**2. Foliations, WZW** *σ***-models and exotic R**<sup>4</sup>

An exotic **<sup>R</sup>**<sup>4</sup> is a topological space with **<sup>R</sup>**4−topology but with a smooth structure different (i.e. non-diffeomorphic) from the standard **R**<sup>4</sup> *std* obtaining its differential structure from the product **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>**. The exotic **<sup>R</sup>**<sup>4</sup> is the only Euclidean space **<sup>R</sup>***<sup>n</sup>* with an exotic smoothness structure. The exotic **R**<sup>4</sup> can be constructed in two ways: by the failure to split arbitrarily a smooth 4-manifold into pieces (large exotic **R**4) and by the failure of the so-called smooth h-cobordism theorem (small exotic **R**4). Here, we deal with the later kind of exotics. We refer the reader to Asselmeyer-Maluga & Brans (2007) for general presentation of various topological and geometrical constructions and their physical perspective. Another useful mathematical books are Gompf & Stipsicz (1999); Scorpan (2005). The reader can find further results in original scientific papers.

Even though there are known, and by now rather widely discussed (see the Introduction), difficulties with making use of different differential structures on **R**<sup>4</sup> (and on other open 4-manifolds) in explicit coordinate-like way (see e.g. Asselmeyer-Maluga & Brans (2007)), it was, however, established, in a series of recent papers, the way how to relate these 4-exotics with some structures on *S*<sup>3</sup> (see e.g. Asselmeyer-Maluga & Król (2009a;b; 2011b)). This *S*<sup>3</sup> is supposed to fulfil specific topological conditions: it has to lie in ambient **R**<sup>4</sup> such that it is a part of the boundary of some compact 4-submanifold with boundary, i.e. Akbulut cork. If so, one can prove that exotic smoothness of the **R**<sup>4</sup> is tightly related to codimension-one foliations of this *S*3, hence, with the 3-rd real cohomology classes of *S*3. Reformulating Theorem 1 we have [Asselmeyer-Maluga & Król (2009a)]:

*The exotic* **R**4*'s, from the radial family of exotic* **R**4*'s embedded in standard* **R**4*, are determined by the codimension-1 foliations,* <sup>F</sup>*'s, with non-vanishing Godbillon-Vey (GV) class in H*3(*S*3, **<sup>R</sup>**) *of a 3-sphere lying at the boundary of the Akbulut corks of* **R**4*'s. The radius in the family, ρ, and value of GV are related by* GV = *ρ*2*. We maintain: the exoticness is localized at a 3-sphere inside the small exotic* **R**<sup>4</sup> *(seen as a submanifold of* **R**4*).*

Let us explain briefly, following Asselmeyer-Maluga & Król (2011d), how the codimension-1 foliations of *S*<sup>3</sup> emerges from the structure of exotic **R**4. The complete construction and proof can be found in Asselmeyer-Maluga & Król (2009a).

4 Will-be-set-by-IN-TECH

effective matter states (without gravity) are described by dual theories with gravity. Hence

In the next section we describe the relation of small exotic **R**<sup>4</sup> with foliations of *S*<sup>3</sup> and WZW models on *SU*(2). Then we show the connections between string theory and exotic **R**4. In particular 4-smoothness underlying spacetime emerges from superstring calculations and it modifies the spectra of charged particles in such spacetime. In Sec. 4 we discuss the Kondo state and show that it generates the same exotic 4-smoothness. Moreover, the Kondo state, when survive the high energy and relativistic limit, would couple to the gravity backgrounds of superstring theory. The backgrounds are precisely those related with exotic smooth **R**<sup>4</sup> as in Sec. 3. We conjecture that one could encounter the experimental trace of existence of exotic

*<sup>p</sup>* in the *k*-channel, *k* > 2, Kondo effect, where the usual fusion rules of the *SU*(2)*<sup>k</sup>* WZW

Next in Sec. 5 we present the connections of branes configurations in superstring theory with

An exotic **<sup>R</sup>**<sup>4</sup> is a topological space with **<sup>R</sup>**4−topology but with a smooth structure different

product **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>**. The exotic **<sup>R</sup>**<sup>4</sup> is the only Euclidean space **<sup>R</sup>***<sup>n</sup>* with an exotic smoothness structure. The exotic **R**<sup>4</sup> can be constructed in two ways: by the failure to split arbitrarily a smooth 4-manifold into pieces (large exotic **R**4) and by the failure of the so-called smooth h-cobordism theorem (small exotic **R**4). Here, we deal with the later kind of exotics. We refer the reader to Asselmeyer-Maluga & Brans (2007) for general presentation of various topological and geometrical constructions and their physical perspective. Another useful mathematical books are Gompf & Stipsicz (1999); Scorpan (2005). The reader can find further

Even though there are known, and by now rather widely discussed (see the Introduction), difficulties with making use of different differential structures on **R**<sup>4</sup> (and on other open 4-manifolds) in explicit coordinate-like way (see e.g. Asselmeyer-Maluga & Brans (2007)), it was, however, established, in a series of recent papers, the way how to relate these 4-exotics with some structures on *S*<sup>3</sup> (see e.g. Asselmeyer-Maluga & Król (2009a;b; 2011b)). This *S*<sup>3</sup> is supposed to fulfil specific topological conditions: it has to lie in ambient **R**<sup>4</sup> such that it is a part of the boundary of some compact 4-submanifold with boundary, i.e. Akbulut cork. If so, one can prove that exotic smoothness of the **R**<sup>4</sup> is tightly related to codimension-one foliations of this *S*3, hence, with the 3-rd real cohomology classes of *S*3. Reformulating Theorem 1 we

*The exotic* **R**4*'s, from the radial family of exotic* **R**4*'s embedded in standard* **R**4*, are determined by the codimension-1 foliations,* <sup>F</sup>*'s, with non-vanishing Godbillon-Vey (GV) class in H*3(*S*3, **<sup>R</sup>**) *of a 3-sphere lying at the boundary of the Akbulut corks of* **R**4*'s. The radius in the family, ρ, and value of GV are related by* GV = *ρ*2*. We maintain: the exoticness is localized at a 3-sphere inside the small*

Let us explain briefly, following Asselmeyer-Maluga & Król (2011d), how the codimension-1 foliations of *S*<sup>3</sup> emerges from the structure of exotic **R**4. The complete construction and proof

*std* obtaining its differential structure from the

non-standard 4-smoothness of **R**4. Discussion and conclusions close the Chapter.

gravity is inherently present in description of such condensed matter states.

model would be modified to these of *SU*(2)*<sup>p</sup>* WZW in high energies.

**2. Foliations, WZW** *σ***-models and exotic R**<sup>4</sup>

(i.e. non-diffeomorphic) from the standard **R**<sup>4</sup>

results in original scientific papers.

have [Asselmeyer-Maluga & Król (2009a)]:

*exotic* **R**<sup>4</sup> *(seen as a submanifold of* **R**4*).*

can be found in Asselmeyer-Maluga & Król (2009a).

**R**<sup>4</sup>

Small exotic **R**<sup>4</sup> is determined by the compact 4-manifold *A* with boundary *∂A* which is homology 3-sphere, and attached several Casson handles CH's. *A* is the Akbulut cork and CH is built from many stages towers of immersed 2-disks. These 2-disks cannot be embedded and the intersection points can be placed in general position in 4D in separated double points. Every CH has infinite many stages of intersecting disks. However, as Freedman proved, CH is topologically the same as (homeomorphic to) open 2-handle, i.e. *<sup>D</sup>*<sup>2</sup> <sup>×</sup> **<sup>R</sup>**2. Now if one replaces CH's, from the above description of small exotic **R**4, by ordinary open 2-handles (with suitable linking numbers in the attaching regions) the resulting object is standard **R**4. The reason is the existence of infinite (continuum) many diffeomorphism classes of CH, even though all are topologically the same.

Consider the following situation: one has two topologically equivalent (i.e. homeomorphic), simple-connected, smooth 4-manifolds *M*, *M*� , which are not diffeomorphic. There are two ways to compare them. First, one calculates differential-topological invariants like Donaldson polynomials Donaldson & Kronheimer (1990) or Seiberg-Witten invariants Akbulut (1996). But there is yet another possibility – one can change a manifold *M* to *M*� by using a series of operations called surgeries. This procedure can be visualized by a 5-manifold *W*, the cobordism. The cobordism *W* is a 5-manifold having the boundary *∂W* = *M* � *M*� . If the embedding of both manifolds *M*, *M*� into *W* induces a homotopy-equivalence then *W* is called an h-cobordism. Moreover, we assume that both manifolds *M*, *M*� are compact, closed (without boundary) and simply-connected. Freedman (1982) showed that every *h*-cobordism implies a homeomorphism, hence *h*-cobordisms and homeomorphisms are equivalent in that case. Furthermore, the following structure theorem for such h-cobordisms holds true [Curtis & Stong (1997)]:

*Let W be a h-cobordism between M*, *M*� *. Then there are contractable submanifolds A* ⊂ *M*, *A*� ⊂ *M*� *together with a sub-cobordism V* ⊂ *W with ∂V* = *A* � *A*� *(the disjoint oriented sum), so that the h-cobordism W* \ *V induces a diffeomorphism between M* \ *A and M*� \ *A*� *.*

Thus, the smoothness of *M* is completely determined (see also Akbulut & Yasui (2008; 2009)) by the contractible submanifold *A* (Akbulut cork) and its embedding *A* → *M* determined by a map *τ* : *∂A* → *∂A* with *τ* ◦ *τ* = *id∂<sup>A</sup>* and *τ* �= ±*id∂A*(*τ* is an involution). Again, according to Freedman (1982), the boundary of every contractible 4-manifold is a homology 3-sphere. This *h*-cobordism theorem is employed to construct an exotic **R**4. First, one considers a neighborhood (tubular) of the sub-cobordism *V* between *A* and *A*� . The interior of *V*, *int*(*V*), (as open manifold) is homeomorphic to **R**4. However, if (and only if) *M* and *M*� are not diffeomorphic (still being homeomorphic), then *int*(*V*) <sup>∩</sup> *<sup>M</sup>* is an exotic **<sup>R</sup>**4.

Next, Bizaca (1994) and Bi ˘ zaca & Gompf (1996) showed how to construct an explicit handle ˘ decomposition of the exotic **R**<sup>4</sup> by using *int*(*V*). The details of the construction can be found in their papers or in the book Gompf & Stipsicz (1999). The idea is simply to use the cork *A* and add some Casson handle to it. The interior of this resulting structure is an exotic **R**4. The key feature here is the appearance of *CH*. Briefly, a Casson handle *CH* is the result of attempts to embed a disk *D*<sup>2</sup> into a 4-manifold. In most cases this attempt fails and Casson (1986) searched for a possible substitute, which is just what we now call a Casson handle. Freedman (1982) showed that every Casson handle *CH* is homeomorphic to the open 2-handle *<sup>D</sup>*<sup>2</sup> <sup>×</sup> **<sup>R</sup>**<sup>2</sup> but in nearly all cases it is not diffeomorphic to the standard handle, Gompf (1984; 1989). The Casson handle is built by iteration, starting from an immersed disk in some 4-manifold *M*, i.e. a map *<sup>D</sup>*<sup>2</sup> <sup>→</sup> *<sup>M</sup>* which has injective differential. Every immersion *<sup>D</sup>*<sup>2</sup> <sup>→</sup> *<sup>M</sup>* is an embedding except on a countable set of points, the double points. One can "kill" one double point by immersing

det *<sup>g</sup>* <sup>=</sup> <sup>−</sup>1. The left invariant 1-form *<sup>g</sup>*−1*dg* generates locally the cotangent space connected to the unit. The forms *<sup>ω</sup><sup>k</sup>* <sup>=</sup> *Tr*((*g*−1*dg*)*k*) are complex *<sup>k</sup>*−forms generating the deRham cohomology of the Lie group. The cohomology classes of the forms *ω*1, *ω*<sup>2</sup> vanish and *<sup>ω</sup>*<sup>3</sup> <sup>∈</sup> *<sup>H</sup>*3(*SU*(2), **<sup>R</sup>**) generates the cohomology group. Then, we obtain as the value for the

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 59

*ω*<sup>3</sup> = 1 .

*Tr*((*G*−1*dG*)3) <sup>∈</sup> **<sup>Z</sup>**

*<sup>k</sup> 's, k* ∈ **Z** *from the radial family determines the WZ term of the k WZW*

*<sup>k</sup>* , then the WZ term

*<sup>k</sup>* with

*<sup>l</sup> , k*, *l* ∈ **Z** *both from the radial family, corresponds*

*<sup>k</sup>* i.e. *<sup>S</sup>*<sup>3</sup> <sup>×</sup> **<sup>R</sup>**. This end cannot be standard smooth

*<sup>k</sup>* . The appearance of the *SU*(2)*<sup>k</sup>* × **R** is

*S*<sup>3</sup>=*SU*(2)

<sup>Ω</sup><sup>3</sup> <sup>=</sup> <sup>1</sup> 8*π*<sup>2</sup> 

This integral can be interpreted as winding number of *g*. Now, we consider a smooth map

*S*3

is the winding number of *G*. Every Godbillon-Vey class with integer value like (1) is generated by a 3-form Ω3. Therefore, the Godbillon-Vey class is the WZ term of the *SU*(2)*k*. Thus, we

This WZ term enables one for the cancellation of the quantum anomaly due to the conformal invariance of the classical *σ*-model on *SU*(2). Thus, we have a method of including this cancellation term from smooth 4-geometry: when a smoothness of the ambient 4-space, in

of the classical *σ*-model with target *S*<sup>3</sup> = *SU*(2), i.e. *SU*(2)*<sup>k</sup>* WZW, is precisely generated by

1 8*π*<sup>2</sup>

*<sup>G</sup>* : *<sup>S</sup>*<sup>3</sup> <sup>→</sup> *SU*(2) with 3-form <sup>Ω</sup><sup>3</sup> <sup>=</sup> *Tr*((*G*−1*dG*)3) so that the integral

which *S*<sup>3</sup> is placed as a part of the boundary of the cork, is this of exotic **R**<sup>4</sup>

this 4-smoothness. As the conclusion, we have the important correlation:

*to the change of the level k of the WZW model on SU*(2)*, i.e. k* WZW →*l* WZW*.*

*<sup>k</sup> to exotic* **<sup>R</sup>**<sup>4</sup>

and it is in fact fake smooth *<sup>S</sup>*<sup>3</sup> <sup>×</sup>Θ*<sup>k</sup>* **<sup>R</sup>**, Freedman (1979). Given the connection of **<sup>R</sup>**<sup>4</sup>

the WZ term as above, we have determined the "quantized" geometry of *SU*(2)*<sup>k</sup>* × **R** as

a source for various further constructions. In particular, we will see that gravitational effects

Let us, following Asselmeyer-Maluga & Król (2011a) (see also Król (2011a;b)), begin with a charged quantum particle, say *e*, moving through non-flat gravitational background, i.e. smooth 4-spacetime manifold. The amount of gravity due to the curvature of this background affects the particle trajectory as predicted by GR. There should exist, however, a high energy limit where gravity contained in this geometrical background becomes quantum rather than classical and the particle may not be described by perturbative field theory any longer. This rather natural, from the point of view of physics, scenario requires, however, quantum gravity calculations which is not in reach in dimension 4. Moreover, mathematics underlying classical

*<sup>k</sup>* on the quantum level are determined via string theory where one replaces consistently

*S*<sup>3</sup>=*SU*(2)

1 8*π*<sup>2</sup>

integral of the generator

obtain the relation:

*model on SU*(2)*.*

of **R**<sup>4</sup>

*The structure of exotic* **R**<sup>4</sup>

*The change of smoothnes of exotic* **R**<sup>4</sup>

Let us consider now the end of the exotic **R**<sup>4</sup>

**3. 10d string theory and 4d-smoothness**

corresponding to the exotic geometry of the end of **R**<sup>4</sup>

flat **<sup>R</sup>**<sup>4</sup> part of the background by curved 4D *SU*(2)*<sup>k</sup>* <sup>×</sup> **<sup>R</sup>**.

another disk into that point. These disks form the first stage of the Casson handle. By iteration one can produce the other stages. Finally, we consider a tubular neighborhood *<sup>D</sup>*<sup>2</sup> <sup>×</sup> *<sup>D</sup>*<sup>2</sup> of this immersed disk, called a kinky handle, on each stage. The union of all neighborhoods of all stages is the Casson handle. So, there are two input data involved with the construction of a *CH*: the number of double points in each stage and their orientation ±. Thus, we can visualize the Casson handle *CH* by a tree: the root is the immersion *<sup>D</sup>*<sup>2</sup> <sup>→</sup> *<sup>M</sup>* with *<sup>k</sup>* double points, the first stage forms the next level of the tree with *k* vertices connected with the root by edges etc. The edges are evaluated using the orientation ±. Every Casson handle can be represented by such an infinite tree. The structure of CH as immersed many-layers 2-disks will be important in Sec. 4 where we will assign fermion fields to CH's.

Next, we turn again to the radial family of small exotic **R**4, i.e. a continuous family of exotic {**R**<sup>4</sup> *<sup>ρ</sup>*}*ρ*∈[0,+∞] with parameter *<sup>ρ</sup>* so that **<sup>R</sup>**<sup>4</sup> *<sup>ρ</sup>* and **<sup>R</sup>**<sup>4</sup> *<sup>ρ</sup>*� are non-diffeomorphic for *ρ* �= *ρ*� . The point is that this radial family has a natural foliation (see Theorem 3.2 in DeMichelis & Freedman (1992)) which can be induced by a polygon *P* in the two-dimensional hyperbolic space **H**2. The area of *P* is a well-known invariant, the Godbillon-Vey class as the element in *H*3(*S*3, **R**). Every GV class determines a codimension-one foliation on the 3-sphere (firstly constructed by Thurston (1972); see also the book Tamura (1992) chapter VIII for the details). This 3-sphere is a part of the boundary *∂A* of the Akbulut cork *A* (there is an embedding *<sup>S</sup>*<sup>3</sup> <sup>→</sup> *<sup>∂</sup>A*). Furthermore, one can show that the codimension-one foliation of the 3-sphere induces a codimension-one foliation of *∂A* so that the area of the corresponding polygons (and therefore the foliation invariants) agree. The Godbillon-Vey invariant [*GV*] <sup>∈</sup> *<sup>H</sup>*3(*S*3, **<sup>R</sup>**) of the foliation is related to the parameter of the radial family by *GV*, [*S*3] = *ρ*<sup>2</sup> using the pairing between cohomology and homology (the fundamental homology class [*S*3] <sup>∈</sup> *<sup>H</sup>*3(*S*3)).

Thus, the relation between an exotic **R**<sup>4</sup> (of Bizaca as constructed from the failure of the smooth h-cobordism theorem) and codimension-one foliation of the *S*<sup>3</sup> emerges. Two non-diffeomorphic exotic **R**<sup>4</sup> imply non-cobordant codimension-one foliations of the 3-sphere described by the Godbillon-Vey class in *H*3(*S*3, **R**) (proportional to the surface of the polygon). 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 obtain the result:

*The exotic* **R**<sup>4</sup> *(of Bizaca) is determined by the codimension-1 foliations with non-vanishing Godbillon-Vey class in H*3(*S*3, **<sup>R</sup>**3) *of a 3-sphere seen as submanifold S*<sup>3</sup> <sup>⊂</sup> **<sup>R</sup>**4*. We say: the exoticness is localized at a 3-sphere inside the small exotic* **R**4*.*

In the particular case of integral *H*3(*S*3, **Z**) one yields the relation of exotic **R**<sup>4</sup> *<sup>k</sup>* , *k*[ ] ∈ *<sup>H</sup>*3(*S*3, **<sup>Z</sup>**), *<sup>k</sup>* <sup>∈</sup> **<sup>Z</sup>** with the WZ term of the *<sup>k</sup>* WZW model on *SU*(2). This is because the integer classes in *H*3(*S*3, **Z**) are of special character. Topologically, this case refers to flat *PSL*(2, **<sup>R</sup>**)−bundles over the space (*S*<sup>2</sup> \ {k punctures}) <sup>×</sup> *<sup>S</sup>*<sup>1</sup> where the gluing of *<sup>k</sup>* solid tori produces a 3-sphere (so-called Heegard decomposition). Then, one obtains the relation [Asselmeyer-Maluga & Król (2009a)]:

$$\frac{1}{(4\pi)^2} \langle GV(\mathcal{F})\_\prime [S^3] \rangle = \frac{1}{(4\pi)^2} \int\_{S^3} GV(\mathcal{F}) = \pm (2-k) \tag{1}$$

in dependence on the orientation of the fundamental class [*S*3]. We can interpret the Godbillon-Vey invariant as WZ term. For that purpose, we use the group structure *SU*(2) = *<sup>S</sup>*<sup>3</sup> of the 3-sphere *<sup>S</sup>*<sup>3</sup> and identify *SU*(2) = *<sup>S</sup>*3. Let *<sup>g</sup>* <sup>∈</sup> *SU*(2) be a unitary matrix with 6 Will-be-set-by-IN-TECH

another disk into that point. These disks form the first stage of the Casson handle. By iteration one can produce the other stages. Finally, we consider a tubular neighborhood *<sup>D</sup>*<sup>2</sup> <sup>×</sup> *<sup>D</sup>*<sup>2</sup> of this immersed disk, called a kinky handle, on each stage. The union of all neighborhoods of all stages is the Casson handle. So, there are two input data involved with the construction of a *CH*: the number of double points in each stage and their orientation ±. Thus, we can visualize the Casson handle *CH* by a tree: the root is the immersion *<sup>D</sup>*<sup>2</sup> <sup>→</sup> *<sup>M</sup>* with *<sup>k</sup>* double points, the first stage forms the next level of the tree with *k* vertices connected with the root by edges etc. The edges are evaluated using the orientation ±. Every Casson handle can be represented by such an infinite tree. The structure of CH as immersed many-layers 2-disks will be important

Next, we turn again to the radial family of small exotic **R**4, i.e. a continuous family of

The point is that this radial family has a natural foliation (see Theorem 3.2 in DeMichelis & Freedman (1992)) which can be induced by a polygon *P* in the two-dimensional hyperbolic space **H**2. The area of *P* is a well-known invariant, the Godbillon-Vey class as the element in *H*3(*S*3, **R**). Every GV class determines a codimension-one foliation on the 3-sphere (firstly constructed by Thurston (1972); see also the book Tamura (1992) chapter VIII for the details). This 3-sphere is a part of the boundary *∂A* of the Akbulut cork *A* (there is an embedding *<sup>S</sup>*<sup>3</sup> <sup>→</sup> *<sup>∂</sup>A*). Furthermore, one can show that the codimension-one foliation of the 3-sphere induces a codimension-one foliation of *∂A* so that the area of the corresponding polygons (and therefore the foliation invariants) agree. The Godbillon-Vey invariant [*GV*] <sup>∈</sup> *<sup>H</sup>*3(*S*3, **<sup>R</sup>**)

pairing between cohomology and homology (the fundamental homology class [*S*3] <sup>∈</sup> *<sup>H</sup>*3(*S*3)). Thus, the relation between an exotic **R**<sup>4</sup> (of Bizaca as constructed from the failure of the smooth h-cobordism theorem) and codimension-one foliation of the *S*<sup>3</sup> emerges. Two non-diffeomorphic exotic **R**<sup>4</sup> imply non-cobordant codimension-one foliations of the 3-sphere described by the Godbillon-Vey class in *H*3(*S*3, **R**) (proportional to the surface of the polygon). This relation is very strict, i.e. if we change the Casson handle, then we must change the

*The exotic* **R**<sup>4</sup> *(of Bizaca) is determined by the codimension-1 foliations with non-vanishing Godbillon-Vey class in H*3(*S*3, **<sup>R</sup>**3) *of a 3-sphere seen as submanifold S*<sup>3</sup> <sup>⊂</sup> **<sup>R</sup>**4*. We say: the exoticness*

*<sup>H</sup>*3(*S*3, **<sup>Z</sup>**), *<sup>k</sup>* <sup>∈</sup> **<sup>Z</sup>** with the WZ term of the *<sup>k</sup>* WZW model on *SU*(2). This is because the integer classes in *H*3(*S*3, **Z**) are of special character. Topologically, this case refers to flat *PSL*(2, **<sup>R</sup>**)−bundles over the space (*S*<sup>2</sup> \ {k punctures}) <sup>×</sup> *<sup>S</sup>*<sup>1</sup> where the gluing of *<sup>k</sup>* solid tori produces a 3-sphere (so-called Heegard decomposition). Then, one obtains the relation

(4*π*)<sup>2</sup>

in dependence on the orientation of the fundamental class [*S*3]. We can interpret the Godbillon-Vey invariant as WZ term. For that purpose, we use the group structure *SU*(2) = *<sup>S</sup>*<sup>3</sup> of the 3-sphere *<sup>S</sup>*<sup>3</sup> and identify *SU*(2) = *<sup>S</sup>*3. Let *<sup>g</sup>* <sup>∈</sup> *SU*(2) be a unitary matrix with

*S*3

polygon. But that changes the foliation and vice verse. Finally, we obtain the result:

In the particular case of integral *H*3(*S*3, **Z**) one yields the relation of exotic **R**<sup>4</sup>

*<sup>ρ</sup>* and **<sup>R</sup>**<sup>4</sup>

*<sup>ρ</sup>*� are non-diffeomorphic for *ρ* �= *ρ*�

*GV*, [*S*3]

*GV*(F) = ±(2 − *k*) (1)

= *ρ*<sup>2</sup> using the

*<sup>k</sup>* , *k*[ ] ∈

.

in Sec. 4 where we will assign fermion fields to CH's.

*is localized at a 3-sphere inside the small exotic* **R**4*.*

1

(4*π*)<sup>2</sup> �*GV*(F), [*S*3]� <sup>=</sup> <sup>1</sup>

[Asselmeyer-Maluga & Król (2009a)]:

*<sup>ρ</sup>*}*ρ*∈[0,+∞] with parameter *<sup>ρ</sup>* so that **<sup>R</sup>**<sup>4</sup>

of the foliation is related to the parameter of the radial family by

exotic {**R**<sup>4</sup>

det *<sup>g</sup>* <sup>=</sup> <sup>−</sup>1. The left invariant 1-form *<sup>g</sup>*−1*dg* generates locally the cotangent space connected to the unit. The forms *<sup>ω</sup><sup>k</sup>* <sup>=</sup> *Tr*((*g*−1*dg*)*k*) are complex *<sup>k</sup>*−forms generating the deRham cohomology of the Lie group. The cohomology classes of the forms *ω*1, *ω*<sup>2</sup> vanish and *<sup>ω</sup>*<sup>3</sup> <sup>∈</sup> *<sup>H</sup>*3(*SU*(2), **<sup>R</sup>**) generates the cohomology group. Then, we obtain as the value for the integral of the generator

$$\frac{1}{8\pi^2} \int\_{S^3 = SU(2)} \omega\_3 = 1\,\text{J}$$

This integral can be interpreted as winding number of *g*. Now, we consider a smooth map *<sup>G</sup>* : *<sup>S</sup>*<sup>3</sup> <sup>→</sup> *SU*(2) with 3-form <sup>Ω</sup><sup>3</sup> <sup>=</sup> *Tr*((*G*−1*dG*)3) so that the integral

$$\frac{1}{8\pi^2} \int\_{S^3 = SU(2)} \Omega\_3 = \frac{1}{8\pi^2} \int\_{S^3} \text{Tr}((G^{-1}dG)^3) \in \mathbb{Z}^3$$

is the winding number of *G*. Every Godbillon-Vey class with integer value like (1) is generated by a 3-form Ω3. Therefore, the Godbillon-Vey class is the WZ term of the *SU*(2)*k*. Thus, we obtain the relation:

*The structure of exotic* **R**<sup>4</sup> *<sup>k</sup> 's, k* ∈ **Z** *from the radial family determines the WZ term of the k WZW model on SU*(2)*.*

This WZ term enables one for the cancellation of the quantum anomaly due to the conformal invariance of the classical *σ*-model on *SU*(2). Thus, we have a method of including this cancellation term from smooth 4-geometry: when a smoothness of the ambient 4-space, in which *S*<sup>3</sup> is placed as a part of the boundary of the cork, is this of exotic **R**<sup>4</sup> *<sup>k</sup>* , then the WZ term of the classical *σ*-model with target *S*<sup>3</sup> = *SU*(2), i.e. *SU*(2)*<sup>k</sup>* WZW, is precisely generated by this 4-smoothness. As the conclusion, we have the important correlation:

*The change of smoothnes of exotic* **R**<sup>4</sup> *<sup>k</sup> to exotic* **<sup>R</sup>**<sup>4</sup> *<sup>l</sup> , k*, *l* ∈ **Z** *both from the radial family, corresponds to the change of the level k of the WZW model on SU*(2)*, i.e. k* WZW →*l* WZW*.*

Let us consider now the end of the exotic **R**<sup>4</sup> *<sup>k</sup>* i.e. *<sup>S</sup>*<sup>3</sup> <sup>×</sup> **<sup>R</sup>**. This end cannot be standard smooth and it is in fact fake smooth *<sup>S</sup>*<sup>3</sup> <sup>×</sup>Θ*<sup>k</sup>* **<sup>R</sup>**, Freedman (1979). Given the connection of **<sup>R</sup>**<sup>4</sup> *<sup>k</sup>* with the WZ term as above, we have determined the "quantized" geometry of *SU*(2)*<sup>k</sup>* × **R** as corresponding to the exotic geometry of the end of **R**<sup>4</sup> *<sup>k</sup>* . The appearance of the *SU*(2)*<sup>k</sup>* × **R** is a source for various further constructions. In particular, we will see that gravitational effects of **R**<sup>4</sup> *<sup>k</sup>* on the quantum level are determined via string theory where one replaces consistently flat **<sup>R</sup>**<sup>4</sup> part of the background by curved 4D *SU*(2)*<sup>k</sup>* <sup>×</sup> **<sup>R</sup>**.
