**1. Introduction**

16 Will-be-set-by-IN-TECH

52 Quantum Gravity

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The search for the theory of quantum gravity (QG) in 4-dimensions (4D) is one of the most significant challenges of temporary physics. The great effort and insights of many theoreticians and experimentalists resulted in the emergence of one of the greatest achievements of 20th century science, i.e. standard model of particles and fields (SM). SM (with its minimal extensions by massive neutrinos and after renormalization) describes and predicts, with enormous accuracy, almost all perturbative aspects and behaviour of interacting quantum fields and particles which place themselves in the realm of electromagnetic, strong and weak nuclear interactions, within the range of energies up to few TeV. However, gravity at quantum level is not covered by this pattern. The oldest, semiclassical, approach to QG relies on the quantization of metric field which is understood as the perturbation of the ground spacetime metric. This is exactly in the spirit of quantum field theory (QFT) as in SM. There should follow various correlation functions of physical processes where gravity at quantum level is present. There should, but actually they do not since the expressions are divergent and the theory is not renormalizable. Even the presence of supersymmetry does not change this substantially. On the other hand, we have a wonderful theory of general relativity (GR) which, however, is a theory of classical gravity and it prevents its quantization in 4D.

Among existing approaches to QG, superstring theory is probably the most advanced and conservative one. It attempts to follow GR and quantum mechanics as much as possible. However, superstring theory has to be formulated in 10 spacetime dimensions and on fixed, not dynamical, background. Many proposals how to reach the observed physics from 10D superstrings were worked out within the years. These are among others, compactiffication, flux stabilization, brane configuration model-buildings, brane worlds, holography or anti-de-Sitter/conformal field theory duality, i.e. AdS/CFT. There exists much ambiguity, however, with determining 4D results by these methods. Some authors estimate that there exist something about 10500 different backgrounds of superstring theory which all could be "good" candidates expressing 4D physics. This means that similar variety of possible models for true physics is predicted by superstring theory. To manage with such huge amount of "good" solutions, there was proposed to use the methods of statistical analysis to such *landscape* of possible backgrounds. Anyway, one could expect better prediction power from the fundamental theory which would unify gravity with other interactions at quantum level. On the other hand, superstring theory presents beautiful, fascinating and extremly rich mathematics which is still not fully comprehended.

Even though neither any explicit exotic metric nor the function on **R**<sup>4</sup> is known, recent relative results made it possible to apply these exotic structures in a variety of contexts relevant to physics. In particular, strong connection with quantum theories and quantization was shown by Asselmeyer-Maluga & Król (2011b). First, we deal here exclusively with *small* exotic **R**4. These arise as the result of failing *h*-cobordism theorem in 4D (see e.g. Asselmeyer-Maluga & Brans (2007)). The others, so called large exotic **R**4, emerge from failing the smooth surgery in 4D. Second, the main technical ingredient of the relative approach to small 4-exotics is the relation of these with some structures defined on a 3-sphere. This *S*<sup>3</sup> should be placed as a part of the boundary of some contractible 4-submanifold of **R**4. This manifold is the Akbulut cork and its boundary is, in general, a closed 3-manifold which has the same homologies as ordinary 3-sphere – homology 3-sphere. Next, we deal with the parameterized by the radii

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 55

standard **R**4. This is the *radial family* of small exotic **R**4's or the deMichellis-Freedman family DeMichelis & Freedman (1992). Let *CS* be the standard Cantor set as a subset of **R**, then the

**Theorem 1** (Asselmeyer-Maluga & Król (2011b))**.** *Let us consider a radial family Rt of small exotic*

*between M and M*<sup>0</sup> *with Akbulut cork A* ⊂ *M and A* ⊂ *M*0*, respectively. Then, the radial family Rt determines a family of codimension-one foliations of ∂A with Godbillon-Vey invariant ρ*2*. Furthermore, given two exotic spaces Rt and Rs, homeomorphic but non-diffeomorphic to each other (and so t* �= *s), then the two corresponding codimension-one foliations of ∂A are non-cobordant to each other.*

This theorem gives a direct relation of small exotic **R**4's - from the radial family, and (codimension one) foliations of some *S*<sup>3</sup> - from the boundary of the Akbulut cork. *M* and *M*<sup>0</sup> are compact non-cobordant 4-manifolds, resulting from the failure of the 4D *h*-cobordism theorem (see the next section). Such relativization of 4-exotics to the foliations of *S*<sup>3</sup> is the source of variety of further mathematical results and their applications in physics. One example of these is the quantization of electric charge in 4D where instead of magnetic monopoles one considers exotic smoothness in some region in spacetime Asselmeyer-Maluga & Król (2009a). More examples of this kind will be presented in the course of this Chapter.

Now we are ready to formulate two important questions as guidelines for this work:

i. What if smooth structure, with respect to which standard model of particles is defined, is not the "correct" one and it does not match with the smooth structure underlying GR and

ii. What if particles and fields, as in standard model of particles, are not fundamental from the point of view of gravity in 4 dimensions? Rather, more natural are effective condensed matter states, and these states should be used in order to unify quantum matter with

This Chapter is thought as giving the explanation for the above questions and for the existence of a fundamental connection between these, differently looking, problems. New point of view on the reconciliation of quantum field theory with general relativity in 4 physical dimensions, emerges. The exact description of quantum matter and fields coupled with QG in 4D, at least in some important cases, is presented. The task to build a final theory of QG in 4D is thus seen from different perspective where rather effective states of condensed matter are well suited for the reconciliation with QG. Such approach is also motivated by the AdS/CFT dualities where

*<sup>ρ</sup>* each of which is the open submanifold of

*<sup>ρ</sup>* ⊂ *CS* ⊂ [0, 1] *induced from the non-product h-cobordism W*

*<sup>ρ</sup>* <sup>∈</sup> **<sup>R</sup>** of *<sup>S</sup>*<sup>4</sup> as a subset of *<sup>R</sup>*4, a family of exotic **<sup>R</sup>**<sup>4</sup>

theories of quantum gravity, in 4 dimensions?

general relativity.

crucial result is:

*<sup>t</sup> with radius <sup>ρ</sup> and t* <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>

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

Therefore Asselmeyer-Maluga & Król (2010) have proposed recently how to find connections of superstring theory with dimension 4 in a new way not relying on the standard techniques. Whole the approach derives from mathematics, especially low dimensional differential topology and geometry. In that approach one considers superstring theory in 10D and supersymmetry as "merely" mathematics describing directly, at least in a variety of important cases, the special smooth geometry on Euclidean, topologically trivial, manifold **R**4. These are various non-diffeomorphic different smooth structures. Smooth manifold **R**<sup>4</sup> with such non-standard smoothness is called exotic **R**<sup>4</sup> and as a smooth Riemannian manifold allows for a variety of metrics. This exotic geometry in turn, is regarded as underlying smoothness for 4-spacetime and is directly related to physics in dimension 4.

The way towards crystallizing such point of view on string theory was laborious and required many important steps. The breakthrough findings in differential topology, from the eighties of the previous century, showed that indeed there are different from the standard one, smoothings of the simplest Euclidean 4-space (see e.g. Asselmeyer-Maluga & Brans (2007)). Spacetime models usually are based on 4D smooth manifolds, hence they are locally described with respect to the standard smooth **R**4. Anything what happens to this fundamental building block might be important at least to classical physics formulated on such spacetime. Indeed, it was conjectured by Brans (1994a;b), and then proved by Asselmeyer (1996) and Sładkowski (2001), that exotic smooth **R**4's can act as sources for the external gravitational field in spacetime. Even mathematics alone, strongly distinguishes these smooth open 4-manifolds: among all **R***<sup>n</sup>* only the case *n* = 4 allows for different smoothings of Euclidean **R***n*. For any other **<sup>R</sup>***n*, *<sup>n</sup>* 4 there exists unique smooth structure. Moreover, there exists infinitely continuum many different smoothings for **R**4. However, mathematical tools suitable for the direct description of, say, metrics or functions on exotic **R**<sup>4</sup> are mostly unknown (see however Asselmeyer-Maluga & Brans (2011)). The main obstruction which prevents progress in our understanding of exotic smoothness on **R**<sup>4</sup> is that there is no known effective coordinate presentation. As the result, no exotic smooth function on any such **R**<sup>4</sup> is known, even though there exist infinite continuum many different exotic **R**4. 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. This is also a strong limitation for the applicability of the structures to physics. Let us note that smooth structures on open 4-manifolds, like on **R**4, are of special character and require special mathematics which, in general, is not completely understood now. The case of *compact* 4-manifolds and their smooth structures is much better recognized also from the point of view of physics (see e.g. Asselmeyer-Maluga (2010); Asselmeyer-Maluga & Brans (2007); Witten (1985)). The famous exception is, however, not resolved yet, negation of the 4D Poincaré conjecture stating that there exists exotic *S*4.

Bizaca (1994) constructed an infinite coordinate patch presentation by using Casson handles. ˘ Still, it seems hopeless to extract physical information from that. The proposition by Król (2004a;b; 2005) indicated that one should use methods of set theory, model theory and categories to grasp properly some results relevant to quantum physics. Such low level constructions modify the smoothness on **R**<sup>4</sup> and the structures survive the modifications as a *classical* exotic **R**4. Thus, functions, although from different logic and category, approach exotic smooth ones, such that some quantum structures emerge due to the rich categorical formalism involved. Still, to apply exotic 4-smoothness in variety of situations one needs more direct relation to existing calculus.

2 Will-be-set-by-IN-TECH

Therefore Asselmeyer-Maluga & Król (2010) have proposed recently how to find connections of superstring theory with dimension 4 in a new way not relying on the standard techniques. Whole the approach derives from mathematics, especially low dimensional differential topology and geometry. In that approach one considers superstring theory in 10D and supersymmetry as "merely" mathematics describing directly, at least in a variety of important cases, the special smooth geometry on Euclidean, topologically trivial, manifold **R**4. These are various non-diffeomorphic different smooth structures. Smooth manifold **R**<sup>4</sup> with such non-standard smoothness is called exotic **R**<sup>4</sup> and as a smooth Riemannian manifold allows for a variety of metrics. This exotic geometry in turn, is regarded as underlying smoothness

The way towards crystallizing such point of view on string theory was laborious and required many important steps. The breakthrough findings in differential topology, from the eighties of the previous century, showed that indeed there are different from the standard one, smoothings of the simplest Euclidean 4-space (see e.g. Asselmeyer-Maluga & Brans (2007)). Spacetime models usually are based on 4D smooth manifolds, hence they are locally described with respect to the standard smooth **R**4. Anything what happens to this fundamental building block might be important at least to classical physics formulated on such spacetime. Indeed, it was conjectured by Brans (1994a;b), and then proved by Asselmeyer (1996) and Sładkowski (2001), that exotic smooth **R**4's can act as sources for the external gravitational field in spacetime. Even mathematics alone, strongly distinguishes these smooth open 4-manifolds: among all **R***<sup>n</sup>* only the case *n* = 4 allows for different smoothings of Euclidean **R***n*. For any other **<sup>R</sup>***n*, *<sup>n</sup>* 4 there exists unique smooth structure. Moreover, there exists infinitely continuum many different smoothings for **R**4. However, mathematical tools suitable for the direct description of, say, metrics or functions on exotic **R**<sup>4</sup> are mostly unknown (see however Asselmeyer-Maluga & Brans (2011)). The main obstruction which prevents progress in our understanding of exotic smoothness on **R**<sup>4</sup> is that there is no known effective coordinate presentation. As the result, no exotic smooth function on any such **R**<sup>4</sup> is known, even though there exist infinite continuum many different exotic **R**4. 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. This is also a strong limitation for the applicability of the structures to physics. Let us note that smooth structures on open 4-manifolds, like on **R**4, are of special character and require special mathematics which, in general, is not completely understood now. The case of *compact* 4-manifolds and their smooth structures is much better recognized also from the point of view of physics (see e.g. Asselmeyer-Maluga (2010); Asselmeyer-Maluga & Brans (2007); Witten (1985)). The famous exception is, however, not resolved yet, negation of the 4D Poincaré conjecture stating that there exists exotic *S*4.

Bizaca (1994) constructed an infinite coordinate patch presentation by using Casson handles. ˘ Still, it seems hopeless to extract physical information from that. The proposition by Król (2004a;b; 2005) indicated that one should use methods of set theory, model theory and categories to grasp properly some results relevant to quantum physics. Such low level constructions modify the smoothness on **R**<sup>4</sup> and the structures survive the modifications as a *classical* exotic **R**4. Thus, functions, although from different logic and category, approach exotic smooth ones, such that some quantum structures emerge due to the rich categorical formalism involved. Still, to apply exotic 4-smoothness in variety of situations one needs

more direct relation to existing calculus.

for 4-spacetime and is directly related to physics in dimension 4.

Even though neither any explicit exotic metric nor the function on **R**<sup>4</sup> is known, recent relative results made it possible to apply these exotic structures in a variety of contexts relevant to physics. In particular, strong connection with quantum theories and quantization was shown by Asselmeyer-Maluga & Król (2011b). First, we deal here exclusively with *small* exotic **R**4. These arise as the result of failing *h*-cobordism theorem in 4D (see e.g. Asselmeyer-Maluga & Brans (2007)). The others, so called large exotic **R**4, emerge from failing the smooth surgery in 4D. Second, the main technical ingredient of the relative approach to small 4-exotics is the relation of these with some structures defined on a 3-sphere. This *S*<sup>3</sup> should be placed as a part of the boundary of some contractible 4-submanifold of **R**4. This manifold is the Akbulut cork and its boundary is, in general, a closed 3-manifold which has the same homologies as ordinary 3-sphere – homology 3-sphere. Next, we deal with the parameterized by the radii *<sup>ρ</sup>* <sup>∈</sup> **<sup>R</sup>** of *<sup>S</sup>*<sup>4</sup> as a subset of *<sup>R</sup>*4, a family of exotic **<sup>R</sup>**<sup>4</sup> *<sup>ρ</sup>* each of which is the open submanifold of standard **R**4. This is the *radial family* of small exotic **R**4's or the deMichellis-Freedman family DeMichelis & Freedman (1992). Let *CS* be the standard Cantor set as a subset of **R**, then the crucial result is:

**Theorem 1** (Asselmeyer-Maluga & Król (2011b))**.** *Let us consider a radial family Rt of small exotic* **R**<sup>4</sup> *<sup>t</sup> with radius <sup>ρ</sup> and t* <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup> *<sup>ρ</sup>* ⊂ *CS* ⊂ [0, 1] *induced from the non-product h-cobordism W between M and M*<sup>0</sup> *with Akbulut cork A* ⊂ *M and A* ⊂ *M*0*, respectively. Then, the radial family Rt determines a family of codimension-one foliations of ∂A with Godbillon-Vey invariant ρ*2*. Furthermore, given two exotic spaces Rt and Rs, homeomorphic but non-diffeomorphic to each other (and so t* �= *s), then the two corresponding codimension-one foliations of ∂A are non-cobordant to each other.*

This theorem gives a direct relation of small exotic **R**4's - from the radial family, and (codimension one) foliations of some *S*<sup>3</sup> - from the boundary of the Akbulut cork. *M* and *M*<sup>0</sup> are compact non-cobordant 4-manifolds, resulting from the failure of the 4D *h*-cobordism theorem (see the next section). Such relativization of 4-exotics to the foliations of *S*<sup>3</sup> is the source of variety of further mathematical results and their applications in physics. One example of these is the quantization of electric charge in 4D where instead of magnetic monopoles one considers exotic smoothness in some region in spacetime Asselmeyer-Maluga & Król (2009a). More examples of this kind will be presented in the course of this Chapter.

Now we are ready to formulate two important questions as guidelines for this work:


This Chapter is thought as giving the explanation for the above questions and for the existence of a fundamental connection between these, differently looking, problems. New point of view on the reconciliation of quantum field theory with general relativity in 4 physical dimensions, emerges. The exact description of quantum matter and fields coupled with QG in 4D, at least in some important cases, is presented. The task to build a final theory of QG in 4D is thus seen from different perspective where rather effective states of condensed matter are well suited for the reconciliation with QG. Such approach is also motivated by the AdS/CFT dualities where

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

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 57

Consider the following situation: one has two topologically equivalent (i.e. homeomorphic),

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*�

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

*together with a sub-cobordism V* ⊂ *W with ∂V* = *A* � *A*� *(the disjoint oriented sum), so that the*

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

(as open manifold) is homeomorphic to **R**4. However, if (and only if) *M* and *M*� are not

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

*h-cobordism W* \ *V induces a diffeomorphism between M* \ *A and M*� \ *A*�

neighborhood (tubular) of the sub-cobordism *V* between *A* and *A*�

diffeomorphic (still being homeomorphic), then *int*(*V*) <sup>∩</sup> *<sup>M</sup>* is an exotic **<sup>R</sup>**4.

, which are not diffeomorphic. There are two

*. Then there are contractable submanifolds A* ⊂ *M*, *A*� ⊂ *M*�

*.*

. The interior of *V*, *int*(*V*),

. If

though all are topologically the same.

*Let W be a h-cobordism between M*, *M*�

& Stong (1997)]:

simple-connected, smooth 4-manifolds *M*, *M*�

effective matter states (without gravity) are described by dual theories with gravity. Hence gravity is inherently present in description of such condensed matter states.

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 **R**<sup>4</sup> *<sup>p</sup>* in the *k*-channel, *k* > 2, Kondo effect, where the usual fusion rules of the *SU*(2)*<sup>k</sup>* WZW model would be modified to these of *SU*(2)*<sup>p</sup>* WZW in high energies.

Next in Sec. 5 we present the connections of branes configurations in superstring theory with non-standard 4-smoothness of **R**4. Discussion and conclusions close the Chapter.
