χ) is called irreducible if it is not a nontrivial sum of two other noncommutative geometries G1 = (A, H1 , D1 , C1 , χ1 ) and G2 = (A, H2 , D2 , C2 , χ2 ), i.e., it does not admit a decomposition of the form G = (A, H1 ⊕ H2 , D1 ⊕ S2 , C1 ⊕ C2 , χ1 ⊕ χ2 ). The main result of this section is the following Theorem 3. Let G = (A, H, D, C, χ) be an irreducible noncommutative spin geometry of dimension n over the algebra A = C ∞ (M ), where M is a compact oriented Riemannian manifold. Then (1) there exists a unique Riemannian metric g = g(D) on the manifold M with the distance function  (13.1) dg (p, q) = sup |f (p) − f (q)| : f ∈ C ∞ (M ), [D, f ] ≤ 1 ; PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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(2) M is a spin manifold, and the Dirac operator D corresponding to this spin structure differs from the original operator D only by the zeroth-order term. In other words, the noncommutative spin geometry over the algebra A = C ∞ (M ) is the Dirac geometry. As in the case of the previous theorem, we give only the idea of the proof of Theorem 3 while referring for details to the book [2]. 13.1. Construction of the volume form. First we define the noncommutative integral by using the following Proposition 17 (cf. [2]). If G is a noncommutative spin geometry of dimension n over the algebra A = C ∞ (M ), then the operator f |D|−n is measurable for every function f ∈ C ∞ (M ). This proposition means, in other words, that the definition of noncommutative integral  f |D|−n = Trω (f |D|−n ) does not depend on the choice of the form ω in the definition of theDixmier trace. It can be shown that the introduced integral is positive definite in the sense that f |D|−n > 0 if f is a positive element of the algebra A. In particular, this integral is nondegenerate, which allows us to introduce its density determining the volume form on M . 13.2. Construction of the spin structure and metric. As a candidate for the spinor module, we take the space H∞ of smooth vectors with respect to the action of the operator D on H. By the finiteness property this space is a finitely generated projective A-module on which the algebra A = C ∞ (M ) acts by the multiplication operators. By the Serre–Swan theorem, H∞ coincides with the module Γ∞ (M, S) of smooth sections of some vector bundle S → M . Proposition 18. There is a unique A-valued pairing {·, ·} on the space H∞ such that  for all ϕ, ψ ∈ H∞ . (ϕ, ψ) = {ψ, ϕ}|D|−n Proof. The space H∞ can be identified with the module mAp for some projector p ∈ Matm (A). On mAp there is a standard Hermitian pairing  aj pjk b∗k , {ap, bp} ≡ ap b∗ = j,k

so we can introduce a new inner product on H∞ by setting  (ϕ, ψ) := {ψ, ϕ} |D|−n . This inner product is equivalent to the original inner product (ϕ, ψ) but, generally speaking, does not coincide with it. However, (ϕ, ψ) = (ϕ, T ψ),

ϕ, ψ ∈ H∞ ,

for some positive invertible operator T ∈ L(H). Then for any f ∈ C ∞ (M ) we have   (ϕ, T f ψ) = (ϕ, f ψ) = {f ψ, ϕ} |D|−n = {ψ, f ∗ ϕ} |D|−n = (f ∗ ϕ, ψ) = (f ∗ ϕ, T ψ) = (ϕ, f T ψ). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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In other words, the operator T commutes with the action of the algebra A, so we can introduce a new inner product on the A-module H∞ by setting {ψ, ϕ} := {T −1 ψ, ϕ} . This A-valued pairing already satisfies the assertion of the proposition. In order to prove the uniqueness of the introduced pairing, note that the difference of two such pairings yields an A-valued bilinear map vanishes on all functionals of the form g → f g|D|−n  that ∞ ∗ with f ∈ C (M ). In particular, f f |D|−n = 0, which is possible only if f f ∗ = 0 in A, i.e., f = 0.  We have identified H∞ with the module of smooth sections Γ∞ (M, S) of the bundle S → M . The first-order condition states that   [D, f ], g = 0 for all f, g ∈ C ∞ (M ). Using the regularity property, one can show that the operator [D, f ] preserves the space H∞ . So the above equality implies that [D, f ] is an operator of zeroth order on the space of sections Γ∞ (M, S), i.e., it belongs to the space Γ∞ (M, End S). In other words, this operator is a matrixvalued multiplication operator in H∞ . Further, for arbitrary f, g ∈ C ∞ (M ) and ψ ∈ H∞ we have [D, f g]ψ = f [D, g]ψ + [D, f ]gψ = f [D, g]ψ + g[D, f ]ψ, i.e., [D, f g] = f [D, g] + g[D, f ]. This means that D is a matrix-valued first-order differential operator acting on smooth sections of the bundle S → M . The principal symbol of this operator is an operator function σ1 (D) defined on the cotangent bundle T ∗ M . Let us compute it using the following formula from the theory of differential operators: σ1 (D)(x, ξ) = lim

t→∞

1 −itf (x) itf (x) e De t

for any function f ∈ C ∞ (M ) such that df (x) = ξ. Using l’Hˆopital’s rule in the limit on the right-hand side, we can rewrite the above formula in the form σ1 (D)(x, ξ) = lim

t→∞

d  −itf (x) itf (x)  e De . dt

Note that for any 1-form η ∈ Ω1 (M ) the map x → σ1 (D)(x, ηx ) defines a smooth section from the space Γ∞ (M, End S). Denote by ρ(η) the section of the form ρ(η)(x) := −iσ1 (D)(x, ηx ).

(13.2)

Then for η = df and ξ = df (x) we have ρ(df )(x) := −iσ1 (D)(x, ξ) = −i lim

t→∞

d  −itf (x) itf (x)  e De = lim e−itf (x) [D, f ]eitf (x) = [D, f ](x), t→∞ dt

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where in the last equality we have used the fact that [D, f ] is a multiplication operator. So, [D, f ] = ρ(df ). Formula (13.2) defines a Clifford action of 1-forms from Ω1 (M ) on the space H∞ . At the same time, −ρ(η)2 generates a nondegenerate metric determined by a quadratic form on T ∗ M : gx (ηx , ηx ) = g−1 (η, η)(x) := −ρ(ηx )2 .

(13.3)

This metric is Riemannian since −ρ(η)2 (x) = σ2 (D 2 )(x, ηx ) coincides with the principal symbol of a positive definite operator. Due to formula (13.3) the Clifford action ρ extends to the whole Clifford bundle Cl(M ). Thus, we have constructed a spinor bundle S → M and a Clifford action ρ : Cl(M ) → End S, i.e., a Spinc structure on M . Consider now the distance function determined by the introduced metric g. Let f ∈ C ∞ (M ) and dg be the distance function on M determined by the metric g. If γ : [0, 1] → M is a piecewise smooth curve with the beginning at a point p and the end at a point q, then 1 f (q) − f (p) = f (γ(1)) − f (γ(0)) =

d f (γ(t)) dt = dt

0

1

1 dfγ(t) (γ(t)) ˙ dt =

0

  gγ(t) gradγ(t) f, γ(t) ˙ dt.

0

Applying the Cauchy inequality to the integrand, we obtain 1 |f (q) − f (p)| ≤

 

dt ≤

gγ(t) gradγ(t) f, γ(t) ˙

0

1



gradγ(t) f |γ(t)| ˙ dt

0

1 |γ(t)| ˙ dt = grad f ∞ (γ),

≤ grad f ∞ 0

where (γ) is the length of the curve γ. So, if grad f ∞ ≤ 1, we will have |f (q) − f (p)| ≤ (γ) for any piecewise smooth curve γ connecting the points p and q. Hence, |f (q) − f (p)| ≤ dg (p, q) and

(13.4)

 sup |f (q) − f (p)| : f ∈ C ∞ (M ), grad f ∞ ≤ 1 ≤ dg (p, q).

In fact, estimate (13.4) holds for any absolutely continuous function f whose gradient is defined almost everywhere as an essentially bounded vector field. In order to show that the supremum in formula (13.4) coincides with dg (p, q), we take the following function as f : f (x) ≡ fp (x) := dg (p, x). This function is Lipschitz continuous with Lipschitz norm equal to 1 (by the triangle inequality), and the supremum in formula (13.4) is attained on this function and is equal to dg (p, q). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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From that we deduce Proposition 19. The distance between the points p and q on the manifold M can be computed by formula (13.1). Proof. Since [D, f ] = ρ(df ), we have ρ(df )2∞ = sup ρ(df )(x)2 = sup gx−1 (df¯(x), df (x)) = sup gx (gradx f¯, gradx f ) = grad f 2∞ , x∈M

x∈M

x∈M

which implies the required assertion.  Thus, the distance on M is completely determined in terms of the operator D. Consider now the spin structure on M determined in terms of the Spinc -structure by the conjugation operator C. For the commutative algebra A = C ∞ (M ), due to the coincidence of the representations π and π o , the equality Cf ∗ C −1 = f should hold, i.e., the operator C defines an antilinear automorphism of the spinor bundle S. If η ∈ Ω1 (M ) is a real 1-form, then the operator C intertwines ρ(η) with −ρ(η). Moreover, C is an antiunitary operator with respect to the pairing {·, ·}. Hence, this operator has the properties listed in Proposition 16 and so indeed defines a spin structure on M . 13.3. Dirac operator. The Dirac operator D for the introduced spin structure on M is, in general, different from the original operator D, but they have the same principal symbol equal to σ1 (D)(x, ηx ) = −iρ(η)(x) = σ1 (D)(x, ηx ). Hence these operators differ only by the zeroth-order term given by a matrix-valued multiplication operator acting in the space H∞ : D = D − m,

where m ∈ Γ∞ (M, End S).

(13.5)

The matrix-valued function m has the same properties as the Dirac operator, namely, m∗ = m,

χm = (−1)n mχ,

CmC −1 = ±m.

(13.6)

Since the operators D and D are elliptic,  the same is true for their powers, so we can consider the noncommutative integrals of the form f |D|−n defined with the help of the Wodzicki residue by the formula  f |D|−n = cn Res(f |D|−n ), where cn = 1/(2[n/2] Ωn ). The operator f |D|−n for f ∈ C ∞ (M ) is of order −n, and its principal symbol has the form f (x)σ−n (f |D|−n ) = f (x)σ−n (Δ−n/2 ) · Id. The Wodzicki density is given by  resx (f |D|−n ) = cn f (x) det gx dn x, √ where cn = 2[n/2] Ωn and νg = det gx dn x is the density of the Riemannian metric g. Hence the integral   −n f |D| = f νg does not depend on the zeroth-order term m. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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On the operators of the form (13.5) with a zeroth-order term m satisfying relations (13.6), we can introduce the action given by a noncommutative integral of the form  S(D) =

|D|−n+2 .

A direct computation of this integral, carried out in [2], shows that this action functional (considered as a function of m) attains its absolute minimum at m = 0 and this minimum is equal to S(D) = −



n−2 24

scalg νg ; M

i.e., it is proportional to the Hilbert–Einstein action. Appendix: HOCHSCHILD HOMOLOGY AND COHOMOLOGY Chain complexes. Let (C• , d) be a chain complex of abelian groups. Definition A.1. A chain map f : (C• , d) → (C• , d ) from a chain complex (C• , d) to a chain complex (C• , d ) is a collection of maps fn : Cn → Cn that make the following diagram commutative: Cn

dn

fn−1

fn

Cn

Cn−1

dn−1

 Cn−1

The maps fn send cycles to cycles and boundaries to boundaries; hence they induce homomorphisms Hn f : Hn (C) → Hn (C  ). Definition A.2. A chain homotopy of two chain maps f, g : (C• , d) → (C• , d ) is a sequence  satisfying the relations of maps hn : Cn → Cn+1 hn−1 dn + dn+1 hn = fn − gn , which can be written in the concise form hd + d h = f − g. If the maps f and g are chain homotopic, then Hn f = Hn g, since if a chain c is closed, dc = 0, then f (c) − g(c) = d h(c). Let (C• (A), b) be a chain complex of the algebras Cn (A) = A⊗(n+1) , where A is a unital algebra, with the boundary map b given on Cn (A) by the formula bn (a0 ⊗ . . . ⊗ an ) =

n−1 

(−1)j a0 ⊗ . . . ⊗ aj aj+1 ⊗ . . . ⊗ an (−1)n an a0 ⊗ a1 ⊗ . . . ⊗ an−1

j=0

and b0 = 0 on C0 (A) = A. For instance, b1 (a0 ⊗ a1 ) = a0 a1 − a1 a0 and b2 (a0 ⊗ a1 ⊗ a2 ) = a0 a1 ⊗ a2 − a0 ⊗ a1 a2 + a2 a0 ⊗ a1 . It is not difficult to show that bn bn−1 = 0 for all n. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Hochschild homology. Definition A.3. The Hochschild homology of the algebra A, denoted by H• (A, A) or HH• (A), is the homology of the complex (C• (A), b). Lemma A.1. HH0 (C) = C, and HHn (C) = 0 for n > 0. Proof. For the algebra A = C we have Cn (C) = C⊗(n+1) ∼ = C and a0 ⊗ a1 ⊗ . . . ⊗ an coincides with the usual product a0 a1 . . . an . The boundary map is defined by the formula  n  1 for even n, (−1)j = bn (1) = 0 for odd n. j=0 So in this case the Hochschild complex is reduced to the exact sequence 1

0

1

0

→C− →C− →C− → C; ... − hence it has trivial homology in all dimensions except for n = 0.  In an analogous way one can define the Hochschild homology of the algebra A with values in an arbitrary A-bimodule E. To this end it is sufficient to set Cn (A, E) := E ⊗A An . The homology of the obtained complex is denoted by H• (A, E). Any algebra homomorphism f : A → A generates a chain map and a homomorphism HH• f : HH• (A) → HH• (A ) of zeroth degree. In other words, HHn is a functor from the category of unital algebras to the category of vector spaces. Example A.1 (homology H0 ). H0 (A, E) = E/[E, A], since the boundary map b : E ⊗ A → E is given by the formula b(a ⊗ a) = sa − as. In particular, HH0 (A) = A/[A, A], which coincides with A if the algebra A is commutative. If E is a symmetric bimodule over the commutative algebra A, then H0 (A, E) = E. Hochschild cohomology. Definition A.4. A Hochschild n-cochain on the algebra A is an (n + 1)-linear functional on the algebra A, or an n-linear form on A with values in the dual space A∗ . Note that A∗ is an A-bimodule with respect to the operation ϕ ∈ A∗ → (bϕc)(a) := ϕ(cab). The coboundary operator b is dual to the boundary operator on homology: n  (−1)j ϕ(a0 , . . . , aj aj+1 , . . . , an+1 ) + (−1)n+1 ϕ(an+1 a0 , . . . , an ). bn ϕ(a0 , . . . , an+1 ) = j=0

The cohomology of the obtained cochain complex is called the Hochschild cohomology of the algebra A and denoted by HH • (A) or H • (A, A∗ ). In particular, a 0-cocycle τ on the algebra A coincides with the trace since τ ∈ A∗ = Hom(A, C) and τ (a0 a1 ) − τ (a1 a0 ) = : b1 τ (a0 , a1 ) = 0. In a more general way, one can define the Hochschild cohomology of the algebra A with values in an arbitrary A-bimodule E. To this end denote by C n (A, E) the vector space of n-linear maps ϕ : An → E considered as an A-bimodule with respect to the operation (bϕc)(a1 , . . . , an ) := bϕ(a1 , . . . , an )c, where b, c ∈ A. The coboundary map in this case is given by the formula n  (−1)j ϕ(a1 , . . . , aj aj+1 , . . . , an+1 ) bn ϕ(a1 , . . . , an+1 ) = a1 ϕ(a2 , . . . , an+1 ) + j=0 n+1

+ (−1)

ϕ(a1 , . . . , an )an+1 .

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ACKNOWLEDGMENTS This work is supported by the Russian Science Foundation under grant 14-50-00005. REFERENCES 1. A. Connes, Noncommutative Geometry (Academic, San Diego, CA, 1994). 2. J. M. Gracia-Bond´ıa, J. C. V´ arilly, and H. Figueroa, Elements of Noncommutative Geometry (Birkh¨ auser, Boston, 2001). 3. M. Khalkhali, Basic Noncommutative Geometry (Eur. Math. Soc., Z¨ urich, 2013). 4. G. Landi, An Introduction to Noncommutative Spaces and Their Geometries (Springer, Berlin, 1997). 5. H. B. Lawson Jr. and M.-L. Michelsohn, Spin Geometry (Princeton Univ. Press, Princeton, NJ, 1989). 6. A. G. Sergeev, “Quantum calculus and quasiconformal mappings,” Mat. Zametki 100 (1), 144–154 (2016) [Math. Notes 100, 123–131 (2016)]. 7. A. G. Sergeev, “Quantization of the Sobolev space of half-differentiable functions,” Mat. Sb. 207 (10), 96–104 (2016) [Sb. Math. 207, 1450–1457 (2016)]. 8. A. G. Sergeev, “Introduction to noncommutative geometry,” E-print (Steklov Math. Inst., Moscow, 2016), http://www.mi.ras.ru/noc/14_15/encgeom_2016.pdf 9. A. G. Sergeev, “Noncommutative geometry and analysis,” in Differential Equations. Mathematical Physics (VINITI, Moscow, 2017), Itogi Nauki Tekhn., Sovrem. Mat. Prilozh., Temat. Obzory 137, pp. 61–81.

Translated by the author

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