Semiquantisation Functor and Poisson-Riemannian Geometry, I
aa r X i v : . [ m a t h . QA ] M a r SEMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIANGEOMETRY, I
EDWIN J. BEGGS & SHAHN MAJID
Abstract.
We study noncommutative bundles and Riemannian geometry atthe semiclassical level of first order in a deformation parameter λ , using afunctorial approach. The data for quantisation of the cotangent bundle isknown to be a Poisson structure and Poisson preconnection and we now showthat this data defines to a functor Q from the monoidal category of clas-sical vector bundles equipped with connections to the monodial category ofbimodules equipped with bimodule connections over the quantised algebra.We adapt this functor to quantise the wedge product of the exterior algebraand in the Riemannian case, the metric and the Levi-Civita connection. Fullmetric compatibility requires vanishing of an obstruction in the classical data,expressed in terms of a generalised Ricci 2-form R , without which our quan-tum Levi-Civita connection is still the best possible. We apply the theory tothe Schwarzschild black-hole and to Riemann surfaces as examples, as well asverifying our results on the 2D bicrossproduct model quantum spacetime. Thequantized Schwarzschild black-hole in particular has features similar to thoseencountered in q -deformed models, notably the necessity of nonassociativity ofany rotationally invariant quantum differential calculus of classical dimensions. Introduction
Noncommutative geometry aims to extend notions of geometry to situations wherethe ‘coordinate algebra’ is noncommutative. Such algebras could arise on ‘quan-tisation’ in the passage from a classical mechanical system to a quantum one or,it is now widely accepted, as a more accurate description of spacetime itself so asto include quantum corrections arising out of quantum gravity. There are differ-ent approaches to the formulation of the right axioms in the noncommutative caseand we mention notably the approach of Connes[10] coming out of cyclic coho-mology, ring-theoretic projective module approach due to Van den Bergh, Staffordand others, e.g. [25], and a constructive approach coming out of quantum groupsbut not limited to them in which the different layers of geometry starting with thedifferential graded algebra are built up, typically guided by quantum symmetryand other considerations. Our work falls within this last approach and particu-larly a set of axioms of ‘noncommutative Riemannian geometry’ using bimoduleconnections[24, 15, 16, 4, 5, 6, 7, 21, 22]. We refer in particular to [7] for a non-trivial 2D example containing a gravitational source and worked out in completedetail in this bimodule connection approach.
Mathematics Subject Classification.
Primary 81R50, 58B32, 83C57.
Key words and phrases. noncommutative geometry, poisson geometry, quantum groups, quan-tum gravity, symplectic connection, torsion, Poisson bracket, monoidal functor.
In this paper we take a step towards the general problem of ‘quantisation’ of all therest of the geometrical structure beyond the algebra itself, within the bimodule con-nections approach. We consider only perturbative phenomena in the sense of orderby order in a deformation parameter λ and hence will miss ‘deep quantum’ effects,but on the plus side this will allow us to construct concrete examples from familiarclassical geometrical data and moreover, by explicitly building structures on theclassical counterparts, we will have a ready-made identification between these andcorresponding quantum objects which would otherwise be open to interpretationsuch as ordering ambiguities. This then provides a route to making experimentalpredictions. Thus, while the deformation problem is by no means adequate for thefull theory of noncommutative geometry, it nevertheless has practical value.In fact we are only going to solve here the problem of quantising Riemannianand other differential geometry to first order in λ . This will already be quite asignificant task as we shall see, but at this level we will arrive at a fairly completeand functorial picture (we would then envision to develop the same ideas orderby order possibly in an A ∞ algebra approach). Physically speaking, the minimalhypothesis is that noncommutative geometry represents an effective descriptionof spacetime to include first planck-scale correction and in that case it may onlybe the first order in λ that are immediately relevant (it is tempting to supposean absolute significance to the noncommutative geometry but that is a furtherassumption). Also, in distance units the value of λ at around 10 − m is extremelysmall making these effects only just now beginning to be measurable in principle,in which case O ( λ ) effects can be expected to be so much even smaller as to bebeyond any possible relevance. This motivates a deeper analysis of the semiclassicallevel where we work to order λ . Mathematically speaking, this means that we areworking at the Poisson level and in principle we could set our our main resultssuch as the functor in Theorem 3.5 in an entirely Poisson setting. However, we willdevelop the theory equivalently in a novel way that keeps better the connectionwith algebra, namely as exact noncommutative geometry but working over the ringof dual numbers C [ λ ]/( λ ) where we set λ =
0. We call the construction of suchnoncommutative geometry more precisely semiquantisation rather than the morefamiliar term ‘semiclassicalisation’.Starting with a classical manifold M expressed algebraically as C ∞ ( M ) , it is well-known that a Poisson bivector ω ij controls its associative deformations and more-over that the ‘quantisation problem’ is then solved notably by Kontsevich[19] at aformal level and in the symplectic case and in the presence of a symplectic connec-tion, more geometrically by Fedosov[14]. These results are, however, only for thealgebra and for actual noncommutative geometry we must ‘quantise’ also differentialstructures, bundles, connections, and so forth. By differential structure we meanthe algebra Ω ( M ) of differential forms and in [17, 2] it was shown that deformationsof the 1-forms Ω ( M ) are controlled to this order by Poisson-preconnections ˆ ∇ onΩ ( M ) (these are only defined along hamiltonian vector fields). The curvature ofthe preconnection entails nonassociativity of the bimodule product (so a breakdownof ( a d b ) c = a (( d b ) c ) for elements a, b, c of the noncommutative algebra) at O ( λ ) .More precisely, for convenience, we will assume an actual connection ∇ with torsionbut in most cases we only make use of the combination ω ij ∇ j which could moregenerally be replaced by a preconnection. Our starting point, implicit in [2], is that EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 3
Poission-compatibility in terms of torsion amounts to the condition for ( ω, ∇ ) ,(1.1) ω ij ; m = ω ki T jkm + ω jk T ikm where T is the torsion of ∇ (see Lemma 3.1). The startling conclusion in Sections3,4 of the paper is that with this assumption we have canonically a functor Q that quantises to lowest order the monoidal category of classical bundles over M (Theorem 3.5) and also the structure of the exterior algebra ( Ω ( M ) , d ) , i.e. thewedge product (and tensor products more generally), see Theorem 4.6.Then in Section 5 we quantise a Riemannian structure consisting of metric g andLevi-Civita connection ̂ ∇ , which will require a further condition on the classicalgeometry (i.e. not every classical Riemannian manifold will be quantisable evenat our first order level). We tend to see that as a good rather than a bad thing,as ultimately constraints of this type could (when the theory is fully developed)explain such things as Einstein’s equation. The thinking is that if classical geome-try emerges out of quantum gravity via noncommuative geometry then the greaterrigidity of the latter can and should imprint constraints on what can emerge atthe classical level and hence explain them. Thus in the 2D toy model in [7] theconstraints of noncommutative geometry forces a curved metric on the chosen space-time algebra and this metric describes either a strongly gravitational source at theorigin in space or a toy model of a big-bang cosmology with fluid matter, dependingon the interpretation and sign of a parameter. Mathematically speaking, the semi-classical analysis allows us to clarify obstructions or ‘anomalies’ to the quantisationprocess and one can take the view just stated that vanishing of the anomaly is aquantisability constraint or, if we wish to consider more general models, one cantake the view that either the anomaly is an order λ effect to live with or that itis a signal that the theory needs to be extended, for example extra dimensions, toabsorb the anomaly.Our quantum wedge product ∧ in Theorem 4.6 consists of a functorial part ∧ Q plusan order λ nonlinear correction that can be attributed to the non-linear nature of theproblem in enforcing the Leibniz rule (the calculus is both used in the functor andnow is being quantised by the functor). As a result the quantum metric g = g Q − λ R consists of a functorial part g Q plus an order λ correction give by a certain 2-form R (which we call the ‘generalised Ricci 2-form’) viewed here as an antisymmetrictensor, see Proposition 5.2. The only requirement at this point beyond (1.1) is(1.2) ∇ g = ∇ , as well as the need for gravity in [7]as forced curvature of the metric. For the Schwarzschild black-hole metric we willfind a similar anomaly to the one for quantum group models, namely that ∇ hasto have curvature. Thus any rotationally invariant deformation of the black-holewill necessarily entail nonassociativity at O ( λ ) if we assume classical dimensions(an alternative is the use of an extra cotangent direction in [22]). This is a tangibleresult of our semiclassical analysis in the present paper.Similarly, both the Poisson-compatible connection ∇ and the classical Levi-Civitaconnection ̂ ∇ get functorially quantised as ∇ Q and ∇ QS respectively, where we EDWIN J. BEGGS & SHAHN MAJID write ̂ ∇ = ∇ + S and quantise each term functorially. As before, the functorial ∇ QS needs an order λ correction, i.e. we construct the quantum Levi-Civita connectionin the form ∇ = ∇ QS + λK for some tensor K which is uniquely determined byrequiring ∇ to be quantum torsion free and requiring merely the symmetric partof ∇ g to vanish; see Theorem 5.7 and our ‘quantum Koszul’ formula (5.12). Theantisymmetric part of ∇ g , however, is a new phenomenon which does not existclassically; an order λ obstruction independent of K and proportional to the lefthand side of the equation(1.3) ̂ ∇R + ω ij g rs S sjn ( R rmki + S rkm ; i ) d x k ⊗ d x m ∧ d x n = , where R is the curvature of ∇ . This additional (1.3) is therefore necessary andsufficient for the existence of a torsion free fully metric compatible ∇ , and whenthis does exist it is given by our above unique ∇ . Otherwise the latter remains‘best possible’ in the sense of killing the part of ∇ g that can be controlled, withthe antisymmetric part remaining as an anomaly. The Schwarzschild black-holewill again be similar to the quantum group case in that there will be this order λ obstruction to full metric compatibility.While the above holds formally over most fields, for physics we want to work over R at the classical level while at the quantum level over C but in a ∗ -algebra settingwhere the classical reality is extended as Hermiticity (we will say ‘reality constraint’as an umbrella terms for the relevant constraint but one could also say loosely‘unitarity’). Thus our quantum metric g will be complex but subject to such a‘reality’ constraint and similarly we would like our quantum connection ∇ to besuitably ‘real’ in the sense of star-preserving[5]. Again our functorial construction ∇ QS works (Theorem 4.14) in that there is always a unique order λ adjustment K to make it star-preserving, leading to a canonical ∇ from this point of view.Fortunately, Corollary 5.9 says that when a quantum torsion free metric compatibleconnection exists it is necessarily star-preserving and coincides with the ∇ givenby this reality/unitarity requirement, so our two constructions coincide in this case.Even when a fully metric compatible one does not exist, our unique star-preserving ∇ still tends to be the same as our unique ‘best possible’ quantum Levi-Civitafound before, at least in nice cases that we have looked at such as the Schwarzschildblack-hole.An alternative to the above straight metric compatibility is[5] to work with the cor-responding sequilinear ‘Hermitian’ metric ( ⋆ ⊗ id ) g and ask for ∇ to be quantumtorsion free and Hermitian-metric compatible. Here again there is no obstructionand we show, Proposition 5.8, that we can always find suitable K . In general these ∇ will not be unique but in some cases there could be a unique one. For example,in the geometry of quantum groups there is a unique perturbative such connectionfor C q ( SU ) with its 3D calculus in [5, Thm. 7.9]. Another weaker notion of metriccompatibility is vanishing cotorsion – meaning ( ∧ ⊗ id ) ∇ g = q -sphere[20] quantum torsion free, cotorsion free turns out tobe the same as quantum torsion free, Hermitian-metric compatible. We will seethat this is also the case for the Schwarzschild black-hole.We remark that one could view ∧ , g , ∇ above as the functor Q applied to pread-justed nonstandard classical maps ∧ ′ , g ′ , ̂ ∇ ′ = ∇+ S ′ differing from the usual ones byan order λ correction, thus g ′ = g − λ R etc. One could then say that to be quantised EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 5 functorially the classical metric needs to acquire an antisymmetric order λ correc-tion (in our conventions λ is imaginary so g ′ remains Hermitian) and similarly for ∧ ′ and S ′ . This interpretation could be a direct path to predictions, although it isnot in our scope to pursue that point of view here.Section 6 turns to examples in the simplest case where S =
0, i.e. where thequantising connection and the Levi-Civita connection coincide. Their quantisa-tion ∇ Q = ∇ QS does not need any order λ adjustment as it is automatically star-preserving and ‘best possible’ in terms of metric compatibility. The constraint (1.1)simplifies to ∇ ω =
0, (1.2) is automatic, while the condition (1.3) for full metric-compatibility simplifies to ∇R =
0. The latter is automatically solved for example if ( ω, g ) is K¨ahler-Einstein. It is similarly solved for any surface of constant curvatureand we give hyperbolic space and sphere in detail. This section thus provides thesimplest class of solutions. The only downside is that the Levi-Civita connectiontypically has significant curvature in examples of interest and if we take this for ourquantizing connection then the quantum differential calculus will be nonassociativeat O ( λ ) .Finally, in Section 7 we given two examples where the quantising connection ∇ isvery different from the Levi-Civita one. The first is the 2D bicrossproduct modelquantum spacetime with curved metric in [7] but analysed now at the semiclassicallevel. Here all our conditions (1.1)-(1.3) hold, ∇ has zero curvature but a lot oftorsion, and application of the general machinery indeed yields a unique torsionfree metric-compatible and star-preserving quantum connection in agreement withone of the two quantum Levi-Civita connections in[7] (the other is ‘deep quantum’with no λ → ω . Thelatter is not covariantly constant so ∇ cannot be the Levi-Civita one, and ratherwe find a 4-functional parameter moduli of rotationally invariant ∇ with torsion.These cannot be adjusted to have zero curvature so there will be nonassociativityat O ( λ ) as explained above. Also as promised, there is a nonzero obstruction tothe antisymmetric part of the metric compatibility and turns out to be ‘topological’in the sense of independent even of the choice of ∇ within the considered moduli.Nevertheless, we find the unique ‘best possible’ quantum Levi-Civita which turnsout also to be the unique star-preserving ∇ .While our focus above has been on the quantum Levi-Civita connection, our func-torial quantisation ∇ Q of any quantising connection ∇ obeying (1.1)–(1.2) also hasnice properties by itself (see Proposition 5.2). Here ∇ Q is quantum star-preservingand, if ∇R =
0, quantum metric compatible but will generally have quantum tor-sion. This quantum connection has a more direct relevance to teleparallel gravity[1]where, when the manifold is parallelisable, one can take ∇ to be the Weitzenb¨ockconnection instead of the Levi-Civita one that we took in Section 6. This has tor-sion but zero curvature and working with it is equivalent to General Relativity butinterpreted differently, with S above now viewed as the contorsion tensor and ourresults similarly viewed as its quantisation Q ( S ) + λK . The Weitzenb¨ock ∇ with itszero curvature corresponds to an associative quantum differential calculus at O ( λ ) ,but as the black-hole example showed one may need to have some small amountof curvature to have a compatible ω , i.e. to be quantisable. This application, aswell as the general theory of the quantum Ricci tensor, quantum Laplacians and EDWIN J. BEGGS & SHAHN MAJID quantum complex structures in the sense of [8] at order λ are deferred to forth-coming work, as is the important case of C P n as an example of a K¨ahler-Einsteinmanifold, the classical Riemannian geometry of which is linked to Berry phase andhigher uncertainty relations in quantum mechanics[9], among other applications.Also, we note that the equation (1.1) has a striking similarity to weak-metric-compatibility [23] g ij ; m = g ki T jkm + g jk T ikm . which applies to metric-connection pairs arising from cleft central extensions of theclassical exterior algebra by an extra closed 1-form θ ′ with θ ′ = θ ′ graded-commutative, whereas in our present paper we extend by λ a central scalar with λ = θ ′ approach was used to associatively quantise theSchwarzschild black-hole in [22] in contrast to our approach now. It would seemthat these two different ideas might be unified into a single construction. This andthe higher order theory are some other directions for further work.2. Preliminaries
Classical differential geometry.
We assume that the reader is comfortablewith classical differential geometry and recall its noncommutative algebraic gen-eralisation in a bimodule approach. For classical geometry suffice it to say thatwe assume M is a smooth manifold with further smooth structures notably theexterior algebra ( Ω ( M ) , d ) but more generally we could start with any graded-commutative classical differential graded algebra with further structure (i.e. thegraded-commutative case of the next section). One small generalisation: we al-low complexifications. However, one could work with real values and a trivial ∗ -operation or one could have the a complex version and, in the classical case, pickout the real part. We use the following categories based on vector bundles on M :Name Objects Morphisms E vector bundles over M bundle maps˜ D ( E, ∇ ) bundle and connection bundle maps D ( E, ∇ ) bundle and connection bundle maps intertwining the connectionsThe condition for the bundle map θ ∶ E → F to intertwine the connections ( E, ∇ E ) and ( F, ∇ F ) is that θ ( ∇ Ei e ) = ∇ F i ( θ ( e )) , where e ∶ M → E is a section of thebundle E . To fit the viewpoint of noncommutative geometry, we will talk aboutsections of the bundles rather than the bundles themselves, and from that point ofview we would write e ∈ E , and consider E as a module over the algebra of smoothfunctions C ∞ ( M ) .We use two tensor products: E ⊗ F will denote the algebraic tensor product, and E ⊗ F will denote the tensor product over C ∞ ( M ) . Thus E ⊗ F obeys therelation e.a ⊗ f = e ⊗ a.f for all a ∈ C ∞ ( M ) , and this corresponds to the usualtensor product of vector bundles, wheras E ⊗ F is much larger. We can use thetensor product to rewrite a connection ( E, ∇ E ) as a map ∇ E ∶ E → Ω ( M ) ⊗ E byusing the formula ∇ E ( e ) = d x i ⊗ ∇ i ( e ) , and the Leibniz rule becomes ∇ E ( a.e ) = d a ⊗ e + a. ∇ E ( e ) . EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 7
As most readers will be more familiar with tensor calculus on manifolds than withthe commutative case of the algebraic version above, we use the former throughoutfor computations in the classical case. We adopt here standard conventions forcurvature and torsion tensors as well Christoffel symbols for a linear connection.On forms and in a local coordinate system we have ∇ j d x i = − Γ ijk d x k while T ∇ = ∧∇ − d ∶ Ω ( M ) → Ω ( M ) has the torsion tensor T ∇ ( d x i ) = − Γ ijk d x j ∧ d x k = T ijk d x k ∧ d x j , T ijk = Γ ijk − Γ ikj . (2.1)Similarly, for the curvature tensor R ∇ ( d x k ) = d x i ∧ d x j ⊗ [ ∇ i , ∇ j ] d x k = R kmij d x j ∧ d x i ⊗ d x m . The summation convention is understood unless specified otherwise.We also recall the interior product ⌟ ∶
Vec M ⊗ Ω n ( M ) → Ω n − ( M ) defined by v ⌟ η being the evaluation for η ∈ Ω ( M ) , or in terms of indices v i η i , extended recursivelyto higher degrees by v ⌟ ( ξ ∧ η ) = ( v ⌟ ξ ) ∧ η + ( − ) ∣ ξ ∣ ξ ∧ ( v ⌟ η ) . Noncommutative bundles and connections.
Here we briefly summarisethe elements of noncommutative differential geometry that we will be concernedwith in our bimodule approach[15, 16, 5, 7]. The following picture can be gener-alised at various places, but for readability we will not refer to this further. Theassociative algebra A (over the complex numbers) plays the role of ‘functions’ onour noncommutative space and need not be commutative.A differential calculus on A consists of n forms Ω n A for n ≥
0, an associative product ∧ ∶ Ω n A ⊗ Ω m A → Ω n + m A and an exterior derivative d ∶ Ω n A → Ω n + A satisfyingthe rules1) Ω A = A (i.e. the zero forms are just the ‘functions’)2) d =
03) d ( ξ ∧ η ) = d ξ ∧ η + ( − ) ∣ ξ ∣ ξ ∧ d η where ∣ ξ ∣ = n if ξ ∈ Ω n A
4) Ω is generated by degree 0,1.These are the rules for a standard differential graded algebra . Note that we do not assume graded commutativity, which would be ξ ∧ η = ( − ) ∣ ξ ∣ ∣ η ∣ η ∧ ξ .A vector bundle is expressed as a (projective) A -module. If E is a left A -modulewe define a left connection ∇ E on E to be a map ∇ E ∶ E → Ω A ⊗ A E obeying theleft Leibniz rule ∇ E ( ae ) = d a ⊗ A e + a ∇ E ( e ) . We say that we have a bimodule connection if E is a bimodule and there is abimodule map σ E ∶ E ⊗ A Ω A → Ω A ⊗ A E, ∇ E ( ea ) = ( ∇ E e ) a + σ E ( e ⊗ A d a ) . If σ E is well-defined then it is uniquely determined, so its existence is a propertyof a left connection on a bimodule. There is a natural tensor product of bimoduleconnections ( E, ∇ E ) ⊗ ( F, ∇ F ) built on the tensor product E ⊗ A F and ∇ E ⊗ A F ( e ⊗ A f ) = ∇ E e ⊗ A f + ( σ E ⊗ id )( e ⊗ A ∇ F f ) . EDWIN J. BEGGS & SHAHN MAJID
There is necessarily an associated σ E ⊗ A F . We denote by E the monoidal categoryof A -bimodules with ⊗ A . We denote by D the monoidal category of pairs ( E, ∇ E ) of bimodules and bimodule connections over A . Its morphisms are bimodule mapsthat intertwine the connections.In the case of a connection on E = Ω n A we define the torsion as T ∇ = ∧∇ − d. Inthe case E = Ω A we define a metric as g ∈ Ω A ⊗ A Ω A and in case of a bimoduleconnection the metric-compatibility tensor is ∇ g ∈ Ω A ⊗ A Ω A ⊗ A Ω A . We alsorequire g to have an inverse ( , ) ∶ Ω A ⊗ A Ω A → A with the usual bimodule mapproperties and this requires that g is central.2.3. Conjugates and star operations.
We suppose that A is a star algebra, i.e.that there is a conjugate linear map a ↦ a ∗ so that ( ab ) ∗ = b ∗ a ∗ and a ∗∗ = a . Alsosuppose that this extends to a star operation on the forms, so that d ( ξ ∗ ) = ( d ξ ) ∗ and ( ξ ∧ η ) ∗ = ( − ) ∣ ξ ∣∣ η ∣ η ∗ ∧ ξ ∗ .Next, for any A -bimodule E , we consider its conjugate bimodule E with elementsdenoted by e ∈ E , where e ∈ E and new right and left actions of A , ea = a ∗ e and ae = ea ∗ . There is a canonical bimodule map Υ ∶ E ⊗ A F → F ⊗ A E given byΥ ( e ⊗ f ) = f ⊗ e . Also, if φ ∶ E → F is a bimodule map we have a bimodule map¯ φ ∶ ¯ E → ¯ F by ¯ φ ( ¯ e ) = φ ( e ) . These constructions are examples of a general notion of abar category[5] but for our purposes the reader should view the conjugate notationas a useful way to keep track of conjugates for noncommutative geometry, and asa book-keeping device to avoid problems. It allows, for example, conjugate linearfunctions to be viewed as linear functions to the conjugate of the original map’scodomain. Bimodules form a bar category as explained and so does the categoryof pairs ( E, ∇ ) . Here ¯ E acquires a right handed connection ¯ ∇ ( ¯ e ) = ( id ⊗ ⋆ − ) Υ ∇ e which we convert to a left connection ∇ ¯ e = ( ⋆ − ⊗ id ) Υ σ − ∇ e. Here ⋆ ∶ Ω A → Ω A is the ∗ -operation viewed formally as a linear map.In general we say that E is a ⋆ -object if there is a linear operation ⋆ ∶ E → E (whichwe can also write as ⋆ ( e ) = e ∗ where e ↦ e ∗ is antilinear) such that ¯ ⋆ ⋆ ( e ) = e forall e ∈ E . Also given ⋆ -objects E , F we say a morphism φ ∶ E → F is ∗ -preserving ¯ φ commutes with ⋆ . If E is a star-object then we define a connection as ∗ -preservingif ( id ⊗ ⋆ ) ∇ ( ⋆ − e ) = ( ⋆ − ⊗ id ) Υ σ − ∇ e and in this case ( E, ∇ ) becomes a ⋆ -object in this bar category. Cleary Ω A itselfis an example of a star-object and so is Ω A in every degree. The product ∧ is anexample of an anti- ∗ -preserving map (i.e. with a minus sign) on products of degree1.In the ∗ -algebra case we say a metric g ∈ Ω A ⊗ A Ω A is ‘real’ in the senseΥ − ( ⋆ ⊗ ⋆ ) g = g If g ij is real symmetric as a matrix valued function, then as the phrase ‘realityproperty’ suggests, this is true classically. We can also work with general metricsequivalently as ‘Hermitian metrics’ G = ( ⋆ ⊗ id ) g ∈ Ω A ⊗ A Ω A and this is ‘real’precisely when Υ − ( id ⊗ bb ) G = G , where bb ∶ Ω A → Ω A is the ‘identity’ map to EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 9 the double conjugate ξ ↦ ξ . In this context it is more natural to formulate metriccompatibility using the ‘Hermitian-metric compatibility tensor’ ( ¯ ∇ ⊗ id + id ⊗ ∇ ) G ∈ Ω A ⊗ A Ω A ⊗ A Ω A . (2.2)If ∇ is ∗ -preserving, then vanishing of this coincides with the regular notion ofmetric compatibility of the corresponding g .2.4. Bundles with extended morphisms.
We will sometimes want to referto extended morphisms and their covariant derivative, which will be particularlyneeded when we come to study quantum curvature in a sequel. For the momentsuffice it to say that if A is an algebra with DGA Ω A , the category of pairs ( E, ∇ E ) where E is a left A -module equipped with a left covariant derivatives, has an ex-tended notion of morphism as a left module map θ ∶ E → Ω n A ⊗ A F for any degree n . We say that θ ∈ Mor n ( E, F ) , where the set of extended morphisms between twoobjects is now a graded vector space. Composition of such an extended morphismwith another, φ ∶ F → Ω m A ⊗ A G , is given by the following formula φ ○ θ = ( id ∧ φ ) θ ∶ E → Ω n + m A ⊗ A G Proposition 2.1. If θ ∶ E → Ω n A ⊗ A F is an extended morphism from ( E, ∇ E ) to ( F, ∇ F ) , then so is ∇ ( θ ) = ∇ [ n ] F ○ θ − ( id ∧ θ ) ∇ E ∶ E → Ω n + A ⊗ A F where ∇ [ n ] F = d ⊗ id + ( − ) n id ∧ ∇ F ∶ Ω n ( A ) ⊗ A F → Ω n + A ⊗ A F .Proof. For a ∈ A and e ∈ E , and setting θ ( e ) = ξ ⊗ f , ∇ [ n ] F ○ θ ( a.e ) − ( id ∧ θ ) ∇ E ( a.e ) = ∇ [ n ] F ( a.ξ ⊗ f ) − ( id ∧ θ )( d a ⊗ e + a. ∇ E ( e )) = d a ∧ ξ ⊗ f + a. ∇ [ n ] F ( ξ ⊗ f ) − d a ∧ θ ( e ) + a. ∇ E ( e ) = a. ( ∇ [ n ] F ○ θ ( e ) − ( id ∧ θ ) ∇ E ( e )) . ◻ Example 2.2.
The A -bimodule A can be given the usual connection d. Considerthe bimodule Ω A with left connection ∇ , and the morphism τ ∈ Mor ( Ω A, A ) given by ξ ↦ ξ ⊗ A
1. Then ∇ ( τ ) is given by ∇ ( τ )( ξ ) = d ξ ⊗ − ( id ∧ τ ) ∇ = ( d ξ − ∧∇ ξ ) ⊗ . This means that −∇ ( τ ) ∈ Mor ( Ω A, A ) is the torsion of the connection on Ω A .2.5. Imposing λ = . We will be working in the setting of a typically noncom-mutative algebra A λ and related structures expanded in a formal power series in λ but truncated to different orders of approximation. It is intuitively clear what thismeans but one way to make it precise is as follows.For a field k , let C be a k -linear Abelian category and let k n = k [ λ ]/( λ n + ) . Thequotient here simply means that we set λ n + =
0. For V ∈ C we let V [ n ] = k n ⊗ k V (sothis consists on n copies of V labelled by powers of λ ). The category C [ n ] consistsof such objects with morphisms those of C extended λ -linearly to become linearover k n . If n > m there is a functor π ∶ C [ n ] → C [ m ] given by the quotient λ m + = k n → k m . If C is monoidal then so is C [ n ] with V [ n ] ⊗ W [ n ] = ( k n ⊗ k V ) ⊗ k n ( k n ⊗ k W ) ≅ k n ⊗ k ( V ⊗ W ) = ( V ⊗ W )[ n ] and this is such that we have a monoidal functor C → C [ n ] for any n . We denote by ⊗ n the tensor productin the category C [ n ] .In our case we are interested in categories where the underlying objects are vectorspaces, so C = Vec. Let A ∈ Vec be an associative algebra. Deforming A to firstorder then means equipping A = A [ ] with an associative product A ⊗ A → A so that A /( λ ) = A . We will use a subscript 1 on categories (typically related to A ) to indicate that we are working in the deformed theory to order λ . Thus E denotes the category of A -bimodules over C [ λ ]/( λ ) , and D the category of pairs ( E, ∇ ) as in Section 2.1 but over A . Aside from A we will not explicitly denotethe change of base on objects, for clarity.We are going to work over k = C with suitable reality conditions but it should beclear that constructions that do not depend on the ∗ -involution work with care overmost fields. 3. Semiquantization of bundles
This section constructs a monoidal functor Q that quantises geometric data on asmooth manifold M to first order in a deformation parameter λ . Here A = C ∞ ( M ) isour initial algebra and its first order quantisation for us means a map A ⊗ A → A as explained in Section 2.5. However, for readability purposes we will also continueto speak in more conventional terms of powerseries in λ with errors O ( λ ) beingignored. In an application where λ was actually a number, the dropping of thesehigher powers would need to be justified by the physics.3.1. Quantising the algebra and modules.
The data we suppose is an antisym-metric bivector ω on M along with a linear connection ∇ subject to the following‘Poisson compatibility’ [2]d ( ω ij ) − ω kj ∇ k ( d x i ) − ω ik ∇ k ( d x j ) = . (3.1) Lemma 3.1.
Let ω be an antisymmetric bivector and ∇ a linear connection, withtorsion tensor T . Then ω obeys (3.1) if and only if ω ij ; m + ω ik T jkm − ω jk T ikm = . In this case ω is a Poisson tensor if and only if ∑ cyclic ( i,j,k ) ω im ω jp T kmp = . Proof.
The first part is essentially in [2] but given here more generally. For the firstpart the explicit version of (3.1) in terms of Christoffel symbols is(3.2) ω ij,m + ω kj Γ ikm + ω ik Γ jkm = . We write the expression on the left as ω ij ,m + ω kj Γ imk + ω ik Γ jmk + ω kj T ikm + ω ik T jkm and we recognise the first three terms as the covariant derivative. For the secondpart, we put (3.2) into the following condition for a Poisson tensor: ∑ cyclic ( i,j,k ) ω im ω jk,m = . ◻ (3.3) EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 11
For example, any manifold with a torsion free connection and ω a covariantly con-stant antisymmetric bivector will do. This happens for example in the case of aK¨ahler manifold, so our results include these.The action of the bivector on a pair of functions is denoted { , } as usual. If ω is aPoisson tensor then this is a Poisson bracket and from the Fedosov and Kontsevichthere is an associative multiplication for functions a ● b = ab + λ { a, b }/ + O ( λ ) . (3.4)We take the same formula in any case and denote by A λ any (possibly not associa-tive) quantisation with this leading order part, which means we fix our associativealgebra A over C [ λ ]/( λ ) and leave higher order unspecified. We will normallyassume that ω is a Poisson tensor because that will be desirable at higher order,but strictly speaking the results in the present paper do not really require this.Similarly, in [2] we found the commutator of a function a and a 1-form ξ ∈ Ω ( M )[ a, ξ ] ● = λ ω ij a ,i ( ∇ j ξ ) + O ( λ ) , (3.5)so we could define the deformed product of a function a and a 1-form ξ as a ● ξ = a ξ + λ ω ij a ,i ( ∇ j ξ )/ + O ( λ ) ,ξ ● a = a ξ − λ ω ij a ,i ( ∇ j ξ )/ + O ( λ ) . (3.6)Again we can drop the corrections and regard these as defining a bimodule structureΩ A ⊗ A → Ω A and A ⊗ Ω A → Ω A where Ω A in this context is over C [ λ ]/( λ ) .Now let ( E, ∇ E ) be a classical bundle and covariant derivative on it, and define,for e ∈ E , a ● e = a ξ + λ ω ij a ,i ( ∇ Ej e ) + O ( λ ) ,e ● a = a ξ − λ ω ij a ,i ( ∇ Ej e ) + O ( λ ) . (3.7)A brief check reveals that the following associative laws hold to errors in O ( λ ) : ( a ● b ) ● e = a ● ( b ● e ) , ( a ● e ) ● b = a ● ( e ● b ) , ( e ● a ) ● b = e ● ( a ● b ) , (3.8)so we have a bimodule structure E ⊗ A → E and A ⊗ E → E . We consider thefollowing categories of modules over A :Name Objects Morphisms˜ E bimodules over A left module maps E bimodules over A bimodule maps D bimodules and connection bimodule maps intertwining the connections Lemma 3.2.
We define the functor Q ∶ ˜ D → ˜ E sending objects to objects accordingto ( . ) and sending bundle maps T ∶ E → F to left module maps Q ( T ) = T + λ ω ij ∇ F i ○ ∇ j ( T ) , where ∇ j ( T ) = ∇ F j ○ T − T ○ ∇ Ej as explained in the Preliminaries. The functorrestricts to Q ∶ D → E as Q ( T ) = T . In general we have Q ( T ○ S ) = Q ( T ) ○ Q ( S ) + λ ω ij ∇ i ( T ) ○ ∇ j ( S ) . Proof.
Take T ∶ E → F a bundle map. We aim for the bimodule properties ( T + λ T )( a ● e ) = a ● ( T + λ T )( e ) , ( T + λ T )( e ● a ) = ( T + λ T )( e ) ● a , (3.9)which to errors in O ( λ ) is T ( a ● e ) + λ T ( a e ) = a ● T ( e ) + λ a T ( e ) ,T ( e ● a ) + λ T ( e a ) = T ( e ) ● a + λ T ( e ) a . (3.10)Using the formula (3.7) for the deformed product gives our conditions as T ( ω ij a ,i ( ∇ Ej e )/ ) + T ( a e ) = ω ij a ,i ( ∇ F j T ( e ))/ + a T ( e ) , − T ( ω ij a ,i ( ∇ Ej e )/ ) + T ( e a ) = − ω ij a ,i ( ∇ F j T ( e ))/ + T ( e ) a . (3.11)It is not possible to satisfy both parts of (3.11) unless T preserves the covariantderivatives, i.e. ∇ F j T ( e ) = T ( ∇ Ej e ) (3.12)and in this case we set T = Q ( T ) = T .More generally, we solve only the first part of (3.11), i.e. a left module map for ( A , ● ) , which needs T ( a e ) − a T ( e ) = ω ij a ,i ( ∇ F j T ( e ) − T ( ∇ Ej e ))/ , (3.13)Define a module map ∇ j ( T ) ∶ E → F by ∇ j ( T )( e ) = ∇ F j T ( e ) − T ( ∇ Ej e ) (3.14)now we have the following choice: T = ω ij ∇ F i ○ ∇ j ( T )/ Q ( T ) . For compositions, Q ( T ○ S ) = T ○ S + λ ω ij ∇ i ○ ∇ j ( T ○ S ) = T ○ S + λ ω ij ∇ i ○ ( ∇ j ( T ) ○ S + T ○ ∇ j ( S )) = Q ( T ) ○ S + λ ω ij ∇ i ○ T ○ ∇ j ( S ) = Q ( T ) ○ S + λ ω ij ∇ i ( T ) ○ ∇ j ( S ) + λ ω ij T ○ ∇ i ○ ∇ j ( S ) . ◻ Note that we distinguish between T ∶ Q ( E ) → Q ( F ) which is the map Q ( e ) ↦ Q ( T ( e )) and Q ( T ) ∶ Q ( E ) → Q ( F ) which is the given quantisation.3.2. Quantising the tensor product.
As we have now described how to deformthe algebra and bimodules, we can now take the “fiberwise” tensor product of twobimodules in the deformed case. This is taken to be Q ( E ) ⊗ Q ( F ) , where similarlyto the definition of ⊗ in Section 2.1, we take e ⊗ a ● f = e ● a ⊗ f for all a ∈ A .Note that as Q is the identity on objects, we could have written E ⊗ F above,but we used the Q to emphasise that the bimodules are taken with the deformedactions. Now we seem to have two ways to quantise the tensor product E ⊗ F , but EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 13 these are related by a natural transformation q in (3.16), this being the definitionof Q being a monoidal functor : E ⊗ F Q ○⊗ / / Q ⊗ Q ' ' Q ( E ⊗ F ) Q ( E ) ⊗ Q ( F ) q E,F O O (3.16)We require the following diagram to commute: Q ( E ) ⊗ Q ( F ) ⊗ Q ( G ) id ⊗ q F,G / / q E,F ⊗ id * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ Q ( E ) ⊗ Q ( F ⊗ G ) q E,F ⊗ G / / Q ( E ⊗ F ⊗ G ) Q ( E ⊗ F ) ⊗ Q ( G ) q E ⊗ F,G ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ Proposition 3.3.
The functor Q ∶ D → E is monoidal with associated naturaltransformation q ∶ Q ⊗ Q Ô⇒ Q ○ ⊗ given by q V,W ( Q ( v ) ⊗ Q ( w )) = Q ( v ⊗ w ) + λ Q ( ω ij ∇ V i v ⊗ ∇ W j w ) . In general we have, for T ∶ E → V and S ∶ F → W , q V,W ( T ⊗ id W ) = ( T ⊗ id W + λ ω ij ∇ i ( T ) ⊗ ∇ W j ) q E,W ,q V,W ( id V ⊗ S ) = ( id V ⊗ S + λ ω ij ∇ V i ⊗ ∇ j ( S )) q V,F . As usual, we have ∇ i ( T ) = ∇ V i ○ T − T ○ ∇ Ei , etc.Proof. We want a natural morphism q V,W ∶ Q ( V ) ⊗ Q ( W ) → Q ( V ⊗ W ) but wesuppress Q since it is the identity on objects, just viewed with a different ● action.For the proposed q to be well-defined we need q V,W ( v ● a ⊗ w ) = q V,W ( v ⊗ a ● w ) , so from (3.7), q V,W (( a v − λ ω ij a ,i ( ∇ V j v )) ⊗ w ) = q V,W ( v ⊗ ( a w + λ ω ij a ,i ( ∇ W j w ))) , which is satisfied by the formula for q V,W .Next, we require each q V,W to be a bimodule map over A . Thus, q V,W ( v ⊗ ( w ● a )) = v ⊗ w a − v ⊗ λ ω ij a ,i ∇ W j w + λ ω ij ∇ V i v ⊗ ∇ W j ( wa ) = v ⊗ w a + λ ω ij ( ∇ V i v ⊗ ∇ W j w ) a + λ ω ij ( ∇ V ⊗ W i ( v ⊗ w )) a ,j q V,W ( v ⊗ w ) ● a = ( v ⊗ w + λ ω ij ∇ V i v ⊗ ∇ W j w ) ● a = ( v ⊗ w ) ● a + λ ω ij ( ∇ V i v ⊗ ∇ W j w ) a = v ⊗ wa − λ ω ij a ,i ∇ V ⊗ W j ( v ⊗ w ) + λ ω ij ( ∇ V i v ⊗ ∇ W j w ) a . (3.17)using the quantum right module structure on V etc (i.e. regarding it as Q ( V ) ) from(3.7). We do not need to use the ● product or non-trivial terms in q if an expressionalready has a λ , as we are working to errors in O ( λ ) . Our two expressions agreeusing antisymmetry of ω . Similarly on the other side, q V,W (( a ● v ) ⊗ w ) = a v ⊗ w + λ ω ij a ,i ( ∇ V j v ) ⊗ w + λ ω ij ∇ V i ( a v ) ⊗ ∇ W j w = v ⊗ w a + λ a ω ij ∇ V i v ⊗ ∇ W j w + λ ω ij a ,i ∇ V ⊗ W j ( v ⊗ w ) a ● q V,W ( v ⊗ w ) = a ● ( v ⊗ w + λ ω ij ∇ V i v ⊗ ∇ W j w ) = av ⊗ w + λ ω ij a ,i ∇ V ⊗ W j ( v ⊗ w ) + λ ω ij ∇ V i v ⊗ ∇ W j w . (3.18)Next, we check that q V,W is functorial. Let T ∶ V → Z be a morphism in D (sointertwining the covariant derivatives) and recall that Q ( T ) is just T . Then q Z,W ( T v ⊗ w ) = = T v ⊗ w + λ ω ij ∇ Zi ( T v ) ⊗ ∇ W j w = T v ⊗ w + λ ω ij ( T ○ ∇ V i v ) ⊗ ∇ W j w = ( T ⊗ id ) q V,W ( v ⊗ w ) , (3.19)and similarly for functoriality on the other side.Finally, it remains to check that q V ⊗ W,Z ○ ( q V,W ⊗ id ) = q V,W ⊗ Z ○ ( id ⊗ q W,Z ) wherethe associators implicit here are all trivial to order λ . This is immediate from theformulae for q working to order λ . Our q are clearly also invertible to this order bythe same formula with − λ . (cid:3) Now we discuss conjugate modules and star operations. For vector bundles withconnection on real manifolds, we define covariant derivatives of conjugates in theobvious manner, ∇ ¯ Ei ( e ) = ∇ Ei e . A star operation on a vector bundle will be con-jugate linear bundle map to itself e ↦ e ∗ , and will be compatible with a connectionif ∇ ¯ Ei ( e ∗ ) = ( ∇ Ei e ) ∗ . It will be convenient to take the linear map to the conjugatebundle ⋆ ∶ E → E defined by ⋆ ( e ) = e ∗ . Proposition 3.4.
Over C , the functor Q ∶ D → E is a bar functor. Hence, if ⋆ ∶ E → E is a star object and compatible with the connection, then Q ( ⋆ ) ∶ Q ( E ) → Q ( E ) is also a star object.Proof. To show we have a functor, we begin by identifying Q ( E ) and Q ( E ) (Re-member that Q ( E ) is simply E but with a different module structure, so Q ( e ) ∈ Q ( E ) is simply e ∈ E as sets.) To do this we need to show that, for all a ∈ A and e ∈ E , a.Q ( e ) = Q ( e ) .a ∗ = Q ( e ● a ∗ ) , a.Q ( e ) = Q ( a ● e ) , so we need to show that a ● e = e ● a ∗ . Now a ● e = a.e + λ ω ij a ,i . ∇ j ( e ) = a.e + λ ω ij a ,i . ∇ j e = e.a ∗ + λ ω ij ∇ j e.a ,i ∗ = e.a ∗ − λ ω ij ∇ j e.a ,i ∗ = e.a ∗ + λ ω ij ∇ i e.a ,j ∗ = e ● a ∗ . Now we have to check the morphisms T ∶ E → F , that Q ( T ) = Q ( T ) . Q ( T )( Q ( e )) = Q ( T )( Q ( e )) = Q ( T ( e ) + λ ω ij ∇ i ( ∇ j ( T ) e )) ,Q ( T )( Q ( e ) = T ( e ) + λ ω ij ∇ i ( ∇ j ( T )( e )) Now we check what ∇ j ( T ) is: ∇ j ( T )( e ) = ∇ j ( T ( e )) − T ( ∇ j ( e )) = ∇ j ( T ( e )) − T ( ∇ j ( e )) = ∇ j ( T ) e . Then we have, as λ is imaginary, Q ( T )( Q ( e ) = T ( e ) + λ ω ij ∇ i ( ∇ j ( T ) e ) = T ( e ) + λ ω ij ∇ i ( ∇ j ( T ) e ) = T ( e ) − λ ω ij ∇ i ( ∇ j ( T ) e ) . EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 15
Thus we have ( Q ( T ) − Q ( T ))( Q ( e )) = Q ( λ ω ij ∇ i ( ∇ j ( T ) e )) , so if T is a morphism in D we get Q ( T ) = Q ( T ) .Now we show that the natural transformation q is compatible with the naturaltransformation Υ in the bar category. This means that the following diagramcommutes: Q ( E ) ⊗ Q ( F ) q E,F / / Υ Q ( E ) ,Q ( F ) (cid:15) (cid:15) Q ( E ⊗ F ) = / / Q ( E ⊗ F ) Υ E,F (cid:15) (cid:15) Q ( F ) ⊗ Q ( E ) = / / Q ( F ) ⊗ Q ( E ) q F,E / / Q ( F ⊗ E ) (3.20)Now we haveΥ E,F ( q E,F ( Q ( e ) ⊗ Q ( f ))) = Υ E,F ( Q ( e ⊗ f + λ ω ij ∇ i e ⊗ ∇ j f )) = Υ E,F ( Q ( e ⊗ f + λ ω ij ∇ i e ⊗ ∇ j f )) = Q ( f ⊗ e − λ ω ij ∇ j f ⊗ ∇ i e ) = Q ( f ⊗ e + λ ω ij ∇ i f ⊗ ∇ j e ) ,q F ,E Υ Q ( E ) ,Q ( F ) ( Q ( e ) ⊗ Q ( f )) = q F ,E ( Q ( f ) ⊗ Q ( e )) = q F ,E ( Q ( f ) ⊗ Q ( e )) = Q ( f ⊗ e + λ ω ij ∇ i f ⊗ ∇ j e ) . ◻ Quantising the quantising covariant derivative.
We now want to extendthe functor Q above to a functor Q ∶ D → D . Theorem 3.5.
Let ( E, ∇ E ) be a classical bundle and connection. Then E with thebimodule structure ● over A has bimodule covariant derivative ∇ Q ( E ) = q − ,E ∇ E − λ ω ij d x k ⊗ [ ∇ Ek , ∇ Ej ] ∇ Ei σ Q ( E ) ( e ⊗ ξ ) = ξ ⊗ e + λ ω ij ∇ j ξ ⊗ ∇ Ei e + λ ω ij ξ j d x k ⊗ [ ∇ Ek , ∇ Ei ] e , where we view contraction with ω as a map Ω ( M ) ⊗ Ω ( M ) → Ω ( M ) . More-over, Q ( E, ∇ E ) = ( Q ( E ) , ∇ Q ( E ) ) is a monoidal functor Q ∶ D → D via q as inProposition 3.3.Proof. We start by considering q − ∇ E ( a ▷ e ) = q − ∇ E ( a e + λ ω ij a ,i ∇ j e / ) = q − ( d a ⊗ e + a d x k ⊗ ∇ k e + λ d ( a ,i ω ij ) ⊗ ∇ j e / + λ ω ij a ,i d x k ⊗ ∇ k ∇ j e / ) = d a ⊗ e + a d x k ⊗ ∇ k e + λ d ( a ,i ω ij ) ⊗ ∇ j e / + λ ω ij a ,i d x k ⊗ ∇ k ∇ j e / − λ ω ij ∇ i d a ⊗ ∇ j e / − λ ω ij ∇ i ( a d x k ) ⊗ ∇ j ∇ k e / = d a ⊗ e + a d x k ⊗ ∇ k e + λ d ( a ,i ω ij ) ⊗ ∇ j e / − λ ω ij ∇ i d a ⊗ ∇ j e / − λ ω ij a ∇ i ( d x k ) ⊗ ∇ j ∇ k e / + λ ω ij a ,i d x k ⊗ [ ∇ k , ∇ j ] e / , and a ▷ q − ∇ E ( e ) = a ▷ q − ( d x k ⊗ ∇ k e ) = a ▷ ( d x k ⊗ ∇ k e − λ ω ij ∇ i ( d x k ) ⊗ ∇ j ∇ k e / ) = a d x k ⊗ ∇ k e − λ ω ij a ∇ i ( d x k ) ⊗ ∇ j ∇ k e / + λ ω ij a ,i ∇ j ( d x k ) ⊗ ∇ k e / , and then q − ∇ E ( a ▷ e ) − a ▷ q − ∇ E ( e ) = d a ⊗ e + λ d ( a ,i ω ij ) ⊗ ∇ j e / + λ ω ij a ,i d x k ⊗ [ ∇ k , ∇ j ] e / − λ ω ij ∇ i d a ⊗ ∇ j e / − λ ω ij a ,i ∇ j ( d x k ) ⊗ ∇ k e / = d a ⊗ e + λ d ( a ,i ω ij ) ⊗ ∇ j e / + λ ω ij a ,i d x k ⊗ [ ∇ k , ∇ j ] e )/ − λ ω ij ∇ i ( a ,k d x k ) ⊗ ∇ j e / − λ ω ik a ,i ∇ k ( d x j ) ⊗ ∇ j e / = d a ⊗ e + λ a ,i d ( ω ij ) ⊗ ∇ j e / + λ ω ij a ,i d x k ⊗ [ ∇ k , ∇ j ] e / − λ ω ij a ,k ∇ i ( d x k ) ⊗ ∇ j e / − λ ω ik a ,i ∇ k ( d x j ) ⊗ ∇ j e / = d a ⊗ e + λ a ,i d ( ω ij ) ⊗ ∇ j e / + λ ω ij a ,i d x k ⊗ [ ∇ k , ∇ j ] e / − λ ω kj a ,i ∇ k ( d x i ) ⊗ ∇ j e / − λ ω ik a ,i ∇ k ( d x j ) ⊗ ∇ j e / = d a ⊗ e + λ a ,i ( d ( ω ij ) − ω kj ∇ k ( d x i ) − ω ik ∇ k ( d x j )) ⊗ ∇ j e / + λ ω ij a ,i d x k ⊗ [ ∇ k , ∇ j ] e / . (3.21)If we have the condition (3.1) then the last long bracket in (3.22) vanishes, giving q − ∇ E ( a ▷ e ) − a ▷ q − ∇ E ( e ) = d a ⊗ e + λ ω ij a ,i d x k ⊗ [ ∇ Ek , ∇ Ej ] e / . (3.22)Now we can set the first order quantisation of the left covariant derivative to be Q ( ∇ E )( e ) = q − ,E ∇ E ( e ) − λ ω ij d x k ⊗ [ ∇ Ek , ∇ Ej ] ∇ Ei ( e ) which we can write as stated.Next, we want to see about a bimodule connection. We compute σ ( e ⊗ d a ) = d a ⊗ e + ∇ [ e, a ] + [ a, ∇ e ] = d a ⊗ e + λ ∇ ( ω ij ∇ i ( e ) a ,j ) + [ a, d x k ⊗ ∇ k e ] = d a ⊗ e + λ d ( ω ij a ,j ) ⊗ ∇ i ( e ) + λ ω ij a ,j d x k ⊗ ∇ k ∇ i ( e ) + λ ω ij a ,i ∇ j ( d x k ) ⊗ ∇ k e + λ ω ij a ,i d x k ⊗ ∇ j ∇ k e = d a ⊗ e + λ d ( ω ij ) a ,j ⊗ ∇ i ( e ) + λ ω ij a ,jk d x k ⊗ ∇ i ( e ) + λ ω ij a ,j d x k ⊗ ∇ k ∇ i ( e ) + λ ω ij a ,i ∇ j ( d x k ) ⊗ ∇ k e + λ ω ji a ,j d x k ⊗ ∇ i ∇ k e = d a ⊗ e + λ d ( ω ij ) a ,j ⊗ ∇ i ( e ) + λ ω ij a ,jk d x k ⊗ ∇ i ( e ) + λ ω ij a ,j d x k ⊗ [ ∇ k , ∇ i ]( e ) + λ ω jk a ,j ∇ k ( d x i ) ⊗ ∇ i e = d a ⊗ e + λ ω ij ∇ j ( a ,k d x k ) ⊗ ∇ i ( e ) − λ ω ij a ,k ∇ j ( d x k ) ⊗ ∇ i ( e ) + λ ω ij a ,j d x k ⊗ [ ∇ k , ∇ i ]( e ) + λ a ,j ( d ( ω ij ) − ω kj ∇ k ( d x i )) ⊗ ∇ i e = d a ⊗ e + λ ω ij ∇ j ( a ,k d x k ) ⊗ ∇ i ( e ) + λ ω ij a ,j d x k ⊗ [ ∇ k , ∇ i ]( e ) + λ a ,j ( d ( ω ij ) − ω kj ∇ k ( d x i ) − ω ik ∇ k ( d x j )) ⊗ ∇ i e , and under condition (3.1) this becomes the given formula. This constructs thequantised covariant derivative. The following two lemmas then verify the desiredcategorical properties so as to complete the proof. (cid:3) EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 17
The following two lemmas complete the proof of the theorem. The first one byshowing that the covariant derivatiive is natural with respect to the tensor prod-uct structure – i.e. that the quantisation of the classical tensor product covariantderivative is the tensor product of the quantised covariant derivatives (using the σ map). This is summarised by Q ( E ) ⊗ Q ( F ) q / / ∇ Q ( E )⊗ Q ( F ) (cid:15) (cid:15) Q ( E ⊗ F ) ∇ Q ( E ⊗ F ) (cid:15) (cid:15) Ω A ⊗ Q ( E ) ⊗ Q ( F ) id ⊗ q / / Ω A ⊗ Q ( E ⊗ F ) (3.23) Lemma 3.6.
For all e ∈ E and f ∈ F , ( id ⊗ q E,F ) ( ∇ Q ( E ) e ⊗ f + ( σ Q ( E ) ⊗ id )( e ⊗ ∇ Q ( F ) f )) = ∇ Q ( E ⊗ F ) q E,F ( e ⊗ f ) Proof.
Begin with ∇ Q ( E ⊗ F ) q E,F ( e ⊗ f ) = ∇ Q ( E ⊗ F ) ( e ⊗ f ) + λ ∇ Q ( E ⊗ F ) ( ω ij ∇ i e ⊗ ∇ j f ) = q − ,E ∇ ( e ⊗ f ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i ( e ⊗ f ) + λ ∇ Q ( E ⊗ F ) ( ω ij ∇ i e ⊗ ∇ j f ) = q − ,E ⊗ F ( d x k ⊗ ( ∇ k e ⊗ f ) + d x k ⊗ ( e ⊗ ∇ k f )) + λ d ( ω ij ) ⊗ ( ∇ i e ⊗ ∇ j f ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ]( ∇ i e ⊗ f + e ⊗ ∇ i f ) + λ ω ij d x k ⊗ ∇ k ( ∇ i e ⊗ ∇ j f ) = d x k ⊗ ( ∇ k e ⊗ f ) + d x k ⊗ ( e ⊗ ∇ k f ) − λ ω ij ∇ i ( d x k ) ⊗ ∇ j (( ∇ k e ⊗ f ) + ( e ⊗ ∇ k f )) + λ ω ij d x k ⊗ ∇ k ( ∇ i e ⊗ ∇ j f ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ]( ∇ i e ⊗ f + e ⊗ ∇ i f ) + λ d ( ω ij ) ⊗ ( ∇ i e ⊗ ∇ j f ) . Also ( id ⊗ q E,F )( ∇ Q ( E ) e ⊗ f ) = ( id ⊗ q E,F )(( q − ,E ∇ E ( e ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i ( e )) ⊗ f ) = ( id ⊗ q E,F )(( d x k ⊗ ∇ k ( e ) − λ ω ij ∇ i ( d x k ) ⊗ ∇ j ∇ k ( e )) ⊗ f ) − λ ω ij d x k ⊗ ([ ∇ k , ∇ j ] ∇ i ( e ) ⊗ f ) = d x k ⊗ ( ∇ k ( e ) ⊗ f ) − λ ω ij ∇ i ( d x k ) ⊗ ( ∇ j ∇ k ( e ) ⊗ f ) − λ ω ij d x k ⊗ ([ ∇ k , ∇ j ] ∇ i ( e ) ⊗ f ) + λ d x k ⊗ ( ω ij ( ∇ i ∇ k ( e ) ⊗ ∇ j f )) . Then ∇ Q ( E ⊗ F ) q E,F ( e ⊗ f ) − ( id ⊗ q E,F )( ∇ Q ( E ) e ⊗ f ) = d x k ⊗ ( e ⊗ ∇ k f ) − λ ω ij ∇ i ( d x k ) ⊗ ( ∇ k e ⊗ ∇ j f ) − λ ω ij ∇ i ( d x k ) ⊗ ∇ j ( e ⊗ ∇ k f ) + λ ω ij d x k ⊗ ∇ k ( ∇ i e ⊗ ∇ j f ) − λ ω ij d x k ⊗ ( ∇ i e ⊗ [ ∇ k , ∇ j ] f ) + λ d ( ω ij ) ⊗ ( ∇ i e ⊗ ∇ j f ) − λ ω ij d x k ⊗ ( ∇ k ∇ j e ⊗ ∇ i f ) − λ ω ij d x k ⊗ ( e ⊗ [ ∇ k , ∇ j ] ∇ i f ) − λ d x k ⊗ ( ω ij ( ∇ i ∇ k ( e ) ⊗ ∇ j f )) = d x k ⊗ ( e ⊗ ∇ k f ) − λ ω ij ∇ i ( d x k ) ⊗ ( ∇ k e ⊗ ∇ j f ) − λ ω ij ∇ i ( d x k ) ⊗ ∇ j ( e ⊗ ∇ k f ) + λ ω ij d x k ⊗ ( ∇ k ∇ i e ⊗ ∇ j f ) + λ ω ij d x k ⊗ ( ∇ i e ⊗ ∇ j ∇ k f ) + λ d ( ω ij ) ⊗ ( ∇ i e ⊗ ∇ j f ) − λ ω ij d x k ⊗ ( ∇ k ∇ j e ⊗ ∇ i f ) − λ ω ij d x k ⊗ ( e ⊗ [ ∇ k , ∇ j ] ∇ i f ) − λ d x k ⊗ ( ω ij ( ∇ i ∇ k ( e ) ⊗ ∇ j f )) = d x k ⊗ ( e ⊗ ∇ k f ) − λ ω ij ∇ i ( d x k ) ⊗ ( ∇ k e ⊗ ∇ j f ) − λ ω ij ∇ i ( d x k ) ⊗ ( ∇ j e ⊗ ∇ k f ) + λ ω ij d x k ⊗ ( ∇ k ∇ i e ⊗ ∇ j f ) + λ ω ij d x k ⊗ ( ∇ i e ⊗ ∇ j ∇ k f ) + λ d ( ω ij ) ⊗ ( ∇ i e ⊗ ∇ j f ) − λ ω ij d x k ⊗ ( ∇ k ∇ j e ⊗ ∇ i f ) − λ ω ij d x k ⊗ ( e ⊗ [ ∇ k , ∇ j ] ∇ i f ) − λ ω ij ∇ i ( d x k ) ⊗ ( e ⊗ ∇ j ∇ k f ) − λ d x k ⊗ ( ω ij ( ∇ i ∇ k ( e ) ⊗ ∇ j f )) = d x k ⊗ ( e ⊗ ∇ k f ) − λ ω ij ∇ i ( d x k ) ⊗ ( ∇ k e ⊗ ∇ j f ) − λ ω ij ∇ i ( d x k ) ⊗ ( ∇ j e ⊗ ∇ k f ) + λ ω ij d x k ⊗ ([ ∇ k , ∇ i ] e ⊗ ∇ j f ) + λ ω ij d x k ⊗ ( ∇ i e ⊗ ∇ j ∇ k f ) + λ d ( ω ij ) ⊗ ( ∇ i e ⊗ ∇ j f ) − λ ω ij d x k ⊗ ( e ⊗ [ ∇ k , ∇ j ] ∇ i f ) − λ ω ij ∇ i ( d x k ) ⊗ ( e ⊗ ∇ j ∇ k f ) . Next ( σ Q ( E ) ⊗ id )( e ⊗ ∇ Q ( F ) f ) = ( σ Q ( E ) ⊗ id )( e ⊗ q − ,E ∇ ( f ) − λ ω ij e ⊗ ( d x k ⊗ [ ∇ k , ∇ j ] ∇ i ( f ))) = ( σ Q ( E ) ⊗ id )( e ⊗ d x k ⊗ ∇ k ( f ) − λ ω ij e ⊗ ∇ i ( d x k ) ⊗ ∇ j ∇ k ( f ) − λ ω ij e ⊗ d x k ⊗ [ ∇ k , ∇ j ] ∇ i ( f )) = d x k ⊗ e ⊗ ∇ k ( f ) − λ ω ij ∇ i ( d x k ) ⊗ e ⊗ ∇ j ∇ k ( f ) − λ ω ij d x k ⊗ e ⊗ [ ∇ k , ∇ j ] ∇ i ( f ) + λ ω ij ∇ j ( d x k ) ⊗ ∇ i e ⊗ ∇ k f + λ ω ik d x l ⊗ [ ∇ l , ∇ i ] e ⊗ ∇ k f , and so ( id ⊗ q E,F )( σ Q ( E ) ⊗ id )( e ⊗ ∇ Q ( F ) f ) = d x k ⊗ ( e ⊗ ∇ k ( f )) − λ ω ij ∇ i ( d x k ) ⊗ ( e ⊗ ∇ j ∇ k ( f )) − λ ω ij d x k ⊗ ( e ⊗ [ ∇ k , ∇ j ] ∇ i ( f )) + λ ω ij ∇ j ( d x k ) ⊗ ( ∇ i e ⊗ ∇ k f ) + λ ω ik d x l ⊗ ([ ∇ l , ∇ i ] e ⊗ ∇ k f ) + λ ω ij d x k ⊗ ( ∇ i e ⊗ ∇ j ∇ k ( f )) From this we have ∇ Q ( E ⊗ F ) q E,F ( e ⊗ f ) − ( id ⊗ q E,F )( ∇ Q ( E ) e ⊗ f ) − ( id ⊗ q E,F )( σ Q ( E ) ⊗ id )( e ⊗ ∇ Q ( F ) f ) = − λ ω ij ∇ i ( d x k ) ⊗ ( ∇ k e ⊗ ∇ j f ) − λ ω ij ∇ i ( d x k ) ⊗ ( ∇ j e ⊗ ∇ k f ) − λ ω ij ∇ j ( d x k ) ⊗ ( ∇ i e ⊗ ∇ k f ) + λ d ( ω ij ) ⊗ ( ∇ i e ⊗ ∇ j f ) = λ ( d ( ω ij ) − ω kj ∇ k ( d x i ) + ω ki ∇ k ( d x j )) ⊗ ( ∇ i e ⊗ ∇ j f ) and this vanishes by (3.1). (cid:3) The second lemma checks functoriality under morphisms.
Lemma 3.7. If T ∶ E → F is a module map and commutes with the covariantderivative, then Q ( ∇ F ) Q ( T ) = ( id ⊗ Q ( T )) Q ( ∇ E ) .Proof. In this case Q ( T ) = T , and ( id ⊗ Q ( T )) ∇ Q ( E ) = ( id ⊗ Q ( T )) q − ,E ∇ E ( e ) − λ ω ij d x k ⊗ T [ ∇ Ek , ∇ Ej ] ∇ Ei ( e ) = q − ,E ( id ⊗ T ) ∇ E ( e ) − λ ω ij d x k ⊗ T [ ∇ Ek , ∇ Ej ] ∇ Ei ( e ) ,Q ( ∇ F ) T = q − ,E ∇ E T ( e ) − λ ω ij d x k ⊗ [ ∇ Ek , ∇ Ej ] ∇ Ei T ( e ) . These are equal, as T commutes with the covariant derivative. (cid:3) EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 19
Lemma 3.8.
Over C , ∇ Q is star preserving to order λ , i.e. the following diagramcommutes: Q ( E ) = Q ( E ) ∇ Q (cid:15) (cid:15) Q ( E ) ⋆ o o ∇ Q / / Q ( Ω ( M )) ⊗ Q ( E ) ⋆⊗ ⋆ (cid:15) (cid:15) Q ( Ω ( M )) ⊗ Q ( E ) σ − QE / / Q ( E ) ⊗ Q ( Ω ( M )) Υ / / Q ( Ω ( M )) ⊗ Q ( E ) (3.24) Proof.
Begin with, using q a natural transformation and (3.20) σ QE Υ − ( ⋆ ⊗ ⋆ ) ∇ Q ( Q ( e )) = σ QE Υ − ( ⋆ ⊗ ⋆ )( q − ( d x p ⊗ ∇ p e ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i e ) = σ QE Υ − ( q − ( d x p ⊗ ∇ p e ∗ ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i e ∗ ) = σ QE q − Υ − ( d x p ⊗ ∇ p e ∗ ) − λ ω ij σ QE ([ ∇ k , ∇ j ] ∇ i e ∗ ⊗ d x k ) = σ QE q − ( ∇ p e ∗ ⊗ d x p ) − λ ω ij σ QE ([ ∇ k , ∇ j ] ∇ i e ∗ ⊗ d x k ) = q − q σ QE q − ( ∇ p e ∗ ⊗ d x p ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i e ∗ = q − ( d x p ⊗ ∇ p e ∗ + λ ω ip d x k ⊗ [ ∇ k , ∇ i ] ∇ p e ∗ ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i e ∗ = q − ( d x p ⊗ ∇ p e ∗ + λ ω ij d x k ⊗ [ ∇ k , ∇ i ] ∇ j e ∗ ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i e ∗ = q − ( d x p ⊗ ∇ p e ∗ − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i e ∗ ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i e ∗ = q − ( d x p ⊗ ∇ p e ∗ − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i e ∗ ) = ∇ Q ( e ∗ ) . ◻ Properties of the generalised braiding.
There is an extension of σ Q ( E ) ∶ E ⊗ Ω A → Ω A ⊗ E (as in Theorem 3.5) to σ Q ( E ) ∶ E ⊗ Ω n A → Ω n A ⊗ E given by σ Q ( E ) ( e ⊗ ξ ) = ξ ⊗ e + λ ω ij ∇ j ξ ⊗ ∇ Ei e + λ ω ij d x k ∧ ( ∂ j ⌟ ξ ) ⊗ [ ∇ Ek , ∇ Ei ] e where ⌟ is the interior product. Now we use [ ∇ k , ∇ i ] ξ = − R snki d x n ∧ ( ∂ s ⌟ ξ ) (3.25)to give, where E is Ω m A with the compatible covariant derivative, σ Q ( η ⊗ ξ ) = ξ ⊗ η + λ ω ij ∇ j ξ ⊗ ∇ i η + λ ω ij d x k ∧ ( ∂ j ⌟ ξ ) ⊗ [ ∇ k , ∇ i ] η = ξ ⊗ η + λ ω ij ∇ j ξ ⊗ ∇ i η − λ ω ij R snki d x k ∧ ( ∂ j ⌟ ξ ) ⊗ d x n ∧ ( ∂ s ⌟ η ) . Now we can take this twice to give σ Q ( η ⊗ ξ ) = η ⊗ ξ + λ ω ij ∇ i η ⊗ ∇ j ξ + λ ω ij ∇ j η ⊗ ∇ i ξ − λ ω ij R snki d x n ∧ ( ∂ s ⌟ η ) ⊗ d x k ∧ ( ∂ j ⌟ ξ ) − λ ω ij R snki d x k ∧ ( ∂ j ⌟ η ) ⊗ d x n ∧ ( ∂ s ⌟ ξ ) = η ⊗ ξ − λ ( ω ij R snki + ω is R jkni ) d x n ∧ ( ∂ s ⌟ η ) ⊗ d x k ∧ ( ∂ j ⌟ ξ ) Proposition 3.9.
The generalised braiding obeys the braid relation, in the sensethat for any ( E, ∇ E ) giving σ Q ( E ) and ( Ω ( M ) , ∇ ) giving σ Q ( σ Q ⊗ id )( id ⊗ σ Q ( E ) )( σ Q ( E ) ⊗ id ) = ( id ⊗ σ Q ( E ) )( σ Q ( E ) ⊗ id )( id ⊗ σ Q ) as a map E ⊗ Ω A ⊗ Ω A → Ω A ⊗ Ω A ⊗ E . Proof.
If we write σ Q ( E ) ( e ⊗ ξ ) = ξ ⊗ e + λ T ( ξ ) ⊗ T ′ ( e ) ,σ Q ( η ⊗ ξ ) = ξ ⊗ η + λ S ( ξ ) ⊗ S ′ ( η ) , then both maps above give the following on being applied to e ⊗ ξ ⊗ η : η ⊗ ξ ⊗ e + λ η ⊗ T ( ξ ) ⊗ T ′ ( e ) + λ T ( η ) ⊗ ξ ⊗ T ′ ( e ) + λ S ( η ) ⊗ S ′ ( ξ ) ⊗ e . ◻ Quantising other connections relative to a given ( E, ∇ E ) . Classically ageneral covariant derivative is given by ∇ S = ∇ E + S , where S ∶ E → Ω ( M ) ⊗ E is avector bundle map. This is adding a left module map to a left covariant derivative,giving another covariant derivative on the same bundle. Proposition 3.10.
For any bundle map S , ∇ QS = ∇ Q ( E ) + q − Q ( S ) , σ QS ( e ⊗ ξ ) = σ Q ( e ⊗ ξ ) + λ ω ij ξ i ∇ j ( S )( e ) defines a bimodule connection on Q ( E ) .Proof. Here q ∇ QS = q ∇ Q ( E ) + S + λ ω ij ∇ i ○ ∇ j ( S ) . From the general equation σ ( e ⊗ d a ) = d a ⊗ e + ∇ [ e, a ] + [ a, ∇ e ] we see that, to order λσ QS ( e ⊗ d a ) = σ Q ( e ⊗ d a ) + S ([ e, a ]) + [ a, S ( e )] = σ Q ( e ⊗ d a ) − λ ω ij a ,i S ( ∇ j e ) + λ ω ij a ,i ∇ j S ( e ) = σ Q ( e ⊗ d a ) + λ ω ij a ,i ∇ j ( S )( e ) . ◻ Now we look at the tensor products and reality of such general connections:
Proposition 3.11.
Given S ∶ E → Ω ( M ) ⊗ E and T ∶ F → Ω ( M ) ⊗ F , define H ∶ E ⊗ F → Ω ( M ) ⊗ E ⊗ F by (where τ is transposition) H = S ⊗ id F + ( τ ⊗ id F )( id E ⊗ T ) . Then q of the tensor product of ∇ QS and ∇ QT is given by q ∇ QH q + λ rem ∶ Q ( E ) ⊗ Q ( F ) → Q ( Ω ( M ) ⊗ E ⊗ F ) where, using T ( f ) = d x k ⊗ T k ( f ) , rem ( e ⊗ f ) = ω ij ( d x k ⊗ [ ∇ Ek , ∇ Ei ] e − ∇ i ( S )( e )) ⊗ T j ( f ) . Proof.
Using q for applying q twice (by Proposition 3.3 order does not matter), q ( ∇ QS ⊗ id F ) = q (( ∇ Q + q − Q ( S )) ⊗ id F ) = q ( ∇ Q ⊗ id F ) + q ( q − Q ( S ) ⊗ id F ) = q ( ∇ Q ⊗ id F ) + q ( Q ( S ) ⊗ id F ) = q ( ∇ Q ⊗ id F ) + ( Q ( S ) ⊗ id F + λ ω ij ∇ i ( Q ( S )) ⊗ ∇ j ) q = q ( ∇ Q ⊗ id F ) + ( Q ( S ) ⊗ id F + λ ω ij ∇ i ( S ) ⊗ ∇ j ) q , where we have used Proposition 3.3. We also need q ( σ QS ⊗ id F )( id E ⊗ ∇ QT ) EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 21 = q ( qσ QS q − ⊗ id F )( q ⊗ id F )( id E ⊗ ∇ QT ) = ( qσ QS q − ⊗ id F + λ ω ij ∇ i ( qσ QS q − ) ⊗ ∇ j ) q ( id E ⊗ ∇ QT ) = ( qσ QS q − ⊗ id F + λ ω ij ∇ i ( qσ QS q − ) ⊗ ∇ j ) q ( id E ⊗ q ∇ QT ) = ( qσ QS q − ⊗ id F + λ ω ij ∇ i ( qσ QS q − ) ⊗ ∇ j ) q ( id E ⊗ ( q ∇ Q + Q ( T ))) . Now λ σ QS = λ τ to order λ , where τ is transposition, so λ ∇ i ( qσ QS q − ) =
0. Then,where we set qσ QS q − = qσ Q q − + λ S ′ , q ( σ QS ⊗ id F )( id E ⊗ ∇ QT ) = ( qσ QS q − ⊗ id F ) q ( id E ⊗ q ∇ Q + id E ⊗ Q ( T )) = (( qσ Q q − + λ S ′ ) ⊗ id F ) q ( id E ⊗ q ∇ Q ) + (( qσ Q q − + λ S ′ ) ⊗ id F )( id E ⊗ Q ( T ) + λ ω ij ∇ i ⊗ ∇ j ( Q ( T ))) q = (( qσ Q q − + λ S ′ ) ⊗ id F ) q ( id E ⊗ q ∇ Q ) + (( qσ Q q − + λ S ′ ) ⊗ id F )( id E ⊗ Q ( T ) + λ ω ij ∇ i ⊗ ∇ j ( T )) q . It follows that the contribution of S and T to q of the tensor product derivative is ( Q ( S ) ⊗ id F ) q + λ ω ij ∇ i ( S ) ⊗ ∇ j + λ ( S ′ ⊗ id F )( id E ⊗ ∇ ) + ( qσ Q q − ⊗ id F )( id E ⊗ Q ( T )) q + λ ω ij ( τ ⊗ id F )( ∇ i ⊗ ∇ j ( T )) + λ ( S ′ ⊗ id F )( id E ⊗ T ) = ( Q ( S ) ⊗ id F ) q + λ ω ij ∇ i ( S ) ⊗ ∇ j + λ ( S ′ ⊗ id F )( id E ⊗ ∇ T ) + ( qσ Q q − ⊗ id F )( id E ⊗ Q ( T )) q + λ ω ij ( τ ⊗ id F )( ∇ i ⊗ ∇ j ( T )) . (3.26)From Theorem 3.5 we have q σ Q ( E ) q − ( e ⊗ ξ ) = ξ ⊗ e + λ ω ij ξ j d x k ⊗ [ ∇ k , ∇ i ] e , and for the moment we write this as q σ Q ( E ) q − = τ + λ σ ′ . Then (3.26) becomes ( S ⊗ id F ) q + λ ω ij ∇ i ( S ) ⊗ ∇ j + λ ( S ′ ⊗ id F )( id E ⊗ ∇ T ) + ( τ ⊗ id F )( id E ⊗ Q ( T )) q + λ ω ij ( τ ⊗ id F )( ∇ i ⊗ ∇ j ( T )) + λ ω ij ∇ i ○ ∇ j ( S ) ⊗ id F + λ ( σ ′ ⊗ id F )( id E ⊗ T ) = ( S ⊗ id F ) q + λ ω ij ∇ i ( S ) ⊗ ∇ j + λ ( S ′ ⊗ id F )( id E ⊗ ∇ T ) + ( τ ⊗ id F )( id E ⊗ T ) q + λ ω ij ( τ ⊗ id F )( ∇ i ⊗ ∇ j ( T )) + λ ω ij ∇ i ○ ∇ j ( S ) ⊗ id F + λ ( σ ′ ⊗ id F )( id E ⊗ T ) + λ ω ij ( τ ⊗ id F )( id E ⊗ ∇ i ○ ∇ j ( T )) . (3.27)Now we use H given above with ∇ j ( H ) = ∇ j ( S ) ⊗ id F + ( τ ⊗ id F )( id E ⊗ ∇ j ( T )) to write (3.27) as Q ( H ) q + λ ω ij ∇ i ( S ) ⊗ ∇ j + λ ( S ′ ⊗ id F )( id E ⊗ ∇ T ) + λ ( σ ′ ⊗ id F )( id E ⊗ T ) . Now from Proposition 3.10 ( S ′ ⊗ id F )( id E ⊗ ∇ T )( e ⊗ f ) = ( S ′ ⊗ id F )( e ⊗ d x k ⊗ ( ∇ k f + T k ( f ))) = ω ij ∇ j ( S )( e ) ⊗ ( ∇ i f + T i ( f )) so we rewrite (3.27) as Q ( H ) q + λ ω ij ∇ j ( S ) ⊗ T i + λ ( σ ′ ⊗ id F )( id E ⊗ T ) . (3.28)Finally for T ( f ) = d x i ⊗ T i ( f ) , ( σ ′ ⊗ id F )( id E ⊗ T )( e ⊗ f ) = ( σ ′ ⊗ id F )( e ⊗ d x p ⊗ T p ( f )) = λ ω ij d x k ⊗ [ ∇ k , ∇ i ] e ⊗ T j ( f ) . ◻ Lemma 3.12.
Over C and if S is real, the difference in going clockwise minusanticlockwise round the diagram Q ( E ) = Q ( E ) ∇ Q S (cid:15) (cid:15) Q ( E ) ⋆ o o ∇ QS / / Q ( Ω ( M )) ⊗ Q ( E ) ⋆⊗ ⋆ (cid:15) (cid:15) Q ( Ω ( M )) ⊗ Q ( E ) Q ( E ) ⊗ Q ( Ω ( M )) σ QS o o Q ( Ω ( M )) ⊗ Q ( E ) Υ − o o (3.29) starting at Q ( e ) ∈ Q ( E ) is λ ω ij ∇ j ( S )( S i ( e ∗ )) − λ ω ij ∇ i ( ∇ j ( S ))( e ∗ ) + λ ω ij d x k ⊗ [ ∇ k , ∇ i ] S j ( e ∗ ) . Proof.
From lemma 3.8 the diagram commutes for S =
0. We look only at thedifference from the S = S case. Going anticlockwise from Q ( E ) weget q − Q ( S ) ⋆ ( e ) = q − Q ( S )( e ∗ ) Going clockwise is more complicated, as two of the arrows involve S . If we set qσ QS q − = qσ Q q − + λ S ′ as in the proof of Proposition 3.11, then to order λ we getthe clockwise contributions σ Q Υ − ( ⋆ ⊗ ⋆ ) q − Q ( S )( Q ( e )) + λ S ′ Υ − ( ⋆ ⊗ ⋆ ) ∇ S ( e ) = σ Q Υ − q − ( ⋆ ⊗ ⋆ ) Q ( S )( Q ( e )) + λ S ′ Υ − ( ⋆ ⊗ ⋆ ) ∇ S ( e ) = σ Q q − Υ − ( ⋆ ⊗ ⋆ ) Q ( S )( Q ( e )) + λ S ′ Υ − ( ⋆ ⊗ ⋆ ) ∇ S ( e ) = q − q σ Q q − Υ − ( ⋆ ⊗ ⋆ ) Q ( S )( Q ( e )) + λ S ′ Υ − ( ⋆ ⊗ ⋆ ) ∇ S ( e ) = q − q σ Q q − Υ − ( ⋆ ⊗ ⋆ )( S ( e ) + λ ω ij ∇ i ( ∇ j ( S )( e ))) + λ S ′ Υ − ( ⋆ ⊗ ⋆ ) ( d x p ⊗ ∇ p e + S ( e )) = q − τ Υ − ( ⋆ ⊗ ⋆ )( S ( e ) + λ ω ij ∇ i ( ∇ j ( S )( e ))) + λ q − σ ′ Υ − ( ⋆ ⊗ ⋆ )( S ( e )) + λ S ′ Υ − ( ⋆ ⊗ ⋆ ) ( d x p ⊗ ∇ p e + S ( e )) , Where we have put q σ Q q − = τ + λ σ ′ . As the classical connections preserve ⋆ and λ is imaginary, we get the following for the clockwise contributions, where S ( e ) = d x p ⊗ S p ( e ) , q − ( S ( e ∗ ) − λ ω ij ∇ i ( ∇ j ( S )( e ∗ ))) + λ q − σ ′ Υ − ( ⋆ ⊗ ⋆ )( d x p ⊗ S p ( e )) + λ S ′ Υ − ( ⋆ ⊗ ⋆ ) ( d x p ⊗ ∇ p e + S ( e )) = q − ( S ( e ∗ ) − λ ω ij ∇ i ( ∇ j ( S )( e ∗ ))) + λ q − σ ′ Υ − ( d x p ⊗ S p ( e ∗ )) + λ S ′ Υ − ( ⋆ ⊗ ⋆ ) ( d x p ⊗ ∇ p e + S ( e )) = q − ( S ( e ∗ ) − λ ω ij ∇ i ( ∇ j ( S )( e ∗ ))) + λ q − σ ′ ( S p ( e ∗ ) ⊗ d x p ) + λ S ′ Υ − ( ⋆ ⊗ ⋆ ) ( d x p ⊗ ( ∇ p e + S p ( e ))) = q − ( S ( e ∗ ) − λ ω ij ∇ i ( ∇ j ( S )( e ∗ ))) + λ q − ( ω ip d x k ⊗ [ ∇ k , ∇ i ] S p ( e ∗ )) + λ S ′ (( ∇ p e ∗ + S p ( e ∗ )) ⊗ d x p ) = q − ( S ( e ∗ ) − λ ω ij ∇ i ( ∇ j ( S )( e ∗ ))) + λ q − ( ω ip d x k ⊗ [ ∇ k , ∇ i ] S p ( e ∗ )) + λ ω pj ∇ j ( S )( ∇ p e ∗ + S p ( e ∗ )) . EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 23
Then the difference, clockwise minus anticlockwise, is to order λ , − λ ω ij ∇ i ( ∇ j ( S )( e ∗ )) + λ ω ip d x k ⊗ [ ∇ k , ∇ i ] S p ( e ∗ ) + λ ω pj ∇ j ( S )( ∇ p e ∗ + S p ( e ∗ )) = − λ ω ij ∇ i ( ∇ j ( S )( e ∗ )) + λ ω ij d x k ⊗ [ ∇ k , ∇ i ] S j ( e ∗ ) + λ ω ij ∇ j ( S )( ∇ i e ∗ + S i ( e ∗ )) = λ ω ij ∇ j ( S )( S i ( e ∗ )) − λ ω ij ∇ i ( ∇ j ( S ))( e ∗ ) + λ ω ij d x k ⊗ [ ∇ k , ∇ i ] S j ( e ∗ ) . ◻ Note that Lemma 3.12 shows that ∇ QS ( ⋆ ) is λ times a module map (i.e. it involvesno derivatives of e ). This means that ∇ QS ( ⋆ ) is also a right module map, and thusit is automatically star-compatible at order λ in the sense described in [5]. We willalso need the following Lemma. Lemma 3.13.
Let ( E, ∇ E ) be a bundle with connection and e ∈ E such that ∇ ( e ) = . Then e is central in the quantised bimodule, the quantised derivative ∇ Q ( e ) = .If in addition T ( e ) = for some T ∶ E → Ω ( M ) ⊗ E then ∇ QT ( e ) = .Proof. For the quantised connection, ∇ Q ( e ) = q − ∇ ( e ) − λ ω ij d x k ⊗ [ ∇ k , ∇ j ] ∇ i ( e ) = . Then classically ∇ T ( e ) = ∇ ( e ) + T ( e ) =
0. Now ∇ j ( T )( e ) = ∇ j ( T ( e )) − T ( ∇ j ( e )) = , so in the quantised case Q ( T )( e ) = T ( e ) + λ ω ij ∇ i ○ ∇ j ( T )( e ) = . Now we have, for the quantisation ∇ QT of ∇ T , ∇ QT ( e ) = ∇ Q ( e ) + q − Q ( T )( e ) = . ◻ Semiquantisation of the exterior algebra
In noncommutative geometry the notion of ‘differential structure’ is largely encodedas a differential graded algebra extending the quantisation of functions to differen-tial forms. The main result in this section is that by adapting the semiquantisationfunctor of Section 3 we have from the same data a canonical semiquantisation offorms of all degree and their wedge product. This is Theorem 4.6.4.1.
Quantizing the wedge product.
Our starting point for the quantum wedgeproduct is the associative product which is given by the functor Q , where we assumethat the connection ∇ on Ω A extends to all orders in the natural way, is thecomposition Q ( Ω ( M )) ⊗ Q ( Ω ( M )) q Ð→ Q ( Ω ( M ) ⊗ Ω ( M )) Q (∧) Ð→ Q ( Ω ( M )) . This gives the formula for ∧ Q ; ξ ∧ Q η = ξ ∧ η + λ ω ij ∇ i ξ ∧ ∇ j η . (4.1)To look at the Leibniz rule we need the following result: Lemma 4.1. d ( ξ ∧ Q η ) − ( d ξ ) ∧ Q η − ( − ) ∣ ξ ∣ ξ ∧ Q d η = − λH ji ∧ ( ∂ i ⌟ ξ ) ∧ ∇ j η + λ ( − ) ∣ ξ ∣ H ij ∧ ∇ i ξ ∧ ( ∂ j ⌟ η ) where H ij ∶= ω is ( T jnm ; s − R jnms ) d x m ∧ d x n . Proof.
Using (3.1) in the following formd ( ω ij ) − ω kj ∇ k ( d x i ) − ω ik ∇ k ( d x j ) = , and also using d ζ = d x k ∧ ∇ k ζ + T skn d x k ∧ d x n ∧ ( ∂ s ⌟ ζ ) and relabeling indices, we findd ( ω ij ∇ i ξ ∧ ∇ j η ) = ω ij ∇ i ( d x k ) ∧ ∇ k ξ ∧ ∇ j η + ω ij ∇ j ( d x k ) ∧ ∇ i ξ ∧ ∇ k η + ω ij d x k ∧ ∇ k ∇ i ξ ∧ ∇ j η + ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ d x k ∧ ∇ k ∇ j η + ω ij T skn d x k ∧ d x n ∧ ( ∂ s ⌟ ∇ i ξ ) ∧ ∇ j η + ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ T skn d x k ∧ d x n ∧ ( ∂ s ⌟ ∇ j η ) = ω ij ∇ i ( d x k ∧ ∇ k ξ ) ∧ ∇ j η + ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ ∇ j ( d x k ∧ ∇ k η ) + ω ij d x k ∧ [ ∇ k , ∇ i ] ξ ∧ ∇ j η + ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ d x k ∧ [ ∇ k , ∇ j ] η + ω ij T skn d x k ∧ d x n ∧ ( ∂ s ⌟ ∇ i ξ ) ∧ ∇ j η + ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ T skn d x k ∧ d x n ∧ ( ∂ s ⌟ ∇ j η ) . From this we getd ( ω ij ∇ i ξ ∧ ∇ j η ) − ω ij ∇ i d ξ ∧ ∇ j η − ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ ∇ j d η = − ω ij ∇ i ( T skn d x k ∧ d x n ∧ ( ∂ s ⌟ ξ )) ∧ ∇ j η − ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ ∇ j ( T skn d x k ∧ d x n ∧ ( ∂ s ⌟ η )) + ω ij d x k ∧ [ ∇ k , ∇ i ] ξ ∧ ∇ j η + ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ d x k ∧ [ ∇ k , ∇ j ] η + ω ij T skn d x k ∧ d x n ∧ ( ∂ s ⌟ ∇ i ξ ) ∧ ∇ j η + ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ T skn d x k ∧ d x n ∧ ( ∂ s ⌟ ∇ j η ) . As ∇ i ( v ⌟ ξ ) = ∇ i ( v ) ⌟ ξ + v ⌟ ∇ i ξ , we getd ( ω ij ∇ i ξ ∧ ∇ j η ) − ω ij ∇ i d ξ ∧ ∇ j η − ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ ∇ j d η = − ω ij T skn ; i d x k ∧ d x n ∧ ( ∂ s ⌟ ξ ) ∧ ∇ j η − ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ T skn ; j d x k ∧ d x n ∧ ( ∂ s ⌟ η ) + ω ij d x k ∧ [ ∇ k , ∇ i ] ξ ∧ ∇ j η + ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ d x k ∧ [ ∇ k , ∇ j ] η . Now we use (3.25) to gived ( ω ij ∇ i ξ ∧ ∇ j η ) − ω ij ∇ i d ξ ∧ ∇ j η − ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ ∇ j d η = − ω ij T skn ; i d x k ∧ d x n ∧ ( ∂ s ⌟ ξ ) ∧ ∇ j η − ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ T skn ; j d x k ∧ d x n ∧ ( ∂ s ⌟ η ) − ω ij d x k ∧ R snki d x n ∧ ( ∂ s ⌟ ξ ) ∧ ∇ j η − ( − ) ∣ ξ ∣ ω ij ∇ i ξ ∧ d x k ∧ R snkj d x n ∧ ( ∂ s ⌟ η ) . ◻ We see that ∧ Q will not in general obey the Leibniz rule for the undeformed d.We have a choice of persisting with a modified Leibniz rule perhaps linking up to EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 25 examples such as [12, 13], or modifying the wedge product. We choose the secondmore conventional option:
Lemma 4.2.
For vector field v , ξ ∈ Ω n ( M ) and covariant derivative ∇ , v ⌟ d ξ + d ( v ⌟ ξ ) = v ⌟ ∇ ξ + ∧ ( ∇ ( v ) ⌟ ξ ) + v j T kji d x i ∧ ( ∂ k ⌟ ξ ) . (The left hand side here is the usual Lie derivative).Proof. First we start with a 1-form ξ , when v ⌟ d ( ξ i d x i ) + d ( v ⌟ ξ i d x i ) = v j ξ i,j d x i − v i ξ i,j d x j + v i ξ i,j d x j + v i,j ξ i d x j = v j ( ξ i,j − Γ kji ξ k ) d x i + ( v i,j + Γ ijk v k ) ξ i d x j + v j T kji ξ k d x i . Now we extend this by induction, for ξ ∈ Ω ( M ) , v ⌟ d ( ξ ∧ η ) = v ⌟ ( d ξ ∧ η − ξ ∧ d η ) = ( v ⌟ d ξ ) ∧ η + d ξ ∧ ( v ⌟ η ) − ( v ⌟ ξ ) d η + ξ ∧ ( v ⌟ d η ) , d ( v ⌟ ( ξ ∧ η )) = d (( v ⌟ ξ ) ∧ η ) − d ( ξ ∧ ( v ⌟ η )) = d ( v ⌟ ξ ) ∧ η + ( v ⌟ ξ ) d ( η ) − d ξ ∧ ( v ⌟ η ) + ξ ∧ d ( v ⌟ η ) . Then, assuming the η ∈ Ω n ( M ) and that the result works for n , v ⌟ d ( ξ ∧ η ) + d ( v ⌟ ( ξ ∧ η )) = ( d ( v ⌟ ξ ) + v ⌟ d ξ ) ∧ η + ξ ∧ ( v ⌟ d η + d ( v ⌟ η )) = ( v ⌟ ∇ ξ + ∧ ( ∇ ( v ) ⌟ ξ ) + v j T kji d x i ∧ ( ∂ k ⌟ ξ )) ∧ η + ξ ∧ ( v ⌟ ∇ η + ∧ ( ∇ ( v ) ⌟ η ) + v j T kji d x i ∧ ( ∂ k ⌟ η )) . ◻ Proposition 4.3.
Let H ij be as in Lemma 4.1. Then ξ ∧ η = ξ ∧ Q η + λ ( − ) ∣ ξ ∣+ H ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) is associative to order λ and the Leibniz rule holds to order λ if and only if H ij = H ji , d H ij + Γ irp d x p ∧ H rj + Γ jrp d x p ∧ H ir = ∀ i, j. Proof.
We write ξ ∧ η = ξ ∧ Q η + λ ξ ̂ ∧ η where ξ ̂ ∧ η = ( − ) ∣ ξ ∣+ H ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) , and for the moment H ij is an arbitrary collection of 2-forms (the first part holdsin general). For the first part, we compute ( ξ ̂ ∧ η ) ∧ ζ = ( − ) ∣ ξ ∣+ H ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) ∧ ζ , ( ξ ∧ η ) ̂ ∧ ζ = ( − ) ∣ ξ ∣+∣ η ∣+ H ij ∧ ( ∂ i ⌟ ( ξ ∧ η )) ∧ ( ∂ j ⌟ ζ ) = ( − ) ∣ ξ ∣+∣ η ∣+ H ij ∧ ( ∂ i ⌟ ξ ) ∧ η ∧ ( ∂ j ⌟ ζ ) + ( − ) ∣ η ∣+ H ij ∧ ξ ∧ ( ∂ i ⌟ η ) ∧ ( ∂ j ⌟ ζ ) , and ξ ∧ ( η ̂ ∧ ζ ) = ( − ) ∣ η ∣+ ξ ∧ H ij ∧ ( ∂ i ⌟ η ) ∧ ( ∂ j ⌟ ζ ) ,ξ ̂ ∧ ( η ∧ ζ ) = ( − ) ∣ ξ ∣+ H ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ ( η ∧ ζ )) = ( − ) ∣ ξ ∣+∣ η ∣+ H ij ∧ ( ∂ i ⌟ ξ ) ∧ η ∧ ( ∂ j ⌟ ζ ) + ( − ) ∣ ξ ∣+ H ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) ∧ ζ . Hence ( ξ ̂ ∧ η ) ∧ ζ + ( ξ ∧ η ) ̂ ∧ ζ = ξ ∧ ( η ̂ ∧ ζ ) + ξ ̂ ∧ ( η ∧ ζ ) , which given that ∧ Q is necessarily associative to order λ by functoriality gives theresult stated.Next, using again the given definition of ξ ̂ ∧ η ,d ( ξ ̂ ∧ η ) = ( − ) ∣ ξ ∣+ d H ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) + ( − ) ∣ ξ ∣+ H ij ∧ d ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) + H ij ∧ ( ∂ i ⌟ ξ ) ∧ d ( ∂ j ⌟ η ) = ( − ) ∣ ξ ∣ ( Γ irt d x t ∧ H rj + Γ jrt d x t ∧ H ir − G ij ) ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) + ( − ) ∣ ξ ∣+ H ij ∧ d ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) + H ij ∧ ( ∂ i ⌟ ξ ) ∧ d ( ∂ j ⌟ η ) . From Lemma 4.2 we use ∂ j ⌟ d ξ + d ( ∂ j ⌟ ξ ) = ∂ j ⌟ ∇ ξ + ∧ ( ∇ ( ∂ j ) ⌟ ξ ) + T kjt d x t ∧ ( ∂ k ⌟ ξ ) = ∇ j ξ + d x t ∧ Γ stj ( ∂ s ⌟ ξ ) + T sjt d x t ∧ ( ∂ s ⌟ ξ ) = ∇ j ξ + d x t ∧ Γ sjt ( ∂ s ⌟ ξ ) to gived ( ξ ̂ ∧ η ) = ( − ) ∣ ξ ∣ ( Γ irt d x t ∧ H rj + Γ jrt d x t ∧ H ir − G ij ) ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) + ( − ) ∣ ξ ∣+ H ij ∧ ( ∇ i ξ + d x t ∧ Γ sit ( ∂ s ⌟ ξ ) − ∂ i ⌟ d ξ ) ∧ ( ∂ j ⌟ η ) + H ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∇ j η + d x t ∧ Γ sjt ( ∂ s ⌟ η ) − ∂ j ⌟ d η ) . Comparing these fragments, we findd ( ξ ̂ ∧ η ) − d ( ξ ) ̂ ∧ η − ( − ) ∣ ξ ∣ ξ ̂ ∧ d ( η ) = H ij ∧ ( ∂ i ⌟ ξ ) ∧ ∇ j η − ( − ) ∣ ξ ∣ H ij ∧ ∇ i ξ ∧ ( ∂ j ⌟ η ) − ( − ) ∣ ξ ∣ G ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) , where G ij ∶= d H ij + Γ irp d x p ∧ H rj + Γ jrp d x p ∧ H ir . Again, this expression holds forany collection H ij .Now comparing with Lemma 4.1 and taking H ij as defined there, we see that theLeibniz rule holds with respect to ∧ if and only if H ij is symmetric and G ij = η in degree 0 so that theinterior product ∂ j ⌟ η = (cid:3) This gives conditions on the curvature and torsion contained in H ij to obtain adifferential graded algebra to order λ .4.2. Results on curvature, torsion and the tensor N . Here we do some calcu-lations in Riemannian geometry with torsion in order to simplify our two conditionsin Proposition 4.3 on the tensor H ij . We use [11] and [26] for the Bianchi identitieswith torsion; ( B ) ∑ cyclic permutations ( abc ) ( T kbc ; a − R kabc − T kai T ibc ) = , ( B ) ∑ cyclic permutations ( abc ) ( R kjbc ; a − R kjai T ibc ) = . We also have to bring out a technical point of the semicolon equals covariant de-rivative notation, which only occurs if we use it more than once. For some ten-sor K (with various indices), we have K ; i = ∇ i K , but K ; ij ≠ ∇ j ∇ i K . This is EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 27 because in K ; ij we take the j th covariant derivative of K ; i including i in the in-dices we take the covariant derivative with respect to, so we get an extra term − Γ pji K ; p which does not appear in ∇ j ∇ i K . Thus in the presence of torsion we get K ; ij − K ; ji = [ ∇ j , ∇ i ] K − T pji K ; p , where the commutator [ ∇ j , ∇ i ] gives the curvature. Lemma 4.4.
Given the compatibility condition (3.1), the 2-forms H ij in Lemma 4.1obey H ij = H ji .Proof. Differentiate the compatibility condition to get0 = ω ij ; mn + ω ik ; n T jkm + ω kj ; n T ikm + ω ik T jkm ; n + ω kj T ikm ; n = ω ij ; mn − ( ω is T ksn + ω sk T isn ) T jkm − ( ω ks T jsn + ω sj T ksn ) T ikm + ω ik T jkm ; n + ω kj T ikm ; n which we rearrange as ω ij ; mn = ω is T ksn T jkm + ω sj T ksn T ikm + ω sk ( T isn T jkm + T ism T jkn ) − ω ik T jkm ; n − ω kj T ikm ; n Now use ω ij ; mn − ω ij ; nm = ω sj R isnm + ω is R jsnm − T pnm ω ij ; p = ω sj R isnm + ω is R jsnm + T pnm ( ω ik T jkp + ω kj T ikp ) , where we have used the compatibility condition again, to get ω sj R isnm + ω is R jsnm = ω is ( T ksn T jkm − T ksm T jkn ) + ω sj ( T ksn T ikm − T ksm T ikn ) − ω is ( T jsm ; n − T jsn ; m ) − ω sj ( T ism ; n − T isn ; m ) − T pnm ( ω ik T jkp + ω kj T ikp ) = ω is ( T ksn T jkm − T ksm T jkn ) + ω sj ( T ksn T ikm − T ksm T ikn ) − ω is ( T jsm ; n − T jsn ; m ) − ω sj ( T ism ; n − T isn ; m ) − T knm ( ω is T jsk + ω sj T isk ) = ω is ( T ksn T jkm − T ksm T jkn − T knm T jsk ) − ω is ( T jsm ; n − T jsn ; m ) + ω sj ( T ksn T ikm − T ksm T ikn − T knm T isk ) − ω sj ( T ism ; n − T isn ; m ) , which we rearrange to give0 = ω is (( T kns T jmk + T ksm T jnk + T kmn T jsk ) − ( T jsm ; n + T jns ; m ) + R jsmn ) + ω sj (( T kns T imk + T ksm T ink + T kmn T isk ) − ( T ism ; n + T ins ; m ) + R ismn ) . Using (B1) gives the symmetry of H ij .0 = ω is ( T jmn ; s − R jmns − R jnsm ) + ω sj ( T imn ; s − R imns − R insm ) . ◻ Lemma 4.5.
Given the compatibility condition (3.1), the 2-forms H ij in Lemma 4.1obey d H ij + Γ irp d x p ∧ H rj + Γ jrp d x p ∧ H ir = .Proof. To calculate d H ij it is important to note that the i, j are fixed indices, andare not summed with the vector or covector basis. This is the reason for the extraChristoffel symbols entering the following expression: ∇ p ( H ij ) = ∇ p ( ω is ( T jnm ; s − R jnms ) d x m ∧ d x n ) = ω is ; p ( T jnm ; s − R jnms ) d x m ∧ d x n + ω is ( T jnm ; sp − R jnms ; p ) d x m ∧ d x n − Γ ipr H rj − Γ jpr H ir . Thus we have, using the compatibility condition,d x p ∧ ∇ p ( H ij ) + Γ irp d x p ∧ H rj + Γ jrp d x p ∧ H ir = ω is ; p ( T jnm ; s − R jnms ) d x p ∧ d x m ∧ d x n + ω is ( T jnm ; sp − R jnms ; p ) d x p ∧ d x m ∧ d x n − T ipr d x p ∧ H rj − T jpr d x p ∧ H ir = − ( ω it T stp + ω ts T itp ) ( T jnm ; s − R jnms ) d x p ∧ d x m ∧ d x n + ω is ( T jnm ; sp − R jnms ; p ) d x p ∧ d x m ∧ d x n − T ipr d x p ∧ H rj − T jpr d x p ∧ H ir . Using (B1) and then differentiating, we see that ∑ cyclic ( pmn ) ( T jnm ; p − R jpnm − T jpr T rnm ) = , ∑ cyclic ( pmn ) ( T jnm ; ps − R jpnm ; s − T jpr ; s T rnm − T jpr T rnm ; s ) = . Since the 3-form has cyclic symmetry in ( pmn ) ,d x p ∧ ∇ p ( H ij ) + Γ irp d x p ∧ H rj + Γ jrp d x p ∧ H ir = − ( ω it T stp + ω ts T itp ) ( T jnm ; s − R jnms ) d x p ∧ d x m ∧ d x n + ω is ( T jnm ; sp − T jnm ; ps + R jpnm ; s + T jpr ; s T rnm + T jpr T rnm ; s − R jnms ; p ) d x p ∧ d x m ∧ d x n − T ipr d x p ∧ H rj − T jpr d x p ∧ H ir = − ( ω it T stp + ω ts T itp ) ( T jnm ; s − R jnms ) d x p ∧ d x m ∧ d x n + ω is ( T rnm R jrps − T jrm R rnps − T jnr R rmps − T jnm ; r T rps + R jpnm ; s + T jpr ; s T rnm + T jpr T rnm ; s − R jnms ; p ) d x p ∧ d x m ∧ d x n − T ipr d x p ∧ H rj − T jpr d x p ∧ H ir = − ( ω is T rsp ( T jnm ; r − R jnmr ) + ω rs T irp ( T jnm ; s − R jnms )) d x p ∧ d x m ∧ d x n + ω is ( T rnm R jrps − T jrm R rnps − T jnr R rmps − T jnm ; r T rps + R jpnm ; s + T jpr ; s T rnm + T jpr T rnm ; s − R jnms ; p ) d x p ∧ d x m ∧ d x n − ω rs ( T jnm ; s − R jnms ) T ipr d x p ∧ d x m ∧ d x n − ω is T jpr ( T rnm ; s − R rnms ) d x p ∧ d x m ∧ d x n = − ( ω is T rsp ( − R jnmr )) d x p ∧ d x m ∧ d x n + ω is ( T rnm R jrps − T jrm R rnps − T jnr R rmps + R jpnm ; s + T jpr ; s T rnm − R jnms ; p ) d x p ∧ d x m ∧ d x n − ω is T jpr ( − R rnms ) d x p ∧ d x m ∧ d x n = ω is ( T rnm R jrps − T jrm R rnps − T jnr R rmps + R jpnm ; s + T jpr ; s T rnm − R jnms ; p + T rsp R jnmr + T jpr R rnms ) d x p ∧ d x m ∧ d x n . Given the overall d x p ∧ d x m ∧ d x n factor, we can make the following substitutions: − R jnms ; p ↦ − R jpns ; m ↦ R jpsn ; m ↦ − R jpsm ; n ↦ R jpms ; n T rsp R jnmr ↦ T rsm R jpnr ↦ − T rms R jpnr ↦ T rns R jpmr ↦ T rns R jpmr ↦ − T rsn R jpmr T jpr R rnms ↦ T jnr R rmps ↦ − T jmr R rnps ↦ T jrm R rnps Using these we can rewrite the previous equations, and then use (B2) to getd x p ∧ ∇ p ( H ij ) + Γ irp d x p ∧ H rj + Γ jrp d x p ∧ H ir = ω is ( T rnm R jrps − T jrm R rnps − T jnr R rmps + T jpr ; s T rnm + R jpnm ; s + R jpsn ; m + R jpms ; n − T rms R jpnr − T rsn R jpmr + T jnr R rmps + T jrm R rnps ) d x p ∧ d x m ∧ d x n = ω is ( T rnm R jrps + T jpr ; s T rnm + T rnm R jpsr ) d x p ∧ d x m ∧ d x n = ω is T rnm ( R jrps + T jpr ; s + R jpsr ) d x p ∧ d x m ∧ d x n . EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 29
We also need to calculate T tvu d x v ∧ d x u ∧ ( ∂ t ⌟ H ij ) = T tvu d x v ∧ d x u ∧ ( ∂ t ⌟ ( ω is ( T jnm ; s − R jnms ) d x m ∧ d x n )) = T pvu ω is ( T jnp ; s − R jnps ) d x v ∧ d x u ∧ d x n − T pvu ω is ( T jpn ; s − R jpns ) d x v ∧ d x u ∧ d x n = T pvu ω is ( T jnp ; s − R jnps + R jpns ) d x v ∧ d x u ∧ d x n = T rpm ω is ( T jnr ; s − R jnrs + R jrns ) d x p ∧ d x m ∧ d x n = T rmn ω is ( T jpr ; s − R jprs + R jrps ) d x p ∧ d x m ∧ d x n = T rmn ω is ( T jpr ; s + R jpsr + R jrps ) d x p ∧ d x m ∧ d x n . The result follows by usingd H ij = d x k ∧ ∇ k H ij + T tkn d x k ∧ d x n ∧ ( ∂ t ⌟ H ij ) . ◻ The last two lemmas prove the following theorem:
Theorem 4.6.
Suppose that the compatibility condition (3.1) holds. Then theconditions on H ij in Proposition 4.3 hold, i.e. we have a differential graded algebra ( ∧ , d ) to order λ . Proposition 4.7.
Over C , the DGA above is a ∗ -DGA.Proof. As both d and ⋆ are undeformed, it is automatic that d ( ξ ∗ ) = ( d ξ ) ∗ . Next η ∗ ∧ ξ ∗ = η ∗ ∧ ξ ∗ + λ ω ij ∇ i η ∗ ∧ ∇ j ξ ∗ + λ ( − ) ∣ η ∣+ H ij ∧ ( ∂ i ⌟ η ∗ ) ∧ ( ∂ j ⌟ ξ ∗ ) = η ∗ ∧ ξ ∗ + λ ω ij ∇ i η ∗ ∧ ∇ j ξ ∗ + λ ( − ) ∣ η ∣+ H ij ∧ ( ∂ i ⌟ η ) ∗ ∧ ( ∂ j ⌟ ξ ) ∗ = ( − ) ∣ ξ ∣ ∣ η ∣ ( ξ ∧ η ) ∗ + ( − ) ∣ ξ ∣ ∣ η ∣ λ ω ij ( ∇ j ξ ∧ ∇ i η ) ∗ + λ ( − ) ∣ η ∣+ +(∣ η ∣− )(∣ ξ ∣− ) H ij ∧ (( ∂ j ⌟ ξ ) ∧ ( ∂ i ⌟ η )) ∗ = ( − ) ∣ ξ ∣ ∣ η ∣ ( ξ ∧ η ) ∗ + ( − ) ∣ ξ ∣ ∣ η ∣ ( λ ω ij ∇ i ξ ∧ ∇ j η ) ∗ + λ ( − ) ∣ ξ ∣ ∣ η ∣+∣ ξ ∣+ ( H ij ∧ ( ∂ j ⌟ ξ ) ∧ ( ∂ i ⌟ η )) ∗ = ( − ) ∣ ξ ∣ ∣ η ∣ ( ξ ∧ η ) ∗ + ( − ) ∣ ξ ∣ ∣ η ∣ ( λ ω ij ∇ i ξ ∧ ∇ j η ) ∗ + ( − ) ∣ ξ ∣ ∣ η ∣ ( λ ( − ) ∣ ξ ∣+ H ij ∧ ( ∂ j ⌟ ξ ) ∧ ( ∂ i ⌟ η )) ∗ ◻ The quantum torsion of the quantising connection.
Here we considerthe quantum torsion of the quantum connection given by applying Theorem 3.5 tothe Poisson-compatible connection ( Ω , ∇ ) itself. This is intimately tied up withthe quantum differential calculus above. Lemma 4.8.
We have ∧ ( σ Q + id ⊗ )( η ⊗ ξ ) = λ ξ j η p ω ji T pnk ; i d x k ∧ d x n , so the quantum torsion is a right module map if and only if ω ji T pnk ; i = .Proof. From Theorem 3.5, ∧ ( σ Q + id ⊗ )( η ⊗ ξ ) = ξ ∧ η + η ∧ ξ + λ ω ij ∇ j ξ ∧ ∇ i η + λ ω ij ξ j d x k ∧ [ ∇ k , ∇ i ] η , and using Proposition 4.3 and the definition of H ij in Lemma 4.1 we get ∧ ( σ Q + id ⊗ )( η ⊗ ξ ) = λ H ij ξ i η j + λ ω ij ξ j d x k ∧ [ ∇ k , ∇ i ] η = λ ξ i η j ω is ( T jnm ; s − R jnms ) d x m ∧ d x n − λ ω ij ξ j R pnki η p d x k ∧ d x n = λ ξ j η p ω ji ( T pnk ; i − R pnki ) d x k ∧ d x n − λ ω ij ξ j R pnki η p d x k ∧ d x n . ◻ Proposition 4.9.
The quantum torsion of the quantising connection on Ω ( M ) is ( ∧ ∇ Q − d )( ξ ) = ( ∧∇ − d )( ξ ) + λ ( ∂ j ⌟ ∇ i ξ ) ω is T jnm ; s d x m ∧ d x n . Proof.
Here all covariant derivatives are the quantising connection on Ω ( M ) : ∧ ∇ Q ξ = ∧ q − ∇ ξ − λ ω ij d x k ∧ [ ∇ k , ∇ j ] ∇ i ξ = ∧ q − ( d x k ⊗ ∇ k ξ ) − λ ω ij d x k ∧ [ ∇ k , ∇ j ] ∇ i ξ = d x k ∧ ∇ k ξ + λ H ij ( ∂ j ⌟ ∇ i ξ ) − λ ω is d x m ∧ [ ∇ m , ∇ s ] ∇ i ξ , and using the definition of H ij in Lemma 4.1 we get ∧ ∇ Q ξ = d x k ∧ ∇ k ξ + λ ( ∂ j ⌟ ∇ i ξ ) ω is ( T jnm ; s − R jnms ) d x m ∧ d x n + λ ω is R jnms d x m ∧ d x n ( ∂ j ⌟ ∇ i ξ ) . ◻ Finally, in classical differential geometry one has for any linear connection that ∇ k ( ξ ∧ η ) = ∇ k ( ξ ) ∧ η + ξ ∧ ∇ k ( η ) . This works because the usual tensor productcovariant derivative on Ω ( M ) ⊗ Ω ( M ) preserves symmetry, so things in the kernelof ∧ stay in the kernel. So given a quantising covariant derivative ∇ on Ω ( M ) , wenaturally get covariant derivatives on all the Ω i ( M ) , which we also call ∇ . We canthen say that the wedge product ∧ ∶ Ω n ( M ) ⊗ Ω n ( M ) → Ω n + m ( M ) intertwinesthe covariant derivatives. We conclude by studying what happens when we quantisethe covariant derivatives. Proposition 4.10. ( id ⊗ ∧ ) ∇ Q ⊗ Q ( ξ ⊗ η ) − ∇ Q ( ξ ∧ η ) = λ ( − ) ∣ ξ ∣ d x k ⊗ ( ∇ k H ij + Γ ikp H pj + Γ jkp H ip ) ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) . Proof:
From modifying (3.17) we get Q ( Ω ( M )) ⊗ Q ( Ω ( M )) q / / ∇ Q ( Ω ( M ))⊗ Q ( Ω ( M )) (cid:15) (cid:15) Q ( Ω ( M ) ⊗ Ω ( M )) ∇ Q ( Ω ( M )⊗ ( M )) (cid:15) (cid:15) Q ( Ω ( M )) ⊗ Q ( Ω ( M )) ⊗ Q ( Ω ( M )) id ⊗ q / / Q ( Ω ( M )) ⊗ Q ( Ω ( M ) ⊗ Ω ( M )) id ⊗ (∧) (cid:15) (cid:15) Q ( Ω ( M )) ⊗ Q ( Ω ( M )) (4.2)As classically ∧ intertwines the covariant derivatives, ( id ⊗ ( ∧ q )) ∇ Q ( Ω ( M ))⊗ Q ( Ω ( M )) = ∇ Q ( Ω ( M )) ( ∧ q ) ∶ Q ( Ω ( M )) ⊗ Q ( Ω ( M )) → Q ( Ω ( M )) ⊗ Q ( Ω ( M )) . In the notation of Proposition 4.3 we now look at ξ ∧ η = ξ ∧ Q η + λ ξ ̂ ∧ η where ξ ̂ ∧ η = ( − ) ∣ ξ ∣+ H ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) . Then we get λ ( id ⊗ ̂ ∧ ) ∇ Q ( Ω ( M ))⊗ Q ( Ω ( M )) ( ξ ⊗ η ) EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 31 = λ ( id ⊗ ̂ ∧ )( d x k ⊗ ( ∇ k ξ ⊗ η + ξ ⊗ ∇ k η )) = λ d x k ⊗ ( ∇ k ξ ̂ ∧ η + ξ ̂ ∧ ∇ k η ) = λ ( − ) ∣ ξ ∣+ d x k ⊗ H ij ∧ (( ∂ i ⌟ ∇ k ξ ) ∧ ( ∂ j ⌟ η ) + ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ ∇ k η )) . Also, using ∇ i ( v ⌟ ξ ) = ∇ i ( v ) ⌟ ξ + v ⌟ ∇ i ξλ ∇ Q ( Ω ( M )) ( ξ ̂ ∧ η ) = λ ( − ) ∣ ξ ∣+ d x k ⊗ ∇ k ( H ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η )) = λ ( − ) ∣ ξ ∣+ d x k ⊗ H ij ∧ (( ∂ i ⌟ ∇ k ξ ) ∧ ( ∂ j ⌟ η ) + ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ ∇ k η )) + λ ( − ) ∣ ξ ∣+ d x k ⊗ ∇ k H ij ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) + λ ( − ) ∣ ξ ∣+ d x k ⊗ H ij ∧ (( Γ pki ∂ p ⌟ ξ ) ∧ ( ∂ j ⌟ η ) + ( ∂ i ⌟ ξ ) ∧ ( Γ pkj ∂ p ⌟ η )) = λ ( id ⊗ ̂ ∧ ) ∇ Q ( Ω ( M ))⊗ Q ( Ω ( M )) ( ξ ⊗ η ) + λ ( − ) ∣ ξ ∣+ d x k ⊗ ( ∇ k H ij + Γ ikp H pj + Γ jkp H ip ) ∧ ( ∂ i ⌟ ξ ) ∧ ( ∂ j ⌟ η ) . ◻ Quantizing other linear connections relative to the background ( Ω , ∇ ) . Here we extend the above to other connections ∇ S = ∇ + S on Ω ( M ) different fromthe quantizing one ∇ , where S ( ξ ) = ξ p S pnm d x n ⊗ d x m for ξ ∈ Ω ( M ) . Quantisationis achieved on the same quantum bundle as defined by ∇ , using Proposition 3.10 Proposition 4.11.
The torsion of ∇ QS is given by T ∇ QS ( ξ ) = T ∇ S ( ξ ) + λ ξ p ; i ω ij ( T pnm ; j − S pnm ; j ) d x m ∧ d x n + λ ξ p ( S pnm H nm + ω ij S pnm ;ˆ i d x n ∧ d x m ) . Note that the hat on ˆ denotes that the j index does not take part in the covariantdifferentiation in the i direction.Proof. We have T ∇ QS ( ξ ) = T ∇ Q ( ξ ) + ∧ q − S ( ξ ) + ∧ λ ω ij ∇ i ○ ∇ j ( S )( ξ ) . Now ∧ q − S ( ξ ) = ξ p S pnm d x n ∧ d x m + λ ξ p S pnm H nm , ∧ λ ω ij ∇ i ○ ∇ j ( S )( ξ ) = ∧ λ ω ij ∇ i ( ξ p S pnm ; j d x n ⊗ d x m ) = λ ω ij ∇ i ( ξ p S pnm ; j d x n ∧ d x m ) = λ ω ij ( ξ p ; i S pnm ; j + ξ p S pnm ;ˆ i ) d x n ∧ d x m . By Proposition 4.9 we get T ∇ QS ( ξ ) = T ∇ ( ξ ) + λ ξ j ; i ω is T jnm ; s d x m ∧ d x n + ∧ S ( ξ ) + λ ξ p S pnm H nm + λ ω ij ( ξ p ; i S pnm ; j + ξ p S pnm ;ˆ i ) d x n ∧ d x m = T ∇ S ( ξ ) + λ ξ p ; i ω ij T pnm ; j d x m ∧ d x n + λ ξ p S pnm H nm + λ ω ij ( ξ p ; i S pnm ; j + ξ p S pnm ;ˆ i ) d x n ∧ d x m = T ∇ S ( ξ ) + ( λ ξ p ; i ω ij T pnm ; j − λ ω ij ξ p ; i S pnm ; j ) d x m ∧ d x n + λ ξ p S pnm H nm + λ ω ij ξ p S pnm ;ˆ i d x n ∧ d x m = T ∇ S ( ξ ) + λ ξ p ; i ω ij ( T pnm ; j − S pnm ; j ) d x m ∧ d x n + λ ξ p ( S pnm H nm + ω ij S pnm ;ˆ i d x n ∧ d x m ) . ◻ Corollary 4.12.
In the case where ∇ S is torsion free, the quantum torsion T ∇ QS ( ξ ) ∶= λ ξ p A pnm d x m ∧ d x n has associated tensor A pnm = ω is ( S pij + S pji ) ( T jnm ; s − R jnms + R jmns ) − ω ij ( T snm R psij − T psm R snij + T psn R smij ) . Proof.
Begin with ∧ ∇ S ( d x p ) = ∧ ∇ ( d x p ) + S pnm d x n ∧ d x m so 0 = T ∇ ( d x p ) + S pnm d x n ∧ d x m and from (2.1) we deduce S pnm d x n ∧ d x m = T pnm d x n ∧ d x m . (4.3)Then Proposition 4.11 gives T ∇ QS ( ξ ) = T ∇ S ( ξ ) + λ ξ p ( S pnm H nm + ω ij T pnm ;ˆ i d x n ∧ d x m ) , and use the formula for the curvature of a tensor and the symmetry of H nm . ◻ We see that quantisation introduces an element of torsion at order λ in the quan-tisation ∇ QS . We similarly look at Lemma 3.12 to measure the deviation of ∇ QS from being star preserving and find an error of order λ : Lemma 4.13.
Over C and if S is real, the difference D aijnm λ ω ij d x n ⊗ d x m ingoing clockwise minus anticlockwise round the diagram in Lemma 3.12 startingfrom Q ( d x a ) is given by D aijnm = S aip S pnm ; j − ( S bnm R abij − S arm R rnij − S anr R rmij ) − S ajr R rmni Proof.
Putting e = e ∗ = d x a in Lemma 3.12, and using ∇ j ( S )( ξ ) = ξ p S pnm ; j d x n ⊗ d x m we get ∇ j ( S )( S i ( d x a )) = ∇ j ( S )( S air d x r ) = S aip S pkm ; j d x k ⊗ d x m , d x k ⊗ [ ∇ k , ∇ i ] S j ( d x a ) = d x k ⊗ [ ∇ k , ∇ i ]( S ajr d x r ) = − S ajr R rmki d x k ⊗ d x m . Now we use the antisymmetry of ω ij to get ω ij ∇ i ( ∇ j ( S ))( d x a ) = ω ij ( S bkm R abij − S arm R rkij − S akr R rmij ) d x k ⊗ d x m . We note that this is equivalent to the derivative of ⋆ ∶ Q ( Ω ( M )) → Q ( Ω ( M )) being ∇ QS ( ⋆ )( Q ( d x a )) = λ ω ij D aijkm d x k ⊗ d x m and we see that this is not necessarily zero. (cid:3) Next we consider adding a correction, so ∇ = ∇ QS + λK where K ∶ Ω ( M ) → Ω ( M ) ⊗ Ω ( M ) is given by K ( ξ ) = ξ p K pnm d x n ⊗ d x m . Theorem 4.14.
Over C and for any real S , there is a unique real K such that ∇ QS + λK is star preserving (namely K anm = ω ij D aijnm ). Moreover, if ∇ S istorsion free, this unique ∇ QS + λK is quantum torsion free. EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 33
Proof.
We look at the following diagram: Q ( Ω ( M )) = Q ( Ω ( M )) λ K (cid:15) (cid:15) Q ( Ω ( M )) ⋆ o o λ K / / Q ( Ω ( M )) ⊗ Q ( Ω ( M )) ⋆⊗ ⋆ (cid:15) (cid:15) Q ( Ω ( M )) ⊗ Q ( Ω ( M )) Q ( Ω ( M )) ⊗ Q ( Ω ( M )) σ QS o o Q ( Ω ( M )) ⊗ Q ( Ω ( M )) Υ − o o where at this order σ QS is simply transposition. Hence for ∇ QS + λK the effect ofadding K is to add − λ ( K anm + ( K anm ) ∗ ) d x n ⊗ d x m . to the difference in Lemma 4.13. This gives the unique value if we assume K is realfor the connection to be ∗ -preserving. Adding K also adds λ ξ a K anm d x n ∧ d x m tothe formula for the torsion in Proposition 4.11 so if K has the unique real valuestated and if ∇ S is torsion free, and using (4.3), K anm d x n ∧ d x m = ω ij D aijnm d x n ∧ d x m = ω ij ( S aip T pnm ; j − ( T bnm R abij − S arm R rnij − S anr R rmij ) − S ajr R rmni ) d x n ∧ d x m = ω ij ( S aip T pnm ; j − ( T bnm R abij − S arm R rnij + S amr R rnij ) − S ajr R rmni ) d x n ∧ d x m = ω ij ( S aip T pnm ; j − ( T bnm R abij + T amr R rnij ) + S air R rmnj ) d x n ∧ d x m = S aip ω ij ( T pnm ; j + R pmnj ) d x n ∧ d x m − ω ij ( T bnm R abij + T amr R rnij ) d x n ∧ d x m = − S aip H ip − ω ij ( T bnm R abij + T amr R rnij ) d x n ∧ d x m . Now Corollary 4.12 gives ∇ QS + λK torsion free. ◻ Thus requiring torsion free and star-preserving gives a unique ‘star-preservingand torsion-preserving’ quantisation of any classical torsion-free connection ∇ S onΩ ( M ) . 5. Semiquantization of Riemannian geometry
We are now in position to semiquantise Riemannian geometry on our above datum ( ω, ∇ ) . We need to proceed carefully, as there are various places where modificationsarise, and there are typically two connections involved. Throughout this sectionsuppose g = g ij d x i ⊗ d x j ∈ Ω ⊗ ( M ) is a Riemannian metric on M . We start withthe quantum metric and the quantisation ∇ Q of the quantizing connection ∇ .5.1. The quantised metric.
We obtain to first order a quantum metric g ∈ Ω ⊗ A characterised by quantum symmetry and centrality. The former is thestatement that g is in the kernel of ∧ ∶ Ω ⊗ A → Ω A and the latter is that g commutes with the elements of the algebra, i.e. a.g = g .a for all a ∈ A . With-out this property, we could not simply apply the metric to a number of tensorproducts over the algebra (i.e. the fiberwise tensor product of bundles), and usingthe metric would become much more complicated.As with the wedge product, we start with a functorial part of the quantum metric(5.1) g Q ∶= q − , Ω ( g ) = g ij d x i ⊗ d x j + λ ω ij ( g ms,i − g ks Γ kim ) d x m ⊗ Γ sjn d x n . and of the quantum connection(5.2) ∇ Q d x i = − ( Γ imn + λ ω sj ( Γ imk,s Γ kjn − Γ ikt Γ ksm Γ tjn − Γ ijk R knms )) d x m ⊗ d x n by application of our functor in Section 3. Lemma 5.1.
If we have ∇ g = , then we also have ∇ Q g Q = as an application ofTheorem 3.5. Moreover, over C , g Q is ‘real’ and ∇ Q is ∗ -preserving.Proof. We consider the metric as a morphism ˜ g ∶ C ∞ ( M ) → Ω ( M ) ⊗ Ω ( M ) in D ,where g = ˜ g ( ) and Ω ( M ) , is equipped with the background quantising connec-tion (assumed now to be metric compatible). Then q − , Ω Q ( ˜ g ) ∶ Q ( C ∞ ( M )) → Q ( Ω ( M )) ⊗ Q ( Ω ( M )) and we evaluate this on 1 to give the element g Q ∈ Ω A ⊗ Ω A . In this case the morphism property of q Ω ( M ) , Ω ( M ) implies (sup-pressing M for clarity) ∇ Q ( Ω )⊗ Q ( Ω ) q − , Ω ○ Q ( ˜ g )( ) = ( id ⊗ q − , Ω ) ∇ Q ( Ω ⊗ Ω ) Q ( ˜ g )( ) and the right hand side is zero since ∇ Ω ⊗ Ω g =
0. One can also see this anotherway, which some readers may prefer: By Lemma 3.6 (which is best summarisedby the commuting diagram (3.23)), as long as the corresponding q s are inserted,the tensor product of the quantised connections is the same as the quantisation ofthe tensor product connection. We take a special case of (3.23), remembering thatΩ A = Q ( Ω ( M )) . Q ( Ω ( M )) ⊗ Q ( Ω ( M )) q / / ∇ Q ⊗ Q (cid:15) (cid:15) Q ( Ω ⊗ ( M )) ∇ Q ( Ω ⊗ ( M )) (cid:15) (cid:15) Ω A ⊗ Q ( Ω ( M )) ⊗ Q ( Ω ( M )) id ⊗ q / / Ω A ⊗ Q ( Ω ⊗ ( M )) (5.3)Now we suppose that classically the quantising connection preserves the classicalRiemannian metric g ∈ Ω ⊗ ( M ) , i.e. that ∇ Ω ⊗ ( M ) g =
0. By Lemma 3.13 we have ∇ Q ( Ω ⊗ ( M )) g =
0, which also gives g central in the quantised system. Also by (5.3)we see that g Q = q − g ∈ Ω A ⊗ Ω A is indeed preserved by the tensor product ofthe quantised connections ∇ Q ⊗ Q . Moreover, we already know from Lemma 3.8 that ∇ Q preserves the star operation hence in this case we also have Hermitian-metriccompatibility with g Q in the sense ( ¯ ∇ Q ⊗ id + id ⊗ ∇ Q )( ⋆ ⊗ id ) g Q = . Over C , reality of g Q in the sense Υ − ( ⋆ ⊗ ⋆ ) g Q = g Q reduces by (3.20) to the clas-sical statement for g and ⋆ ⊗ ⋆ , which is trivial certainly if the classical coefficients g ij are real and symmetric. (cid:3) However, g Q is not necessarily ‘quantum symmetric’. We can correct for this by anadjustment at order λ . Proposition 5.2.
Let ( ω, ∇ ) be a Poisson tensor with Poisson-compatible connec-tion and define the associated ‘generalised Ricci 2-form’ and adjusted metric R = g ij H ij , g = g Q − λq − R where on the right the 2-form is lifted to an antisymmetric tensor. Suppose that ∇ g = . EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 35 (1)
If the lowered T ijk is totally antisymmetric then d R = . (2) ∧ ( g ) = , and q ∇ Q g = − λ ∇ R . Here ∇ Q g = if and only if ∇ R = . (3) Over C , g is ‘real’ (and ∇ Q is star-preserving).Proof. (1) We use Lemma 4.5 in the following formula,d ( g ij H ij ) = g ij,p d x p ∧ H ij + g ij d H ij = g ij,p d x p ∧ H ij − g ij ( Γ irp d x p ∧ H rj + Γ jrp d x p ∧ H ir ) = ( g ij,p − g rj Γ rip − g ir Γ rjp ) d x p ∧ H ij . If ∇ preserves the metric we also have0 = ∇ p ( g ij d x i ⊗ d x j ) = ( g ij,p − g rj Γ rpi − g ir Γ rpj ) d x i ⊗ d x j , and using this, if the lowered T ijk is totally antisymmetricd ( g ij H ij ) = ( g rj Γ rpi + g ir Γ rpj − g rj Γ rip − g ir Γ rjp ) d x p ∧ H ij = ( g rj T rpi + g ir T rpj ) d x p ∧ H ij = ( T jpi + T ipj ) d x p ∧ H ij = . (5.4)(2) Clearly ∧ ( g Q ) = λ R so ∧ ( g ) =
0. Likewise q ∇ Q g = q ∇ Q g Q − λ ∇ R = − λ ∇ R by Lemma 5.1, where the last term here is viewed as an element of Ω ( M ) ⊗ byan antisymmetric lift. The antisymmetric lift commutes with ∇ so ∇ g = ∇ R = R as a 2-form. To give the formulae here more explicitly, weremember our 2-form conventions so that(5.5) R = R nm d x m ∧ d x n , R mn = g ij ω is ( T jnm ; s − R jnms + R jmns ) . in which case, g = g Q + λ R mn d x m ⊗ d x n . (3) Over C , we also have the condition Υ − ( ⋆ ⊗ ⋆ ) g = g , as the correction isboth imaginary and antisymmetric. ∇ Q is still star-preserving because that state-ment is not dependent on the metric (which means that it is also Hermitian-metriccompatible with the corresponding Hermitian metric ( ⋆ ⊗ id ) g ). (cid:3) In general we may not have either of these properties of R but we do have ∇ Q g being order λ and that is enough to make g commute with elements of A to order λ which is what we wanted to retain at this point. The terminology for R comesfrom the K¨ahler case which is a subcase of the following special case. Corollary 5.3.
If the quantising connection ∇ is the Levi-Civita one,(1) Poisson-compatibility reduces to ω covariantly constant.(2) ∇ Q is quantum torsion free and R = ω ji R inmj d x m ∧ d x n is closed.(3) ∇ Q g = , i.e. ∇ Q is a quantum-Levi-Civita connection for g , if and only if ∇ R = .Proof. This is a special caae of Proposition 5.2. For the quantum torsion we useProposition 4.9 where the torsion T of ∇ is currently being assumed to be zero. Inthis case d R = T = ∇ R ≠ ∇ Q g is order λ by Lemma 5.1. (cid:3) Relating general ∇ and the Levi-Civita ̂ ∇ . In general the quantisingconnection ∇ may not be the same as the classical Levi-Civita connection ̂ ∇ forour chosen metric on M . In this section we write the latter in the general form ∇ S = ∇ + S for some S ∶ Ω ( M ) → Ω ( M ) ⊗ Ω ( M ) and we assume that thequantising connection ∇ obeys ∇ g =
0. The quantising connection has torsion T and we lower its indices by the Riemannian metric T abc = g ad T dbc . It is well-known(see [18]) that given an arbitrary torsion T , there is a unique metric compatiblecovariant derivative ∇ with that torsion, given byΓ abc = ̂ Γ abc + g ad ( T dbc − T bcd − T cbd ) . (5.6)Here Γ abc in our case is the Christoffel symbols for the quantising connection and ̂ Γ abc is the Christoffel symbols for the Levi-Civita connection so that ∇ S ( d x a ) =− ̂ Γ abc d x b ⊗ d x c . Hence(5.7) S abc = g ad ( T dbc − T bcd − T cbd ) . As a quick check of conventions, note that this formula is consistent with (4.3).Throughout this section T is arbitrary which fixes ∇ such that this is metric com-patible, and S is the above function of T so that ∇ S = ̂ ∇ , the Levi-Civita connection. Lemma 5.4.
The curvatures are related by ̂ R lijk = R lijk − S lki ; j + S lji ; k − T mjk S lmi + S mki S ljm − S mji S lkm , where semicolon is derivative with respect to ∇ .Proof. This is elementary: ̂ Γ mji = Γ mji − S mji so that ̂ R lijk = ̂ Γ lki,j − ̂ Γ lji,k + ̂ Γ mki ̂ Γ ljm − ̂ Γ mji ̂ Γ lkm = R lijk − S lki,j + S lji,k − Γ mki S ljm + Γ mji S lkm − S mki Γ ljm + S mji Γ lkm + S mki S ljm − S mji S lkm = R lijk − S lki ; j + S lji ; k − T mjk S lmi + S mki S ljm − S mji S lkm . ◻ This gives a different point of view on some of the formulae below, if we wish torewrite expressions in terms of the Levi-Civita connection. In the same vein:
Proposition 5.5.
Suppose that a connection ∇ is metric-compatible. Then ( ∇ , ω ) are Poisson-compatible if and only if (̂ ∇ k ω ) ij + ω ir S jrk − ω jr S irk = or equivalently ω jm S imk = ((̂ ∇ k ω ) ij − (̂ ∇ r ω ) mj g ri g mk + (̂ ∇ r ω ) im g rj g mk ) . Proof.
The compatibility condition gives0 = (̂ ∇ m ω ) ij + ω ik ( T jkm + g jd ( T dmk − T mkd − T kmd )) + ω kj ( T ikm + g id ( T dmk − T mkd − T kmd )) = (̂ ∇ m ω ) ij + ω ik g jd ( T dkm − T mkd − T kmd ) + ω kj g id ( T dkm − T mkd − T kmd ) = (̂ ∇ m ω ) ij + ω ik g jd ( T dkm + T mdk − T kmd ) + ω kj g id ( T dkm + T mdk − T kmd ) EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 37 = (̂ ∇ m ω ) ij + ( ω ik g jd − ω jk g id )( T dkm + T mdk − T kmd ) which is the first condition stated in terms of S . From this, (̂ ∇ m ω ) ij g ir g js = − ω ik g ir S skm + ω jk g js S rkm . Now define − Θ mrs ∶= (̂ ∇ m ω ) ij g ir g js − ω jk g js S rkm = − ω ik g ir S skm − ω jk g js S rkm and note that Θ mrs is symmetric on swapping r, s . Rearranging this gives ω jk g js S rkm = (̂ ∇ m ω ) ij g ir g js + Θ mrs ,ω jk S rkm = (̂ ∇ m ω ) ij g ir + Θ mrs g sj . (5.8)We can also write Θ mrs = − (̂ ∇ m ω ) ij g ir g js + ω jk g js S rkm , and from this we get the following condition, which we repeat with permuted indicesΘ mrs + Θ rms = − (̂ ∇ m ω ) ij g ir g js − (̂ ∇ r ω ) ij g im g js , Θ rsm + Θ srm = − (̂ ∇ r ω ) ij g is g jm − (̂ ∇ s ω ) ij g ir g jm , Θ smr + Θ msr = − (̂ ∇ s ω ) ij g im g jr − (̂ ∇ m ω ) ij g is g jr . (5.9)Taking the first line of (5.9), subtracting the second and adding the third givesΘ mrs = (̂ ∇ r ω ) ij g is g jm + (̂ ∇ s ω ) ij g ir g jm . (5.10)Now we rewrite (5.8) as ω jk S rkm = (̂ ∇ m ω ) ij g ir + ((̂ ∇ r ω ) it g is g tm + (̂ ∇ s ω ) it g ir g tm ) g sj = (̂ ∇ m ω ) ij g ir − (̂ ∇ r ω ) ij g im + (̂ ∇ s ω ) it g ir g tm g sj which we write as stated. (cid:3) Metric compatibility in the general case.
Now we look for a quantumLevi Civita connection in the general case where the quantising connection ∇ isnot the Levi-Civita one. As in Section 5.1 we assume a metric g ∈ Ω ⊗ ( M ) and ∇ g = S be a function of T so that ∇ S = ∇ + S = ̂ ∇ ,the classical Levi-Civita connection for g . We do the straight metric compatibilityin this section (which makes sense over any field) and the Hermitian version in thenext section (recall that the two versions of the metric-compatibility coincide if thequantum connection is star-preserving). Lemma 5.6.
For ∇ S the Levi-Civita connection, the quantum metric compatibilitytensor and quantum torsion T ∇ QS = λ ξ p A pnm d x m ∧ d x n are given respectively by q ∇ QS ⊗ QS ( g Q ) = − λ ω ij g rs S sjn ( R rmki + S rkm ; i )( d x k ⊗ d x m ⊗ d x n ) A pnm = − ω ij ( g pd ( T isd + T sid ) ( T snm ; j − R snmj + R smnj ) + T snm R psij − T psm R snij + T psn R smij ) . Proof.
We look at Proposition 3.11, and set H = S ⊗ id F + ( τ ⊗ id )( id ⊗ S ) ∶ Ω ⊗ ( M ) → Ω ( M ) ⊗ Ω ⊗ ( M ) . As classically both ∇ and ∇ S preserve g , we get H ( g ) =
0. By Lemma 3.13 again,we get Q ( H )( g ) = ∇ QH ( g ) =
0. Now applying Proposition 3.11 gives q ∇ QS ⊗ QS ( q − g ) = ( q ∇ QH q + λ rem )( q − g ) = λ rem ( q − g ) , (5.11) where, using S ( f ) = d x k ⊗ S k ( f ) rem ( e ⊗ f ) = ω ij ( d x k ⊗ [ ∇ k , ∇ i ] e − ∇ i ( S )( e )) ⊗ S j ( f ) . We now have, by (5.11), q ∇ QS ⊗ QS ( q − g ) = λ rem ( q − g ) = λ rem ( g rs d x r ⊗ d x s ) = λ ω ij g rs ( d x k ⊗ [ ∇ k , ∇ i ]( d x r ) − ∇ i ( S )( d x r )) ⊗ S j ( d x s ) = λ ω ij g rs S sjn ( d x k ⊗ [ ∇ k , ∇ i ]( d x r ) − ∇ i ( S )( d x r )) ⊗ d x n which we write as stated. For the torsion we used S pij + S pji = − g pd ( T ijd + T jid ) inCorollary 4.12 and relabelled. (cid:3) We see that the quantisation ∇ QS given by the procedure outlined in Section 3.5is only quantum metric compatible to an error of order λ . However we have thefreedom to add an order λ correction to g Q as above and an order λ correction tothe proposed quantum connection: Theorem 5.7.
Let ∇ S be the Levi-Civita connection. There is a unique quantumconnection of the form ∇ = ∇ QS + λK such that the quantum torsion and merelythe symmetric part of ∇ g vanish. The antisymmetric part, ( id ⊗ ∧ ) q ∇ g = − λ ̂ ∇ R − λ ω ij g rs S sjn ( R rmki + S rkm ; i ) d x k ⊗ d x m ∧ d x n , is independent of K . A fully metric compatible torsion free ∇ exists if and only ifthe above expression vanishes, in which case it is given by the unique ∇ discussed.Proof. We write K ( ξ ) = ξ p K pnm d x n ⊗ d x m , then (where semicolon is given by thequantising connection) the results in the preceding lemma are clearly adjusted to q ∇ ( g ) = − λ ω ij g rs S sjn ( R rmki + S rkm ; i ) d x k ⊗ d x m ⊗ d x n − λ ∇ S ⊗ S ( g ij ω is ( T jnm ; s − R jnms + R jmns ) d x m ⊗ d x n ) + λ ( g pn K pkm + g mp K pkn ) d x k ⊗ d x m ⊗ d x n T ∇ ( ξ ) = λ ξ p ( K pnm − K pmn − A pnm ) d x n ∧ d x m . Looking at the first expression reveals that the second term is purely antisymmetricin nm , whereas the third term (the only one to contain the order λ correction K abc )is purely symmetric in nm . Hence there is nothing we can do by adding K abc tomake the part of the metric compatibility tensor which is antisymmetric in nm vanish, it will have the value stated, but we show that we can choose K abc to makethe part which is symmetric in nm vanish, namely by setting g np K pkm + g mp K pkn = B knm where B knm = ω ij g rs ( S sjn ( R rmki + S rkm ; i ) + S sjm ( R rnki + S rkn ; i )) while for vanishing torsion, clearly we need K pnm − K pmn = A pnm . If we set K nkm = g np K pkm then these conditions become K nkm + K mkn = B knm , K knm − K kmn = g kp A pnm . Now K nkm = B knm − K mkn = B knm + g mp A pnk − K mnk , EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 39 and continuing in this manner six times gives a unique value of K , K nkm = ( B knm − B nkm + B mnk + g mp A pnk + g kp A pnm + g np A pkm ) . (5.12)where B knm − B nkm + B mnk = ω ij g rs ( S sjn ( R rmki + S rkm ; i ) + S sjm ( R rnki + S rkn ; i )) − ω ij g rs ( S sjk ( R rmni + S rnm ; i ) + S sjm ( R rkni + S rnk ; i )) + ω ij g rs ( S sjn ( R rkmi + S rmk ; i ) + S sjk ( R rnmi + S rmn ; i )) = ω ij g rs ( S sjn ( R rmki + R rkmi + S rmk ; i + S rkm ; i ) + S sjm ( R rnki − R rkni − S rnk ; i + S rkn ; i ) + S sjk ( R rnmi − R rmni − S rnm ; i + S rmn ; i )) = ω ij g rs ( S sjn ( R rmki + R rkmi − g rd ( T mkd ; i + T kmd ; i )) + S sjm ( R rnki − R rkni + T rkn ; i ) + S sjk ( R rnmi − R rmni + T rmn ; i )) using S rmk + S rkm = − g rd ( T mkd + T kmd ) . (cid:3) This clearly reduces to Corollary 5.3 in the case where T = T for the quantizing connection provided onlythat ( ∇ , ω ) are Poisson-compatible. We might hope to use this freedom to set R = λ , and/or wemight hope to choose T so that the the antisymmetric part of the quantum metriccompatibility tensor also vanishes. In the nontrivial example black-hole below thisexpression will not even depend on T within the class discussed, i.e. can have moreof a topological character. Hence we can’t always obtain full metric compatibilitybut rather can have an unavoidable quantum correction. In that case we still havea ‘best possible’ choice of ∇ given by the formula (5.12).5.4. Hermitian-metric compatibility.
Here we again assume that our Poisson-compatible connection ∇ obeys ∇ g = ∇ S = ∇ + S is the Levi-Civitaconnection for g . We set ∇ = ∇ QS + λK , for some real K , and ask this time that ∇ is Hermitian-metric compatible with the Hermitian metric ( ⋆ ⊗ id ) g correspondingto g . This is a potentially different condition from straight metric compatibilityunless ∇ is star-preserving, in which case it is equivalent. Proposition 5.8.
Over C and with ∇ S the Levi-Civita connection, the conditionfor ∇ QS + λK to be Hermitian-metric compatible with g is, where ˆ; denotes theLevi-Civita derivative, K npm − K mpn = R nm ˆ; p + ω ij ( g rm ∇ i ( ∇ j ( S )) rpn − g nr ∇ i ( ∇ j ( S )) rpm ) . This can always be solved simultaneously with vanishing of the quantum torsion.Proof. (1) If we write the quantum correction to the metric in Proposition 5.2 as g = g Q − λ g c , then Hermitian-metric compatibility tensor for ∇ QS becomes (( id ⊗ ⋆ − ) Υ q − Q ( S ) ⊗ id + id ⊗ q − Q ( S ))( ⋆ ⊗ id ) g Q − λ ( ¯ ∇ S ⊗ id + id ⊗ ∇ S )( ⋆ ⊗ id ) g c (5.13)From Proposition 3.4 we can write this as ( q − ( id ⊗ ⋆ − ) Υ Q ( S ) ⊗ id + id ⊗ q − Q ( S ))( ⋆ ⊗ id ) g Q − λ ( ¯ ∇ S ⊗ id + id ⊗ ∇ S )( ⋆ ⊗ id ) g c . The definition of Q ( S ) gives ( q − ( id ⊗ ⋆ − ) Υ S ⊗ id + id ⊗ q − S )( ⋆ ⊗ id ) g Q + λ ω ij ( − ( id ⊗ ⋆ − ) Υ ∇ i ○ ∇ j ( S ) ⊗ id + id ⊗ ∇ i ○ ∇ j ( S ))( ⋆ ⊗ id ) g − λ (( id ⊗ ⋆ − ) Υ ∇ S ⊗ id + id ⊗ ∇ S )( ⋆ ⊗ id ) g c . Now we use Proposition 3.3 and apply q , noting that q ( q − ( id ⊗ ⋆ − ) Υ S ⊗ id + id ⊗ q − S )( ⋆ ⊗ id ) g Q = q (( id ⊗ ⋆ − ) Υ S ⊗ id + id ⊗ S )( ⋆ ⊗ id ) q − g = (( id ⊗ ⋆ − ) Υ S ⊗ id + id ⊗ S )( ⋆ ⊗ id ) g + λ ω ij ( ∇ i (( id ⊗ ⋆ − ) Υ S ) ⊗ ∇ j + ∇ i ⊗ ∇ j ( S ))( ⋆ ⊗ id ) g = (( id ⊗ ⋆ − ) Υ S ⊗ id + id ⊗ S )( ⋆ ⊗ id ) g + λ ω ij (( id ⊗ ⋆ − ) Υ ∇ i ( S ) ⊗ ∇ j + ∇ i ⊗ ∇ j ( S ))( ⋆ ⊗ id ) g = (( id ⊗ ⋆ − ) Υ S ⊗ id + id ⊗ S )( ⋆ ⊗ id ) g − λ ω ij (( id ⊗ ⋆ − ) Υ ∇ i ( S ) ○ ∇ j ⊗ id + id ⊗ ∇ j ( S ) ○ ∇ i )( ⋆ ⊗ id ) g as g is preserved by ∇ so that ( ∇ i ⊗ id ) g = − ( id ⊗ ∇ i ) g .Then q applied to (5.13) gives (( id ⊗ ⋆ − ) Υ S ⊗ id + id ⊗ S )( ⋆ ⊗ id ) g − λ ω ij ( − ( id ⊗ ⋆ − ) Υ ∇ j ( S ) ○ ∇ i ⊗ id + id ⊗ ∇ j ( S ) ○ ∇ i )( ⋆ ⊗ id ) g + λ ω ij ( − ( id ⊗ ⋆ − ) Υ ∇ i ○ ∇ j ( S ) ⊗ id + id ⊗ ∇ i ○ ∇ j ( S ))( ⋆ ⊗ id ) g − λ (( id ⊗ ⋆ − ) Υ ∇ S ⊗ id + id ⊗ ∇ S )( ⋆ ⊗ id ) g c = (( id ⊗ ⋆ − ) Υ S ⊗ id + id ⊗ S )( ⋆ ⊗ id ) g + λ ω ij ( − ( id ⊗ ⋆ − ) Υ ∇ i ( ∇ j ( S )) ⊗ id + id ⊗ ∇ i ( ∇ j ( S )))( ⋆ ⊗ id ) g − λ (( id ⊗ ⋆ − ) Υ ∇ S ⊗ id + id ⊗ ∇ S )( ⋆ ⊗ id ) g c . (5.14)Now set g = g nm d x n ⊗ d x m and ∇ i ( ∇ j ( S ))( d x a ) = ∇ i ( ∇ j ( S )) anm d x n ⊗ d x m , andusing the reality of S the first two lines of the result of (5.14) become ( g rm S rpn + g nr S rpm ) d x n ⊗ d x p ⊗ d x m + λ ω ij ( − g rm ∇ i ( ∇ j ( S )) rpn + g nr ∇ i ( ∇ j ( S )) rpm ) d x n ⊗ d x p ⊗ d x m , and the first line of this vanishes as ∇ S preserves g . Now we write g c = − R nm d x n ⊗ d x m where R nm is antisymmetric giving q ( ¯ ∇ QS ⊗ id + id ⊗ ∇ QS )( ⋆ ⊗ id ) g = − λ C npm d x n ⊗ d x p ⊗ d x m ; C npm = − R nm ˆ; p + ω ij ( g rm ∇ i ( ∇ j ( S )) rpn − g nr ∇ i ( ∇ j ( S )) rpm ) . (2) Now we look at ∇ = ∇ QS + λK , then clearly ( ¯ ∇ ⊗ id + id ⊗ ∇ )( ⋆ ⊗ id ) g = λ ( g na K apm − g ma K apn − C npm ) d x n ⊗ d x p ⊗ d x m so we need to solve K npm − K mpn = C npm to preserve the Hermitian metric, andalso K knm − K kmn = g ks A snm if we want to have zero torsion as in the previoussection. These equations have a required compatibility condition C npm + C mnp + C pmn + g ms A spn + g ps A snm + g ns A smp = . We use the formula (5.7) for S abc in terms of the torsion to write C npm = − R nm ˆ; p + ω ij ( g rm S rpn ;ˆ ji − g nr S rpm ;ˆ ji ) = − R nm ˆ; p + ω ij (( T mpn − T pnm − T npm ) − ( T npm − T pmn − T mpn )) ;ˆ ji EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 41 = − R nm ˆ; p + ω ij ( T mpn − T pnm − T npm ) ;ˆ ji , and taking the cyclic sum gives C npm + C mnp + C pmn = − R nm ˆ; p − R pn ˆ; m − R mp ˆ; n + ω ij ( T mpn + T pnm − T npm ) ;ˆ ji . We have − g pa A anm = ω ij ( T isp + T sip ) ( T snm ; j − R snmj + R smnj ) + ω ij g pa ( T snm R asij − T asm R snij + T asn R smij ) = ω ij ( T isp + T sip ) ( T snm ; j − R snmj + R smnj ) + ω ij T pnm ;ˆ ji , so now the cyclic sum becomes C npm + C mnp + C pmn = − R nm ˆ; p − ω ij ( T isp + T sip ) ( T snm ; j − R snmj + R smnj ) − R pn ˆ; m − ω ij ( T ism + T sim ) ( T spn ; j − R spnj + R snpj ) − R mp ˆ; n − ω ij ( T isn + T sin ) ( T smp ; j − R smpj + R spmj ) . (5.15)This is totally antisymmetric in npm , so we may equivalently consider the 3-form α = ( − R nm ˆ; p − ω ij ( T isp + T sip ) ( T snm ; j − R snmj + R smnj )) d x p ∧ d x n ∧ d x m = d x p ∧ ( − R nm ˆ; p d x n ∧ d x m ) + ( T isp + T sip ) d x p ∧ H is = d x p ∧ ̂ ∇ p ( g ij H ij ) + T isp d x p ∧ H is where we use H ij = ω is ( T jnm ; s − R jnms ) d x m ∧ d x n and the symmetry of H ij .Now we have as in (5.4) (but not requiring this to be zero)d R = ( T jpi + T ipj ) d x p ∧ H ij = − T ijp d x p ∧ H ij , so vanishing of α = d x p ∧ ̂ ∇ p ( R ) − d R is the condition for a joint solution. But thisis zero as the Levi-Civita connection is torsion free. ◻ Note that Proposition 5.8 does not say that such a torsion free quantum connectionpreserving the Hermitian metric is unique. If we take the collection of K ijk forpermutations of the ijk , then the equations fix what the relative value of, forexample K ijk − K kij will be, but we can add an overall factor to each of theseclasses under the permutation group S . Corollary 5.9.
Over C and with ∇ S the Levi-Civita connection, if a torsion freemetric compatible quantum connection of the form ∇ = ∇ QS + λK exists, it isstar-preserving and coincides with the unique star-preserving quantum connectionin Theorem 4.14.Proof. From Lemma 4.13 and Theorem 4.14 the star preserving connection is givenby K anm = ω ij D aijnm , or K anm = ω ij ( S aip S pnm ; j − ( S bnm R abij − S arm R rnij − S anr R rmij ) − S ajr R rmni ) = ω ij ( S aip S pnm ; j + S aip R pmnj − ( S bnm R abij − S arm R rnij − S anr R rmij )) = ω ij S aip ( S pnm ; j + R pmnj ) − ω ij ([ ∇ i , ∇ j ] S ) anm = ω ij S aip ( S pnm ; j + R pmnj ) − ω ij ∇ i ( ∇ j ( S )) anm . From this we get K npm = g an ω ij S ais ( S spm ; j + R smpj ) − g nr ω ij ∇ i ( ∇ j ( S )) rpm . From Proposition 5.8 the condition for ∇ = ∇ QS + λK to be Hermitian-metriccompatible is the following, where ˆ; denotes Levi-Civita derivative K npm − K mpn = − R nm ˆ; p + ω ij ( g rm ∇ i ( ∇ j ( S )) rpn − g nr ∇ i ( ∇ j ( S )) rpm ) , so on substituting for K npm we find the single condition ̂ ∇ R = − ω ij g rs S sjn ( R rmki + S rkm ; i ) d x k ⊗ d x m ∧ d x n . This is the same as the condition for existence of a fully metric compatible torsionfree connection of our assumed form in Theorem 5.7. So, if such a connectionexists, our star-preserving one gives it. The converse direction is also proved, butobvious (if our star-preserving connection is Hermitian-metric compatible then itis also straight metric compatible and hence the stated condition must hold byTheorem 5.7.) (cid:3) Quantized Surfaces and K¨ahler-Einstein manifolds
We have seen that our theory applies in particular to any Riemannian manifoldequipped with a covariantly constant Poisson-bivector, with the choice ∇ = ∇ LC .We then always have a quantum differential algebra by Theorem 4.6 and Corol-lary 5.3 says that the nicest case is when the ω -contracted Ricci tensor is covariantlyconstant. In this case we have a quantum symmetric g and a quantum-Levi-Civitaconnection for it. Proposition 6.1.
In the case of a K¨ahler manifold, R in Corollary 5.3 is the Ricci2-form. A sufficient condition for this to be covariantly constant is for the metricto be K¨ahler-Einstein.Proof. Here ω ij = − g ik J kj = J ki g kj where J = − id and R = R nm d x m ∧ d x n inour conventions so in Corollary 5.3 we have R nm = ω ji R inmj = g kj ω ji R inmk =− J ji R inmj . Now we use standard complexified local coordinates z a , ¯ z a in which J ab = ıδ ab and J ¯ a ¯ b = − ıδ ¯ a ¯ b . The only nonzero elements of Riemann are then of theform R ¯ abcd = − R b ¯ acd , R a ¯ b ¯ c ¯ d = − R ¯ ba ¯ c ¯ d . Hence R ¯ nm = − ıR a ¯ nma = ıR ¯ nama = ı Ricci ¯ nm and similarly R n ¯ m = − ı Ricci n ¯ m =− R ¯ mn by symmetry of Ricci. Then R ij = − J ik Ricci kj in our conventions for 2-formcomponents. Equivalently, R = R a ¯ b d z ¯ b ∧ d z a + R ¯ ba d z a ∧ d¯ z b = ı Ricci a ¯ b d z a ∧ d¯ z b as usual. Clearly in the K¨ahler-Einstein case we have also that Ricci = αg for someconstant α . Then R ij = − J ik αg kj = − αω ij in terms of the inverse ω ij of the Poissontensor, or R = αω / ω = ω ij d x i ∧ d x j . This iscovariantly constant by our assumption of Poisson-compatibility by Lemma 3.1. (cid:3) Note that the Ricci 2-form here is closed and represents the 1st Chern class. Itis known that every K¨ahler manifold with c ≤ C P n with itsFubini-study metric. Also note that on a K¨ahler manifold the J is also covariantlyconstant and we may hope to have a noncommutative complex structure in thesense of [8] to order λ . This will be considered elsewhere. EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 43
Any orientable surface can be given the structure of a K¨ahler manifold so that theabove applies. In fact we do not make use above of the full K¨ahler structure andin the case of an orientable surface we can consider any metric and Poisson tensor ω = − Vol − as obtained from the volume form, which will be covariantly constant.The generalised Ricci 2-form is then a constant multiple of S Vol where S is theRicci scalar (this follows from the Ricci tensor being gS / R will be covariantly constant if and only if S is constant, i.e. the case of constantcurvature.Some general formulae for any surface are as follows, in local coordinates ( x, y ) .Here Vol = √ det ( g ) d x d y where g = ( g ij ) is the metric. The Poisson tensor ω =− Vol − is then ω = w ( ∂∂x ⊗ ∂∂y − ∂∂y ⊗ ∂∂x ) ; ω = w ( x, y ) ∶= √ det ( g ) which of course gives our product as x ● x = x , y ● y = y , x ● y = xy + λ w , y ● x = xy − λ w , or commutation relations [ x, y ] ● = λw on the generators. Similarly,the bimodule commutation relations from the form of ω are [ f, ξ ] ● = λw ( ∂f∂x ∇ y − ∂f∂y ∇ x ) ξ where ∇ x , ∇ y are the covariant derivatives along ∂∂x and ∂∂y respectively. In termsof Γ we have [ f, d x j ] ● = λw ( f , Γ j m d x m − f , Γ j m d x m ) or on generators and with ǫ = [ x i , d x j ] ● = − λwǫ in Γ jnm d x m . There are similar expressions for ● itself in terms of the the classical product plushalf of the relevant commutator.Next, Ricci = S g implies by symmetries of the Riemann tensor that R = S ( g ) =∶ ρ ( x, y ) say, with other components determined by its symmetries. In this case R = − R = − ω is R i s = wρ, R = − wρ d x d y = − S H ij = − ω is R inms d x m d x n which we compute first with j lowered by the metric as H = H = − wρ x d y = − S , H = H = H ij = − S g ij Vol . By Theorem 4.6 we necessarily have a differential graded algebra to order λ . HereProposition 4.3 in our case becomesd x i ● d x j = d x i ∧ d x j + λ w ( Γ i Γ j − i Γ j + Γ i Γ j ) d x ∧ d y − λS g ij Vol so that the anticommutation relations for the quantum wedge product have theform { d x i , d x j } ● = λ ( w ( Γ i Γ j − i Γ j + Γ i Γ j ) − S g ij ) Vol . Finally the quantized metric, from (5.1) and since ∇ g = g = g Q + λ R ̃ Vol = ˜ g + λw ǫ ij g ma Γ aib Γ bjn d x m ⊗ d x n + λS ̃ Vol(6.1)where the first two terms are g Q and˜ g ∶= g ij d x i ⊗ d x j , ̃ Vol ∶= w ( d x ⊗ d y − d y ⊗ d x ) are shorthand notations. Similarly, the connection ∇ Q is computed from the localformula (5.2). As explained, we will have ∇ Q g = λ if and only if S isconstant. We compute further details for the two basic examples.6.1. Quantised hyperbolic space.
As the basic example we look at the Poincar´eupper half plane with its hyperbolic metric M = {( x, y ) ∈ R ∣ y > } , g = y ( d x ⊗ d x + d y ⊗ d y ) which is readily found to have nonzero Christoffel symbolsΓ = Γ = Γ = − y − , Γ = y − or Γ i j = − ǫ ij y − and Γ i j = − δ ij y − . The bivector ω = − ω = y is easily seen to bethe unique solution to (3.1) up to normalisation. This is the inverse of the volumeform Vol = y − d x d y .Clearly from the Poisson tensor ω = y ( ∂∂x ⊗ ∂∂y − ∂∂y ⊗ ∂∂x ) we have [ x, y ] ● = λy , which relations also occur for the standard bicrossproductmodel spacetime in 2-dimensions in terms of inverted coordinates in [7]. Alsonote that [ x, y − ] ● = λ . Note that although the relations do extend to an obviousassociative algebra A λ , this is not unique and not immediately relevant.Next, from Γ we see that [ f, d x ] ● = λy ( ∂f∂x d x − ∂f∂y d y ) , [ f, d y ] ● = λy ( ∂f∂y d x + ∂f∂x d y ) or on generators we have [ x, d x ] ● = [ y, d y ] ● = λy d x, [ x, d y ] ● = − [ y, d x ] ● = λy d y. There are similar expressions for ● itself in terms of the the classical product.The Ricci scalar here is S = − R = Vol , H ij = g ij Voland from the latter we obtaind x i ● d x j = d x i ∧ d x j + λ y ( Γ i Γ j − i Γ j + Γ i Γ j ) d x ∧ d y + λ δ ij d x ∧ d y EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 45 which from the form of Γ simplifies further tod x i ● d x j = d x i ∧ d x j − λ δ ij d x ∧ d y, { d x i , d x j } ● = − λδ ij d x ∧ d y which has a ‘Clifford algebra-like’ form. The result here is the same as obtained byapplying d to the bimodule relations, i.e. is consistent with the maximal prolonga-tion of the first order calculus.Finally, we have our constructions of noncommutative Riemannian geometry. Inour case ǫ ij g ma Γ aib Γ bjn = g Q has he same form as classically but with ⊗ and g = d x i y ⊗ d x i − λ ̃ Vol(sum over i ). Similarly, one may compute using the form of Γ in (5.2) that ∇ Q d x i = d yy ⊗ d x i + d xy ⊗ ǫ ij d x j which again has the same form as classically. There is an associated generalisedbrading σ Q making this a bimodule connection. As per our general theory, ∇ Q isquantum torsion free and metric compatible with g .All constructions above are invariant under SL ( R ) and hence under the modulargroup and other discrete subgroups. Indeed, the metric is well known to be in-variant. The volume form can also easily be seen to be and correspondingly ω isinvariant. As these are the only inputs into the theory it follows that the deformedstructures are likewise compatible with this action. The quotient of the construc-tions corresponds to replacing the Poincar´e upper half plane by a Riemann surfaceof constant negative curvature, constructed as quotient. Therefore this is achievedin principle. We might reasonably then expect a role for modular forms in thedeeper aspects of the noncommutative geometry.6.2. Quantised sphere.
The case of a surface of constant positive curvature, thesphere, is the n = C P n which will be covered elsewhere in holomorphiccoordinates. Here we give it is as an example of the analysis for surfaces above.We work in the upper hemisphere in standard cartesian coordinates, with similarformulae for the lower hemisphere, so M = {( x, y ) ∣ x + y < } , z = √ − x − y ,g = z (( − y ) d x ⊗ d x + xy ( d x ⊗ d y + d y ⊗ d x ) + ( − x ) d y ⊗ d y ) which is readily found to have symmetric Christoffel symbolsΓ = xz ( − y ) , Γ = xz ( − x ) , Γ = x yz Γ = yz ( − y ) , Γ = yz ( − x ) , Γ = xy z . or compactly Γ ijk = x i g jk . The inverse of the volume form Vol = z − d x d y gives the Poisson bivector ω = z ( ∂∂x ⊗ ∂∂y − ∂∂y ⊗ ∂∂x ) so we have relations [ x, y ] ● = λz, [ z, x ] ● = λy, [ y, z ] ● = λx, the standard relations of the fuzzy sphere. In this case there is an associativequantisation to all orders as the enveloping algebra U ( su ) modulo a constantvalue of the quadratic Casimir. It is known that this algebra does not admit anassociative 3D rotationally invariant calculus[3] so there won’t be a zero-curvaturePoisson-compatible preconnection. At present we use the Levi-Civita connectionaccording to Corollary 5.3. Then from Γ we have [ f, d x j ] ● = − λz x j f ,i ǫ ik g km d x m for the bimodule relations of the quantum differential calculus, where ǫ = [ x, d x i ] ● = − λ x i z ( xy d x + ( − x ) d y ) , [ y, d x i ] ● = λ x i z (( − y ) d x + xy d y ) . Next, the Ricci scalar of the unit sphere is S = R = − Vol , H ij = − g ij Vol , where g ij = ( − x − xy − xy − y ) . From this and Γ we obtaind x i ∧ ● d x j = d x i ∧ d x j + λ ( x i x j − g ij ) Vol , { d x i , d x j } ● = λ ( x i x j − g ij ) Volfor the exterior algebra relations. One can verify that this is the maximal prolon-gation of the bidmodule relations.Finally, we note that from the form of g that x a g ai = x i z − and ω ab g ai g bj = ǫ ij z − .The first of these and the form of Γ gives us g ij,k = Γ aki g aj + Γ akj g ia = x a ( g ki g aj + g kj g ia ) = z − ( x j g ki + x i g kj ) using metric compatibility. One may then compute the quantum metric and con-nection from (6.1) and (5.2) respectively as g = ˜ g + λ z x m d x m ⊗ x a ǫ an d x n + λ ̃ Vol ∇ Q d x i = − x i ˜ g − λx i ̃ Vol − λ z x m d x m ⊗ ( ǫ ib g bn + x i x b z ǫ bn ) d x n = − x i g − λ z x m d x m ⊗ ǫ ib g bn d x n = − x i ● g (sum over m ). Here on the left x i ˜ g is a shorthand notation for the previously definedelement of Ω A ⊗ Ω A but now with an extra classical x i in the definition. Onecan think if it as made with the classical product when the classical and quantumvector spaces are identified, and ditto for x i g . The expression x i ● g is computedwith the quantised product but only on the first tensor factor of g ∈ Ω A ⊗ Ω A (since this is the relevant bimodule structure). EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 47
In fact, this example at the level of first order differentials was proposed in [3] wherewe showed that the Levi-Civita connection arises as a cochain twist of the classicalexterior algebra by a certain action of the Lorenz group. This could potentially beused to construct the full noncommutative nonassociative Riemannian geometry bytwisting[6].7. ∇ far from Levi-Civita: bicrossproduct and black-hole models In this section we give two contrasting examples where where we cannot take ∇ = ̂ ∇ .The black-hole for a natural rotationally invariant Poisson bracket will provide anexample where some of the obstructions in the general theory hold, i.e. we cannotfind a quantising connection ∇ (expressed by choice of its torsion T ) such that thequantum ‘Levi-Civita’ connection of the form ∇ = ∇ QS + λK is star preserving,torsion free and metric compatible at the same time; one or more of these featuresnecessarily gets an order λ correction. Here ∇ S = ̂ ∇ is the classical Levi-Civita.On the other hand, we will find that the generalised Ricci 2-form R = g = g Q , the functorial one. We’ll find in the black-hole casethat ∇ necessarily has curvature and hence the quantum differential calculus willbe nonassociative.Before doing that we give an easier warm-up example which also illustrates all oursemiclassical theory and where the algebraic version is already exactly solved bycomputer algebra[7]. In this 2D example all the obstructions vanish and there isa unique quantum ∇ that is star-preserving, torsion free and metric compatible.Here the existing differential calculus, derived from the theory of quantum groups,gives ω, ∇ while ∇ g = ∇ has torsion but nocurvature and yet the 2-form R ≠
0, in contrast to the Schwarzschild black-holecase. On the other hand this 2D model still has a physical interpretation as a toymodel with strong gravitational source, so strong that even light can’t escape (sosomething like the inside of a black hole but with decaying rather than zero Riccitensor). We refer to [7] for details and for a different, cosmological, interpretationas well.7.1.
The 2D bicrossproduct model.
Setting x = t and x = r , we have ω =− ω = r as the semiclassical data behind the bicrossproduct model commutationrelations [ t, r ] ● = λr . It is known that this model has a standard 2D differentialcalculus with nonzero relations [ r, d t ] ● = λ d r, [ t, d t ] ● = λ d t, which has as its underlying semiclassical data a connection with Christoffel symbolsΓ = − r − and Γ = r − and all other Christoffel symbols zero. This has torsion T = − T = r − and T ij = T i p = T i ,p + Γ ipn T n − Γ np T in − Γ np T i n = δ i ( T ,p + Γ pn T n − Γ np T n − Γ np T n ) = δ i ( T ,p + Γ p T − Γ p T ) = δ i δ p T , = r − δ i δ p and that the curvature is zero, as it should since the standard calculus is associativeto all orders. To see this, without loss of generality, we look at j = k = R li = ∂ Γ l i ∂x − ∂ Γ l i ∂x + Γ m i Γ l m − Γ m i Γ l m = δ l ( ∂ Γ i ∂t − ∂ Γ i ∂r + Γ i Γ − Γ i Γ ) = δ l ( − ∂ Γ i ∂t − Γ i Γ ) = δ l δ i ( − ∂ Γ ∂r − Γ Γ ) = . Next we compute, H ij ∶= ω is ( T jnm ; s − R jnms ) d x m ∧ d x n = δ j ω is T nm ;1 δ s d x m ∧ d x n = δ j ω i T nm ;1 d x m ∧ d x n = δ j δ i ω T nm ;1 d x m ∧ d x n = δ j δ i ω ( T d x ∧ d x + T d x ∧ d x ) = δ j δ i ω ( T − T ) d x ∧ d x = δ j δ i ω T d r ∧ d t = δ j δ i ( − r ) r − d r ∧ d t = δ j δ i r − d t ∧ d r . The wedge product obeying the Leibniz rule in Theorem 4.6 is then; ξ ∧ η = ξ ∧ η + λ ω ij ∇ i ξ ∧ ∇ j η + ( − ) ∣ ξ ∣+ λ r − d t ∧ d r ∧ ( ∂ ⌟ ξ ) ∧ ( ∂ ⌟ η ) . (7.1)For ξ and η being either d r or d t , the only potentially deformed case isd t ∧ d t = λ ω ij ∇ i ( d t ) ∧ ∇ j ( d t ) + λ r − d t ∧ d r ∧ ( ∂ ⌟ d t ) ∧ ( ∂ ⌟ d t ) = λ ( ω ∇ ( d t ) ∧ ∇ ( d t ) + ω ∇ ( d t ) ∧ ∇ ( d t )) + λ r − d t ∧ d r = λ ω ( ∇ ( d t ) ∧ ∇ ( d t ) − ∇ ( d t ) ∧ ∇ ( d t )) + λ r − d t ∧ d r = . The exterior algebra among these basis elements is therefore undeformed, in agree-ment with the noncommutative algebraic picture where this is known (and holdsto all orders).Our goal is to study the semiclassical geometry of this model using our functorialmethods. First of all, the above connection is not compatible with the flat metric,but is compatible with the metric g = g ij d x i ⊗ d x j = b r d t ⊗ d t − b r t ( d t ⊗ d r + d r ⊗ d t ) + ( + b t ) d r ⊗ d r . where b is a non-zero real parameter. This is our semiclassical analogue of theobstruction discovered in [7]. For our purposes it is better to write the metric asthe following, where v = r d t − t d rg = d r ⊗ d r + b v ⊗ v . Note that ∇ applied to both d r gives zero. We quantise the classical bicrossproductspacetime with this metric. First q − ( g ) = d r ⊗ d r + b v ⊗ v . From the expression for H ij , we have R = g ij H ij = b r d t ∧ d r = b v ∧ d r = ± √∣ b ∣ Voland according to our general scheme, we take g = d r ⊗ d r + b v ⊗ v + bλ ( d r ⊗ v − v ⊗ d r ) . EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 49
To compare with [7], if we let(7.2) ν ∶= r ● λ d t − t ● λ d r = v + λ d r, ν ∗ ∶= ( d t ) ● λ r − ( d r ) ● λ t = v − λ d r and identify these with v, v ∗ in [7] (apologies for the clash of notation) then thequantum metric there gives the same answer as g above, i.e. this is the leading orderpart of the noncommutative geometry. From Theorem 3.5 we get ∇ Q vanishing onboth d r and v , and for all 1-forms ξ , σ Q ( d r ⊗ ξ ) = ξ ⊗ d r and σ Q ( v ⊗ ξ ) = ξ ⊗ v .Next we express the classical Levi-Civita connection for the above metric in theform ∇ S . We use (5.7) together with the only nonvanishing downstairs torsionsbeing T = − T = b r and T = − T = − b t and S abc = g a ( T bc − T bc − T cb ) + g a ( T bc − T bc − T cb ) , to give S a = − g a T , S a = − g a T , S a = g a T , S a = g a T . The upstairs metric is given by g = ( + b t )/( b r ) , g = g = t r − , g = .S a = b t g a , S a = b r g a , S a = − b t g a , S a = − b r g a , which we write compactly, along with its covariant derivative, as S aij = bǫ im x m ǫ jn g an , S aij ; k = bǫ ik ǫ jm g am where ǫ = g is preserved by ∇ (as expressed by ; i ). Wealso have ̂ ∇ R = R was a multiple of the volume form, and R = ∇ , so the obstruction in Theorem 5.7 for a torsion free metriccompatible quantum connection is ̂ ∇ R + ω ij g rs S sjn ( R rmki + S rkm ; i ) d x k ⊗ d x m ∧ d x n = ω ij g rs S sjn S rkm ; i d x k ⊗ d x m ∧ d x n = S and its covariant derivative. Hence The-orem 5.7 tells us that there is a unique such quantum connection of the form ∇ = ∇ QS + λK . Corollary 5.9 tells us that this is also the unique star-preservingconnection of this form. In short, all obstructions vanish and we have a uniquequantum Levi-Civita connection with all our desired properties.It only remains to compute ∇ . We take the liberty of changing the basis to writefor K real K ( v ) = K vvv v ⊗ v + K vrv d r ⊗ v + K vvr v ⊗ d r + K vrr d r ⊗ d r ,K ( d r ) = K rvv v ⊗ v + K rrv d r ⊗ v + K rvr v ⊗ d r + K rrr d r ⊗ d r . Proposition 7.1.
The unique star-preserving quantum connection of the form ∇ =∇ QS + λK is also torsion free and metric compatible (‘quantum Levi-Civita’) andgiven by non-zero components K rvr = K vvv = − b r − in our basis, leading to ∇ d r = bvr ⊗ v − bλr v ⊗ d r, ∇ v = − vr ⊗ d r − bλr v ⊗ v . Proof.
Note that v ∗ = v and d r ∗ = d r and also that Theorem 4.14 tells us thevalue of K which can be computed out as the value stated. But we still need tocompute ∇ and, moreover, since this is an illustrative example we will also verifyits properties directly as a nontrivial check of all our main theorems.First we compute S as an operator from the components stated above (or one canreadily compute the classical Levi-Civita connection and find S as the differencebetween this and ∇ ). Either way, S ( d r ) = b r − v ⊗ v, S ( d t ) = b t r − v ⊗ v − r − v ⊗ d r, S ( v ) = − r − v ⊗ d r Next we compute ∇ QS and its associated generalised braiding. In the following cal-culation, ∇ , ∇ denote the components ∇ i of the classical connection ∇ (apologiesfor the clash of notation). We have ∇ ( S ) = ∇ ( S )( v ) = ∇ ( S ( v )) = ∇ ( − r − v ⊗ d r ) = r − v ⊗ d r , ∇ ( S )( d r ) = ∇ ( S ( d r )) = ∇ ( b r − v ⊗ v ) = − b r − v ⊗ v . From Proposition 3.10, σ QS ( v ⊗ ξ ) = σ Q ( v ⊗ ξ ) + λ ω ξ ∇ ( S )( v ) = ξ ⊗ v − λ r ξ ∇ ( S )( v ) = ξ ⊗ v − λ ξ r − v ⊗ d r ,σ QS ( d r ⊗ ξ ) = ξ ⊗ d r − λ r ξ ∇ ( S )( d r ) = ξ ⊗ d r + λ ξ b r − v ⊗ v . and Q ( S )( v ) = S ( v ) + λ ω ij ∇ i ( ∇ j ( S )( v )) = S ( v ) + λ ω ∇ ( ∇ ( S )( v )) = S ( v ) = − r − v ⊗ d r ,Q ( S )( d r ) = S ( d r ) + λ ω ij ∇ i ( ∇ j ( S )( d r )) = S ( d r ) + λ ω ∇ ( ∇ ( S )( d r )) = S ( d r ) = b r − v ⊗ v . Then ∇ QS ( v ) = ∇ Q ( v ) + q − Q ( S )( v ) = − q − ( r − v ⊗ d r ) = − r − v ⊗ d r , ∇ QS ( d r ) = ∇ Q ( d r ) + q − Q ( S )( d r ) = b q − ( r − v ⊗ v ) = b r − v ⊗ v . We can add this to the K obtained from Theorem 4.14 to obtain the result statedfor the quantum Levi-Civita connection.For illustrative purposes let us also see directly why this adjustment is necessaryand that it succeeds. Firstly, to be star preserving we need ( id ⊗ ⋆ ) ∇ QS ( ξ ) = ( ⋆ − ⊗ id ) Υ σ − QS ∇ QS ( ξ ∗ ) for our two cases, ξ = v and ξ = d r . It is more convenientto rearrange this as ∇ QS ( ξ ∗ ) = σ QS Υ − ( ⋆ ⊗ ⋆ ) ∇ QS ( ξ ) . We do this for the two cases, where b is real σ QS Υ − ( ⋆ ⊗ ⋆ ) ∇ QS ( v ) = σ QS Υ − ( ⋆ ⊗ ⋆ )( − r − v ⊗ d r ) = − σ QS Υ − ( r − v ⊗ d r ) = − σ QS ( d r ⊗ r − v ) = − ( r − v ⊗ d r + λ b r − v ⊗ v ) ,σ QS Υ − ( ⋆ ⊗ ⋆ ) ∇ QS ( d r ) = σ QS Υ − ( ⋆ ⊗ ⋆ )( b r − v ⊗ v ) = b σ QS Υ − ( r − v ⊗ v ) EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 51 = b σ QS ( v ⊗ r − v ) = b ( r − v ⊗ v − λ r − v ⊗ d r ) . The difference in going clockwise minus anticlockwise round the diagram in Lemma 3.12is now σ QS Υ − ( ⋆ ⊗ ⋆ ) ∇ QS ( v ) − ∇ QS ( v ) = − ( λ b r − v ⊗ v ) ,σ QS Υ − ( ⋆ ⊗ ⋆ ) ∇ QS ( d r ) − ∇ QS ( d r ) = b ( − λ r − v ⊗ d r ) . Thus ∇ QS is not star preserving. However we follow Theorem 4.14 to see that ∇ QS + λK is star preserving if and only if the only nonzero K abc in this basis are K vvv = K rvr = − b r − . This completes the direct derivation of the values in this example.Although implied by our theory, let us also see how the quantum torsion and metriccompatibility conditions get to hold. We calculate the torsions from (7.1) by T ∇ QS ( v ) = − r − v ∧ d r − d v = − r − v ∧ d r − d v = ,T ∇ QS ( d r ) = b r − v ∧ v = b λ d t ∧ d r . The condition for ∇ QS + λK to be torsion free is that0 = λ ( ∧ K ( v )) , = λ ( ∧ K ( d r )) + b λ d t ∧ d r , which becomes K vrv = K vvr , K rvr − K rrv + b r − = ∇ QS ( v ) = ( id ⊗ ⋆ − ) Υ ∇ QS ( v ) = − ( id ⊗ ⋆ − ) Υ r − v ⊗ d r = − r ⊗ r − v , ¯ ∇ QS ( d r ) = ( id ⊗ ⋆ − ) Υ ∇ QS ( d r ) = b ( id ⊗ ⋆ − ) Υ r − v ⊗ v = b v ⊗ r − v . Now we apply ∇ QS to the Hermitian metric ( ⋆ ⊗ id ) g as follows: ( ¯ ∇ QS ⊗ id + id ⊗ ∇ QS )( ⋆ ⊗ id ) g = . Hence the condition for ∇ = ∇ QS + λK to preserve the Hermitian metric is0 = ( id ⊗ ⋆ − ) Υ λ K ( d r ) ⊗ d r + b ( id ⊗ ⋆ − ) Υ λ K ( v ) ⊗ v + d r ⊗ λ K ( d r ) + b v ⊗ λ K ( v ) . We split this into two parts, depending on whether we end in d r or v , to give, as λ is imaginary,0 = K rvr d r ⊗ v + K rrr d r ⊗ d r + b K vvr v ⊗ v + b K vrr v ⊗ d r − ( id ⊗ ⋆ − ) Υ K ( d r ) = ( b K vvr − K rvv ) v ⊗ v + ( b K vrr − K rrv ) v ⊗ d r , = K rvv d r ⊗ v + K rrv d r ⊗ d r + b K vvv v ⊗ v + b K vrv v ⊗ d r − b ( id ⊗ ⋆ − ) Υ K ( v ) = ( K rvv − b K vvr ) d r ⊗ v + ( K rrv − b K vrr ) d r ⊗ d r . Thus the conditions for ∇ to preserve the Hermitian quantum metric reduce to K rvv = b K vvr and K rrv = b K vrr , which again holds for our found values.Finally we consider straight quantum metric compatibility ∇ g . Again, this hasto follow from Hermitian-metric compatibiity since ∇ is star-preserving, but wecheck directly what is needed as per Theorem 5.7. First we apply ∇ QS to g , ∇ QS ( d r ⊗ d r ) = ∇ QS ( d r ) ⊗ d r + ( σ QS ⊗ id )( d r ⊗ ∇ QS ( d r )) = b r − v ⊗ v ⊗ d r + b σ QS ( d r ⊗ r − v ) ⊗ v = b r − v ⊗ ( v ⊗ d r + d r ⊗ v ) + b λ r − v ⊗ v ⊗ v , ∇ QS ( d r ⊗ v ) = ∇ QS ( d r ) ⊗ v + ( σ QS ⊗ id )( d r ⊗ ∇ QS ( v )) = b r − v ⊗ v ⊗ v − σ QS ( d r ⊗ r − v ) ⊗ d r = r − v ⊗ ( b v ⊗ v − d r ⊗ d r ) − λ b r − v ⊗ v ⊗ d r , ∇ QS ( v ⊗ d r ) = ∇ QS ( v ) ⊗ d r + ( σ QS ⊗ id )( v ⊗ ∇ QS ( d r )) = − r − v ⊗ d r ⊗ d r + b σ QS ( v ⊗ r − v ) ⊗ v = r − v ⊗ ( b v ⊗ v − d r ⊗ d r ) − b λ r − v ⊗ d r ⊗ v , ∇ QS ( v ⊗ v ) = ∇ QS ( v ) ⊗ v + ( σ QS ⊗ id )( v ⊗ ∇ QS ( v )) = − r − v ⊗ d r ⊗ v − σ QS ( v ⊗ r − v ) ⊗ d r = − r − v ⊗ ( d r ⊗ v + v ⊗ d r ) + λ r − v ⊗ d r ⊗ d r . Adding these with the appropriate weights gives ∇ QS ( g ) = b λ r − v ⊗ v ⊗ v + b λ r − v ⊗ d r ⊗ d r . Hence the condition for ∇ = ∇ QS + λK to preserve g is0 = b λ r − v ⊗ v ⊗ v + b λ r − v ⊗ d r ⊗ d r + λ ( τ ⊗ id )( d r ⊗ K ( d r )) + b λ ( τ ⊗ id )( v ⊗ K ( v )) + λ K ( d r ) ⊗ d r + b λ K ( v ) ⊗ v . Again splitting into the endings, the derivative of g is the following ⊗ d r b r − v ⊗ d r + K ( d r ) + ( τ ⊗ id )( d r ⊗ ( K rrr d r + K rvr v )) + b ( τ ⊗ id )( v ⊗ ( K vrr d r + K vvr v )) = b r − v ⊗ d r + K rvv v ⊗ v + K rrv d r ⊗ v + K rvr v ⊗ d r + K rrr d r ⊗ d r + ( τ ⊗ id )( d r ⊗ ( K rrr d r + K rvr v )) + b ( τ ⊗ id )( v ⊗ ( K vrr d r + K vvr v )) = ( K rvv + b K vvr ) v ⊗ v + ( K rrv + b K vrr ) d r ⊗ v + ( b r − + K rvr ) v ⊗ d r + K rrr d r ⊗ d r , plus the following ⊗ v b r − v ⊗ v + b K ( v ) + ( τ ⊗ id )( d r ⊗ ( K rrv d r + K rvv v )) + b ( τ ⊗ id )( v ⊗ ( K vrv d r + K vvv v )) = b r − v ⊗ v + b ( K vvv v ⊗ v + K vrv d r ⊗ v + K vvr v ⊗ d r + K vrr d r ⊗ d r ) + ( τ ⊗ id )( d r ⊗ ( K rrv d r + K rvv v )) + b ( τ ⊗ id )( v ⊗ ( K vrv d r + K vvv v )) = ( b r − + b K vvv ) v ⊗ v + b K vrv d r ⊗ v + ( b K vvr + K rvv ) v ⊗ d r + ( b K vrr + K rrv ) d r ⊗ d r . Thus the condition for the derivative of g to be zero is K rrr = K vrv = , K rvr = K vvv = − b r − , K rrv = − b K vrr , K rvv = − b K vvr . Combined with the condition for vanishing quantum torsion again gives us ourstated values. (cid:3)
One can also check that this quantum connection is indeed the part to order λ ofthe full connection found in [7] by algebraic methods, provided we make the identi-fication (7.2). In summary, all steps can be made to work in the 2D bicrossproductmodel quantum spacetime including a quantum metric g and quantisation of theLevi-Civita connection so as to be ∗ -preserving and at the same time compatiblewith g and torsion free. That this was possible was not in doubt but we see indetail how it arises at the semiclassical level. EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 53
Semiquantisation of the Schwarzschild black hole.
We take polar coor-dinates plus t for 4-dimensional space, where φ is the angle of rotation about the z -axis and θ is the angle to the z -axis. We take any static isotropic form of metric(including the Schwarzschild case) g = − e N ( r ) d t ⊗ d t + e P ( r ) d r ⊗ d r + r ( d θ ⊗ d θ + sin ( θ ) d φ ⊗ d φ ) (7.3)The Levi-Civita Christoffel symbols are zero except for ̂ Γ = ̂ Γ = N ′ , ̂ Γ = P ′ , ̂ Γ = N ′ e N − P ̂ Γ = − r e − P , ̂ Γ = − r e − P sin ( θ ) , ̂ Γ = ̂ Γ = ̂ Γ = ̂ Γ = r − ̂ Γ = − sin ( θ ) cos ( θ ) , ̂ Γ = ̂ Γ = cot ( θ ) . (7.4)We shall only consider rotationally invariant Poisson tensors ω . Consider a bivectorand rotation invariance in the spherical polar coordinate system. To generate theLie algebra of the rotation group, we only need two infinitesimal rotations, about the z axis and about the y axis. For the first, denoting change under the infinitesimalrotation by δ , we get δ ( θ ) = δ ( φ ) =
1, and δ ( d θ ) = δ A ( d φ ) =
0. The infinitesimalrotation about the y axis is rather more complicated in polar coordinates: δ ( θ ) = cos φ , δ ( φ ) = − cot θ sin φ , δ ( d θ ) = − sin φ d φ ,δ ( d φ ) = − cot θ cos φ d φ + csc θ sin φ d θ . It is now easily checked that a rotation invariant 2-form on the sphere is, up to amultiple, sin θ d θ ∧ d φ . It follows that a rotation invariant bivector on the sphereis, up to a multiple, given in polars by ω = csc θ . Proposition 7.2. If ω is rotationally invariant and independent of x , then only ω = − ω = k ( r ) and ω = − ω = f ( r )/ sin θ are non-zero. The condition to be apoisson tensor is that ω ω , = , i.e. k ( r ) f ′ ( r ) = .Proof. We now suppose that ω is rotationally invariant as a bivector field. Toanalyse this is it useful to use our Minkowski-polar coordinates to view E i = ω i as a spatial vector in polar coordinates and to view ω ij where i, j ≠ B . Now consider their values at the northpole of a sphere of radius r . Under rotation about the z -axis the north pole doesnot move so there is no orbital angular momentum. There is, however, rotationof the vector indices unless both E, B point along the z -axis. This applies equallyat any point of the sphere, i.e. E, B must point radially. Equation (3.3) gives thePoisson result. (cid:3)
We now write the Christoffel symbols Γ abc for the quantising connection ∇ in termsof its torsion T and use Mathematica to get the following result: Proposition 7.3.
Assume time independence and axial symmetry (i.e. that thetorsions T ijk are independent of the coordinates t and φ ). Then the general solutionfor the Poisson-compatibility and metric-compatiblity conditions for ( ∇ , ω ) is givenby ω = / sin θ (up to a constant multiple set to one), ω = , and the followingrestrictions on T ijk , apart from the obvious T ijk = − T ikj : T = T + T T = T + T T = T = T = T = − T T = r T = − T T = T = T = r sin ( θ ) T = T and T are non-zero, we cannot take for ∇ the Levi-Civita connection.We get the following value of H ij , independently of any choice in the torsions: H ij = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ − sin θ d θ ∧ d φ i = j = csc θ d θ ∧ d φ i = j =
30 otherwise . From this R = g ij H ij =
0, so the correction to the metric is zero, g = g Q .Moreover, we find in Theorem 5.7 that (with semicolons WRT the quantising con-nection) that antisymmetric part of ∇ g is proportional to ω ij g rs S sjn ( R rmki + S rkm ; i ) − ω ij g rs S sjm ( R rnki + S rkn ; i ) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ − r sin θ ( k, m, n ) = ( , , ) & ( k, m, n ) = ( , , ) r sin θ ( k, m, n ) = ( , , ) & ( k, m, n ) = ( , , ) ∇ . Thus there is an obstruction and no adjustment ∇ exactlypreserves the metric.We now specialise to the case where the T ijk are rotationally symmetric, which givesthe following as the only non-zero torsions, apart from the obvious T ijk = − T ikj : T = f ( r ) T = f ( r ) T = − T = − f ( r ) sin θT = r T = r sin ( θ ) T = − T = − f ( r ) sin θ where f ( r ) , f ( r ) , f ( r ) , f ( r ) are arbitrary functions of r only.Finallly, we specialise further to the Schwarzschild case, where e N = c ( − r s / r ) and e P = ( − r s / r ) − , where r s is the Schwarzschild radius. A short calculationwith Mathematica then gives Lemma 7.4.
For the Schwarzschild metric the non-zero R ijkl , up to the obvious R ijkl = − R ijlk are R = R = − f ′ ( r ) + c r s r − c ( − r s / r ) R = sin θ ( f ( r ) − r f ′ ( r )) r − R = − csc θ ( f ( r ) − r f ′ ( r )) r − R = − R = sin θ .In particular, the curvature cannot vanish entirely. We also have (using row i column j notation) S ij = ⎛⎜⎜⎜⎝ − e − N f ( r ) − e − N f ( r ) ⎞⎟⎟⎟⎠ , S ij = ⎛⎜⎜⎜⎝ − e − P f ( r ) − e − P f ( r ) e − P r
00 0 0 e − P r sin ( θ )⎞⎟⎟⎟⎠ ,S ij = ⎛⎜⎜⎜⎜⎝ − f ( r ) sin ( θ ) r − f ( r ) sin ( θ ) r − r ⎞⎟⎟⎟⎟⎠ , S ij = ⎛⎜⎜⎜⎜⎝ f ( r ) csc ( θ ) r
00 0 csc ( θ ) f ( r ) r
00 0 0 00 − r ⎞⎟⎟⎟⎟⎠ , EMIQUANTISATION FUNCTOR AND POISSON-RIEMANNIAN GEOMETRY, I 55 and the Christoffel symbols for the quantising connection are areΓ ij = ⎛⎜⎜⎜⎜⎝ N ′ ( r ) − e − N f ( r ) N ′ ( r ) − e − N f ( r ) ⎞⎟⎟⎟⎟⎠ , Γ ij = ⎛⎜⎜⎜⎜⎝ − f ( r ) sin ( θ ) r r − f ( r ) sin ( θ ) r − cos ( θ ) sin ( θ )⎞⎟⎟⎟⎟⎠ , Γ ij = ⎛⎜⎜⎜⎝ e − P ( e N N ′ ( r ) − f ( r )) − e − P f ( r ) − N ′ ( r ) ⎞⎟⎟⎟⎠ , Γ ij = ⎛⎜⎜⎜⎜⎝ f ( r ) csc ( θ ) r
00 0 csc ( θ ) f ( r ) r r ( θ ) ( θ ) ⎞⎟⎟⎟⎟⎠ . Clearly we can chose the functions here to minimise but not eliminate either thetorsion or the curvature.Now we consider a quantum connection of the form ∇ QS + λK and ask to whatextent it preserves the metric, given that we know that it cannot be fully metric-compatible with g . Proposition 7.5.
For the Schwarzschild metric, the unique K that gives a torsionfree quantum connection with the symmetric part of the metric-compatibility tensorvanishing (in Theorem 5.7) coincides with the unique real K that gives a star-preserving quantum conneciton (in Theorem 4.14). Its non-zero components are K = K = − e − P r sin θ / . Proof.
The unique star-preserving connection (which is also torsion free) is com-puted by Mathematica from the theorem as stated. For the torsion free connectionwith vanishing symmetric part of the metric compatibility tensor a Mathematicacomputation gives A pnm = B knm for thequantity in the proof of Theorem 5.7 are B = B = B = B = − r sin θ . Following the proof of Theorem 5.7, the conditions to preserve the symmetric partof the metric and be torsion free are, respectively K nkm + K mkn = B knm , K knm − K kmn = g kp A pnm . leading us in our case to K ijk = K = K = − r sin θ . (7.5)This is the same solution as stated after raising an index. (cid:3) This would therefore be a good candidate for ‘best possible’ quantum Levi-Civita’connection as it is suitably ‘real’ (unitary in a suitable context) as expressed bybeing ∗ -preserving and also comes as close as possible to metric-compatible giventhe order λ obstruction.Meanwhile, from Proposition 5.8 the condition for preserving the correspondingHermitian metric is K npm = K mpn . If we combine this with quantum torsionfreeness (as in the proof above), the condition for torsion free Hermitian-metriccompatibiity is that K ijk is totally symmetric in ijk . This means that there isa 4-functional moduli of Hermitian-metric compatible connections (but necessarilydisjoint from the point in Proposition 7.5, since that point is star-preserving but not fully metric compatible). Also, from (5.12), in the case where R =
0, the cotorsion(as in the approach used in [20]) is ( ∧ ⊗ id ) q ∇ ( g ) = − λ ω ij g rs S sjn ( R rmki + S rkm ; i ) d x k ∧ d x m ⊗ d x n + λ ( g pn K pkm + g mp K pkn ) d x k ∧ d x m ⊗ d x n , and from this the condition for vanishing cotorsion comes down to K nkm + K mkn − K nmk − K kmn =
0. Adding to this the condition for vanishing torsion, K nkm = K nmk ,we see that K mkn = K kmn , so the resulting K kmn are totally symmetric. Thus thespace of torsion free Hermitian-metric compatible ∇ in the case of the black-holeis the same as the space of torsion and cotorsion free ones.In summary we have shown that we inevitably have curvature of ∇ and hence anonassociative calculus at order λ if we try to quantise the black-hole and keep ro-tational invariance, an anomaly in line with experience in quantum group models[2].As with those models, the alternative is to quantise associatively but have an ex-tra cotangent dimension as in the wave-operator quantisation of the black holerecently achieved to all orders in [22]. And by perturbing ∇ QS we can either havea unique ‘unitary’ connection in the sense of star-preserving (and best possible butnot fully metric compatible) or a moduli of Hermitian-metric compatible (but notstar-preserving) connections perhaps fixed by further requirements. Which type of K to take would depend on the application or experimental assumption as to whichfeature we want to fix in the quantisation process. References [1] R. Aldrovandi and J.G. Pereira,
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Dept of Mathematics, Swansea University, Singleton Parc, Swansea SA2 8PP, +, Schoolof Mathematical Sciences, Queen Mary, University of London, Mile End Rd, LondonE1 4NS, UK
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