Families of Monads and Instantons from a Noncommutative ADHM Construction
aa r X i v : . [ m a t h . QA ] A ug FAMILIES OF MONADS AND INSTANTONS FROMA NONCOMMUTATIVE ADHM CONSTRUCTION
SIMON BRAIN AND GIOVANNI LANDI
Dedicated to Alain Connes
Abstract.
We give a θ -deformed version of the ADHM construction of instantonswith arbitrary topological charge on the sphere S . Classically, the instanton gaugefields are constructed from suitable monad data; we show that in the deformed casethe set of monads is itself a noncommutative space. We use these monads to constructnoncommutative ‘families’ of instantons (i.e. noncommutative families of anti-self-dualconnections) on the deformed sphere S θ . We also compute the topological charge of eachof the families. Finally we discuss what it means for such families to be gauge equivalent. Contents
1. Introduction 22. The Twistor Fibration 32.1. The noncommutative Hopf fibration 32.2. The basic instanton 52.3. Noncommutative twistor space 63. The Quantum Conformal Group 73.1. The quantum groups SL θ (2 , H ) and Sp θ (2) 83.2. Quantum conformal transformations 94. A Noncommutative ADHM construction 114.1. A noncommutative space of monads 124.2. The subspace of self-dual monads 154.3. ADHM construction of noncommutative instantons 184.4. ADHM construction of charge one instantons 225. Gauge Equivalence of Noncommutative Instantons 245.1. Gauge equivalence for families of instantons 245.2. Morita equivalent geometries and gauge theory 265.3. Gauge theory from quantum symmetries 27References 28 Date : v1: 7 January 2009; v2: 6 August 2010.2000
Mathematics Subject Classification.
Primary 58B34; Secondary 14D21, 81T13.Partially supported by:the ‘Italian project Cofin06 - Noncommutative geometry, quantum groups and applications’. . Introduction
The purpose of the present article is to generalise the ADHM method for constructinginstantons on the four-sphere S to the framework of noncommutative geometry, by givinga construction of instantons on the noncommutative four-sphere S θ of [9].Instantons arise in physics as anti-self-dual solutions of the Yang-Mills equations.Mathematically they are connections with anti-self-dual curvature on smooth G -bundlesover a four-dimensional compact manifold. Since the very beginning they have been ofcentral importance for both disciplines, an importance that has only grown over the years.Of particular interest are instantons on SU(2)-bundles over the Euclidean four-sphere S . Thanks to the ADHM method of [2], the full solution to the problem of constructingsuch instantons on S has long been known and, as a consequence, the moduli space M k of instantons with topological charge equal to k is known to be a manifold of dimension8 k −
3. Starting with a trivial vector bundle over S , the ADHM strategy is to constructan orthogonal projection to some (non-trivial) sub-bundle E in such a way that theprojection of the trivial connection to E has anti-self-dual curvature.The geometric ingredient which implements the classical ADHM construction is thePenrose twistor fibration CP → S . The total space CP of the fibration is called thetwistor space of S and may be thought of as the bundle of projective spinors over S (although it has its origins elsewhere [21]). The pull-back of an instanton bundle alongthis fibration is a holomorphic vector bundle over CP equipped with a set of realityconditions which identify it as such a pull-back [24]. In this way, the construction ofinstantons is equivalent to the construction of holomorphic bundles over twistor space.Using powerful results from algebraic geometry, one gives an explicit description of allrelevant holomorphic vector bundles over a complex projective space ([12, 4], cf . also[20]). Each of them arises as the cohomology of a monad : a suitable complex of vectorbundles 0 → A σ −→ B τ −→ C → σ is injective and τ is surjective. The ADHM construction tells us how to converta given monad into an orthogonal projection of vector bundles as described above andguarantees that the resulting connection has anti-self-dual curvature.Following the general strategy of the classical case, our goal is to give a deformed versionof the ADHM method and hence a construction of instantons on the noncommutativefour-sphere S θ . The techniques involved lend themselves rather neatly to the frameworkof noncommutative geometry; the construction of vector bundles and connections byorthogonal projection is particularly natural in light of the Serre-Swan theorem [11],which trades vector bundles for finitely generated projective modules.The paper is organised as follows. Sect. 2 reviews the noncommutative spaces inquestion, namely the θ -deformed versions of the four-sphere S θ and its twistor space CP θ .We recall also the construction of the basic instanton and the principal bundle on whichit is defined, as well as the details of the noncommutative twistor fibration. Sect. 3 recallsthe construction of the quantum group SL θ (2 , H ) of conformal transformations of S θ andthe quantum subgroup Sp θ (2) of isometries. The main purpose of these two sections isto gather together into one place the relevant contributions from [9, 14, 15, 16, 5] andto establish notation; in doing so we also make some novel improvements to previous ersions. Sect. 4 presents the deformed ADHM construction itself. We show that in thedeformed case the set of all monads is parameterised by a collection of noncommutativespaces f M θ ; k indexed by k a positive integer. We use each of these spaces to constructa noncommutative ‘family’ of instantons whose topological charge we show to be equalto k . Finally in Sect. 5 we discuss what it means for families of instantons to be gaugeequivalent. In particular, we show that the quantum symmetries of the sphere S θ generategauge degrees of freedom, a feature which is a consequence of the noncommutativity andis not present in the classical construction. For further discussion in this direction werefer to [6]. 2. The Twistor Fibration
The use of the twistor fibration in the ADHM construction is crucial: this fibrationcaptures in its geometry the very nature of the anti-self-duality equations, with the resultthat an instanton bundle is reinterpreted via pull-back in terms of holomorphic data ontwistor space [24] ( cf . also [1]). In particular, this means that twistor space plays therole of an ‘auxiliary space’ on which the ADHM construction takes place, before passingback down to the base space S (we refer to [19] for more on the ADHM constructionfrom a twistor perspective).We start by recalling the details of the algebra inclusion A ( S θ ) ֒ → A ( S θ ) as a non-commutative principal bundle with undeformed structure group SU(2); associated to thisprincipal bundle there is in particular a basic instanton bundle [14]. Next we give a de-scription of the noncommutative twistor space in terms of its coordinate algebra A ( CP θ ),as well as a dualised description of the twistor fibration, now appearing [5] as an algebrainclusion A ( S θ ) ֒ → A ( CP θ ).2.1. The noncommutative Hopf fibration.
With λ = exp (2 π i θ ) the deformationparameter, the coordinate algebra A ( S θ ) of the noncommutative four-sphere S θ is the ∗ -algebra generated by a central real element x and elements α , β , α ∗ , β ∗ , modulo therelations(1) αβ = λβα, α ∗ β ∗ = λβ ∗ α ∗ , β ∗ α = λαβ ∗ , βα ∗ = λα ∗ β, together with the sphere relation(2) α ∗ α + β ∗ β + x = 1 . Similarly, the coordinate algebra of the noncommutative seven-sphere A ( S θ ) is generatedas a ∗ -algebra by the elements { z j , z ∗ j | j = 1 , . . . , } and is subject to the commutationrelations(3) z j z l = η jl z l z j , z j z ∗ l = η lj z ∗ l z j , z ∗ j z ∗ l = η jl z ∗ l z ∗ j , as well as the sphere relation(4) z ∗ z + z ∗ z + z ∗ z + z ∗ z = 1 . ompatibility with the SU(2) principal bundle structure requires the deformation matrix( η jk ) be given by(5) ( η jk ) = µ µ µ ¯ µµ ¯ µ µ µ , µ = exp (i πθ ) . The values of the deformation parameters λ , µ are precisely those which allow an em-bedding of the classical group SU(2) into the group Aut A ( S θ ). We denote by A ( C θ ) thealgebra generated by the { z j , z ∗ j } subject to the relations (3); the quotient by the addi-tional sphere relation yields the algebra A ( S θ ). The algebra inclusion A ( S θ ) ֒ → A ( S θ ) isgiven explicitly by α = 2( z z ∗ + z ∗ z ) , β = 2( z z ∗ − z ∗ z ) , x = z z ∗ + z z ∗ − z z ∗ − z z ∗ . (6)One easily verifies that for the right SU(2)-action on A ( S θ ) given on generators by(7) ( z , z ∗ , z , z ∗ ) ( z , z ∗ , z , z ∗ ) (cid:18) w w (cid:19) , w = (cid:18) w − ¯ w w ¯ w (cid:19) ∈ SU(2) , the invariant subalgebra is generated as expected by α , β , x and their conjugates, so oneindeed has Inv SU(2) A ( S θ ) = A ( S θ ) . When θ = 0 we recover the usual algebras of functions on the classical spheres S and S . The inclusion A ( S ) ֒ → A ( S ) is just a dualised description of the standard SU(2)Hopf fibration S → S .These noncommutative spheres have canonical differential calculi arising as deforma-tions of the classical ones. Explicitly, one has a first order differential calculus Ω ( S θ ) on A ( S θ ) spanned as an A ( S θ )-bimodule by { d z j , d z ∗ j , j = 1 , . . . , } , subject to the relations z i d z j = η ij d z j z i , z i d z ∗ j = η ji d z ∗ j z i , with η ij as before. One also has relationsd z i d z j + η ij d z j d z i = 0 , d z i d z ∗ j + η ji d z ∗ j d z i = 0 , allowing one to extend the first order calculus to a differential graded algebra Ω( S θ ).There is a unique differential d on Ω( S θ ) such that d : z j d z j . Furthermore, Ω( S θ ) hasan involution given by the graded extension of the map z j z ∗ j . The story is similar forthe four-sphere, in that the differential graded algebra Ω( S θ ) is generated in degree oneby d α , d α ∗ , d β , d β ∗ , d x , subject to the relations α d β = λ (d β ) α, β ∗ d α = λ (d α ) β ∗ , d α d β + λ d β d α = 0 , d β ∗ d α + λ d α d β ∗ = 0 . The above are the same as the relations (1) or (3) but with d inserted. As vector spaces,the graded components Ω k ( S θ ) and Ω k ( S θ ) of k -forms on the noncommutative spheresare identical to their classical counterparts, although the algebra relations between formsare twisted. In particular this means that the Hodge ∗ -operator on S θ , ∗ θ : Ω k ( S θ ) → Ω − k ( S θ ) , s defined by the same formula as it is classically. One still has that ∗ θ = 1, whence thereis a direct sum decomposition of two-formsΩ ( S θ ) = Ω ( S θ ) ⊕ Ω − ( S θ ) , with Ω ± ( S θ ) := { ω ∈ Ω ( S θ ) | ∗ θ ω = ± ω } the spaces of self-dual and anti-self-dualtwo-forms.2.2. The basic instanton.
Amongst the nice properties of the classical Hopf fibrationis that its canonical connection is an anti-instanton: its curvature is a self-dual two-formwith values in the Lie algebra su (2) of the structure group. This property holds also inthe noncommutative case, giving a simple example of a noncommutative instanton. Ithas an elegant description [14] in terms of the function algebras A ( S θ ), A ( S θ ) as follows.One takes the pair of elements of the right A ( S θ )-module A ( S θ ) := C ⊗ A ( S θ ) givenby | ψ i = (cid:0) z z z z (cid:1) t , | ψ i = (cid:0) − z ∗ z ∗ − z ∗ z ∗ (cid:1) t . With the natural Hermitian structure on A ( S θ ) given by h ξ | η i = P i ξ ∗ i η i , one sees that h ψ j | ψ l i = δ jl . It is convenient to introduce the matrix-valued function Ψ on S θ given by(8) Ψ = (cid:0) | ψ i | ψ i (cid:1) = (cid:18) z z z z − z ∗ z ∗ − z ∗ z ∗ (cid:19) t . From orthonormality of the columns one has that Ψ ∗ Ψ = 1 and hence the matrix(9) q := ΨΨ ∗ = 12 x α − ¯ µ β ∗ x β µ α ∗ α ∗ β ∗ − x − µ β ¯ µ α − x is a self-adjoint idempotent of rank two, i.e. q ∗ = q = q and Tr q = 2. The action (7) ofSU(2) on A ( S θ ) now takes the formΨ Ψ w, w ∈ SU(2) , from which the SU(2)-invariance of the entries of q is immediately deduced. We may alsowrite the commutation relations of A ( S θ ) in the useful form(10) Ψ ia Ψ jb = η ij Ψ jb Ψ ia , a, b = 1 , i, j = 1 , , , . If ρ is the defining representation of SU(2) on C , the finitely generated projective right A ( S θ )-module E := q A ( S θ ) is isomorphic to the module of equivariant maps from A ( S θ )to C , E ∼ = { φ ∈ A ( S θ ) ⊗ C | ( w ⊗ id) φ = (id ⊗ ρ ( w − )) φ for all w ∈ SU(2) } . The module E has the role of the module of sections of the ‘associated vector bundle’ E = S θ × SU(2) C . With the projection q = ΨΨ ∗ there comes the canonical Grassmannconnection defined on the module E by ∇ := q ◦ d : E → E ⊗ A ( S θ ) Ω ( S θ ) . The curvature of ∇ is ∇ = q (d q ) , which may be shown to be self-dual with respect tothe Hodge operator, ∗ θ ( q (d q ) ) = q (d q ) . he complementary projector p = 1 − q yields a connection whose curvature is anti-self-dual, ∗ θ ( p (d p ) ) = − p (d p ) , and hence an instanton on the noncommutative four-sphere,which we call the basic instanton . Noncommutative index theory computes its ‘topologicalcharge’ to be equal to − e , e ) of C , equivariant maps are written as φ = P a φ a ⊗ e a .On them, one has explicitly that ∇ ( φ a ) = d φ a + X b ω ab φ b , where the connection one-form ω = ω ab is found to be(11) ω ab = X j ((Ψ ∗ ) aj dΨ jb − d(Ψ ∗ ) aj Ψ jb ) . From this it is easy to see that ω ab = − ( ω ∗ ) ba and P a ω aa = 0, so that ω is an element ofΩ ( S θ ) ⊗ su (2).2.3. Noncommutative twistor space.
It is well-known that, as a real six-dimensionalmanifold, the space CP may be identified with the set of all 4 × C . Thus the coordinate algebra A ( CP )of CP has a defining matrix of generators(12) Q = t x x x x ∗ t y y x ∗ y ∗ t y x ∗ y ∗ y ∗ t , with t ∗ j = t j , j = 1 , . . . , Q = P j t j = 1, as well as the relations coming fromthe condition Q = Q , that is to say P j Q kj Q jl = Q kl . The noncommutative twistoralgebra A ( CP θ ) is obtained by deforming these relations: with deformation parameter λ = exp (2 π i θ ), one has that t , . . . , t are central, that x x = ¯ λx x , x x = ¯ λx x , x x = ¯ λx x as well as the auxiliary relations y y = ¯ λy y , y y = ¯ λy y , y y = ¯ λy y , x ( y , y , y ) = (¯ λ y , ¯ λy , λy ) x ,x ( y , y , y ) = (¯ λy , y , λy ) x , x ( y , y , y ) = (¯ λy , ¯ λy , y ) x , and similar relations obtained by taking the adjoint under ∗ of those above (we refer to[5] for further details). To proceed further it is useful to note that classically CP is thequotient of the sphere S by the action of the diagonal U(1) subgroup of SU(2). Thisremains true in the noncommutative case and one identifies the generators of A ( CP θ ) as(13) Q jl = z j z ∗ l , via the generators { z j , z ∗ j } of A ( S θ ). Indeed, from equation (13) one could infer therelations on the generators of A ( CP θ ) from those on the generators of A ( S θ ). By itsvery definition A ( CP θ ) is the invariant subalgebra of A ( S θ ) under this U(1)-action andequation (13) defines an inclusion of algebras A ( CP θ ) ֒ → A ( S θ ) , iving a noncommutative principal bundle with structure group U(1). We thus havealgebra inclusions(14) A ( S θ ) ֒ → A ( CP θ ) ֒ → A ( S θ ) , with the left-hand arrow still to be determined. As in the classical case, this inclusionis not a principal fibration (the ‘typical fibre’ is a copy of the undeformed CP ) but wemay nevertheless express the generators of A ( CP θ ) in terms of the generators of A ( S θ ).For this we need the non-degenerate map on A ( C θ ) given on generators by(15) J ( z , z , z , z ) := ( − z ∗ , z ∗ , − z ∗ , z ∗ )and extended as an anti-algebra map. Classically, in doing so we would be identifyingthe set of quaternions H with the set of 2 × C of the form c + c j ∈ H (cid:18) c − ¯ c c ¯ c (cid:19) ∈ M ( C ) , and the map J corresponds to right multiplication by the quaternion j . In the deformedcase, this very same identification defines the algebra A ( H θ ) to be equal to the algebra A ( C θ ) equipped with the map J [16].Using the identification of generators (13) the map J extends to an automorphism of A ( CP θ ), given in terms of the matrix generators in equation (12) by J ( t ) = t , J ( t ) = t , J ( t ) = t , J ( t ) = t ,J ( x ) = − x , J ( y ) = − y , J ( x ∗ ) = − x ∗ , J ( y ∗ ) = − y ∗ ,J ( x ) = µ y ∗ , J ( x ) = − y ∗ , J ( x ∗ ) = ¯ µ y , J ( x ∗ ) = − y ,J ( y ) = ¯ µ x ∗ , J ( y ) = − x ∗ , J ( y ∗ ) = µ x , J ( y ∗ ) = − x , as required for J to respect the algebra relations of A ( CP θ ). The subalgebra fixed by themap J is precisely A ( S θ ); in fact one has an algebra inclusion A ( S θ ) ֒ → A ( CP θ ) givenon generators by(16) x t + t − , α x + µ y ∗ ) , β − x ∗ + y ) , with µ = √ λ = exp ( π i θ ). In the notation of equation (8) we have Q = | ψ ih ψ | , and wenote also that | ψ i = | J ψ i , so that equation (16) is just the statement that q = | ψ ih ψ | + | ψ ih ψ | = | ψ ih ψ | + | J ψ ih J ψ | = Q + J ( Q ) . This gives us the promised algebraic description of the twistor fibration (14): the genera-tors of A ( S θ ) are identified with the degree one elements of A ( CP θ ) of the form Z + J ( Z ).3. The Quantum Conformal Group
Next, we briefly review the construction of the quantum groups which describe thesymmetries of the spheres S θ and S θ (and the symmetries of the Hopf fibration definedin Sect. 2.1). .1. The quantum groups SL θ (2 , H ) and Sp θ (2) . To begin, we need a noncommuta-tive analogue of the set of all linear transformations of the quaternionic vector space H θ defined above. To this end, we define a transformation bialgebra for the algebra A ( H θ )to be a bialgebra B such that there is a ∗ -algebra map ∆ L : A ( C θ ) → B ⊗ A ( C θ ) commut-ing with the map J of equation (15). The set of all transformation bialgebras for A ( H θ )forms a category in the natural way; we define the bialgebra A (M θ (2 , H )) as the universalinitial object in the category, meaning that whenever B is a transformation bialgebra for A ( H θ ) there is a morphism of transformation bialgebras A (M θ (2 , H )) → B [16]. Usingthe universality property, one finds that A (M θ (2 , H )) is the associative algebra generatedby the entries of the following 4 × A = (cid:18) a ij b ij c ij d ij (cid:19) = a − a ∗ b − b ∗ a a ∗ b b ∗ c − c ∗ d − d ∗ c c ∗ d d ∗ . With our earlier notation, we think of this matrix as generated by four quaternion-valuedfunctions, writing a = ( a ij ) = (cid:18) a − a ∗ a a ∗ (cid:19) and similarly for the other entries b, c, d . The coalgebra structure on A (M θ (2 , H )) is givenby ∆( A ij ) = X l A il ⊗ A lj , ǫ ( A ij ) = δ ij for i, j = 1 , . . . ,
4, and its ∗ -structure is evident from the matrix (17). The coaction ∆ L is determined to be(18) ∆ L : A ( C θ ) → A (M θ (2 , H )) ⊗ A ( C θ ) , ∆ L (Ψ ia ) = X j A ij ⊗ Ψ ja , where Ψ is the matrix in equation (8) (although here we do not assume the sphere relationand instead think of the entries of Ψ as generators of the algebra A ( C θ )). The relationsbetween the generators of A (M θ (2 , H )) are found from the requirement that ∆ L make A ( C θ ) into an A (M θ (2 , H ))-comodule algebra. One computes(19) ∆ L (Ψ ia Ψ jb ) = X km ( A im A jl − η ij η lm A jl A im ) ⊗ Ψ ma Ψ lb and, since the products Ψ ma Ψ lb may be taken to be all independent as k, l, a, b vary, wemust have that(20) A im A jl = η ij η lm A jl A im for i, j, l, m = 1 , . . . ,
4. It is not difficult to see that the algebra generated by the a ij iscommutative, as are the algebras generated by the b ij , c ij , d ij , although overall the alge-bra is noncommutative due to some non-trivial relations among components in differentblocks.Of course, A (M θ (2 , H )) is not quite a Hopf algebra since it does not have an antipode.We obtain a Hopf algebra by passing to the quotient of A (M θ (2 , H )) by the Hopf ∗ -idealgenerated by the element D −
1, where D = det A is the formal determinant of thematrix A in (17). We denote the quotient by A (SL θ (2 , H )), the coordinate algebra on thequantum group SL θ (2 , H ) of matrices in M θ (2 , H ) with determinant one, and continue o write the generators of the quotient as A ij . The algebra A (SL θ (2 , H )) inherits a ∗ -bialgebra structure from that of A (M θ (2 , H )) and we use the determinant to define anantipode S : A (SL θ (2 , H )) → A (SL θ (2 , H )) as in [16]. The datum ( A (SL(2 , H )) , ∆ , ǫ, S )constitutes a Hopf ∗ -algebra.The Hopf algebra A (Sp θ (2)) is the quotient of A (SL θ (2 , H )) by the two-sided ∗ -Hopfideal generated by X l ( A ∗ ) li A lj − δ ij , i, j = 1 , . . . , . In this algebra we have the relations A ∗ A = AA ∗ = 1, or equivalently S ( A ) = A ∗ . ThisHopf algebra is the coordinate algebra on the quantum group Sp θ (2), the subgroup ofSL θ (2 , H ) of unitary matrices.Finally there is an inclusion of algebras A ( S θ ) ֒ → A (Sp θ (2)) given on generators bythe ∗ -algebra map(21) z a , z a , z c , z c . This means that we may identify the first two columns of the matrix A with the matrixΨ of equation (8). Similarly there is an algebra inclusion A ( S θ ) ֒ → A (Sp θ (2)) given by(22) x a a ∗ − a a ∗ + c c ∗ − c c ∗ , α a c ∗ − a ∗ c , β
7→ − a ∗ c + a c ∗ . These inclusions yield algebra isomorphisms of A ( S θ ) and A ( S θ ) with certain subalge-bras of A (Sp θ (2)) of coinvariants under coactions by appropriate sub-Hopf algebras, thusrealising the noncommutative spheres as quantum homogeneous spaces for Sp θ (2). Werefer to [16] for details of these constructions.3.2. Quantum conformal transformations.
We now review how the quantum groupsobtained in the previous section (co)act on the spheres S θ and S θ as ‘quantum symme-tries’. The coaction(23) ∆ L : A ( C θ ) → A (SL θ (2 , H )) ⊗ A ( C θ ) , ∆ L (Ψ ia ) = X j A ij ⊗ Ψ ja , is by construction a ∗ -algebra map and so, if we assume that the quantity r := X j z ∗ j z j is invertible with inverse r − , then we may also define an inverse for the quantity ρ := ∆ L (cid:16)X j z ∗ j z j (cid:17) by ρ − := ∆ L ( r − ). Inverting r corresponds to deleting the origin in C θ and we definethe coordinate algebra of the corresponding subset of C θ by A ( C θ ) := A ( C θ )[ r − ] , the algebra A ( C θ ) with r − adjoined. Extending ∆ L as a ∗ -algebra map gives a well-defined coaction ∆ L : A ( C θ ) → A (SL θ (2 , H )) ⊗ A ( C θ )for which A ( C θ ) is an A (SL θ (2 , H ))-comodule algebra.Writing A ( e C θ ) := ∆ L ( A ( C θ )) for the image of A ( C θ ) under ∆ L , both ρ and ρ − are central in the algebra A ( e C θ ), since r and r − are central in A ( C θ ). ow the coaction ∆ L descends to a coaction of the Hopf algebra A (Sp θ (2)),(24) ∆ L : A ( C θ ) → A (Sp θ (2)) ⊗ A ( C θ ) , by the same formula (18) now viewed for the quotient A (Sp θ (2)). In particular, for thiscoaction one has(Ψ ∗ Ψ) ab X ijl ( A ∗ ) li A ij ⊗ (Ψ ∗ ) al Ψ jb = X jl δ lj ⊗ (Ψ ∗ ) al Ψ jb = 1 ⊗ (Ψ ∗ Ψ) ab , since the generators A ij satisfy the relations P i ( A ∗ ) li A ij = δ lj in the algebra A (Sp θ (2)).Then both A ( S θ ) and A ( S θ ) are A (Sp θ (2))-comodule algebras, since this coaction pre-serves the sphere relations (2) and (4).In contrast, the spheres S θ and S θ are not preserved under the coaction of the largerquantum group SL θ (2 , H ). Although defined on the algebra A ( C θ ), the coaction ∆ L of A (SL θ (2 , H )) is not well-defined on the seven-sphere A ( S θ ) since it does not preserve thesphere relation r = 1 of equation (4). By definition, we have instead that ∆ L ( r ) = ρ ,meaning that the coaction of A (SL θ (2 , H )) ‘inflates’ the sphere A ( S θ ) [16]. Since r isa central element of A ( C θ ), we may evaluate it as a positive real number. The resultis the coordinate algebra of a noncommutative sphere S θ,r of radius r ; as this radiusvaries in A ( C θ ), it sweeps out a family of seven-spheres. Similarly, evaluation of thecentral element ρ in A ( e C θ ) yields the coordinate algebra of a noncommutative sphere e S θ,ρ of radius ρ and, as the value of ρ varies in A ( e C θ ), it sweeps out another familyof seven-spheres. The coaction ∆ L of A (SL θ (2 , H )) on A ( C θ ) serves to map the familyparameterised by r onto the family parameterised by ρ .A similar fact is found for the generators α , β , x of the four-sphere algebra A ( S θ ). Thecoaction of A (SL θ (2 , H )) does not preserve the sphere relation but gives instead that∆ L ( α ∗ α + β ∗ β + x ) = ρ , and the four-sphere S θ is also inflated. Let us write A ( Q θ ) for the subalgebra of A ( C θ )generated by α , β , x and their conjugates. Then as r varies in A ( Q θ ), we get a family ofnoncommutative four-spheres. Similarly, we define ˜ α := ∆ L ( α ), ˜ β := ∆ L ( β ), ˜ x := ∆ L ( x )and so forth, and write A ( e Q θ ) for the subalgebra of A ( e C θ ) that they generate. It isprecisely the SU(2)-invariant subalgebra of A ( e C θ ), and as ρ varies in A ( e Q θ ) we getanother family of noncommutative four-spheres. The coaction of the quantum group A (SL θ (2 , H )) maps the family parameterised by r onto the family parameterised by ρ .Thus there is a family of SU(2)-principal fibrations given by the algebra inclusion A ( Q θ ) ֒ → A ( C θ ), the family being parameterised by the function r . For a fixed valueof r we get an SU(2) principal bundle S θ,r → S θ,r . Similarly, the algebra inclusion A ( e Q θ ) ֒ → A ( e C θ ) defines a family of SU(2)-principal fibrations parameterised by thefunction ρ . The above construction shows that the coaction of the quantum group A (SL θ (2 , H )) carries the former family of principal fibrations onto the latter.All of this means that, as things stand, we cannot use the presentations of A ( S θ )and A ( S θ ) of Sect. 2.1 to give a well-defined coaction of A (SL θ (2 , H )), since the sphererelations we use to define them are not preserved by the coaction. Rather we should workwith the families of spheres all at once (this is the price we have to pay for working withthe coaction of a Hopf algebra rather than the action of a group). To do this, we note hat the algebra A ( S θ ) may be identified with the subalgebra of A ( C θ ) generated by r − α, r − β , r − x , together with their conjugates, since the sphere relation(25) ( r − α )( r − α ) ∗ + ( r − β )( r − β ) ∗ + ( r − x ) = 1is automatically satisfied in A ( C θ ). The result of doing so is that we have a well-definedcoaction, ∆ L : A ( S θ ) → A (SL θ (2 , H )) ⊗ A ( S θ ) , defined on the generators r − α , r − β , r − x and their conjugates, with the sphere relation(25) now preserved by ∆ L . In this way, we think of SL θ (2 , H )) as the quantum group ofconformal transformations of S θ .In these new terms, the construction of the defining projector for A ( S θ ) needs to bemodified only slightly. We now take the normalised matrix(26) Ψ = r − (cid:18) z z z z − z ∗ z ∗ − z ∗ z ∗ (cid:19) t , at the price of including the generator r − as well (not a problem in the smooth closure[16]). Thanks to the relation (25), we still have Ψ ∗ Ψ = 1 and the required projector is(27) q := ΨΨ ∗ = r − r + x α − ¯ µ β ∗ r + x β µ α ∗ α ∗ β ∗ r − x − µ β ¯ µ α r − x . By the above discussion, the coaction ∆ L of A (SL θ (2 , H )) is now well-defined on thealgebra generated by the entries of this matrix. Writing e Ψ ia := ∆ L (Ψ ia ), the image of q under ∆ L is computed to be(28) ˜ q := e Ψ e Ψ ∗ = ρ − ρ + ˜ x α − ¯ µ ˜ β ∗ ρ + x ˜ β µ ˜ α ∗ ˜ α ∗ ˜ β ∗ ρ − x − µ ˜ β ¯ µ ˜ α ρ − ˜ x . The entries of these projectors generate respectively subalgebras of A ( C θ ) and A ( e C θ ),each parameterising the families of noncommutative four-spheres discussed above.Finally, we observe that similar statements may be made about the U(1)-principalfibration S θ → CP θ . We do not need a sphere relation in order to define the coordinatealgebra A ( CP θ ): in Sect. 2.3 it was merely convenient to do so. Instead, we may identify A ( CP θ ) as the U(1)-invariant subalgebra of A ( C θ ) generated by elements t = r − z z ∗ , x = r − z z ∗ , x = r − z z ∗ , x = r − z z ∗ and so forth.4. A Noncommutative ADHM construction
There is a well-known solution to the problem of constructing instantons on the classicalfour-sphere S which goes under the name of ADHM construction. Techniques of linearalgebra are used to construct vector bundles over twistor space CP , which are in turnput together to construct a vector bundle over S equipped with an instanton connection.It is known that all such connections are obtained in this way [2, 3]. ur goal here is to generalise the ADHM method to a deformed version which con-structs instantons on the noncommutative sphere S θ . The classical construction maybe obtained from our deformed version by setting θ = 0. As usual our approach stemsfrom writing the classical construction in a dualised language which does not dependon the commutativity of the available function algebras, although here the situation isnot as straightforward as one might first expect. The deformed construction is rathermore subtle than it is in the commutative case and produces noncommutative ‘families’of instantons.4.1. A noncommutative space of monads.
The algebra A ( C θ ) has a natural Z -grading given by assigning to its generators the degreesdeg( z j ) = 1 , deg( z ∗ j ) = − , j = 1 , . . . , , which results in a decomposition A ( C θ ) = ⊕ n ∈ Z A n . Then for each r ∈ Z there is a‘degree shift’ map from A ( C θ ) to itself whose image we denote A ( C θ )( r ); by definitionthe degree n component of A ( C θ )( r ) is A r + n .Similarly, if a given A ( C θ )-module E is Z -graded, we denote the degree-shifted modulesby E ( r ), r ∈ Z . In particular, for each finite dimensional vector space H the correspondingfree right module H ⊗ A ( C θ ) is Z -graded by the grading on A ( C θ ), and the shift mapson A ( C θ ) induce the shift maps on H ⊗ A ( C θ ).The input data for the classical ADHM construction of SU(2) instantons with topo-logical charge k is a monad , by which we mean a sequence of free right modules over thealgebra A ( C ),(29) H ⊗ A ( C )( − σ z −→ K ⊗ A ( C ) τ z −→ L ⊗ A ( C )(1) , where H , K and L are complex vector spaces of dimensions k , 2 k + 2 and k respectively.The arrows σ z and τ z are A ( C )-module homomorphisms assumed to be such that σ z isinjective, τ z is surjective and that the composition τ z σ z = 0. This is the usual approach inalgebraic geometry [20], although here we work with A ( C )-modules, i.e. global sectionsof vector bundles, rather than with locally-free sheaves.The degree shifts signify we think of σ z and τ z respectively as elements of H ∗ ⊗ K ⊗ A and K ∗ ⊗ L ⊗ A , where A is the degree one component of A ( C ) (the vector spacespanned by the generators z , . . . , z ). This means that alternatively we may think ofthem as linear maps(30) σ z : H × C → K, τ z : K × C → L, thus recovering the more explicit geometric approach of [2].Our goal in this section is to give a description of a monad of the form (29) in analgebraic framework which allows the possibility of the algebra A ( C θ ) being noncom-mutative. In this setting, we require the maps σ z and τ z to be parameterised by thenoncommutative space C θ rather than by the classical space C , as was the case in equa-tion (30). Our first task then is to find an analogue of the space of linear module maps H ⊗ A ( C θ )( − → K ⊗ A ( C θ ).Following a general strategy [23], we define A ( f M θ ( H, K )) to be the universal algebrafor which there is a morphism of right A ( C θ )-modules, σ z : H ⊗ A ( C θ )( − → A ( f M θ ( H, K )) ⊗ K ⊗ A ( C θ ) , hich is linear in the generators z , . . . , z of A ( C θ ). By this we mean that when-ever B is an algebra satisfying these properties there exists a morphism of algebras φ : A ( f M θ ( H, K )) → B and a commutative diagram H ⊗ A ( C θ )( − σ z −−−→ A ( f M θ ( H, K )) ⊗ K ⊗ A ( C θ ) y id y φ ⊗ id H ⊗ A ( C θ )( − σ ′ z −−−→ B ⊗ K ⊗ A ( C θ )of right A ( C θ )-modules, with σ ′ z denoting the corresponding map for the algebra B .Choosing a basis ( u , . . . , u k ) for the vector space H and a basis ( v , . . . , v k +2 ) for thevector space K , the algebra A ( f M θ ( H, K )) is generated by the matrix elements { M αab | a = 1 , . . . , k + 2 , b = 1 , . . . , k, α = 1 , . . . , } , which define a map σ z , expressed on simple tensors by(31) σ z : u b ⊗ Z X a,α M αab ⊗ v a ⊗ z α Z, Z ∈ A ( C θ ) . In more compact notation, for each α we arrange these elements into a (2 k + 2) × k matrix M α = ( M αab ), so that with respect to the above bases, σ z may be written(32) σ z = X α M α ⊗ z α . To find the relations in the algebra A ( f M θ ( H, K )), let us write (ˆ u , . . . , ˆ u k ) for the basisof H ∗ which is dual to ( u , . . . , u k ) and write (ˆ v , . . . , ˆ v k +2 ) for the basis of K ∗ dual to( v , . . . , v k +2 ). Then the map (31) has an equivalent dual description (also denoted σ z )in terms of the dual vector spaces H ∗ , K ∗ as(33) σ z : ˆ v a ⊗ Z X b,α M αab ⊗ ˆ u b ⊗ z α Z, and extended as an A ( C θ )-module map. The functionals ˆ u b , ˆ v a together with theirconjugates ˆ u ∗ b , ˆ v ∗ a generate the coordinate algebras of H and K respectively. It is onlynatural to require that (33) be an algebra map. Proposition 4.1.
With ( η αβ ) the matrix (5) of deformation parameters, the matrix ele-ments M αab enjoy the relations (34) M αab M βcd = η βα M βcd M αab for each a, c = 1 , . . . , k + 2 , each b, d = 1 , . . . , k and each α, β = 1 , . . . , .Proof. The requirement that (33) is an algebra map means that in degree one we need σ z (ˆ v a ˆ v c ) = σ z (ˆ v c ˆ v a ) for all a, c = 1 , . . . , k + 2, which translates into the statement that X b,d,α,β M αab M βcd ⊗ ˆ u b ˆ u d ⊗ z α z β = X b,d,α,β M βcd M αab ⊗ ˆ u d ˆ u b ⊗ z β z α for all a, c = 1 , . . . , k + 2. Using in turn the relations (3) and the fact that the generatorsˆ u b , ˆ u d commute for all values of b, d , this equation may be rearranged to give X b,d,α,β (cid:16) M αab M βcd − η βα M βcd M αab (cid:17) ⊗ ˆ u b ˆ u d ⊗ z α z β = 0 . Since for b ≤ d and α ≤ β the quantities ˆ u b ˆ u d ⊗ z α z β may all be taken to be independent,we must have that their coefficients are all zero, leading to the stated relations. (cid:3) he above proposition simply says that the entries of a given matrix M α all commute,whereas the relations between the entries of the matrices M α and M β are determinedby the deformation parameter η βα . Hence the algebra A ( f M θ ( H, K )) is generated bythe M αab subject to the relations (34). The algebra A ( f M θ =0 ( H, K )) is commutative andparameterises the space of all possible maps σ z , since for each point x ∈ f M θ =0 ( H, K )there is an evaluation map, ev x : A ( f M θ =0 ( H, K )) → C , which yields an A ( C )-module homomorphism(ev x ⊗ id) σ z : H ⊗ A ( C )( − → K ⊗ A ( C ) , (ev x ⊗ id) σ z := X ev x ( M αab ) ⊗ z α . When θ is different from zero, there need not be enough evaluation maps available. Nev-ertheless, we think of A ( f M θ ( H, K )) as a noncommutative family of maps parameterisedby the noncommutative space f M θ ( H, K ). Remark 4.2.
Since we constructed A ( f M θ ( H, K )) through the minimal requirement that σ z is an algebra map, it is indeed the universal algebra with the required properties. Thismeans that our interpretation of A ( f M θ ( H, K )) as a noncommutative family of maps is inagreement with the approaches of [23, 25, 22] for quantum families of maps parameterisedby noncommutative spaces. Moreover, it also agrees with the definition of algebras ofrectangular quantum matrices discussed in [17]. It may also be viewed as a kind of‘comeasuring’ as introduced in [18], but now for modules instead of algebras.Thus we have a noncommutative analogue of the space of all maps σ z . A similarconstruction works for the maps τ z : there is a universal algebra A ( f M θ ( K, L )) generatedby matrix elements N αba for labels b = 1 , . . . , k , a = 1 , . . . , k + 2 and α = 1 , . . . ,
4, herecoming from a map(35) τ z : v a ⊗ Z X b,α N αba ⊗ w b ⊗ z α Z, having chosen a basis ( w , . . . , w k ) for the vector space L . Dually, the requirement that τ z be an algebra map from the coordinate algebra of L to the coordinate algebra of K results in relations for the generators of the algebra A ( f M θ ( K, L )),(36) N αba N βdc = η βα N βdc N αba , which are the parallel of conditions (34) for the algebra A ( f M θ ( H, K )).To complete the monad picture we finally require that the composition of the maps σ z and τ z be zero. In the dualised format the composition is easily dealt with as thecomposition as a map from the coordinate algebra of L to that of H , with the productappearing as part of a general procedure for ‘gluing’ quantum matrices [17]. By this wemean that the composition ϑ z := τ z ◦ σ z is given in terms of an algebra-valued k × k matrix, the product of a k × (2 k + 2) matrix with a (2 k + 2) × k matrix. Explicitly, themap is ϑ z : H ⊗ A ( C θ )( − → A ( f M θ ( H, L )) ⊗ L ⊗ A ( C θ )(1) ,ϑ z : ˆ w a ⊗ Z X b,α,β T α,βab ⊗ ˆ w b ⊗ z α z β Z, here A ( f M θ ( H, L )) is the coordinate algebra generated by the matrix elements T α,βab for α, β = 1 , . . . , a, b = 1 , . . . , k . The matrix multiplication ( τ z , σ z ) ϑ z now appearsas a ‘coproduct’ A ( f M θ ( H, L )) → A ( f M θ ( K, L )) ⊗ A ( f M θ ( H, K )) ,T α,βcd := X b N αcb ⊗ M βbd , α, β = 1 , . . . , , c, d = 1 , . . . , k. The condition τ z σ z = 0 is thus that the image of this map in A ( f M θ ( K, L )) ⊗A ( f M θ ( H, K ))is zero; this is established by the following proposition.
Proposition 4.3.
The condition τ z σ z = 0 is equivalent to the requirement that (37) X r ( N αbr M βrd + η βα N βbr M αrd ) = 0 for all b, d = 1 , . . . , k and all α, β = 1 , . . . , .Proof. In terms of algebra-valued matrices the map τ z σ z is computed as the compositionof the duals of the maps (31) and (35), following the discussion above, to be equal to( τ z σ z ) bd = X r,α,β N αbr M βrd ⊗ z α z β . Equating to zero the coefficients of the linearly independent generators z α z β for α ≤ β gives the relations as stated. (cid:3) The conditions in equation (37) may be expressed more compactly in terms of productsof matrices as N α M β + η βα N β M α = 0 , for α, β = 1 , . . . , α and β in this expression). Definition 4.4.
Define A ( f M θ ; k ) to be the algebra generated by the matrix elements M αab and N βba subject to the relations M αab M βcd = η βα M βcd M αab , N αba N βdc = η βα N βdc N αba , as well as the relations X r ( N αdr M βrb + η βα N βbr M αrd ) = 0for all α, β = 1 , . . . ,
4, all b, d = 1 , . . . , k and all a, c = 1 , . . . , k + 2.The noncommutative algebra A ( f M θ ; k ) is by construction universal amongst all al-gebras having the property that the resulting maps σ z and τ z are algebra maps whichcompose to zero. Our interpretation is that for fixed k the collection of monads over C θ is parameterised by the noncommutative space which is ‘dual’ to this algebra.4.2. The subspace of self-dual monads.
In the classical case, the input datum of amonad is by itself insufficient to construct bundles over the four-sphere S . To achievethis, one must incorporate the quaternionic structure afforded by the map J as in (15)(in the classical limit) and ensure that the monad is compatible with this extra structure.The same is true in the noncommutative case, as we shall see presently.Given the pair of maps constructed in the previous section, σ z : H ⊗ A ( C θ )( − → A ( f M θ ( H, K )) ⊗ K ⊗ A ( C θ ) ,τ z : K ⊗ A ( C θ ) → A ( f M θ ( K, L )) ⊗ L ⊗ A ( C θ )(1) , e firstly note that the anti-algebra map J in (15) induces a new pair of maps, σ J ( z ) : H ⊗ J (cid:0) A ( C θ )( − (cid:1) → A (cid:16) f M θ ( H, K ) (cid:17) ⊗ K ⊗ J (cid:0) A ( C θ ) (cid:1) ,σ J ( z ) := X α M α ⊗ J ( z α ) , (38)and τ J ( z ) : K ⊗ J (cid:0) A ( C θ ) (cid:1) → A (cid:16) f M θ ( K, L ) (cid:17) ⊗ L ⊗ J (cid:0) A ( C θ )(1) (cid:1) ,τ J ( z ) := X α N α ⊗ J ( z α ) . (39)Here, J ( A ( C θ )) is the left A ( C θ )-module induced by the anti-algebra map J and σ J ( z ) , τ J ( z ) are homomorphisms of left A ( C θ )-modules. We may also take the adjoints of theabove maps. To make sense of this, we need to add to our picture the matrix elements M αab ∗ , so that the adjoint of σ z is(40) σ ⋆z : v a ⊗ Z X b,α M αab ∗ ⊗ u b ⊗ z ∗ α Z, Z ∈ A ( C θ ) , where a = 1 , . . . , k + 2, b = 1 , . . . , k and α = 1 , . . . ,
4. Let us denote by M α † the k × (2 k + 2) matrix with entries ( M α † ) ba = M αab ∗ . Then with respect to the above choiceof bases, the adjoint map σ ⋆z may be written more compactly as σ ⋆z = X α M α † ⊗ z ∗ α . Similarly, we add the matrix elements N αdc ∗ and write ( N α † ) cd = N αdc ∗ , so that the adjointof τ z is τ ⋆z : w b ⊗ Z X a,α N αba ∗ ⊗ v a ⊗ z ∗ α Z, or τ ⋆z = P α N α † ⊗ z ∗ α in compact notation. The elements M αab ∗ are the generators ofthe algebra A ( f M θ ( K ∗ , H ∗ )), whereas the elements N αdc ∗ are the generators the algebra A ( f M θ ( L ∗ , K ∗ )). Applied to equations (38) and (39), all of this yields a pair of homo-morphisms of right A ( C θ )-modules σ ⋆J ( z ) : K ∗ ⊗ J (cid:0) A ( C θ ) (cid:1) ∗ → A (cid:16) f M θ ( K ∗ , H ∗ ) (cid:17) ⊗ H ∗ ⊗ J (cid:0) A ( C θ ) (cid:1) ∗ (1) ,τ ⋆J ( z ) : L ∗ ⊗ J (cid:0) A ( C θ ) (cid:1) ∗ ( − → A (cid:16) f M θ ( L ∗ , K ∗ ) (cid:17) ⊗ K ∗ ⊗ J (cid:0) A ( C θ ) (cid:1) ∗ , defined respectively by σ ⋆J ( z ) = X α M α † ⊗ J ( z α ) ∗ , τ ⋆J ( z ) = X α N α † ⊗ J ( z α ) ∗ . Of course, we may identify the vector spaces H and L ∗ through the basis isomorphism u b ˆ w b for each b = 1 , . . . , k . Similarly the isomorphism v a ˆ v a for a = 1 , . . . , k + 2gives an identification of the vector space K with its dual K ∗ . Also, the right module J ( A ( C θ )) ∗ may be identified with A ( C θ ) by the composition of the map J with theinvolution ∗ (noting that this identification is not the identity map). Through these dentifications, we may think of σ ⋆J ( z ) and τ ⋆J ( z ) as module homomorphisms τ ⋆J ( z ) : H ⊗ A ( C θ )( − → A (cid:16) f M θ ( H, K ) (cid:17) ⊗ K ⊗ A ( C θ ) ,σ ⋆J ( z ) : K ⊗ A ( C θ ) → A (cid:16) f M θ ( K, L ) (cid:17) ⊗ L ⊗ A ( C θ )(1) . It is straightforward to check that we now have σ ⋆J ( z ) τ ⋆J ( z ) = 0 and so all of this meansthat the maps σ ⋆J ( z ) and τ ⋆J ( z ) also give a parameterisation of the noncommutative space ofmonads, albeit a different parameterisation from the one we started with. In the classicalcase the above procedure applied to a given monad again yields a monad, although it isnot necessarily the one we started with. If fact, in the classical case, one is interestedonly in the subset of monads which are invariant under the above construction, namelythe monad obtained by applying J and dualising is required to be isomorphic to the onewe start with (this is the sense in which we require monads to be compatible with J ). Wecall such monads self-dual . In our algebraic framework, where we work not with specificmonads but rather with the (possibly noncommutative) space f M θ ; k of all monads, thisextra requirement is encoded as follows. Proposition 4.5.
The space of self-dual monads is parameterised by the algebra A ( f M SDθ ; k ) ,the quotient of the algebra A ( f M θ ; k ) by the further relations (41) N = − M † , N = M † , N = − M † , N = M † . Proof.
The condition that the maps σ z and τ z should parameterise self-dual monads isthat σ z = τ ⋆J ( z ) , equivalently that τ z = − σ ⋆J ( z ) . In terms of the matrices M α , N α , theformer condition reads(42) X α M α ⊗ z α = X α N α † ⊗ J ( z α ) ∗ . Equating coefficients of generators of A ( C θ ) in each of these equations yields the extrarelations as stated. (cid:3) Remark 4.6.
The identification of the vector space K with its dual K ∗ means that themodule K ⊗ A ( C θ ) acquires a bilinear form given by(43) ( ξ, η ) := h J ξ | η i = X a ( J ξ ) ∗ a η a for ξ = ( ξ a ) and η = ( η a ) ∈ K ⊗ A ( C θ ), with h · | · i the canonical Hermitian structure on K ⊗ A ( C θ ). The monad condition, which now reads0 = τ z σ z = − σ ⋆J ( z ) σ z , translates into the more practical condition that the columns of the matrix σ z (equiva-lently the rows of τ z ) are orthogonal with respect to the form ( · , · ).Moreover, we see that0 = τ z + J ( z ) σ z + J ( z ) = τ z σ z + τ z σ J ( z ) + τ J ( z ) σ z + τ J ( z ) σ J ( z ) = τ z σ J ( z ) + τ J ( z ) σ z = − σ ⋆J ( z ) σ J ( z ) + σ ⋆z σ z so that in the matrix algebra M k ( C ) ⊗ A ( f M SDθ ; k ) ⊗ A ( C θ ) we have also σ ⋆J ( z ) σ J ( z ) = σ ⋆z σ z . emark 4.7. The above identifications of vector spaces H ∼ = L ∗ and K ∼ = K ∗ yield anidentification of A ( f M θ ( H, K )) with A ( f M θ ( L ∗ , K ∗ )) and hence a reality structure on thegenerators M αab . It follows that the space of self-dual monads is parameterised by a totalof 4 k (2 k + 2) generators M αab . As already remarked, the condition σ ⋆J ( z ) σ z = 0 is equiv-alent to demanding that the columns of σ z are pairwise orthogonal with respect to thebilinear form ( · , · ) and, since σ z has k columns, this yields k ( k −
1) such orthogonalityconditions. Now as in Prop. 4.3 we may equate to zero the coefficients of the products z α z β for α ≤ β , and we note that there are 10 such coefficients in each orthogonalitycondition. This yields a total of 5 k ( k −
1) constraints on the generators M αab .4.3. ADHM construction of noncommutative instantons.
We are ready for theconstruction of charge k noncommutative bundles with instanton connections. As inprevious sections, we have the (2 k + 2) × k algebra-valued matrices σ z = M ⊗ z + M ⊗ z + M ⊗ z + M ⊗ z ,σ J ( z ) = − M ⊗ z ∗ + M ⊗ z ∗ − M ⊗ z ∗ + M ⊗ z ∗ which, as already observed, have the properties σ ⋆J ( z ) σ z = 0 and σ ⋆J ( z ) σ J ( z ) = σ ⋆z σ z . Lemma 4.8.
The entries of the matrix ρ := σ ⋆z σ z = σ ⋆J ( z ) σ J ( z ) commute with those ofthe matrix σ z .Proof. One finds that the ( µ, ν ) entry of ρ is( ρ ) µν = X r,α,β ( M α † ) µr M βrν ⊗ z ∗ α z β and that the a, b entry of σ z is ( σ z ) ab = X γ M γab ⊗ z γ . It is straightforward to check that these elements always commute using the relations (3)for A ( C θ ) and the relations of Prop. 4.1 for A ( f M θ ( H, K )). The essential feature is thatevery factor of η βα coming from the relations between the M α ’s is cancelled by a factorof η αβ coming from the relations between the z α ’s. (cid:3) We need to enlarge slightly the matrix algebra M k ( C ) ⊗ A ( f M SDθ ; k ) ⊗ A ( C θ ) by adjoiningan inverse element ρ − for ρ , together with a square root ρ − . That these matrices maybe inverted is an assumption, even in the commutative case where doing so corresponds tothe deletion of the non-generic points of the moduli space; these correspond to so-called‘instantons of zero size’.From the previous lemma the matrix ρ , which is self-adjoint by construction, hasentries in the centre of the algebra A ( f M SDθ ; k ) ⊗ A ( C θ ), so these new matrices ρ − and ρ − must also be self-adjoint with central entries. We collect the matrices σ z , σ J ( z ) togetherinto the (2 k + 2) × k matrix(44) V := (cid:0) σ z σ J ( z ) (cid:1) and we have by construction that V ∗ V = ρ (cid:18) I k I k (cid:19) , here I k denotes the k × k identity matrix. This of course means that the quantity(45) Q := V ρ − V ∗ = σ z ρ − σ ⋆z + σ J ( z ) ρ − σ ⋆J ( z ) is automatically a projection: Q = Q = Q ∗ . For convenience we denote Q z := σ z ρ − σ ⋆z , Q J ( z ) := σ J ( z ) ρ − σ ⋆J ( z ) , which are themselves projections, in fact orthogonal ones, Q J ( z ) Q z = 0, due to the factthat σ ⋆J ( z ) σ z = 0. Lemma 4.9.
The trace of the projection Q z is equal to k ; likewise for Q J ( z ) .Proof. We compute the trace as follows:Tr Q z = X µ ( σ z ρ − σ ⋆z ) µµ = X µ,r,s ( σ z ) µr ( ρ − ) rs ( σ ⋆z ) sµ = X µ,r,s ( ρ − ) rs ( σ z ) µr ( σ z ) ⋆sµ = X µ,r,s ( ρ − ) rs ( σ z ) ⋆sµ ( σ z ) µr = X r,s ( ρ − ) rs ( σ ⋆z σ z ) sr = Tr I k = k. In the third equality we have used the fact that, as said, the entries of ρ − commutewith those of σ z , whereas in the fourth equality we have used the fact that every elementof A ( f M SDθ ; k ) ⊗ A ( C θ ) commutes with its own adjoint. An analogous chain of equalityestablishes the same result for the projection Q J ( z ) . (cid:3) As a consequence the projection Q has trace 2 k . Proposition 4.10.
The operator P := I k +2 − Q is a projection in the algebra M k +2 (cid:16) A ( f M SDθ ; k ) ⊗ A ( S θ ) (cid:17) with trace equal to .Proof. The entries of the projection Q z are in the subalgebra of A ( f M SDθ ; k ) ⊗ A ( C θ ) madeup of U(1)-invariants which, by the discussion of Sect. 3.2, is precisely A ( f M SDθ ; k ) ⊗A ( CP θ ).Now recall from Sect. 2.3 that the degree one elements of A ( CP θ ) of the form Z + J ( Z )generate the J -invariant subalgebra, which may be identified with A ( S θ ). The entries of Q z being linear in the generators of A ( CP θ ), it follows that the projection Q has entries in A ( f M SDθ ; k ) ⊗ A ( S θ ). The same is true of the complementary projection P as well. Finally,since the projection Q has trace 2 k , the trace of the projector P is just 2. (cid:3) We think of the projective right A ( S θ )-module E := P A ( S θ ) k +2 as defining a family ofrank two vector bundles over S θ parameterised by the noncommutative space f M SDθ ; k . Weequip this family of vector bundles with the associated family of Grassmann connections ∇ := P ◦ (id ⊗ d), after extending the exterior derivative from A ( S θ ) to A ( f M SDθ ; k ) ⊗ A ( S θ )by id ⊗ d. Moreover, we need also to extend the Hodge ∗ -operator as id ⊗ ∗ θ . Proposition 4.11.
The curvature F = P ((id ⊗ d) P ) of the Grassmann connection ∇ = P ◦ (id ⊗ d) is anti-self-dual, that is to say (id ⊗ ∗ θ ) F = − F . roof. When θ = 0 this construction is the usual ADHM construction and it is known [2]( cf . also [19]) that it produces connections whose curvature is an anti-self-dual two-form:(id ⊗ ∗ θ ) P ((id ⊗ d) P ) = − P ((id ⊗ d) P ) . As observed in Sect. 2.1, the Hodge ∗ -operator is defined by the same formula as itis classically and, as vector spaces, the self-dual and anti-self-dual two-forms Ω ± ( S θ )are the same as their undeformed counterparts Ω ± ( S ). This identification survives thetensoring by A ( f M SDθ ; k ) which yields A ( f M SDθ ; k ) ⊗ Ω ± ( S θ ) to be isomorphic, as vector spaces,to A ( f M SDk ) ⊗ Ω ± ( S ). Thus the anti-self-duality holds also when θ = 0. (cid:3) Remark 4.12.
One may alternatively verify the anti-self-duality via a complex structure.Indeed, there is an (almost) complex structure γ : Ω ( CP θ ) → Ω ( CP θ ) given by γ (d z l ) :=(d ◦ J )( z l ), l = 1 , . . . ,
4, the operator J being the one defined in (15), for which we declarethe forms d z l to be holomorphic and the forms d z ∗ l to be anti-holomorphic. For instance,on generators of A ( CP θ ) we haved( z j z ∗ l ) = η lj z ∗ l d z j + z j d z ∗ l from the Leibniz rule and the relations (3), and we write d = ∂ + ¯ ∂ with respect tothis decomposition into holomorphic and anti-holomorphic forms. Since as vector spacesthe various graded components of the differential algebra Ω( CP θ ) are undeformed, theseoperators ∂ , ¯ ∂ extend to a full Dolbeault complex with ∂ = ¯ ∂ = ¯ ∂∂ + ∂ ¯ ∂ = 0. Thealgebra inclusion A ( S θ ) ֒ → A ( CP θ ) extends to an inclusion of differential graded algebrasΩ( S θ ) ֒ → Ω( CP θ ) and the Hodge operator ∗ θ is, as in the classical case, defined in such away that a two-form ω ∈ Ω ( S θ ) is anti-self-dual if and only if its image in Ω ( CP θ ) is oftype (1 , P ((id ⊗ d) P ) is anti-self-dual, we use thisinclusion of forms (i.e. we express everything in terms of d z j , d z ∗ j ) and check that eachcomponent F ad = P ab ((id ⊗ d) P bc ) ∧ (id ⊗ d) P cd ) of the curvature is a sum of terms oftype (1 , E := P A ( S θ ) k +2 given above. To this end we observe that the matrix σ z has k linearlyindependent columns (since if not, it would not be injective) and that the columns of σ J ( z ) are obtained from those of σ z by applying the map J . Clearly we are free to rearrange thecolumns of the matrix V (since this will not alter the class of the projection P ), whencewe may as well arrange them asV = (cid:0) σ z (1) J ( σ z (1) ) σ z (2) J ( σ z (2) ) · · · σ z ( k ) J ( σ z ( k ) ) (cid:1) , where σ z ( l ) denotes the l -th column of σ z and J ( σ z ( l ) ) denotes the l -th column of σ J ( z ) .For fixed l , we denote the entries of the column σ z ( l ) (together with their conjugates) by w µ ( l ) := X α M αµl ⊗ z α , ( w µ ( l ) ) ∗ := X α M αµl ∗ ⊗ z ∗ α , µ = 1 , . . . , k + 2 . The entries of the column J ( σ z ( l ) ) are obtained from those of σ z ( l ) by applying the map J , and one clearly has J (( w µ ( l ) ) ∗ ) = ( J ( w µ ( l ) )) ∗ . In the classical limit θ = 0, we couldevaluate the parameters M αab as fixed numerical values: this would identify the columns ( l ) z and J ( σ ( l ) z ) as spanning a quaternionic line in H k +1 , where the latter is defined by the2 k + 2 complex coordinates w ( l ) µ and their conjugates, equipped with an anti-involution J . In the noncommutative case, although we lack the evaluation of the parameters M αab ,we continue to interpret the columns σ ( l ) z and J ( σ ( l ) z ) as spanning a ‘one-dimensional’quaternionic line.As already observed in Rem. 4.6, the columns of σ z are orthogonal, as are the columnsof σ J ( z ) ; whence the rank 2 k projection Q in (45) decomposes as a sum of projections Q = Q + · · · + Q k , where Q l := e Ψ i e Ψ l ∗ is the rank two projection defined by the (2 k + 2) × e Ψ l comprised of the columns σ z ( l ) and J ( σ z ( l ) ), appropriately normalised by ρ − . Explicitly,this matrix is e Ψ l := (cid:0)P r,α ( M αµr ⊗ z α )( ρ − ) rl P s,β ( M βµs ⊗ J ( z β ))( ρ − ) sl (cid:1) µ =1 ,..., k +2 , and a direct check yields e Ψ l ∗ e Ψ l = I so that Q l is indeed a projection for each l = 1 , . . . , k .Hence the matrix V in (44) has 2 k columns which we interpret as spanning k quaternioniclines, with the same being true of the normalised matrix V ρ − . The computation of thetopological charge of the projection Q therefore boils down to the computation of thecharge of each of the projections Q l , for l = 1 , . . . , k . Lemma 4.13.
For each l = 1 , . . . , k the projection Q l is Murray-von Neumann equivalentto the projection ⊗ q in the algebra M k +2 ( A ( f M SDθ ; k ) ⊗ A ( S θ )) , where q is the basicprojection defined in equation (27).Proof. From equations (26) and (27) we know that q = ΨΨ ∗ . Then, for each l = 1 , . . . , k define a partial isometry V l in M k +2 , (cid:16) A ( f M SDθ ; k ) ⊗ A ( S θ ) (cid:17) by V l := e Ψ l (1 ⊗ Ψ ∗ ) , V ∗ l := (1 ⊗ Ψ) e Ψ ∗ l . Straightforward computations show that V l V l ∗ = Q l and V l ∗ V l = 1 ⊗ q . (cid:3) We invoke the strategy of [16] to compute the topological charge of the family ofbundles defined by each Q l . Indeed, the charge of the projection q was shown in [14]be equal to 1, given as a pairing between the second Chern class ch ( q ), which lives inthe cyclic homology group HC ( A ( S θ )), with the fundamental class of S θ , which livesin the cyclic cohomology HC ( A ( S θ )). Although the class ch ( Q l ), being an elementin HC ( A ( f M SDθ ; k ) ⊗ A ( S θ )), may not a priori be paired with the fundamental class of S θ , Kasparov’s KK-theory is used to show that in fact there is a well-defined pairingbetween the K-theory K (cid:16) A ( f M SDθ ; k ) ⊗ A ( S θ ) (cid:17) and the K-homology K ( A ( S θ )). Since bythe previous lemma the projections 1 ⊗ q and Q l define the same class in the K-theory of A ( f M SDθ ; k ) ⊗ A ( S θ ), it follows as in [16] that the topological charge of each projection Q l is equal to 1. Proposition 4.14.
The family of bundles E = P A ( S θ ) k +2 has topological charge equalto − k . roof. By the argument given above, the projections Q l have topological charge equal to1 for each l = 1 , . . . , k . The projection Q = Q + · · · + Q k therefore has charge k , whence P must have charge − k . (cid:3) We finish this section by remarking that the construction given above in the section hasan interpretation in terms of ‘universal connections’, as described in [3]. As already said,the classical quaternion vector space H k +1 may be identified with the complex vector space C k +2 equipped with the quaternionic structure J . Points of the Grassmannian manifoldGr k ( H k +1 ) of quaternionic k -dimensional subspaces of H k +1 may thus be identified with2 k -dimensional subspaces of C k +2 which are invariant under the involution J . Followingthe general strategy of [5] for the coordinatisation of Grassmannians, the algebra offunctions on Gr k ( H k +1 ) is given by functions taking values in the set of rank 2 k projectors P = ( P µν ) on C k +2 which are J -invariant, viz. A (Gr k ( H k +1 )) := C h P µν | X λ P µλ P λν = P µν , ( P µν ) ∗ = P νµ , X µ P µµ = 2 k, J ( P µν ) = P µν i , where µ, ν = 1 , . . . , k + 2. In the classical case, when θ = 0, the projection Q in (45)realises A ( S θ =0 ) as a subalgebra of A (Gr k ( H k +1 )), whence this construction should beviewed as the dual of an embedding S ֒ → Gr k ( H k +1 ), as given in [3]. We expect that, inthe deformed case, the projection Q views A ( S θ ) as a subalgebra of a suitably-deformedversion of A (Gr k ( H k +1 )). For fixed k , the set of monads is bound to parameterise the setof such ‘algebra embeddings’.4.4. ADHM construction of charge one instantons.
As a way of illustration webriefly verify that the above ADHM construction of noncommutative families of instan-tons gives back the family constructed in [16] when performed for the charge one case.The starting point is the basic instanton on S θ described in Sect. 2.2 and which arisesvia a monad construction as follows. The monad we consider is the sequence(46) A ( C θ )( − σ z −→ C ⊗ A ( C θ ) τ z −→ A ( C θ )(1) , where the arrows are the maps σ z = (cid:0) z z z z (cid:1) t , τ z = σ ⋆J ( z ) = (cid:0) − z z − z z (cid:1) . Since τ z σ z = σ ⋆J ( z ) σ z = 0, it is clear that this is a monad with k = 1; by construction it isself-dual. In the present case ρ = σ ⋆z σ z = P j z ∗ j z j = r , which we already assumed wasinvertible (corresponding to the deletion of the origin in C θ ). One computes thatVV ∗ = r − r + x α β r + x − µβ ∗ ¯ µα ∗ α ∗ − ¯ µβ r − x β ∗ µα r − x which is just the projector q of equation (27). This is the ‘tautological’ monad construc-tion given in [5]. The anti-self-dual version is the projector P = 1 − VV ∗ , in agreementwith the ADHM construction above.The monad (46) may be rewritten in the form(47) σ z = (1 , , , t ⊗ z + (0 , , , t ⊗ z + (0 , , , t ⊗ z + (0 , , , t ⊗ z , ith τ z defined as its dual. With the strategy of [16] one generates new instantons bycoacting on the generators z , . . . , z with the quantum conformal group A (SL θ (2 , H )).Using the formula (18) for the coaction, the monad map (47) transforms into(48) σ ∆ L ( z ) = a a c c ⊗ z + − a ∗ a ∗ − c ∗ c ∗ ⊗ z + b b d d ⊗ z + − b ∗ b ∗ − d ∗ d ∗ ⊗ z , and these four column vectors are the columns of the matrix (17) which defines thealgebra A (SL θ (2 , H )). If we write c M = (cid:0) a a c c (cid:1) t , c M = (cid:0) − a ∗ a ∗ − c ∗ c ∗ (cid:1) t , c M = (cid:0) b b d d (cid:1) t , c M = (cid:0) − b ∗ b ∗ − d ∗ d ∗ (cid:1) t , then we have the algebra relations c M αj c M βl = η jl η βα c M βl c M αj coming from the relations (20)for A (SL θ (2 , H )). We thus think of the algebra generated by the c M αj as parameterisingthe set of charge one instantons, since the map σ ∆ L ( z ) may be used to construct the family(28) of projections with topological charge equal to 1 and hence a family of Grassmannconnections with anti-self-dual curvature, just as in [16].In contrast, the ADHM construction of Sect. 4.3 for the case k = 1 says that the chargeone monads are parameterised by the algebra A ( f M θ ; k ) generated by the matrix elements M αj , with j, α = 1 , . . . ,
4, subject in particular to the relations M αj M βl = η βα M βl M αj .We see that these two approaches seem to give different parameterisations of the set ofmonads for the case k = 1, and hence of the set of charge one instantons. The discrepancyhas its root in the fact that the ADHM construction requires generators lying in the samerow of the matrix ( A ij ) to commute, whereas the ‘coaction approach’ given above saysthat such generators do not commute.However, the discrepancy fades away when we pass to the ‘true’ parameter space for thefamilies. On the one hand, as observed in [16], the coaction (24) of the quantum subgroup A (Sp θ (2)) of A (SL θ (2 , H )) leaves the basic one-form (11) invariant. We think of the lattercoaction as generating gauge-equivalent instantons, so that the ‘true’ parameter space forthis family is rather the subalgebra of A (SL θ (2 , H )) of coinvariants under the coaction of A (Sp θ (2)). The generators of this algebra are computed to be b m αβ := X l c M αl ∗ c M βl , α, β = 1 . . . , , whose relations are easily found to be(49) b m αβ b m µν = η βµ η νβ η µα η αν b m µν b m αβ , and which certainly do not depend on the rows of the matrix ( A ij ). On the other hand,gauge equivalence for the ADHM family parameterised by the M αi is generated by theaction of the classical group Sp(2) (we borrow this result from Prop. 5.2 in the nextsection), and here the invariant subalgebra is generated by elements of the form m αβ := X l M αl ∗ M βl , α, β = 1 . . . , . The relations in this algebra are just as in equation (49), so that these two families ofcharge one instantons are just the same. . Gauge Equivalence of Noncommutative Instantons
Classically, a way to think of a gauge transformation of a vector bundle E over S is asa unitary change of basis in each fibre E x in a way which depends smoothly on x ∈ S .Two connections on E are said to be gauge equivalent if they are related by a gaugetransformation in this way. Now, rather than being interested in the set of all instantonson S , one is interested in the collection of gauge equivalence classes, that is to say classesof instantons modulo gauge transformations.It is therefore necessary to have an analogue of the notion of gauge equivalence alsofor the noncommutative families of instantons constructed previously. In fact, noncom-mutative geometry is a very natural setting for the study of gauge transformations, aswe shall see in this section; we refer in particular to [7, 8] ( cf . also [13]).5.1. Gauge equivalence for families of instantons.
Recall that a first order differen-tial calculus on a unital ∗ -algebra A is a pair (Ω A, d A ), where Ω A is an A - A -bimodulegiving the space of one-forms and d A : A → Ω A is a linear map satisfying the Leibnizrule, d A ( xy ) = x (d A y ) + (d A x ) y for all x, y ∈ A. One also assumes that the map x ⊗ y → x (d A y ) is surjective. One names Ω A a ∗ -calculusif for x j , y j ∈ A one has that P j x j d y j = 0 implies P j d( y ∗ j ) x ∗ j = 0: it follows from thiscondition that there is [26] a unique ∗ -structure on Ω A such that (d A a ) ∗ = d A ( a ∗ ) forall a ∈ A . The differential calculi on A ( S θ ) and A ( S θ ) in Sect. 2.1 are examples of firstorder differential ∗ -calculi on noncommutative spaces.Let us fix a choice of ∗ -calculus on A . Then let E be a finitely generated projective right A -module endowed with an A -valued Hermitian structure denoted by h·|·i . A connectionon E is a linear map ∇ : E → E ⊗ A Ω A satisfying the Leibniz rule ∇ ( ξx ) = ( ∇ ξ ) x + ξ ⊗ d A x for all ξ ∈ E , x ∈ A. The connection ∇ is said to be compatible with the Hermitian structure on E if it obeys h∇ ξ | η i + h ξ |∇ η i = d A h ξ | η i for all ξ ∈ E , x ∈ A. On E there is at least one compatible connection, the so-called Grassmann connection ∇ .If P ∈ End A ( E ), P = P = P ∗ , is the projection which defines E as a direct summand ofa free module, that is, E = P ( C N ⊗ A ), then ∇ = P ◦ d. Any other connection on E isof the form ∇ = ∇ + ω , where ω is an element of Hom A ( E , E ⊗ A Ω A ).The gauge group of E is defined to be U ( E ) := { U ∈ End A ( E ) | h U ξ | U η i = h ξ | η i for all ξ, η ∈ E } . If ∇ is a compatible connection on E , each element U of the gauge group U ( E ) induces a‘new’ connection by the action ∇ U := U ∇ U ∗ . Of course, ∇ U is not really a different connection, it simply expresses ∇ in terms ofthe transformed bundle U E , hence one says that a pair of connections ∇ , ∇ on E are gauge equivalent if they are related by such a gauge transformation U . In terms of thedecomposition ∇ = ∇ + ω , one finds that ∇ U = ∇ + ω U , where ω U := U ( ∇ U ∗ ) + U ωU ∗ . choice of gauge would be a choice of partial isometry Ψ : E → A N such that Ψ ∗ Ψ = Id E and ΨΨ ∗ = P . Any other gauge is then given by an element U of the gauge group of E :the partial isometry Ψ gets replaced by U Ψ, for which we indeed have( U Ψ) ∗ ( U Ψ) = Ψ ∗ Ψ = Id E . and the projection P gets transformed to( U Ψ)( U Ψ) ∗ = U (ΨΨ ∗ ) U ∗ = U P U ∗ , an operation that does not change the equivalence class of P . In the fixed gauge theGrassmann connection ∇ = P ◦ d naturally acts on ‘equivariant maps’ ϕ = Ψ F where F ∈ A N . The result is an ‘equivariant one-form’, ∇ (Ψ F ) = (ΨΨ ∗ )d(Ψ F ) = Ψ (cid:16) d F + Ψ ∗ d(Ψ) F (cid:17) , and identifies the gauge potential to be given by A = (Ψ ∗ (dΨ) − ( dΨ ∗ )Ψ) . Under the transformation Ψ U Ψ, the gauge potential transforms as expected:Ψ ∗ dΨ Ψ ∗ (dΨ) + Ψ ∗ U ∗ (d U )Ψ . We now turn back to the construction of instantons. Gauge equivalence being defined asabove by unitary module endomorphisms means that we are free to act on the right A ( C θ )-module K = K ⊗ A ( C θ ) by a unitary element of the matrix algebra M k +2 ( C ) ⊗ A ( C θ ).In order to preserve the instanton construction, we must do so in a way preservingthe bilinear form ( · , · ) of equation (43) which comes from the identification of K withits dual K ∗ . Hence the map σ z in (31) (or in (33)) is defined up to a transformation A ∈ End A ( C θ ) ( K ), which is unitary and is required to commute with the quaternionstructure J . Similarly, we are free to change basis in the modules H = H ⊗ A ( C θ ) and L = L ⊗ A ( C θ ), provided we preserve the fact that we identify J ( H ) ⋆ and L . This meansthat the map τ z of (35) is defined up to an invertible transformation B ∈ End A ( C θ ) ( H ).All this is saying is that the monad maps σ z and τ z were expressed as matrices withrespect to a choice of basis for each of the vector spaces H , K and L ; it is natural toquestion the extent to which the resulting Grassmann connection ∇ depends on the choiceof these bases. We denote by GL( H ) the set of automorphisms of H and by Sp( K ) theset of all unitary endomorphisms of K respecting the quaternion structure:Sp( K ) := { A ∈ End A ( C θ ) ( K ) | h Aξ | Aξ i = h ξ | ξ i , J ( Aξ ) = AJ ( ξ ) for all ξ ∈ K} . Given A ∈ Sp( K ) and B ∈ GL( H ), the gauge freedom is to map σ z Aσ z B . Proposition 5.1.
For all B ∈ GL( H ) , under the transformation σ z σ z B the projection P of Prop. 4.10 is left invariant.Proof. One first checks that ρ ( σ z B ) ⋆ ( σ z B ) = B ⋆ ρ B under this transformation, sothat Q z σ z B ( B ⋆ ρ B ) − B ⋆ σ ⋆z = σ z B ( B − ρ − ( B ⋆ ) − ) B ⋆ σ ⋆z = Q z , whence the projection P is unchanged. (cid:3) Proposition 5.2.
For all A ∈ Sp( K ) , under the transformation σ z Aσ z the projection P of Prop. 4.10 transforms as P A P A ⋆ . roof. Replacing σ z by Aσ z leaves ρ invariant (since A is unitary) and so has the effectthat Q z Aσ z ρ − σ ⋆z A ⋆ = A Q z A ⋆ , whence it follows that P is mapped to A P A ⋆ . (cid:3) These results give the general gauge freedom on monads, although from the point ofview of computing the number of constraints on the algebra generators M αab we need onlyconsider the effect of these transformations on the vector spaces H , K and L , i.e. itis enough to consider the groups of ‘constant’ automorphisms. This means the groupSp( K ) = Sp( k + 1) ⊂ Sp( K ) and the group GL( k, R ) ⊂ GL( H ) (the latter because wemust preserve the identification of J ( H ) ⋆ with L , and complex linear transformations of H would interfere with the tensor product in J ( H ) ⋆ ). In fact, it is known in the classicalcase that these constant transformations are sufficient to generate all gauge symmetriesof the instanton bundles produced by the ADHM construction.We conclude that in the noncommutative case as well the gauge equivalence imposesan additional ( k +1)(2( k +1)+1) constraints due to Sp( k +1) and a further k constraintsdue to GL( k, R ). From Rem. 4.7, the total number of generators minus the total numberof constraints is thus computed to be(8 k + 8 k ) − k ( k − − (3 k + 5 k + 3) = 8 k − , just as for the classical case, a result which is somehow reassuring.5.2. Morita equivalent geometries and gauge theory.
It is a known idea thatMorita equivalent algebras describe the same topological space. The simplest case isthat of a one-point space X = {∗} : the matrix algebras M n ( C ) for any positive integer n all have the same one-point spectrum. More generally, if X is a compact Hausdorff space,the algebras C ( X ) ⊗ M n ( C ) are all Morita equivalent and all have the same spectrum X .With this in mind, gauge theory arises naturally out of the consideration of how totransfer differential structures between Morita equivalent algebras. If one takes suchstructures to be defined by a Dirac operator and associated spectral triple, then themethod for doing this is discussed in [7, 8]. Here we discuss a more general framework,where algebras may be equipped with differential calculi not necessarily coming from aspectral triple.Let A be a unital ∗ -algebra and suppose that the ∗ -algebra B is Morita equivalent to A via the B - A -bimodule E , that is to say B ≃ End A ( E ). In addition, on E there arecompatible A -valued and B -valued Hermitian structures . Then a choice of a connection ∇ on E , viewed as a right A -module, yields a differential calculus on B . First of all, theoperator on B given by d ∇ B ( x ) := [ ∇ , x ] , x ∈ B, is easily seen to be a derivation: d ∇ B ( xy ) = x (d ∇ B y ) + (d ∇ B x ) y , for x, y ∈ B . The B - B -bimodule Ω B of one-forms is then defined byΩ B := B (cid:0) d ∇ B ( B ) (cid:1) B. We shall also require the Hermitian structures to be self-dual, i.e. every right A -module homomor-phism ϕ : E → A is represented by an element of η ∈ E by the assignment ϕ ( · ) = h η |·i . A similarproperty holds for the second Hermitian structure as well. or this to define a ∗ -calculus we need that the connection ∇ be compatible with the A -valued Hermitian structure on E in the sense that h∇ ξ | η i + h ξ |∇ η i = d A h ξ | η i for all a ∈ A and ξ, η ∈ E . If this compatibility condition is satisfied, the assumption P j x j d ∇ B y j = 0 translates into P j ( x j ∇ y j ) ξ = P j x j y j ( ∇ ξ ) for all x j , y j ∈ B and all ξ ∈ E . This implies, for all ξ, η ∈ E and all x j , y j ∈ B , that X j h d ∇ B ( y ∗ j ) x ∗ j ξ | η i = X j h∇ ( x ∗ j y ∗ j ξ ) − x ∗ j ∇ ( y ∗ j ξ ) | η i = X j −h x ∗ j y ∗ j ξ |∇ η i + d A h x ∗ j y ∗ j ξ | η i + h y ∗ j ξ |∇ ( x j η ) i − d A h y ∗ j ξ | x j η i = X j −h x ∗ j y ∗ j ξ |∇ η i + h ξ | y j x j ∇ η i , whence it follows that P j d ∇ B ( y ∗ j ) x ∗ j = 0 as it should for a ∗ -calculus. We interpret thepassage d A → d ∇ B as an inner fluctuation of the geometry which results in a ‘Moritaequivalent’ first order calculus (Ω B, d ∇ B ), now for the algebra B .A natural application is to think of the algebra A as being Morita equivalent to itself,so that E = A as a right A -module and B = A . In this case, any Hermitian connectionon E is necessarily of the form(50) ∇ ξ = d A ξ + ωξ, for ξ ∈ E , with ω = − ω ∗ ∈ Ω A a skew-adjoint one-form. The corresponding differential on B = A is computed to be(d ∇ A b ) ξ = [ ∇ , b ] ξ = ∇ ( bξ ) − b ∇ ξ = d A ( bξ ) + ωbξ − b d A ξ − bωξ = (d A b ) ξ + [ ω, b ] ξ, using the Leibniz rule for d A . The passaged A → d ∇ A = d A + [ ω, · ]is once again interpreted as an inner fluctuation of the geometry, although when A iscommutative there are no non-trivial inner fluctuations and thus no new degrees of free-dom generated by the above self-Morita mechanism. However, in the noncommutativesituation there is an interesting special case where ω is taken to be of the form ω = u ∗ d A u ,for u a unitary element of the algebra A . Such a fluctuation is unitarily equivalent toacting on A by the inner automorphism α u : A → A, α u ( a ) = uau ∗ , since for all a ∈ A we have that d ∇ A ( a ) = u ∗ d A ( α u ( a )) u . It therefore follows that innerfluctuations defined by inner automorphisms generate gauge theory on A .5.3. Gauge theory from quantum symmetries.
We now consider a slightly differenttype of gauge equivalence for our instanton construction which is not present in theclassical case and is a purely quantum ( i.e. noncommutative) phenomenon.We consider the case where A is a comodule ∗ -algebra under a left coaction of a Hopfalgebra H , so that A is isomorphic to its image B = ∆ L ( A ). To transfer a calculus n A to one on B , a possible strategy is as follows. We take the B - A -bimodule to be E := B = ∆ L ( A ) with left B -action and right A -action defined by b ⊲ ξ := bξ, ξ ⊳ a = ξ ∆ L ( a )for ξ ∈ E , a ∈ A , b ∈ B . We also assume that the calculus Ω A is left H -covariant, sothat ∆ L extends to a coaction on Ω A as a bimodule map such that d A is an intertwiner,whence the above bimodule structure on E extends to one-forms in the natural way. Thisalso canonically equips B with a ∗ -calculus Ω B , where the differential is d B = id ⊗ d A .We choose an arbitrary Hermitian connection on the right A -module E for the calculus(Ω A, d A ), which is necessarily of the form ∇ ξ = (id ⊗ d A ) ξ + ˜ ωξ, ξ ∈ E with ˜ ω = ∆ L ( ω ) for some ω = − ω ∗ ∈ Ω A a skew-adjoint one-form. The correspondingdifferential on B is again defined by(d ∇ B b ) ξ = [ ∇ , b⊲ ] ξ = ∇ ( b ⊲ ξ ) − b ⊲ ∇ ξ = d A ( b ⊲ ξ ) + ω ( b ⊲ ξ ) − b ⊲ (d A ξ + ωξ ) , and works out to be d B b = (id ⊗ d A ) b + [˜ ω, b ] . Note also that for all b ∈ B we have b = ∆ L ( a ) for some a ∈ A and so it follows thatd B b = ∆ L (d A a ) + ∆ L ([ ω, a ]) , so that the coaction commutes with inner fluctuations. Moreover, in the case where A isnoncommutative, there are non-trivial inner automorphisms of A and hence non-trivialgauge degrees of freedom which carry over from A to ∆ L ( A ).In particular, we apply this to the case A = A ( S θ ), with H = A (SL θ (2 , H )) the quan-tum conformal group of S θ . The above discussion means that the coaction of A (SL θ (2 , H ))on A ( S θ ) by conformal transformations in itself generates gauge freedom. The naturalway to extend the exterior derivative d A on A ( S θ ) to ∆ L ( A ( S θ )) is as id ⊗ d A : thiscorresponds to taking ˜ ω = 0 and is the choice made in [16]. However, in general we havethe freedom to make the transitiond A → (id ⊗ d A ) + [˜ ω, · ]for some ˜ ω = ∆ L ( u ∗ d A u ), where u is some unitary element of A ( S θ ). Since the groupof inner automorphisms of A is trivial when A is commutative, this is a feature of gaugetheory which is certainly not present in the classical case and is unique to the noncom-mutative paradigm. More on this will be reported elsewhere. References [1] Atiyah M.F., Ward R.S.: Instantons and Algebraic Geometry. Commun. Math. Phys. , 117–124(1977)[2] Atiyah M.F., Drinfel’d V.G., Hitchin N.J., Manin Yu.I.: Construction of Instantons. Phys. Lett. , 185–187 (1978)[3] Atiyah M.F.: Geometry of Yang-Mills Fields . Fermi Lectures, Scuola Normale Pisa, 1979[4] Barth W.: Moduli of Vector Bundles on the Projective Plane. Invent. Math. , 63–91 (1977)[5] Brain S.J., Majid S.: Quantisation of Twistor Theory by Cocycle Twist. Commun. Math. Phys. , 713–774 (2008)
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