Exotic smooth , noncommutative algebras and quantization
The paper shows deep connections between exotic smoothings of small , noncommutative algebras of foliations and quantization. At first, based on the close relation of foliations and noncommutative -algebras we show that cyclic cohomology invariants characterize some small exotic . Certain exotic smooth ’s define a generalized embedding into a space which is -theoretic equivalent to a noncommutative Banach algebra.
Furthermore, we show that a factor von Neumann algebra is naturally related with nonstandard smoothing of a small and conjecture that this factor is the unique hyperfinite factor . We also show how an exotic smoothing of a small is related to the Drinfeld-Turaev (deformation) quantization of the Poisson algebra of complex functions on the space of flat connections over a surface , and that the result of this quantization is the skein algebra for the deformation parameter . This skein algebra is retrieved as a factor of horocycle flows which is Morita equivalent to the factor von Neumann algebra which in turn determines the unique factor as crossed product . Moreover, the structure of Casson handles determine the factor algebra too. Thus, the quantization of the Poisson algebra of closed circles in a leaf of the codimension 1 foliation of gives rise to the factor associated with exotic smoothness of .
Finally, the approach to quantization via exotic 4-smoothness is considered as a fundamental question in dimension 4 and compared with the topos approach to quantum theories.
- 1 Introduction
- 2 Exotic and codimension-1 foliation
- 3 Exotic and operator algebras
- 4 The connection between exotic smoothness and quantization
- 5 Conclusions
- A Non-cobordant foliations of detected by the Godbillon-Vey class
Even though non-standard smooth ’s exist as a 4-dimensional smooth manifolds, one still misses a direct coordinate-like presentation which would allow the usage of global, exotic smooth functions. Such functions are smooth in the exotic smoothness structure but fail to be differentiable in a standard way determined by the topological product of axes. On the way toward uncovering peculiarities of exotic ’s we obtained some unexpected connections of these with quantum theories and string theory formalism, see AsselmeyerKrol2009 (); AsselmeyerKrol2009a (). It seems however that any complete knowledge of the connection especially to quantum theories is still missing. The whole complex of problems is strongly connected to a future theory of quantum gravity where one has to include exotic smoothness structures in the formal functional integration over the space of metrics or connections. The presented paper aims to fill this gap and shed light on the above-mentioned connection with quantum theories. The relation with string theory will be addressed in our forthcoming paper. At first we will show how exotic smooth ’s are related with certain noncommutative Banach and -algebras. Then we will discuss the main result of the paper: the explanation how the factor von Neumann algebra corresponds to small exotic and the explicit quantization procedure of the Poisson algebra of loops on surfaces driven by a small exotic . Moreover, the geometric content of the quantization and the structure of this factor is carefully worked out. We observe and discuss the possible relation of this kind of quantization driven by 4-exotics, with the topos approach to (quantum) theories of physics.
We have observed in our previous paper AsselmeyerKrol2009 (), that the exoticness of some small exotic ’s is localized on some compact submanifolds and depend on the embedding of the submanifolds. Or, a small exotic structures of the is determined by the so-called Akbulut cork (a 4-dimensional compact contractible submanifold of with a boundary), and its embedding given by an attached Casson handle. The boundary of the cork is a homology 3-sphere containing a 3-sphere such that the codimension-1 foliations are determined by the foliations of . We have explained this important point in details in our previous papers AsselmeyerKrol2009 (); AsselmeyerKrol2009a (). Following this topological situation we want to localize exotica on a compact submanifold of such that we are able to relate the topological data, like foliations and Casson handles, with some other structures on the submanifold. The changes of the structures were grasped already by techniques from conformal field theory and WZW models and gerbes on groupoids and can be related with string theory and correspond to the changes of exotic smoothness on (see AsselmeyerKrol2009 (); AsselmeyerKrol2009a ()). We follow this philosophy further in this paper. Especially we will find:
a compact submanifold of such that its -theory is deformed towards the -theory of some noncommutative algebra when changing the standard smoothness on toward exotic one. This compact submanifold is again the part of the boundary of the Akbulut cork of exotic , i.e. . We describe the noncommutative Banach algebra whose -theory is the deformed -theory of the commutative algebra , when the smoothness is changed from standard to exotic. We say that such an exotic contains an embedded or that exotic deforms the -theory of the 3-sphere. This is the content of section 3.2 and theorem 3.2.
Next, given a codimension-1 foliation of where is an integrable subbundle , one can associate a - algebra to the leaf space of the foliation. Hurder and Katok HurKat:84 () showed that every algebra of a foliation with non-trivial Godbillon-Vey invariant contains a factor subalgebra. Based on the relation of codimension-1 foliations of and small exotic smooth structures on , as established in AsselmeyerKrol2009 (), one has a factor algebra corresponding to the exotic ’s. In subsection 3.3 we construct the - algebra of the foliation with a nontrivial Godbillon-Vey invariant and relate the exotic to the factor von Neumann algebra. Furthermore we conjecture that the factor is in fact the unique hyperfinite factor . In subsection 3.4 we relate the smoothing of with the cyclic cohomology invariants of the - algebra . This is done via the KK theory of Kasparov , allowing the definition of the K theory of the leaf space of the foliation, , and the analytic assembly map of Connes and Baum using finally the Connes pairing between K theory and cyclic cohomology.
In section 4 we turn to the quantization procedure as related to nonstandard smoothings of . Based on the dictionary between operator algebra and foliations one has the corresponding relation of small exotic ’s and operator algebras. This is a noncommutative algebra which can be seen as the algebra of quantum observables of some theory.
First, in subsection 3.3.2 we recognized the algebra as the hyperfinite factor von Neumann algebra. From Tomita-Takesaki theory it follows that any factor algebra decomposes as a crossed product into where is a factor . Via Connes procedure one can relate the factor foliation to a factor foliation. Then we obtain a foliation of the horocycle flow on the unit tangent bundle over some genus surface which determines the factor . This foliation is in fact determined by the horocycles which are closed circles.
Next we are looking for a classical algebraic structure which would give the above mentioned noncommutative algebra of observables as a result of quantization. The classical structure is recovered by the idempotent of the algebra and has the structure of a Poisson algebra. The idempotents were already constructed in subsection 3.3.1 as closed curves in the leaf of the foliation of . As noted by Turaev Turaev1991 (), closed curves in a surface induces a Poisson algebra: Given a surface let be the space of flat connections on bundles on ; this space carries a Poisson structure as is shown in subsection 4.2. The complex functions on can be considered as the algebra of classical observables forming the Poisson algebra .
Next in the subsection 4.3 we find a quantization procedure of the above Poisson algebra which is the Drinfeld-Turaev deformation quantization. It is shown that the result of this quantization is the skein algebra for the deformation parameter ( corresponds to the commutative Poisson structure on ).
This skein algebra is directly related to the factor von Neumann algebra derived from the foliation of . In fact the skein algebra is constructed in subsection 4.4 as the factor algebra Morita equivalent to the factor which in turn determines the factor of the foliation.
Finally, in subsection 4.6 we argue that the appearance of the factor and quantization in context of small exotic ’s can be lifted to a general and fundamental tool applicable for local relativistic algebraic quantum field theories in dimension 4. This is quite similar to the topos approach to (quantum) theories as has been recently actively developed (see e.g. Isham:08 (); Krol2006a (); Landsman:07 ()). We show that under some natural suppositions regarding the role of the factor and the quantization of the Poisson algebra of complex functions on , in field theories, our 4-exotics approach can be seen as a way how to reformulate quantum theories and make them classical. The price to pay is the change of smoothing of and this works in dimension 4. In the topos approach one is changing logic and set theory into intuitionistic ones valid in topoi and this is quite universal procedure. However, we think that the specific choice of dimension 4 by our 4-exotics formalism can be particularly well suited to the task of quantization of gravity in dimension 4 (cf. Krol:04a (); Krol:2005 (); AssRos:05 ()).
Then we have a closed circle: the exotic determines a codimension-1 foliation of a 3-sphere which produces a factor used in the algebraic quantum field theory (see Borchers2000 ()) as vacuum. In AsselmeyerKrol2009 () we constructed a relation between the leaf space of the foliation and the disks in the Casson handle. Thus the complicated structure of the leaf space is directly related to the complexity of the Casson handle making them to a noncommutative space.
2 Exotic and codimension-1 foliation
Here we present the main line of argumentation in our previous paper AsselmeyerKrol2009 ():
In Bizacas exotic one starts with the neighborhood of the Akbulut cork in the K3 surface . The exotic is the interior of .
This neighborhood decomposes into and a Casson handle representing the non-trivial involution of the cork.
From the Casson handle we construct a grope containing Alexanders horned sphere.
Akbuluts construction gives a non-trivial involution, i.e. the double of that construction is the identity map.
From the grope we get a polygon in the hyperbolic space .
This polygon defines a codimension-1 foliation of the 3-sphere inside of the exotic with an wildly embedded 2-sphere, Alexanders horned sphere. This foliation agrees with the corresponding foliation of the homology 3-sphere . This codimension-1 foliations of is partly classified by the Godbillon-Vey class lying in which is isomorphic to .
Finally we get a relation between codimension-1 foliations of the 3-sphere and exotic .
This relation is very strict, i.e. if we change the Casson handle
then we must change the polygon. But that changes the foliation and
vice verse. Finally we obtained the result:
The exotic (of Bizaca) is determined by the codimension-1 foliations with non-vanishing Godbillon-Vey class in of a 3-sphere seen as submanifold .
3 Exotic and operator algebras
In our previous paper AsselmeyerKrol2009 () we uncover a relation between an exotic (small) and non-cobordant codimension-1 foliations of the classified by Godbillon-Vey class as element of the cohomology group . By using the -gerbes it was possible to interpret the integer elements as characteristic class of a -gerbe over . We also discuss a possible deformation to include the full real case as well. Here we will use the idea to relate an operator algebra to the foliation. Then the invariants of the operator algebra will reflect the invariants of the foliation.
3.1 Twisted -theory and algebraic -theory
It is known that ordinary -theory of a (compact) manifold is the algebraic -theory of the (commutative) algebra of continuous, complex valued functions on , (Connes94 (), Chap. 2, Sec.1). Then twisted -theory of (w.r.t. the twisting ) can be similarly determined as an algebraic -theory of some generalized algebra. In fact this algebra must be a noncommutative (non-unital) Banach algebra which we are going to describe now AtiyahSegal2004 ().
To describe twisted -theory on a manifold one can follow the idea of ordinary non-twisted -theory to construct a (representation) space such that where is the space of homotopy classes of continuous maps. As shown by Atiyah Atiyah1967 (), the representation space for the -theory of is the topological space of Fredholm operators in an infinite dimensional Hilbert space , with the norm topology . Thus
This is based on the fact that a family of deformations of Fredholm operators parametrized by whose kernel and co-kernel have locally constant dimensions, are given by vector bundles on . The formal difference of these vector bundles determines the element of .
The twisting of this -theory can be performed by the elements of . One of the interpretations of these integral 3-rd cohomology classes is by the projective, infinite dimensional bundles on , namely given a class we can represent it by a projective bundle whose class is . Here is the group of unitary operators on . Let us see briefly how it is possible AtiyahSegal2004 (); AsselmeyerKrol2009a ().
The classifying space of the third cohomology group of is the Eilenberg-MacLane space . The projective unitary group on , , can now be determined. A model for is the classifying space of , i.e., . This means that and the realization of is as follows:
Isomorphism classes of principal bundles over correspond one to one to the classes from .
Now let us associate to a class (torsion or not) a bundle representing the class. Let acts on by conjugations . We can form an associated bundle
Let denote the space of all homotopy classes of sections of the bundle . Then one can define the twisted -theory:
The twisted by -theory of , i.e. is given by the homotopy classes of the sections of , i.e.
Now we are searching for bundles of algebras on whose sections would be an algebra such that its -theory is the . One natural candidate is the bundle of endomorphisms of the projective -bundle representing the class . However, this choice gives rise to the trivial -theory AtiyahSegal2004 (). Instead one should take the algebra of compact operators in with the norm topology. This is a noncommutative Banach algebra without unity. Therefore, we attach to the projective bundle , representing , the bundle of non-unital algebras. The fiber of the bundle at is the algebra of compact operators acting on . The algebra of sections of the bundle is an noncommutative Banach non-unital algebra . Now one obtains the result (AtiyahSegal2004 (), Definition 3.4 and Theorem 3.2 of Segal2001 ()):
Let be a compact manifold and , the group is canonically isomorphic to the algebraic -theory of the noncommutative non-unital Banach algebra .
3.2 Exotic and noncommutative Banach algebras
The interpretation of twisted -theory for a manifold as the algebraic -theory of some noncommutative Banach algebra can be seen as deformation of a space towards the noncommutative space using the twisting of -theory. We are interested in the following special smooth submanifolds of :
We say that a compact submanifold of a manifold is embedded in as a noncommutative subspace or embedded in a generalized smooth sense when the following two conditions hold
is embedded in in ordinary sense,
when a smooth structure on is changed, the -theory of is deformed toward the algebraic - theory of some noncommutative Banach algebra.
From AsselmeyerKrol2009a (), Theorem 2, we know that small exotic ’s (corresponding to the integral 3-rd cohomologies of ) deform -theory of toward twisted -theory. Twisted -theory of the compact manifold by , is in turn the -theory of the noncommutative Banach algebra as we explained in the previous subsection.
In our case of we have the twisted -theory with twist :
which can be interpreted as the algebraic -theory of the noncommutative algebra. Finally we can formulate:
Small exotic , corresponding to the integral 3-rd cohomologies of , deform the -theory of toward the -theory of some noncommutative algebra. Thus some small non-standard smooth structures of deform embedded towards a noncommutative space.
The deformation is performed by using the commutative algebra of complex-valued continuous functions on and changing it to the noncommutative non-unital Banach algebra of sections of the bundle on :
Here is the projective bundle on whose Dixmier-Douady class is and the deformed exotic smooth is determined by the same class (see Subsec. 3.1). We see that the exotic smoothness of is localized on making it a noncommutative space, while the standard smooth corresponds rather to the ordinary, non-twisted space and the commutative algebra .
The above case of small exotic and embedded in it, is the example of the generalized embedding in a sense of our definition 2 and this is the only known example to us of this phenomenon. In fact, any other , excludes any generalized smooth embedding. There is a possibility to be more explicit in the description of this generalized embeddings, namely by generalized Hitchin’s structures on and their relation to exotics AsselmeyerKrol2009 (). However we do not address this issue here.
3.3 Leaf space and factor -algebras
Given a foliation of a manifold , i.e. an integrable subbundle of the tangent bundle . The leaves of the foliation are the maximal connected submanifolds with . We denote with the set of leaves or the leaf space. Now one can associate to the leaf space a algebra by using the smooth holonomy groupoid of the foliation. For a codimension-1 foliation there is the Godbillon-Vey invariant GodVey:71 () as element of . Hurder and Katok HurKat:84 () showed that the algebra of a foliation with non-trivial Godbillon-Vey invariant contains a factor subalgebra. In the following we will construct this algebra and discuss the factor case.
3.3.1 The smooth holonomy groupoid and its algebra
Let be a foliated manifold. Now we shall construct a von Neumann algebra canonically associated to and depending only on the Lebesgue measure class on the space of leaves of the foliation. The classical point of view, , will only give the center of . According to Connes Connes94 (), we assign to each leaf the canonical Hilbert space of square-integrable half-densities . This assignment, i.e. a measurable map, is called a random operator forming a von Neumann . The explicit construction of this algebra can be found in Connes1984 (). Here we remark that is also a noncommutative Banach algebra which is used above. Alternatively we can construct as the compact endomorphisms of modules over the algebra of the foliation also known as holonomy algebra. From the point of view of K theory, both algebras and are Morita-equivalent to each other leading to the same groups. In the following we will construct the algebra by using the holonomy groupoid of the foliation.
Given a leaf of and two points of this leaf, any simple path from to on the leaf uniquely determines a germ of a diffeomorphism from a transverse neighborhood of to a transverse neighborhood of . The germ of diffeomorphism thus obtained only depends upon the homotopy class of in the fundamental groupoid of the leaf , and is called the holonomy of the path . The holonomy groupoid of a leaf is the quotient of its fundamental groupoid by the equivalence relation which identifies two paths and from to (both in ) iff . The holonomy covering of a leaf is the covering of associated to the normal subgroup of its fundamental group given by paths with trivial holonomy. The holonomy groupoid of the foliation is the union of the holonomy groupoids of its leaves.
Recall a groupoid is a category where every morphism is invertible. Let be a set of objects and the set of morphisms of , then the structure maps of reads as:
where is the composition of the composable two morphisms (target of the first is the source of the second), is the inversion of an arrow, the source and target maps respectively, assigns the identity to every object. We assume that are smooth manifolds and all structure maps are smooth too. We require that the maps are submersions, thus is a manifold as well. These groupoids are called smooth groupoids.
Given an element of , we denote by the origin of the path and its endpoint with the range and source maps . An element of is thus given by two points and of together with an equivalence class of smooth paths: the , with and , tangent to the bundle (i.e. with , ) identifying and as equivalent iff the holonomy of the path at the point is the identity. The graph has an obvious composition law. For , the composition makes sense if . The groupoid is by construction a (not necessarily Hausdorff) manifold of dimension . We state that is a smooth groupoid, the smooth holonomy groupoid.
Then the algebra of the foliation is the algebra of the smooth holonomy groupoid . For completeness we will present the explicit construction (see Connes94 () sec. II.8). The basic elements of ) are smooth half-densities with compact supports on , , where for is the one-dimensional complex vector space , where , and is the one-dimensional complex vector space of maps from the exterior power ,, to such that
For , the convolution product is given by the equality
Then we define via a operation making into a algebra. For each leaf of one has a natural representation of on the space of the holonomy covering of . Fixing a base point , one identifies with and defines the representation
The completion of with respect to the norm
makes it into a algebra . Among all elements of the algebra, there are distinguished elements, idempotent operators or projectors having a geometric interpretation in the foliation. For later use, we will construct them explicitly (we follow Connes94 () sec. closely). Let be a compact submanifold which is everywhere transverse to the foliation (thus ). A small tubular neighborhood of in defines an induced foliation of over with fibers . The corresponding algebra is isomorphic to with the algebra of compact operators. In particular it contains an idempotent , , where is a minimal projection in . The inclusion induces an idempotent in . Now we consider the range map of the smooth holonomy groupoid defining via a submanifold. Let be a section (with compact support) of the bundle of half-density over so that the support of is in the diagonal in and
Then the equality
defines an idempotent . Thus, such an idempotent is given by a closed curve in transversal to the foliation.
3.3.2 Some information about the factor case
In our case of codimension-1 foliations of the 3-sphere with nontrivial Godbillon-Vey invariant we have the result of Hurder and Katok HurKat:84 (). Then the corresponding von Neumann algebra contains a factor algebra. At first we will give an overview about the factor .
Remember a von Neumann algebra is an involutive subalgebra of the algebra of operators on a Hilbert space that has the property of being the commutant of its commutant: . This property is equivalent to saying that is an involutive algebra of operators that is closed under weak limits. A von Neumann algebra is said to be hyperfinite if it is generated by an increasing sequence of finite-dimensional subalgebras. Furthermore we call a factor if its center is equal to . It is a deep result of Murray and von Neumann that every factor can be decomposed into 3 types of factors . The factor case divides into the two classes and with the hyperfinite factors the complex square matrices and the algebra of all operators on an infinite-dimensional Hilbert space . The hyperfinite factors are given by , the Clifford algebra of an infinite-dimensional Euclidean space , and . The case remained mysterious for a long time. Now we know that there are three cases parametrized by a real number : the Krieger factor induced by an ergodic flow , the Powers factor for and the Araki-Woods factor for all with . We remark that all factor cases are induced by infinite tensor products of the other factors. One example of such an infinite tensor space is the Fock space in quantum field theory.
But now we are interested in an explicit construction of a factor von Neumann algebra of a foliation. The interesting example of this situation is given by the Anosov foliation of the unit sphere bundle of a compact Riemann surface of genus endowed with its Riemannian metric of constant curvature . In general the manifold is the quotient of the semi-simple Lie group , the isometry group of the hyperbolic plane , by the discrete cocompact subgroup , and the foliation of is given by the orbits of the action by left multiplication on of the subgroup of upper triangular matrices of the form
The von Neumann algebra of this foliation is the (unique) hyperfinite factor of type . In the appendix A we describe the construction of the codimension-1 foliation on the 3-sphere . The main ingredient of this construction is the convex polygon in the hyperbolic plane having curvature . The whole construction don’t depend on the number of vertices of but on the volume only. Thus without loss of generality, we can choose the even number for of vertices for . As model of the hyperbolic plane we choose the usual upper half-plane model where the group (the real Möbius transformations) and the hyperbolic group (the group of all orientation-preserving isometries of ) act via fractional linear transformations. Then the polygon is a fundamental polygon representing a Riemann surface of genus . Via the procedure above, we can construct a foliation on with . This foliation is also induced from the foliation of (as well as the foliation of the ) via the left action above. The difference between the foliation on and on is given by the different usage of the polygon . Thus the von Neumann algebra of the codimension-1 foliation of the 3-sphere contains a factor algebra in agreement with the results in HurKat:84 (). Currently we don’t know whether it is the hyperfinite factor . The Reeb components of this foliation of (see appendix A) are represented by a factor algebra and thus don’t contribute to the Godbillon-Vey class. Putting all things together we will get
A small exotic as represented by a codimension-1 foliation of the 3-sphere with non-trivial Godbillon-Vey invariant is also associated to a von Neumann algebra induced by the foliation which contains a factor algebra.
We conjecture that this factor algebra is the hyperfinite factor . Now one may ask, what is the physical meaning of the factor ? Because of the Tomita-Takesaki-theory, factor algebras are deeply connected to the characterization of equilibrium temperature states of quantum states in statistical mechanics and field theory also known as Kubo-Martin-Schwinger (KMS) condition. Furthermore in the quantum field theory with local observables (see Borchers Borchers2000 () for an overview) one obtains close connections to Tomita-Takesaki-theory. For instance one was able to show that on the vacuum Hilbert space with one vacuum vector the algebra of local observables is a factor algebra. As shown by Thiemann et. al. Thiemann2006 () on a class of diffeomorphism invariant theories there exists an unique vacuum vector. Thus the observables algebra must be of this type.
3.4 Cyclic cohomology and K-theoretic invariants of leaf spaces
As Connes Connes1984 () showed, interesting K-theoretic invariants of foliations can be described by Kasparov’s KK theory. For completeness we will give a short introduction. Then we will apply the theory to our case.
Lets begin with KK theory. Let and be graded -algebras. is the set of all triples , where is a countably generated graded Hilbert module over , is a graded -homomorphism from to the bounded operators over , and is an operator in of degree , such that , , and are all compact operators (i.e. in ) for all (i.e. is a Fredholm operator). The elements of are called Kasparov modules for . is the set of triples in for which, , and are for all . The elements of are called degenerate Kasparov modules. Then one defines a homotopy between two triples for by a triple in relating the two triples in an obvious way. The KK groups are given by
the equivalence classes. Especially we get for the usual K homology and for the K theory . A delicate part of KK theory is the existence of a intersection (or cup) product so that elements in and combine to an element in . See chapter VIII in the nice book Bla:86 ().
Let be the module on of a foliation for a manifold (associated to the von Neumann algebra ). Let be a transversal of the foliation constructed above (i.e. a submanifold with transversal to the foliation). It defines a tranverse bundle which is associated to a bundle over the classifying space of the holonomy groupoid of the foliation . Using KK theory we can define the K theory of the leaf space via the algebra . By geometric methods Baum and Connes were able to construct a purely geometric group together with an isomorphism (analytic assembly map, see Connes1984 ()). On the foliation determines a Haefliger structure via the holonomy, i.e. we have a continuous map (unique up to homotopy) mapping elements of to elements in . The last K group is related to the usual cohomology via Chern-Weil theory.
Now we specialize to the codimension-1 foliation over . Then we have especially the relation between and and by the analytic assembly map above, we have
where the last equality is the Bott isomorphism. Via Connes pairing between K theory and cyclic cohomology we have a map
to the cyclic cohomology of the foliation. Thus we have shown
Let a small exotic be determined by a codimension-1 foliation on the 3-sphere . Every such foliation defines an element in the K theory determining an element in the cyclic cohomology of the foliation (the transverse fundamental class).
It is interesting to note that we already had found an interpretation of an element in the cyclic cohomology. In AssRos:05 () we studied the change of the connection by changing the smoothness structure on a compact manifold to get a singular connection representing the change. This singular connection had a close connection to a cyclic cohomology class used to proof that the set of singular forms builds an algebra, the Temperley-Lieb algebra or the factor algebra. In subsection 4.4 we will show that there is a subfactor in the von Neumann algebra of the foliation which contains a factor algebra.
4 The connection between exotic smoothness and quantization
In this section we describe a deep relation between quantization and the codimension-1 foliation of the determining the smoothness structure on a small exotic .
4.1 From exotic smoothness to operator algebras
In subsection 3.3.1 we constructed (following Connes Connes94 ()) the smooth holonomy groupoid of a foliation and its operator algebra . The correspondence between a foliation and the operator algebra (as well as the von Neumann algebra) is visualized by table 1.
|closed curve transversal to foliation||projector (idempotent operator)|
|holonomy||linear functional (state)|
|local chart||center of algebra|
As extract of our previous paper AsselmeyerKrol2009 (), we obtained a relation between exotic ’s and codimension-1 foliations of the 3-sphere . Furthermore we showed in subsection 3.3.2 that the codimension-1 foliation of consists partly of Anosov-like foliations and Reeb foliations. Using Tomita-Takesaki-theory, one has a continuous decomposition (as crossed product) of any factor algebra into a factor algebra together with a one-parameter group111The group is the group of positive real numbers with multiplication as group operation also known as Pontrjagin dual. of automorphisms of , i.e. one obtains
But that means, there is a foliation induced from the foliation of the producing this factor. As we saw in subsection 3.3.2 one has a codimension-1 foliation as part of the foliation of the whose von Neumann algebra is the hyperfinite factor . Connes Connes94 () (in section I.4 page 57ff) constructed the foliation canonically associated to having the factor as von Neumann algebra. In our case it is the horocycle flow: Let the polygon on the hyperbolic space determining the foliation of the (see appendix A). is equipped with the hyperbolic metric together with the collection of unit tangent vectors to . A horocycle in is a circle contained in which touches at one point. Then the horocycle flow is the flow moving an unit tangent vector along a horocycle (in positive direction at unit speed). As above the polygon determines a surface of genus with abelian torsion-less fundamental group so that the homomorphism determines an unique (ergodic invariant) Radon measure. Finally the horocycle flow determines a factor foliation associated to the factor foliation. We remark for later usage that this foliation is determined by a set of closed curves (the horocycles).
Using results of our previous papers, we have the following picture:
Every small exotic is determined by a codimension-1 foliation (unique up to cobordisms) of some homology 3-sphere (as boundary of a contractable submanifold , the Akbulut cork).
This codimension-1 foliation on determines via surgery along a link uniquely a codimension-1 foliation on the 3-sphere and vice verse.
This codimension-1 foliation on has a leaf space which is determined by the von Neumann algebra associated to the foliation.
The von Neumann algebra contains a hyperfinite factor algebra as well as a factor algebra coming from the Reeb foliations.
Thus by this procedure we get a noncommutative algebra from an exotic . The relation to the quantum theory will be discussed now. We remark that we have already a quantum theory represented by the von Neumann algebra . Thus we are in the strange situation to construct a (classical) Poisson algebra together with a quantization to get an algebra which we already have.
4.2 The observable algebra and Poisson structure
In this section we will describe the formal structure of a classical theory coming from the algebra of observables using the concept of a Poisson algebra. In quantum theory, an observable is represented by a hermitean operator having the spectral decomposition via projectors or idempotent operators. The coefficient of the projector is the eigenvalue of the observable or one possible result of a measurement. At least one of these projectors represent (via the GNS representation) a quasi-classical state. Thus to construct the substitute of a classical observable algebra with Poisson algebra structure we have to concentrate on the idempotents in the algebra.
In subsection 3.3.1, an idempotent was constructed in the algebra of the foliation and geometrically interpreted as closed curve transversal to the foliation. Such a curve meets every leaf in a finite number of points. Furthermore the 3-sphere is embedded in the some 4-space with the tubular neighborhood . Then we have a thickened curve or a closed curve on a surface. Thus we have to consider closed curves in surfaces. Now we will see that the set of closed curves on a surface has the structure of a Poisson algebra.
Let us start with the definition of a Poisson algebra. Let be a commutative algebra with unit over or . A Poisson bracket on is a bilinearform fulfilling the following 3 conditions:
Then a Poisson algebra is the algebra . Now we consider a surface together with a closed curve . Additionally we have a Lie group given by the isometry group. The closed curve is one element of the fundamental group . From the theory of surfaces we know that is a free abelian group. Denote by the free -module ( a ring with unit) with the basis , i.e. is a freely generated -modul. Recall Goldman’s definition of the Lie bracket in (see Goldman1984 ()). For a loop we denote its class in by . Let be two loops on lying in general position. Denote the (finite) set by . For denote by the intersection index of and in . Denote by the product of the loops based in . Up to homotopy the loop is obtained from by the orientation preserving smoothing of the crossing in the point . Set
According to Goldman Goldman1984 (), Theorem 5.2, the bilinear pairing given by (4) on the generators is well defined and makes to a Lie algebra. The algebra of symmetric tensors is then a Poisson algebra (see Turaev Turaev1991 ()).
The whole approach seems natural for the construction of the Lie algebra but the introduction of the Poisson structure is an artificial act. From the physical point of view, the Poisson structure is not the essential part of classical mechanics. More important is the algebra of observables, i.e. functions over the configuration space forming the Poisson algebra. Thus we will look for the algebra of observables in our case. For that purpose, we will look at geometries over the surface. By the uniformization theorem of surfaces, there is three types of geometrical models: spherical , Euclidean and hyperbolic . Let be one of these models having the isometry group . Consider a subgroup of the isometry group acting freely on the model forming the factor space . Then one obtains the usual (closed) surfaces , , and its connected sums like the surface of genus (). For the following construction we need a group containing the isometry groups of the three models. Furthermore the surface is part of a 3-manifold and for later use we have to demand that has to be also a isometry group of 3-manifolds. According to Thurston Thu:97 () there are 8 geometric models in dimension 3 and the largest isometry group is the hyperbolic group isomorphic to the Lorentz group It is known that every representation of can be lifted to the spin group . Thus the group fulfilling all conditions is identified with . This choice fits very well with the 4-dimensional picture.
Now we introduce a principal bundle on , representing a geometry on the surface. This bundle is induced from a bundle over having always a flat connection. Alternatively one can consider a homomorphism represented as holonomy functional
with the path ordering operator and as flat connection (i.e. inducing a flat curvature ). This functional is unique up to conjugation induced by a gauge transformation of the connection. Thus we have to consider the conjugation classes of maps
forming the space of gauge-invariant flat connections of principal bundles over . Now (see Skovborg2006 ()) we can start with the construction of the Poisson structure on The construction based on the Cartan form as the unique bilinearform of a Lie algebra. As discussed above we will use the Lie group but the whole procedure works for every other group too. Now we consider the standard basis
of the Lie algebra with . Furthermore there is the bilinearform written in the standard basis as
Now we consider the holomorphic function and define the gradient along at the point as with and
The calculation of the gradient for the trace along a matrix
is given by
Given a representation of the fundamental group and an invariant function extendable to . Then we consider two conjugacy classes represented by two transversal intersecting loops and define the function by . Let be the intersection point of the loops and a path between the point and the fixed base point in . The we define and . Finally we get the Poisson bracket
where is the sign of the intersection point . Thus the space has a natural Poisson structure (induced by the bilinear form on the group) and the Poisson algebra of complex functions over them is the algebra of observables.
4.3 Drinfeld-Turaev Quantization
Now we introduce the ring of formal polynomials in with values in . This ring has a topological structure, i.e. for a given power series the set forms a neighborhood. Now we define a Quantization of a Poisson algebra as a algebra together with the -algebra isomorphism so that
the modul is isomorphic to for a vector space
let and be , then
One speaks of a deformation of the Poisson algebra by using a deformation parameter to get a relation between the Poisson bracket and the commutator.
Now we have the problem to find the deformation of the Poisson algebra