**4. Quantum effective spin matter and exotic R**<sup>4</sup> **– the Kondo effect**

Gravitational interaction is very exceptional among all interactions in Nature. On the one hand gravity is the geometry of spacetime on which fields propagate and interactions take place. On the other hand, gravity couples with any kind of energy and matter. Further, it is the only interaction which restrains quantization.

Based on the entanglement of ideas presented so far, we want to argue that gravity is present in some states of magnetic effective quantum matter in a *nonstandard* way. The latter means that some states of spin matter, already at low temperatures, are coupled with 4D gravity via special 4-geometry *directly*, rather than, by energy-momentum tensor. This coupling can be extended over quantum regime of gravity, at least in some cases, and relates effective rather than fundamental fields and particles from SM. The coupling is understood as the presence of a non-flat 4-geometry which becomes dominating in some limits. The special 4-geometry is, again, exotic smoothness of Euclidean 4-space **R**4, thus becoming a guiding principle for presented approach to QG. The presence of gravity in the description of nonperturbative, strongly entangled states of 4D matter field is not a big surprise, as recent vital activity on the methods of AdS/CFT correspondence shows. However, our approach is different and makes use of inherently 4-dimensional new geometrical findings, which, at this stage of development, do not refer to AdS/CFT techniques (cf. Król (2005)).

In the thirties of the last century strange behaviour of conducting electrons occurring in some metallic alloys was observed. Namely the resistivity *ρ*(*T*) in these alloys in the presence of magnetic spin *s* impurities, growth substantially when the temperature is lowering below the critical temperature *TK* called the Kondo temperature. *TK* is as low as a few K.

Kondo proposed in 1964 a simple phenomenological Hamiltonian Affleck (1995):

$$H = \sum\_{\stackrel{\rightarrow}{k},a} \psi\_{\stackrel{\rightarrow}{k}}^{+a} \psi\_{\stackrel{\rightarrow}{k}a} \,\,\epsilon(k) + \lambda \stackrel{\rightarrow}{S} \cdot \sum\_{\stackrel{\rightarrow}{k},\stackrel{\rightarrow}{k'}} \psi\_{\stackrel{\rightarrow}{k}}^{+} \frac{\overrightarrow{\sigma}}{2} \psi\_{\stackrel{\rightarrow}{k'}}^{\rightarrow}.\tag{24}$$

explaining the growth of the resistivity *ρ*(*T*). Here *ψ* is the annihilation operator for the conduction electron of spin *<sup>α</sup>* and momentum <sup>→</sup> *k* , the antiferromagnetic interaction term is that between spin *<sup>s</sup>* impurity <sup>→</sup> *S* with spins of conducting electrons, at <sup>→</sup> *x* = 0; <sup>→</sup> *σ* is the vector of Pauli matrices. From this Hamiltonian one can derive, in the Born approximation, that *ρ*(*T*) ∼ *λ* + *νλ*<sup>2</sup> ln *<sup>D</sup> <sup>T</sup>* <sup>+</sup> ...<sup>2</sup> where *D* is the 'width of the band' parameter and the second term is divergent in *T* = 0. Thus, this divergence explains the growth of the resistivity. The Hamiltonian (24) can be also derived from the more microscopic Anderson model Anderson (1961). The Kondo antiferromagnetic coupling appears as the tunnelling of electrons thus screening the spin impurity (see eg. Potok et al. (2007)).

The exact low *T* behavior was proposed by Affleck (1995); Affleck & Ludwig (1991; 1993; 1994) and Potok et al. (2007) by the use of boundary conformal field theory (BCFT). This insightful use of the CFT methods makes it possible to work out the connection with smooth 4-geometry.

Let us see in brief how the structure of the *SU*(2)*<sup>k</sup>* WZW model is well suited to the description of the *k*-channel Kondo effect. Recall that Kac-Moody algebra *SU*(2)*<sup>k</sup>* is spanned

term is expressed by the spinor field *ψ* describing the immersion of *D*<sup>2</sup> into **R**3, which extends

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 69

 *∂*(**R**<sup>4</sup> *<sup>k</sup>*\CH)

This can be extended to 4-dimensional Einstein-Hilbert action with the source depending on

Again, it was shown in Asselmeyer-Maluga & Brans (2011) that the spinor field *ψ* extends over whole 4-manifold such that the 4D Dirac equations are fulfilled. This way we have fermion fields which are determined by CH. Moreover, this fermions plays a role of gravity sources as

effective *q*(*r*)-many infinite branches. Each such branch generates a fermion field. Attaching

Let us assign now the simplest possible CH to every CH in the handlebody of exotic **R**4, such that replacing the original CH by this simple one does not change the exotic smoothness. This is the model handlebody we refer to in the context of the Kondo effect (see Fig. 3 for the

The *k*-channel Kondo state, in the *k*-channel Kondo effect, is the entangled state of conducting electrons in *k* bands and the magnetic spin *s* impurity. The physics of resulting state is described by BCFT by the Verlinde fusion rules in *SU*(2)*<sup>k</sup>* WZW model. To have the WZ term in this WZW model one certainly needs *p* = *k*. This kWZ term is generated by exotic **R**<sup>4</sup>

as we explained in Sec. 2. The draft of the dependance of the number of infinite branches on

shows the example of the ramified structure of CH's in the precise language of the graphical

*One assigns the 4-smooth geometry on* **R**<sup>4</sup> *to the k-channel Kondo effect such that k corresponds to*

*p*, *k* ∈ **N***. The change between the physical Kondo states, from this emerging in k*<sup>1</sup> *channel Kondo effect to this with k*<sup>2</sup> *channels, k*<sup>1</sup> �= *k*2*, corresponds to the change between 4-geometries, from exotic*

Whether actually *p* = *k* or not is the question about the level of the *SU*(2) WZW model and the corresponding fusion rules in use. If *k* = *p* the exotic geometry gives the same fusion rules as the Affleck proposed. In the case *k* �= *p* and *k* < *p* in the *k*-channel Kondo effect the fusion rules derived from the exotic geometry are those of the *SU*(2)*<sup>p</sup>* WZW model. It would be interesting to decide experimentally, which fusion rules apply for bigger *k*. Probably in higher energies, if the Kondo state survives, the proper fusion rules are those derived from

are generated potentially by (each infinite branch of) Casson handles from the handlebody of

*<sup>p</sup>*. This reflects the situation that electrons in different conduction bands (channels)

*<sup>p</sup>*<sup>2</sup> *, p*<sup>1</sup> �= *p*2*, p*1, *p*<sup>2</sup> ∈ **N***, such that p*<sup>1</sup> = *p*1(*k*1) *and p*<sup>2</sup> = *p*2(*k*2) *as above.*

*the number of infinite branches of CH's in the handlebody. This 4-geometry is* **R**<sup>4</sup>

(*R* + *ψγμDμψ*)

*ψγμDμψ*

*<sup>p</sup>* we have *r* Casson handles in its handlebody. These *r* CH's generate

*<sup>p</sup>*. Hence, *p* is the function of *q* in general, *p* = *p*(*q*(*r*)).

<sup>√</sup>*g∂d*3*<sup>x</sup>* . (27)

√*gd*4*x* . (28)

*<sup>k</sup>* , is presented in Fig. 3a. Fig. 3b

*<sup>p</sup> where p* = *p*(*k*)*,*

*k*

*KCH*√*g∂d*3*<sup>x</sup>* <sup>=</sup>

*<sup>k</sup>* ) =

in 28. In fact every infinite branch of the CH determines some 4D fermion.

**R**<sup>4</sup> *<sup>k</sup>*\CH

to the immersion of *<sup>D</sup>*<sup>2</sup> <sup>×</sup> (0, 1) into **<sup>R</sup>**4: *∂*(**R**<sup>4</sup> *<sup>k</sup>*\CH)

*k* :

*S*CH <sup>4</sup> (**R**<sup>4</sup>

the function of the number of CH's in the handlebody of **R**<sup>4</sup>

Kirby calculus (see e.g. Gompf & Stipsicz (1999)).

The general correspondence appears:

**R**<sup>4</sup> *<sup>p</sup>*<sup>1</sup> *to* **<sup>R</sup>**<sup>4</sup>

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

the CH, hence on exotic **R**<sup>4</sup>

Second, given exotic **R**<sup>4</sup>

the CH's to the cork results in exotic **R**<sup>4</sup>

examples of the simplest CH's).

on 3-components currents <sup>→</sup> J *<sup>n</sup>*, *n* = ... − 2, −1, 0, 1, 2, ...:

$$[\mathcal{J}^a\_{n'} \mathcal{J}^b\_m]\_k = i\epsilon^{abc} \mathcal{J}^c\_{n+m} + \frac{1}{2}kn \delta^{ab} \delta\_{n,-m} \,. \tag{25}$$

Next, we decompose the currents <sup>→</sup> <sup>J</sup> *<sup>n</sup>* as <sup>→</sup> <sup>J</sup> *<sup>n</sup>* <sup>=</sup> <sup>→</sup> *<sup>J</sup> <sup>n</sup>* <sup>+</sup> <sup>→</sup> *<sup>S</sup>* such that <sup>→</sup> *J <sup>n</sup>* obey the same Kac-Moody algebra, i.e. [*J<sup>a</sup> <sup>n</sup>*, *J<sup>b</sup> <sup>m</sup>*]*<sup>k</sup>* = *i�abc J<sup>c</sup> <sup>n</sup>*+*<sup>m</sup>* + <sup>1</sup> <sup>2</sup> *knδabδn*,−*<sup>m</sup>* and usual relations for <sup>→</sup> *S*, i.e. [*Sa*, *Sb*] = *i�abcSc*, [*Sa*, *J<sup>b</sup> <sup>n</sup>*] = 0. From the point of view of field theories describing the interacting currents with spins, <sup>→</sup> J *<sup>n</sup>* corresponds to the effective infrared fixed point of the theory of interacting spins <sup>→</sup> *<sup>S</sup>* with <sup>→</sup> *<sup>J</sup> <sup>n</sup>*where the coupling constant *<sup>λ</sup>* is taken as <sup>2</sup> <sup>3</sup> for *k* = 1. The interacting Hamiltonian of the theory, for *k* = 1, reads:

$$H\_8 = c \left( \frac{1}{3} \sum\_{-\infty}^{+\infty} \stackrel{\rightarrow}{f}\_{-n} \cdot \stackrel{\rightarrow}{f}\_n + \lambda \sum\_{-\infty}^{+\infty} \stackrel{\rightarrow}{f}\_n \cdot \stackrel{\rightarrow}{S} \right) \,. \tag{26}$$

For *λ* = <sup>2</sup> <sup>3</sup> , one completes the square and the algebra (25) for the currents <sup>→</sup> J *<sup>n</sup>* follows. Then, the new Hamiltonian, where <sup>→</sup> *S* is now effectively absent (still for *k* = 1), is given by *H* = *c*� ∑+<sup>∞</sup> −∞ <sup>→</sup> J −*n* · → <sup>J</sup> *<sup>n</sup>* <sup>−</sup> <sup>3</sup> 4 (*c*, *c*� are some constants).

A similar procedure holds for arbitrary integer *k* where the spin part of the Hamiltonian reads: *Hs*,*<sup>k</sup>* = <sup>1</sup> 2*π*(*k*+2) → *J* 2 + *λ* → *J* · → *S δ*(*x*) and the infrared effective fixed point is now reached for *k* = 2 <sup>2</sup>+*<sup>k</sup>* . The spins <sup>→</sup> *S* reappear as the boundary conditions in the boundary CFT represented by the WZW model on *SU*(2). This model defines the Verlinde fusion rules and is determined by these. The following *fusion rules hypothesis*, was proposed by Affleck (1995), which explains the creation and nature of the multichannel Kondo states:

*The infrared fixed point, in the k-channel spin-s Kondo problem, is given by fusion with the spin-s primary for s k*/2 *or with the spin k*/2 *primary for s* > *k*/2*.* Thus, the level *k* Kac-Moody algebra, as in the level *k* WZW *SU*(2) model, governs the behaviour of the Kondo state in the presence of *k* channels of conducting electrons and magnetic impurity of spin *s*.

This is also the reason why, already in low temperatures, entangled magnetic matter of impurities and conduction electrons indicates the correlation with exotic 4-geometry. First, every CH generates a fermion field. Every small exotic **R**<sup>4</sup> can be represented as handlebody where Akbulut cork has several CH's attached. The important thing is that the handlenbody has a boundary and only after removing it the interior is diffeomorphic to, say, exotic **R**<sup>4</sup> *<sup>k</sup>* . Let us remove a single CH from the handlebody **R**<sup>4</sup> *<sup>k</sup>* . The result is **<sup>R</sup>**<sup>4</sup> *<sup>k</sup>* \ CH. The boundary of it reads *∂*(**R**<sup>4</sup> *<sup>k</sup>* \ CH). The contribution to the Einstein action **R**<sup>4</sup> *<sup>k</sup>*\CH *<sup>R</sup>*√*gd*4*<sup>x</sup>* from this boundary is the suitable surface term:

$$\int\_{\partial(\mathbb{R}^4\_k \backslash \mathbf{CH})} \mathcal{R}\sqrt{\mathcal{g}}d^4\mathbf{x} + \int\_{\partial(\mathbb{R}^4\_k \backslash \mathbf{CH})} \mathcal{K}\_{\mathbf{CH}}\sqrt{\mathcal{g}\_{\partial}}d^3\mathbf{x}$$

where *KCH* is the trace of the 2-nd fundamental form and *g<sup>∂</sup>* the metric on the boundary Asselmeyer-Maluga & Brans (2011). But as shown in Asselmeyer-Maluga & Brans (2011) this 16 Will-be-set-by-IN-TECH

*<sup>n</sup>*+*<sup>m</sup>* +

*<sup>J</sup> <sup>n</sup>* <sup>+</sup> <sup>→</sup>

1 2

*<sup>n</sup>*] = 0. From the point of view of field theories describing the interacting currents

J *<sup>n</sup>* corresponds to the effective infrared fixed point of the theory of interacting

+∞ ∑ −∞ → *J <sup>n</sup>* · → *S* 

*S* is now effectively absent (still for *k* = 1), is given by *H* =

*S δ*(*x*) and the infrared effective fixed point is now reached for *k* =

*S* reappear as the boundary conditions in the boundary CFT represented by

*<sup>k</sup>* . The result is **<sup>R</sup>**<sup>4</sup>

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

*KCH*

<sup>√</sup>*g∂d*3*<sup>x</sup>*

*<sup>S</sup>* such that <sup>→</sup>

<sup>2</sup> *knδabδn*,−*<sup>m</sup>* and usual relations for <sup>→</sup>

*knδabδn*,−*<sup>m</sup>* . (25)

*J <sup>n</sup>* obey the same Kac-Moody

<sup>3</sup> for *k* = 1. The interacting

. (26)

J *<sup>n</sup>* follows. Then,

*<sup>k</sup>* . Let

*<sup>k</sup>* \ CH. The boundary of it

*<sup>k</sup>*\CH *<sup>R</sup>*√*gd*4*<sup>x</sup>* from this boundary

*S*, i.e. [*Sa*, *Sb*] =

J *<sup>n</sup>*, *n* = ... − 2, −1, 0, 1, 2, ...:

*<sup>m</sup>*]*<sup>k</sup>* <sup>=</sup> *<sup>i</sup>�abc*<sup>J</sup> *<sup>c</sup>*

<sup>J</sup> *<sup>n</sup>* <sup>=</sup> <sup>→</sup>

<sup>J</sup> *<sup>n</sup>* as <sup>→</sup>

*<sup>J</sup> <sup>n</sup>*where the coupling constant *<sup>λ</sup>* is taken as <sup>2</sup>

+∞ ∑ −∞

(*c*, *c*� are some constants).

→ *<sup>J</sup>* <sup>−</sup>*<sup>n</sup>* · → *J <sup>n</sup>* + *λ*

<sup>3</sup> , one completes the square and the algebra (25) for the currents <sup>→</sup>

A similar procedure holds for arbitrary integer *k* where the spin part of the Hamiltonian reads:

the WZW model on *SU*(2). This model defines the Verlinde fusion rules and is determined by these. The following *fusion rules hypothesis*, was proposed by Affleck (1995), which explains

*The infrared fixed point, in the k-channel spin-s Kondo problem, is given by fusion with the spin-s primary for s k*/2 *or with the spin k*/2 *primary for s* > *k*/2*.* Thus, the level *k* Kac-Moody algebra, as in the level *k* WZW *SU*(2) model, governs the behaviour of the Kondo state in the

This is also the reason why, already in low temperatures, entangled magnetic matter of impurities and conduction electrons indicates the correlation with exotic 4-geometry. First, every CH generates a fermion field. Every small exotic **R**<sup>4</sup> can be represented as handlebody where Akbulut cork has several CH's attached. The important thing is that the handlenbody has a boundary and only after removing it the interior is diffeomorphic to, say, exotic **R**<sup>4</sup>

> *∂*(**R**<sup>4</sup> *<sup>k</sup>*\CH)

where *KCH* is the trace of the 2-nd fundamental form and *g<sup>∂</sup>* the metric on the boundary Asselmeyer-Maluga & Brans (2011). But as shown in Asselmeyer-Maluga & Brans (2011) this

presence of *k* channels of conducting electrons and magnetic impurity of spin *s*.

*R*√*gd*4*x* +

*<sup>k</sup>* \ CH). The contribution to the Einstein action

*<sup>n</sup>*+*<sup>m</sup>* + <sup>1</sup>

 1 3

[<sup>J</sup> *<sup>a</sup> <sup>n</sup>* , <sup>J</sup> *<sup>b</sup>*

*Hs* = *c*

the creation and nature of the multichannel Kondo states:

us remove a single CH from the handlebody **R**<sup>4</sup>

 *∂*(**R**<sup>4</sup> *<sup>k</sup>*\CH)

*<sup>m</sup>*]*<sup>k</sup>* = *i�abc J<sup>c</sup>*

Hamiltonian of the theory, for *k* = 1, reads:

on 3-components currents <sup>→</sup>

algebra, i.e. [*J<sup>a</sup>*

*i�abcSc*, [*Sa*, *J<sup>b</sup>*

with spins, <sup>→</sup>

*<sup>S</sup>* with <sup>→</sup>

spins <sup>→</sup>

For *λ* = <sup>2</sup>

<sup>→</sup> J −*n* ·

2*π*(*k*+2)

<sup>2</sup>+*<sup>k</sup>* . The spins <sup>→</sup>

*c*� ∑+<sup>∞</sup> −∞

2

*Hs*,*<sup>k</sup>* = <sup>1</sup>

reads *∂*(**R**<sup>4</sup>

is the suitable surface term:

Next, we decompose the currents <sup>→</sup>

*<sup>n</sup>*, *J<sup>b</sup>*

the new Hamiltonian, where <sup>→</sup>

→ <sup>J</sup> *<sup>n</sup>* <sup>−</sup> <sup>3</sup> 4 

→ *J* 2 + *λ* → *J* · → term is expressed by the spinor field *ψ* describing the immersion of *D*<sup>2</sup> into **R**3, which extends to the immersion of *<sup>D</sup>*<sup>2</sup> <sup>×</sup> (0, 1) into **<sup>R</sup>**4:

$$\int\_{\partial(\mathbb{R}^4\_k \backslash \mathbf{CH})} K\_{\mathbb{CH}} \sqrt{\mathfrak{g}\_{\partial}} d^3 \mathbf{x} = \int\_{\partial(\mathbb{R}^4\_k \backslash \mathbf{CH})} \psi \gamma^\mu D\_\mu \overline{\psi} \sqrt{\mathfrak{g}\_{\partial}} d^3 \mathbf{x} \,. \tag{27}$$

This can be extended to 4-dimensional Einstein-Hilbert action with the source depending on the CH, hence on exotic **R**<sup>4</sup> *k* :

$$S\_4^{\rm CH}(\mathbb{R}\_k^4) = \int\_{\mathbb{R}\_k^4 \backslash \rm CH} (\mathbb{R} + \psi \gamma^\mu D\_\mu \overline{\psi}) \sqrt{\chi} d^4 \mathbf{x} \,. \tag{28}$$

Again, it was shown in Asselmeyer-Maluga & Brans (2011) that the spinor field *ψ* extends over whole 4-manifold such that the 4D Dirac equations are fulfilled. This way we have fermion fields which are determined by CH. Moreover, this fermions plays a role of gravity sources as in 28. In fact every infinite branch of the CH determines some 4D fermion.

Second, given exotic **R**<sup>4</sup> *<sup>p</sup>* we have *r* Casson handles in its handlebody. These *r* CH's generate effective *q*(*r*)-many infinite branches. Each such branch generates a fermion field. Attaching the CH's to the cork results in exotic **R**<sup>4</sup> *<sup>p</sup>*. Hence, *p* is the function of *q* in general, *p* = *p*(*q*(*r*)).

Let us assign now the simplest possible CH to every CH in the handlebody of exotic **R**4, such that replacing the original CH by this simple one does not change the exotic smoothness. This is the model handlebody we refer to in the context of the Kondo effect (see Fig. 3 for the examples of the simplest CH's).

The *k*-channel Kondo state, in the *k*-channel Kondo effect, is the entangled state of conducting electrons in *k* bands and the magnetic spin *s* impurity. The physics of resulting state is described by BCFT by the Verlinde fusion rules in *SU*(2)*<sup>k</sup>* WZW model. To have the WZ term in this WZW model one certainly needs *p* = *k*. This kWZ term is generated by exotic **R**<sup>4</sup> *k* as we explained in Sec. 2. The draft of the dependance of the number of infinite branches on the function of the number of CH's in the handlebody of **R**<sup>4</sup> *<sup>k</sup>* , is presented in Fig. 3a. Fig. 3b shows the example of the ramified structure of CH's in the precise language of the graphical Kirby calculus (see e.g. Gompf & Stipsicz (1999)).

The general correspondence appears:

*One assigns the 4-smooth geometry on* **R**<sup>4</sup> *to the k-channel Kondo effect such that k corresponds to the number of infinite branches of CH's in the handlebody. This 4-geometry is* **R**<sup>4</sup> *<sup>p</sup> where p* = *p*(*k*)*, p*, *k* ∈ **N***. The change between the physical Kondo states, from this emerging in k*<sup>1</sup> *channel Kondo effect to this with k*<sup>2</sup> *channels, k*<sup>1</sup> �= *k*2*, corresponds to the change between 4-geometries, from exotic* **R**<sup>4</sup> *<sup>p</sup>*<sup>1</sup> *to* **<sup>R</sup>**<sup>4</sup> *<sup>p</sup>*<sup>2</sup> *, p*<sup>1</sup> �= *p*2*, p*1, *p*<sup>2</sup> ∈ **N***, such that p*<sup>1</sup> = *p*1(*k*1) *and p*<sup>2</sup> = *p*2(*k*2) *as above.*

Whether actually *p* = *k* or not is the question about the level of the *SU*(2) WZW model and the corresponding fusion rules in use. If *k* = *p* the exotic geometry gives the same fusion rules as the Affleck proposed. In the case *k* �= *p* and *k* < *p* in the *k*-channel Kondo effect the fusion rules derived from the exotic geometry are those of the *SU*(2)*<sup>p</sup>* WZW model. It would be interesting to decide experimentally, which fusion rules apply for bigger *k*. Probably in higher energies, if the Kondo state survives, the proper fusion rules are those derived from exotic **R**<sup>4</sup> *<sup>p</sup>*. This reflects the situation that electrons in different conduction bands (channels) are generated potentially by (each infinite branch of) Casson handles from the handlebody of

**5. From smooth geometry of string backgrounds to quantum D-branes**

strings we have additionally gauge field *F<sup>a</sup>*

quantum D-branes regim in string theory.

as in (3). Moreover, supposing dilaton is constant and *F<sup>a</sup>*

are discussed in what follows.

exotic **R**4's.

One could wonder what is, if any, suitable sense assigned to geometry of spacetime in various string constructions or backgrounds. As we know the geometry of GR, hence, classical gravity, is the one of (pseudo)-Riemannian differentiable manifolds. String theory has GR (10D Einstein equations) as its classical gravitational limit; however, string theory is the theory of QG and the spacetime geometry should be modified. What is the fate of this (pseudo) Riemannian geometry when gravity is quantized? To answer this question we should find correct classical limit for some quantum string constructions. The proper way is to consider the string backgrounds. These are semi-classical solutions in string theory or supergravity, around which one develops a purturbative theory. GR is not the only ingredient of classical geometry in string theory. There are other fields which are equally fundamental. In type II we have metric *Gμν*, antisymmetric *H*-field, i.e. three-form *Hμνρ*, and dilaton Φ. In heterotic

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 71

The presence of *B*-field such that *H* is represented by the non-zero cohomology class (see below), is a highly non-trivial fact and indicates that the correct, semi-classical, geometry for string theory is one based on *abelian gerbes* as supplementing Riemannian geometry Król (2010a;b); Segal (2001). Small exotic **R**4's show strong connections with abelian gerbes on *S*<sup>3</sup> Asselmeyer-Maluga & Król (2009a) which has many important consequences. Some of them

Another crucial feature is the role assigned to D- and NS-branes. Closed string theory, as we made use of it in Sec. 3, is not complete in the sense that there are possible boundary conditions, Dirichlet (D) or Neveu-Schwarz (NS), for open strings, already appearing in closed string theories. These boundary conditions determine geometric subspaces on which open strings can end. In that sense open string theory complements the closed one and predicts the existence of D- or NS-branes. This tame picture of branes as subspaces has only very limited validity. In the quantum regime, or even in the non-zero string coupling *gs*, the picture of D-branes as above fails Aspinwall (2004). Nevertheless, interesting proposals were presented recently. They are based on the ideas from non-commutative geometry and aim toward replacing D-branes and spacetime by corresponding (sub) *C*-algebras Brodzki et al. (2008a;b); Szabo (2008). Surprisingly, such an *C*-algebraic setting again shows deep connections with

The appearance of the codimension-one foliations of *S*<sup>3</sup> in the structure of small exotic **R**4, is the key for the whole spectrum of the connections of exotics, beginning with differential geometry and topology, up to non-commutative geometry. This opens very atractive possibilities for exploring both, 1) the classical limit of string geometry, as above and 2)

Let us comment on 1) above. The presence of non-zero *B*-field in a string background is crucial from the point of view of resulting geometry: in *σ*-model the *B*-field modifies metric

of (8) (the *β*-function), enforces the background be non-flat, unless *H* = *dB* is zero. Given *S*<sup>3</sup> part of the linear dilaton background as in Sec. 3, we have non-trivial *H*-field on it. The topological classification of *H*-fields is given by 3-rd de Rham cohomology classes on background manifold *M*, *H*3(*M*, **R**). In order to avoid anomalies we restrict to the integral case *H*3(*S*3, **Z**) for *M* = *S*3. These classes however are equally generated by exotic **R**<sup>4</sup>

(see Sec. 2). On the other hand, the classification of D-branes in string backgrounds is

*μν* and the calculations of Sec. 3 made use of these.

*μν* vanishes, the second equation

*<sup>k</sup>* , *k* ∈ **Z**

Fig. 3. (a) (Redrawn from Asselmeyer-Maluga & Brans (2011)) Two CH's, the upper one with 3 infinite branches, the lower one is the simplest CH with the single infinite labeled branch with a single intersection point at every stage. This CH appears in the simplest possible exotic smooth **R**4. (b) Schematic structure of the *r* = 4 CH's in the handlebody of exotic **R**<sup>4</sup> *p*. Each infinite branch generates a fermion field hence, this exotic **R**<sup>4</sup> *<sup>p</sup>* can model the Kondo state in the *k* = 6-channels Kondo effect.

the exotic **R**<sup>4</sup> *<sup>p</sup>*, though not every CH generates the actual channel contributing to the Kondo effect. The higher energy more potential CH contributes to the actual electron bands. Then, the fusion rules are given by the exotic geometry, i.e. *SU*(2)*<sup>p</sup>* WZW model. In suitable high energies 4-geometry (Casson handles) acts as anihilation or creation operator for fermions (electrons). This is the content of our *relativistic fusion rule* (RFR) hypothesis. The experimental confirmation of such discrepancy (between the levels of the WZW model) in high energies in Kondo effect for *p* > 2 channels, would serve as indication for the role of 4-exotic geometry in the relativistic limit of the Kondo state.

Let us illustrate this hypothesis and consider the simplest CH and the simplest exotic **R**<sup>4</sup> described by Bizaca & Gompf (1996). Suppose this exotic ˘ **R**<sup>4</sup> <sup>1</sup> is the member of the radial family and its radius, hence GV invariant of the foliation of *S*3, is equal to 1. The corresponding WZ term would be then derived from the *SU*(2)<sup>1</sup> WZW model. Thus, in this case, there is precisely one channel of conducting electrons in the Kondo effect. More complicated exotic **R**<sup>4</sup> <sup>2</sup> could have two CH's in the handlebody and the radii equal to <sup>√</sup>2. Two channels of conducting electrons give rise to the *SU*(2)<sup>2</sup> WZ fusion rules. However, more complicated exotic **R**<sup>4</sup> *p*, *p* > 2, could spoil this 1 to 1 correspondence between number of CH's and the number of channels in the Kondo effect.

We have derived the trace of the (exotic) 4-geometry in the low energy Kondo effect. This geometry is probably not physically valid at energies of the Kondo effect (as gravity is not). However, exotic **R**<sup>4</sup> *<sup>k</sup>* in high energy (and relativistic) limit can become dominating or giving viable physical contributions. These contributions appear when geometric CH's become the actual sources for fermions in KE, thus, changing its CFT structure. In fact the appearance of non-flat **R**<sup>4</sup> *<sup>k</sup>* when describing the *p*-channel Kondo effect, indicates a new fundamental link between *matter, geometry* and *gravity* in dimension 4.

18 Will-be-set-by-IN-TECH

(a) (b)

Fig. 3. (a) (Redrawn from Asselmeyer-Maluga & Brans (2011)) Two CH's, the upper one with 3 infinite branches, the lower one is the simplest CH with the single infinite labeled branch with a single intersection point at every stage. This CH appears in the simplest possible exotic smooth **R**4. (b) Schematic structure of the *r* = 4 CH's in the handlebody of exotic **R**<sup>4</sup>

effect. The higher energy more potential CH contributes to the actual electron bands. Then, the fusion rules are given by the exotic geometry, i.e. *SU*(2)*<sup>p</sup>* WZW model. In suitable high energies 4-geometry (Casson handles) acts as anihilation or creation operator for fermions (electrons). This is the content of our *relativistic fusion rule* (RFR) hypothesis. The experimental confirmation of such discrepancy (between the levels of the WZW model) in high energies in Kondo effect for *p* > 2 channels, would serve as indication for the role of 4-exotic geometry in

Let us illustrate this hypothesis and consider the simplest CH and the simplest exotic **R**<sup>4</sup>

and its radius, hence GV invariant of the foliation of *S*3, is equal to 1. The corresponding WZ term would be then derived from the *SU*(2)<sup>1</sup> WZW model. Thus, in this case, there is precisely one channel of conducting electrons in the Kondo effect. More complicated exotic **R**<sup>4</sup>

have two CH's in the handlebody and the radii equal to <sup>√</sup>2. Two channels of conducting electrons give rise to the *SU*(2)<sup>2</sup> WZ fusion rules. However, more complicated exotic **R**<sup>4</sup>

*p* > 2, could spoil this 1 to 1 correspondence between number of CH's and the number of

We have derived the trace of the (exotic) 4-geometry in the low energy Kondo effect. This geometry is probably not physically valid at energies of the Kondo effect (as gravity is not).

viable physical contributions. These contributions appear when geometric CH's become the actual sources for fermions in KE, thus, changing its CFT structure. In fact the appearance of

*<sup>k</sup>* in high energy (and relativistic) limit can become dominating or giving

*<sup>k</sup>* when describing the *p*-channel Kondo effect, indicates a new fundamental link

*<sup>p</sup>*, though not every CH generates the actual channel contributing to the Kondo

Each infinite branch generates a fermion field hence, this exotic **R**<sup>4</sup>

described by Bizaca & Gompf (1996). Suppose this exotic ˘ **R**<sup>4</sup>

between *matter, geometry* and *gravity* in dimension 4.

state in the *k* = 6-channels Kondo effect.

the relativistic limit of the Kondo state.

channels in the Kondo effect.

However, exotic **R**<sup>4</sup>

non-flat **R**<sup>4</sup>

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

*p*.

<sup>2</sup> could

*p*,

*<sup>p</sup>* can model the Kondo

<sup>1</sup> is the member of the radial family
