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

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

from *SU*(2)*<sup>k</sup>* × **R***φ*, so *k* is even. Here *SU*(2)*<sup>k</sup>* is the affine Kac-Moody algebra at level *k* and **R***<sup>φ</sup>* the linear dilaton, both appearing in the exact string background realized by the superconformal 2D field theory (see, e.g. Kiritsis & Kounnas (1995b)). The deformation of such curved 4D part of the background will be performed in heterotic superstring theory in the language of *σ*-model. The deformations will correspond to the introducing almost constant magnetic field *H* and its gravitational backreaction on the 4D curved part of the background. First let us describe undeformed theory. The action for heterotic *σ*-model in this

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 61

*∂x*0*∂x*<sup>0</sup> + *ψ*0*∂ψ*<sup>0</sup> +

*SU*(2) = *S*3, *R*(2) is the 2D worldsheet curvature, *g* is the determinant of the target metric and *Q* is the dilaton charge with *x*<sup>0</sup> the coordinate of **R***φ*. The bosonic *σ*-model action reads in

One can decompose (see e.g. Prezas & Sfetsos (2008)) the supersymmetric WZW model into the bosonic *SU*(2)*k*−<sup>2</sup> with affine currents *<sup>J</sup><sup>i</sup>* and three free fermions *<sup>ψ</sup>a*, *<sup>a</sup>* <sup>=</sup> 1, 2, 3 in the adjoint representation of *SU*(2). As the result the supersymmetric N = 1 affine

<sup>3</sup> <sup>+</sup> *<sup>ψ</sup>*+*ψ*−, <sup>J</sup> <sup>±</sup> <sup>=</sup> *<sup>J</sup>*

Let us redefine the indices in the fermion fields as: <sup>+</sup> <sup>→</sup> 1, − → 2, then <sup>J</sup> <sup>3</sup> <sup>=</sup> *<sup>J</sup>*<sup>3</sup> <sup>+</sup> *<sup>ψ</sup>*1*ψ*2.

From the point of view of the *σ*-model, the vertex for the magnetic field *H* on 4-dimensional **<sup>R</sup>***<sup>φ</sup>* <sup>×</sup> *SU*(2)*<sup>k</sup>* part of the background is the exact marginal operator given by *Vm* <sup>=</sup> *<sup>H</sup>*(*J*<sup>3</sup> <sup>+</sup>

The shape of these operators follow from the fact that, in general, the marginal deformations of the WZW model can be constructed as bilinears in the currents *J*, *J* of the model [Orlando

> *i*,*j cijJ i* (*z*)*J<sup>j</sup>*

, *J<sup>j</sup>* are left and right-moving affine currents respectively, Orlando (2007).

Here, following Kiritsis & Kounnas (1995b), we consider covariantly constant magnetic field

*jk* and constant curvature <sup>R</sup>*il* <sup>=</sup> *�ijk�lmn*R*jmkn* in the 4-dimensional background as

O(*z*, *<sup>z</sup>*) = ∑

. Similarly, the vertex for the corresponding gravitational part is *Vgr* <sup>=</sup> <sup>R</sup>(*J*<sup>3</sup> <sup>+</sup>

(*ψ*<sup>1</sup> <sup>±</sup> *<sup>i</sup>ψ*2) and the corresponding change of the affine bosonic currents *<sup>J</sup>*<sup>±</sup> <sup>=</sup> *<sup>J</sup>*<sup>1</sup> <sup>±</sup> *i J*2,

*d*2*z*(*Gμν* + *Bμν*)*∂xμ∂x<sup>ν</sup>* +

*G*<sup>00</sup> = 1, *Gαα* = *Gββ* = *Gγγ* = *<sup>k</sup>*

3 ∑ *a*=1

1 4*π*

<sup>4</sup> , *<sup>G</sup>αγ* <sup>=</sup> *<sup>k</sup>*

<sup>2</sup> *�abcψbψc*. After introducing the complex fermions combination

<sup>±</sup> <sup>±</sup> <sup>√</sup>

*ψa∂ψ<sup>a</sup>* <sup>+</sup> *<sup>Q</sup>* 4*π*

*∂α∂α* + *∂β∂β* + *∂γ∂γ* + 2*cosβ∂α∂γ* in Euler angles of

<sup>√</sup>*gR*(2)

<sup>4</sup> cos *β*

<sup>4</sup> cos *<sup>β</sup>*, <sup>Φ</sup> <sup>=</sup> *Qx*<sup>0</sup> <sup>=</sup> *<sup>x</sup>*<sup>0</sup> <sup>√</sup>*k*+<sup>2</sup> . (4)

<sup>√</sup>*gR*(2)

Φ(*x*) (3)

2*ψ*3*ψ*<sup>±</sup> (5)

(*z*) (6)

*x*<sup>0</sup> (2)

*SO*(3)*k*/2 × **R***<sup>φ</sup>* background is:

where I*SO*(3)(*α*, *<sup>β</sup>*, *<sup>γ</sup>*) = <sup>1</sup>

currents are <sup>J</sup> *<sup>a</sup>* <sup>=</sup> *<sup>J</sup><sup>a</sup>* <sup>−</sup> *<sup>i</sup>*

the supersymmetric affine currents read:

<sup>I</sup>*SO*(3)(*α*, *<sup>β</sup>*, *<sup>γ</sup>*) + <sup>1</sup>

*<sup>S</sup>* <sup>=</sup> <sup>1</sup> 2*π* 

2*π d*2*z* 

2*π d*2*z* 

so comparing with (2) gives the non-zero background fields as:

<sup>J</sup> <sup>3</sup> <sup>=</sup> *<sup>J</sup>*

, and represents truly marginal deformations too.

*Bαγ* = *<sup>k</sup>*

*<sup>S</sup>*<sup>4</sup> <sup>=</sup> *<sup>k</sup>* 4

general:

*ψ*± = √ 1 2

*ψ*1*ψ*2)*J a*

*ψ*1*ψ*2)*J* 3

(2007)]:

where *J<sup>i</sup>*

*<sup>i</sup>* <sup>=</sup> *�ijkFa*

*Ha*

Fig. 1. a) *j*<sup>1</sup> is the change of the standard smooth **R**<sup>4</sup> to the exotic **R**<sup>4</sup> *<sup>k</sup>* , end(**R**4) assigns the standard end to **<sup>R</sup>**4, *GV*(F*S*<sup>3</sup> ) generates the WZk-term from exotic **<sup>R</sup>**<sup>4</sup> *<sup>k</sup>* via *GV* invariant of the codim.-1 foliation of *S*3. b) The change of string backgrounds s.t. flat **R**<sup>4</sup> part is replaced by the linear dilaton background *SU*(2)*<sup>k</sup>* × **R***<sup>φ</sup>*

gravity is of (pseudo-)Riemannian smooth geometry and should change to a new 'geometry, or to, unknown at all, mathematics, when the transformation of GR to QG is performed.

As discussed already in the Introduction, it was proposed in Asselmeyer-Maluga & Król (2010; 2011c;d; 2012); Król (2011a;b) that 4-dimensional effects of string theory should be seen via its connections with exotic smoothness of topologically trivial **R**4. Several results were derived as if superstring theory were formulated on backgrounds which contain 4-dimensional part which is exotic **R**<sup>4</sup> rather than standard smooth **R**4. This serves as a new window to 4-dimensional physics. The argumentation dealt with exact string backgrounds in any order of *α*� . The existence of such backgrounds is rather exceptional in superstring theory (see e.g. Orlando (2006; 2007)) and this always indicates important and exactly calculable effects of the theory. This is precisely the tool which we want to apply to the above stated problems, i.e. the description of both, 4D QG effects due to gravity present in background spacetime, and mathematics behind the shift GR → QG in 4D.

Exotic **R**<sup>4</sup> *<sup>k</sup>* is a smooth Riemannian manifold, however, its structure essentially deals with non-commutative geometry and quantization Asselmeyer-Maluga & Król (2011b). The connection with string exact backgrounds was also recognized in Asselmeyer-Maluga & Król (2010; 2011c). Thus, under the topological assumptions discussed in Sec. 2, the following correspondence emerges:

The change of the smoothness from the standard **R**<sup>4</sup> to exotic **R**<sup>4</sup> *<sup>k</sup>* , corresponds to the change of exact string backgrounds from **<sup>R</sup>**<sup>4</sup> <sup>×</sup> *<sup>K</sup>*<sup>6</sup> to *SU*(2)*<sup>k</sup>* <sup>×</sup> **<sup>R</sup>***<sup>φ</sup>* <sup>×</sup> *<sup>K</sup>*6.

Let us note that only because of the *exotic* smooth structure of **R**4, the link to string backgrounds exists. If smoothness of **R**<sup>4</sup> were standard, only separated regimes of 4-geometry (GR) and superstrings (QG) would appear. In superstring theory one understands fairy well how to change the exact background containing flat **R**<sup>4</sup> to this with curved 4-dimensional part: **<sup>R</sup>**<sup>4</sup> <sup>×</sup> *<sup>K</sup>*<sup>6</sup> <sup>→</sup> *SU*(2)*<sup>k</sup>* <sup>×</sup> **<sup>R</sup>***<sup>φ</sup>* <sup>×</sup> *<sup>K</sup>*6. This requires supersymmetry in 10 dimensions. The presence of supersymmetry is, however, just a technical mean allowing for the consistent shifts between the backgrounds and performing the QG calculations effectively.

## **3.1 The magnetic deformation of 4D part of the string background**

To be specific let us consider the *SO*(3)*k*/2 × **R***<sup>φ</sup>* as the 4D part of the 10D string background which replaces the flat **<sup>R</sup>**<sup>4</sup> part. This *SO*(3)*k*/2 <sup>×</sup> **<sup>R</sup>***<sup>φ</sup>* geometry is the result of the projection 8 Will-be-set-by-IN-TECH

b) flat **R**<sup>4</sup>

sustring

 *SU*(2)*<sup>k</sup>* × **R***<sup>φ</sup>*

*<sup>k</sup>* , end(**R**4) assigns the

*<sup>k</sup>* via *GV* invariant of the

*<sup>k</sup>* , corresponds to the

*GV*(F*S*<sup>3</sup> )

codim.-1 foliation of *S*3. b) The change of string backgrounds s.t. flat **R**<sup>4</sup> part is replaced by

gravity is of (pseudo-)Riemannian smooth geometry and should change to a new 'geometry, or to, unknown at all, mathematics, when the transformation of GR to QG is performed.

As discussed already in the Introduction, it was proposed in Asselmeyer-Maluga & Król (2010; 2011c;d; 2012); Król (2011a;b) that 4-dimensional effects of string theory should be seen via its connections with exotic smoothness of topologically trivial **R**4. Several results were derived as if superstring theory were formulated on backgrounds which contain 4-dimensional part which is exotic **R**<sup>4</sup> rather than standard smooth **R**4. This serves as a new window to 4-dimensional physics. The argumentation dealt with exact string backgrounds in any order

. The existence of such backgrounds is rather exceptional in superstring theory (see e.g. Orlando (2006; 2007)) and this always indicates important and exactly calculable effects of the theory. This is precisely the tool which we want to apply to the above stated problems, i.e. the description of both, 4D QG effects due to gravity present in background spacetime, and

*<sup>k</sup>* is a smooth Riemannian manifold, however, its structure essentially deals with

non-commutative geometry and quantization Asselmeyer-Maluga & Król (2011b). The connection with string exact backgrounds was also recognized in Asselmeyer-Maluga & Król (2010; 2011c). Thus, under the topological assumptions discussed in Sec. 2, the following

Let us note that only because of the *exotic* smooth structure of **R**4, the link to string backgrounds exists. If smoothness of **R**<sup>4</sup> were standard, only separated regimes of 4-geometry (GR) and superstrings (QG) would appear. In superstring theory one understands fairy well how to change the exact background containing flat **R**<sup>4</sup> to this with curved 4-dimensional part: **<sup>R</sup>**<sup>4</sup> <sup>×</sup> *<sup>K</sup>*<sup>6</sup> <sup>→</sup> *SU*(2)*<sup>k</sup>* <sup>×</sup> **<sup>R</sup>***<sup>φ</sup>* <sup>×</sup> *<sup>K</sup>*6. This requires supersymmetry in 10 dimensions. The presence of supersymmetry is, however, just a technical mean allowing for the consistent shifts between

To be specific let us consider the *SO*(3)*k*/2 × **R***<sup>φ</sup>* as the 4D part of the 10D string background which replaces the flat **<sup>R</sup>**<sup>4</sup> part. This *SO*(3)*k*/2 <sup>×</sup> **<sup>R</sup>***<sup>φ</sup>* geometry is the result of the projection

The change of the smoothness from the standard **R**<sup>4</sup> to exotic **R**<sup>4</sup>

change of exact string backgrounds from **<sup>R</sup>**<sup>4</sup> <sup>×</sup> *<sup>K</sup>*<sup>6</sup> to *SU*(2)*<sup>k</sup>* <sup>×</sup> **<sup>R</sup>***<sup>φ</sup>* <sup>×</sup> *<sup>K</sup>*6.

the backgrounds and performing the QG calculations effectively.

**3.1 The magnetic deformation of 4D part of the string background**

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

*<sup>S</sup>*<sup>3</sup> <sup>×</sup> **<sup>R</sup>** *<sup>j</sup>*<sup>2</sup> *SU*(2)*<sup>k</sup>* <sup>×</sup> **<sup>R</sup>***<sup>φ</sup>*

Fig. 1. a) *j*<sup>1</sup> is the change of the standard smooth **R**<sup>4</sup> to the exotic **R**<sup>4</sup>

standard end to **<sup>R</sup>**4, *GV*(F*S*<sup>3</sup> ) generates the WZk-term from exotic **<sup>R</sup>**<sup>4</sup>

a) flat **<sup>R</sup>**<sup>4</sup> *<sup>j</sup>*<sup>1</sup>

the linear dilaton background *SU*(2)*<sup>k</sup>* × **R***<sup>φ</sup>*

mathematics behind the shift GR → QG in 4D.

end(**R**<sup>4</sup>)

of *α*�

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

correspondence emerges:

from *SU*(2)*<sup>k</sup>* × **R***φ*, so *k* is even. Here *SU*(2)*<sup>k</sup>* is the affine Kac-Moody algebra at level *k* and **R***<sup>φ</sup>* the linear dilaton, both appearing in the exact string background realized by the superconformal 2D field theory (see, e.g. Kiritsis & Kounnas (1995b)). The deformation of such curved 4D part of the background will be performed in heterotic superstring theory in the language of *σ*-model. The deformations will correspond to the introducing almost constant magnetic field *H* and its gravitational backreaction on the 4D curved part of the background. First let us describe undeformed theory. The action for heterotic *σ*-model in this *SO*(3)*k*/2 × **R***<sup>φ</sup>* background is:

$$\mathbf{S}\_{4} = \frac{k}{4} \mathbf{I}\_{\mathrm{SO}(3)}(\mathbf{a}, \boldsymbol{\theta}, \boldsymbol{\gamma}) + \frac{1}{2\pi} \int d^2 z \left[ \partial \mathbf{x}^0 \overline{\partial} \mathbf{x}^0 + \boldsymbol{\psi}^0 \partial \boldsymbol{\psi}^0 + \sum\_{a=1}^3 \boldsymbol{\psi}^a \partial \boldsymbol{\psi}^a \right] + \frac{Q}{4\pi} \int \sqrt{g} \mathbf{R}^{(2)} \mathbf{x}^0 \tag{2}$$

where I*SO*(3)(*α*, *<sup>β</sup>*, *<sup>γ</sup>*) = <sup>1</sup> 2*π d*2*z ∂α∂α* + *∂β∂β* + *∂γ∂γ* + 2*cosβ∂α∂γ* in Euler angles of *SU*(2) = *S*3, *R*(2) is the 2D worldsheet curvature, *g* is the determinant of the target metric and *Q* is the dilaton charge with *x*<sup>0</sup> the coordinate of **R***φ*. The bosonic *σ*-model action reads in general:

$$S = \frac{1}{2\pi} \int d^2 z (G\_{\mu\nu} + B\_{\mu\nu}) \partial x^{\mu} \overline{\partial} x^{\nu} + \frac{1}{4\pi} \int \sqrt{g} \mathcal{R}^{(2)} \Phi(\mathbf{x}) \tag{3}$$

so comparing with (2) gives the non-zero background fields as:

$$\begin{aligned} \mathbf{G\_{00}} &= 1, \; \mathbf{G\_{a\alpha}} = \mathbf{G\_{\beta\beta}} = \mathbf{G\_{\gamma\gamma}} = \frac{k}{4}, \; \mathbf{G\_{\alpha\gamma}} = \frac{k}{4} \cos \beta\\ \mathbf{B\_{\alpha\gamma}} &= \frac{k}{4} \cos \beta, \; \Phi = \mathbf{Q} \mathbf{x}^0 = \frac{\mathbf{x}^0}{\sqrt{k+2}}. \end{aligned} \tag{4}$$

One can decompose (see e.g. Prezas & Sfetsos (2008)) the supersymmetric WZW model into the bosonic *SU*(2)*k*−<sup>2</sup> with affine currents *<sup>J</sup><sup>i</sup>* and three free fermions *<sup>ψ</sup>a*, *<sup>a</sup>* <sup>=</sup> 1, 2, 3 in the adjoint representation of *SU*(2). As the result the supersymmetric N = 1 affine currents are <sup>J</sup> *<sup>a</sup>* <sup>=</sup> *<sup>J</sup><sup>a</sup>* <sup>−</sup> *<sup>i</sup>* <sup>2</sup> *�abcψbψc*. After introducing the complex fermions combination *ψ*± = √ 1 2 (*ψ*<sup>1</sup> <sup>±</sup> *<sup>i</sup>ψ*2) and the corresponding change of the affine bosonic currents *<sup>J</sup>*<sup>±</sup> <sup>=</sup> *<sup>J</sup>*<sup>1</sup> <sup>±</sup> *i J*2, the supersymmetric affine currents read:

$$\mathcal{J}^3 = \mathfrak{J}^3 + \mathfrak{\psi}^+ \mathfrak{\psi}^- \text{ , } \mathcal{J}^\pm = \mathfrak{J}^\pm \pm \sqrt{2} \mathfrak{\psi}^3 \mathfrak{\psi}^\pm \tag{5}$$

Let us redefine the indices in the fermion fields as: <sup>+</sup> <sup>→</sup> 1, − → 2, then <sup>J</sup> <sup>3</sup> <sup>=</sup> *<sup>J</sup>*<sup>3</sup> <sup>+</sup> *<sup>ψ</sup>*1*ψ*2.

From the point of view of the *σ*-model, the vertex for the magnetic field *H* on 4-dimensional **<sup>R</sup>***<sup>φ</sup>* <sup>×</sup> *SU*(2)*<sup>k</sup>* part of the background is the exact marginal operator given by *Vm* <sup>=</sup> *<sup>H</sup>*(*J*<sup>3</sup> <sup>+</sup> *ψ*1*ψ*2)*J a* . Similarly, the vertex for the corresponding gravitational part is *Vgr* <sup>=</sup> <sup>R</sup>(*J*<sup>3</sup> <sup>+</sup> *ψ*1*ψ*2)*J* 3 , and represents truly marginal deformations too.

The shape of these operators follow from the fact that, in general, the marginal deformations of the WZW model can be constructed as bilinears in the currents *J*, *J* of the model [Orlando (2007)]:

$$\mathcal{O}(z,\overline{z}) = \sum\_{i,j} c\_{ij} J^i(z) \overline{J^j}(\overline{z}) \tag{6}$$

where *J<sup>i</sup>* , *J<sup>j</sup>* are left and right-moving affine currents respectively, Orlando (2007).

Here, following Kiritsis & Kounnas (1995b), we consider covariantly constant magnetic field *Ha <sup>i</sup>* <sup>=</sup> *�ijkFa jk* and constant curvature <sup>R</sup>*il* <sup>=</sup> *�ijk�lmn*R*jmkn* in the 4-dimensional background as

corresponding exact background of string theory via *σ*-model calculations, Hassan & Sen (1993); Kiritsis & Kounnas (1995b). Again, the fields in this background which solve the

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 63

*G*<sup>00</sup> = 1, *Gββ* = *<sup>k</sup>*

4

(*λ*<sup>2</sup>+1)<sup>2</sup>−(8*H*2*λ*<sup>2</sup>+(*λ*<sup>2</sup>−1)<sup>2</sup>) cos2 *<sup>β</sup>*) (*λ*<sup>2</sup>+1+(*λ*<sup>2</sup>−1) cos *β*)<sup>2</sup>

(*λ*<sup>2</sup>+1)<sup>2</sup>−(8*H*2*λ*<sup>2</sup>−(*λ*<sup>2</sup>−1)<sup>2</sup>) cos<sup>2</sup> *<sup>β</sup>*) (*λ*<sup>2</sup>+1+(*λ*<sup>2</sup>−1) cos *β*)<sup>2</sup>

<sup>4</sup>*λ*<sup>2</sup>(1−2*H*<sup>2</sup>) cos *<sup>β</sup>*+(*λ*<sup>4</sup>−1) sin<sup>2</sup> *<sup>β</sup>* (*λ*<sup>2</sup>+1+(*λ*<sup>2</sup>−1) cos *β*)<sup>2</sup>

> *<sup>λ</sup>*<sup>2</sup>−1+(*λ*<sup>2</sup>+1) cos *<sup>β</sup>* (*λ*<sup>2</sup>+1+(*λ*<sup>2</sup>−1) cos *β*)<sup>2</sup>

(*λ*<sup>2</sup>+1+(*λ*<sup>2</sup>−1) cos *β*)<sup>2</sup>

(*λ*<sup>2</sup>+1+(*λ*<sup>2</sup>−1) cos *β*)<sup>2</sup>

*<sup>λ</sup>* + (*<sup>λ</sup>* <sup>−</sup> <sup>1</sup>

*<sup>λ</sup>* ) cos *β* 

*A<sup>α</sup>* = *H* cos *β* , *A<sup>β</sup>* = 0 , *A<sup>γ</sup>* = *H* . (12)

det*GGμν*(*∂ν* <sup>−</sup> *ieAν*) . (13)

<sup>3</sup> <sup>+</sup> <sup>O</sup>(*R*−1) (15)

by introducing new

2

 *eH*−*m*<sup>2</sup> *R*

(11)

(14)

<sup>√</sup>*<sup>k</sup> <sup>H</sup><sup>λ</sup>* cos *<sup>β</sup>*

<sup>√</sup>*<sup>k</sup> <sup>H</sup><sup>λ</sup>*

*λ* + <sup>1</sup>

The dependence on *λ* shows the existence of gravitational backreaction which was absent in

In the case of field theory in 4 dimensions we introduce the magnetic field on *S*<sup>3</sup> which agrees

√

where *<sup>R</sup>* is the radius of *<sup>S</sup>*3, *<sup>j</sup>* <sup>∈</sup> **<sup>Z</sup>** and <sup>−</sup>*<sup>j</sup>* <sup>≤</sup> *<sup>m</sup>* <sup>≤</sup> *<sup>j</sup>*, as is the case for *SO*(3). In the flat limit we

where magnetic field is pointing into 3-rd direction and the re-scaling of *eH* is performed as *eH* <sup>=</sup> *eH*˜ <sup>+</sup> *<sup>κ</sup><sup>R</sup>* <sup>+</sup> <sup>O</sup>(1), *<sup>m</sup>* <sup>=</sup> *eHR*˜ <sup>2</sup> + (*p*<sup>3</sup> <sup>+</sup> *<sup>κ</sup>*) + <sup>O</sup>(1). This follows from rewriting

*<sup>R</sup>*<sup>2</sup> [*n*(*n* + 1) + |*m*|(2*n* + 1)] +

*<sup>j</sup>*(*<sup>j</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *<sup>m</sup>*<sup>2</sup> + (*eH* <sup>−</sup> *<sup>m</sup>*)<sup>2</sup>

<sup>2</sup> log

**3.2 Field theory vs. string theory spectra of charged particles in standard 4-space**

effective field theory equations (8), are [Kiritsis & Kounnas (1995b)]:

*Gαα* = *<sup>k</sup>* 4

*Gγγ* = *<sup>k</sup>* 4

> *Gαγ* = *<sup>k</sup>* 4

> > *Bαγ* = *<sup>k</sup>* 4

*A<sup>α</sup>* = 2*g*

*A<sup>γ</sup>* = 2*g*

Φ = √ *t <sup>k</sup>*+<sup>2</sup> <sup>−</sup> <sup>1</sup>

with the magnetic part of the string background (10) as:

**<sup>H</sup>** <sup>=</sup> <sup>1</sup>

The energy spectrum for **H** is then given by:

the spectrum (14) as Δ*En*,*<sup>m</sup>* = <sup>1</sup>

parameter *n*: *j* = |*m*| + *n* for |*m*|,*n* ∈ **N**.

The Hamiltonian for a particle with electric charge *e* moving on *S*3, is

where we assume at the beginning that *Gμν* is standard metric on *S*3.

*R*2 

retrieve the Landau spectrum in 3-dimensional space of spinless particles:

Δ*En*,*p*<sup>3</sup> = *eH*˜ (2*n* + 1) + *p*<sup>2</sup>

<sup>Δ</sup>*Ej*,*<sup>m</sup>* <sup>=</sup> <sup>1</sup>

<sup>√</sup>det*<sup>G</sup>* (*∂μ* <sup>−</sup> *ieAμ*)

the purely magnetic deformed background (10).

above of closed superstring theory. When this chromo-magnetic field is in the *μ* = 3 direction the following deformation is proportional to (*J*<sup>3</sup> + *ψ*1*ψ*2)*J* and the right moving current *J* is normalized as < *J*(1)*J*(0) >= *kg*/2. Rewriting the currents in the Euler angles, i.e. *J*<sup>3</sup> = *k*(*∂γ* + cos *β∂α*), *J* <sup>3</sup> = *<sup>k</sup>*(*∂α* + cos *β∂γ*), we obtain for the perturbation of the (heterotic) action in (2), the following expression:

$$\delta \mathbb{S}\_4 = \frac{\sqrt{k \mathbb{k}\_\mathcal{S}} H}{2\pi} \int d^2 z (\partial \gamma + \cos \beta \partial \alpha) \overline{\mathcal{I}} \,. \tag{7}$$

The new *σ*-model with the action *S*<sup>4</sup> + *δS*<sup>4</sup> is again conformally invariant with all orders in *α*� since:

*S*<sup>4</sup> + *δS*<sup>4</sup> = *<sup>k</sup>* <sup>4</sup> <sup>I</sup>*SO*(3)(*α*, *<sup>β</sup>*, *<sup>γ</sup>*) + *<sup>δ</sup>S*<sup>4</sup> <sup>+</sup> *kg* 4*π d*2*z∂φ∂φ* = *<sup>k</sup>* <sup>4</sup> I*SO*(3)(*α*, *β*, *γ* + 2 *kg <sup>k</sup> Hφ*) + *kg* (1−2*H*<sup>2</sup>) 4*π d*2*z∂φ∂φ*. This shows that, in fact the magnetic deformation is exactly marginal. Here we have chosen for the currents *J* and *J*, *∂φ* and *∂φ* correspondingly, as their bosonizations.

The background corresponding to the perturbation (7) is determined by background fields, i.e. a graviton *Gμν*, gauge fields *F<sup>a</sup> μν*, an antisymmetric field (three form) *Hμνρ* and a dilaton Φ, which, in turn, are solutions to the following equations of motion:

$$\begin{aligned} \frac{3}{2} \left[ 4(\nabla \Phi)^2 - \frac{10}{3} \Box \Phi - \frac{2}{3} \mathcal{R} + \frac{1}{12g^2} F^a\_{\mu \nu} F^{a, \mu \nu} \right] &= \mathcal{C} \\ R\_{\mu \nu} - \frac{1}{4} H^2\_{\mu \nu} - \frac{1}{2g^2} F^a\_{\mu \rho} F^{a \rho}\_{\nu} + 2 \nabla\_{\mu} \nabla\_{\nu} \Phi &= 0 \\ \nabla^{\mu} \left[ e^{-2\Phi} H\_{\mu \nu \rho} \right] &= 0 \\ \nabla^{\nu} \left[ e^{-2\Phi} F^a\_{\mu \nu} \right] - \frac{1}{2} F^{a \nu \rho} H\_{\mu \nu \rho} e^{-2\Phi} &= 0 \end{aligned} \tag{8}$$

These are derived from the variations of the following effective 4-dimensional gauge theory action:

$$S = \int d^4x \sqrt{G}e^{-2\Phi} \left[ R + 4(\nabla\Phi)^2 - \frac{1}{12}H^2 - \frac{1}{4g^2}F^a\_{\mu\nu}F^{a,\mu\nu} + \frac{\mathcal{C}}{3} \right] \tag{9}$$

where *C* is the l.h.s. of the first equation in (8). Here *gstr* = 1, the gauge coupling *g*<sup>2</sup> = 2/*kg*, *F<sup>a</sup> μν* <sup>=</sup> *∂μA<sup>ν</sup>* <sup>−</sup> *∂νA<sup>μ</sup>* <sup>+</sup> *<sup>f</sup> abcAb μA<sup>c</sup> <sup>ν</sup>*, *<sup>H</sup>μνρ* <sup>=</sup> *∂μBνρ* <sup>−</sup> <sup>1</sup> 2*g*<sup>2</sup> *Aa μF<sup>a</sup> νρ* <sup>−</sup> <sup>1</sup> <sup>3</sup> *<sup>f</sup> abcAa μA<sup>b</sup> νA<sup>c</sup> ρ* + permutations. *f abc* are structure constants of the gauge group and *A<sup>a</sup> <sup>μ</sup>* is the effective gauge field. One can observe that the term in the square bracket in *Hμνρ* is the Chern-Simons term for the gauge potential *A<sup>a</sup> μ*.

Now, the background complying with these equations and which respects the deformation (7), reads:

$$\begin{aligned} \text{G}\_{00} &= 1, \,\text{G}\_{\beta\beta} = \frac{k}{4}, \,\text{G}\_{a\gamma} = \frac{k}{4} (1 - 2H^2) \cos\beta\\ \text{G}\_{aa} &= \frac{k}{4} (1 - 2H^2 \cos^2\beta), \,\text{G}\_{\gamma\gamma} = \frac{k}{4} (1 - 2H^2), \,\text{B}\_{a\gamma} = \frac{k}{4} \cos\beta\\ \text{A}\_{\theta} &= \text{g} \sqrt{k}H \cos\beta, \,\text{A}\_{\gamma} = \text{g} \sqrt{k}H, \,\text{A} = \frac{\text{x}^0}{\sqrt{k+2}}. \end{aligned} \tag{10}$$

where *H* is the magnetic field as in (7).

Similarly, when gravitational marginal deformations as in the vertex *Vgr* <sup>=</sup> <sup>R</sup>(*J*<sup>3</sup> <sup>+</sup> *<sup>ψ</sup>*1*ψ*2)*<sup>J</sup>* 3 are included, where R is the curvature parameter of the deformation, one can derive 10 Will-be-set-by-IN-TECH

above of closed superstring theory. When this chromo-magnetic field is in the *μ* = 3 direction the following deformation is proportional to (*J*<sup>3</sup> + *ψ*1*ψ*2)*J* and the right moving current *J* is normalized as < *J*(1)*J*(0) >= *kg*/2. Rewriting the currents in the Euler angles, i.e. *J*<sup>3</sup> =

<sup>3</sup> = *<sup>k</sup>*(*∂α* + cos *β∂γ*), we obtain for the perturbation of the (heterotic) action

*d*2*z∂φ∂φ* = *<sup>k</sup>*

*μν*, an antisymmetric field (three form) *Hμνρ* and a dilaton Φ,

= *C*

*μνFa*,*μν* +

*C* 3 

<sup>4</sup> cos *β*

<sup>3</sup> *<sup>f</sup> abcAa*

*<sup>μ</sup>* is the effective gauge

*μA<sup>b</sup> νA<sup>c</sup> ρ* +

*μνFa*,*μν*

*d*2*z∂φ∂φ*. This shows that, in fact the magnetic deformation is exactly marginal.

<sup>3</sup>*<sup>R</sup>* <sup>+</sup> <sup>1</sup>

 = 0

<sup>12</sup>*g*<sup>2</sup> *<sup>F</sup><sup>a</sup>*

<sup>2</sup> *<sup>F</sup>a*,*νρHμνρe*−2<sup>Φ</sup> <sup>=</sup> <sup>0</sup>

<sup>12</sup> *<sup>H</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup>

*<sup>ν</sup>*, *<sup>H</sup>μνρ* <sup>=</sup> *∂μBνρ* <sup>−</sup> <sup>1</sup>

<sup>4</sup>*g*<sup>2</sup> *<sup>F</sup><sup>a</sup>*

2*g*<sup>2</sup> *Aa μF<sup>a</sup> νρ* <sup>−</sup> <sup>1</sup>

<sup>4</sup> (<sup>1</sup> <sup>−</sup> <sup>2</sup>*H*2) cos *<sup>β</sup>*

<sup>4</sup> (<sup>1</sup> <sup>−</sup> <sup>2</sup>*H*2), *<sup>B</sup>αγ* <sup>=</sup> *<sup>k</sup>*

<sup>√</sup>*kH* , <sup>Φ</sup> <sup>=</sup> *<sup>x</sup>*<sup>0</sup> <sup>√</sup>*k*+<sup>2</sup> .

*<sup>ν</sup>* + 2∇*μ*∇*ν*Φ = 0

*d*2*z*(*∂γ* + cos *β∂α*)*J* . (7)

<sup>4</sup> I*SO*(3)(*α*, *β*, *γ* + 2

*kg*

*<sup>k</sup> Hφ*) +

(8)

(9)

(10)

3

*k*(*∂γ* + cos *β∂α*), *J*

*S*<sup>4</sup> + *δS*<sup>4</sup> = *<sup>k</sup>*

bosonizations.

since:

*kg* (1−2*H*<sup>2</sup>) 4*π*

action:

*g*<sup>2</sup> = 2/*kg*, *F<sup>a</sup>*

(7), reads:

in (2), the following expression:

i.e. a graviton *Gμν*, gauge fields *F<sup>a</sup>*

*S* = *d*4*x* √ *Ge*−2<sup>Φ</sup> 

for the gauge potential *A<sup>a</sup>*

3 2  *δS*<sup>4</sup> =

<sup>4</sup> <sup>I</sup>*SO*(3)(*α*, *<sup>β</sup>*, *<sup>γ</sup>*) + *<sup>δ</sup>S*<sup>4</sup> <sup>+</sup> *kg*

which, in turn, are solutions to the following equations of motion:

<sup>4</sup>(∇Φ)<sup>2</sup> <sup>−</sup> <sup>10</sup>

<sup>4</sup> *<sup>H</sup>*<sup>2</sup> *μν* <sup>−</sup> <sup>1</sup> <sup>2</sup>*g*<sup>2</sup> *<sup>F</sup><sup>a</sup> μρFa<sup>ρ</sup>*

permutations. *f abc* are structure constants of the gauge group and *A<sup>a</sup>*

*G*<sup>00</sup> = 1, *Gββ* = *<sup>k</sup>*

<sup>4</sup> (<sup>1</sup> <sup>−</sup> <sup>2</sup>*H*<sup>2</sup> cos<sup>2</sup> *<sup>β</sup>*), *<sup>G</sup>γγ* <sup>=</sup> *<sup>k</sup>*

<sup>√</sup>*kH* cos *<sup>β</sup>* , *<sup>A</sup><sup>γ</sup>* <sup>=</sup> *<sup>g</sup>*

<sup>∇</sup>*<sup>μ</sup>*

*<sup>R</sup>μν* <sup>−</sup> <sup>1</sup>

∇*ν e*−2Φ*F<sup>a</sup> μν* − 1

*μν* <sup>=</sup> *∂μA<sup>ν</sup>* <sup>−</sup> *∂νA<sup>μ</sup>* <sup>+</sup> *<sup>f</sup> abcAb*

*μ*.

*A<sup>α</sup>* = *g*

*Gαα* = *<sup>k</sup>*

where *H* is the magnetic field as in (7).

*kkgH* 2*π*

4*π*

<sup>3</sup> <sup>Φ</sup> <sup>−</sup> <sup>2</sup>

The new *σ*-model with the action *S*<sup>4</sup> + *δS*<sup>4</sup> is again conformally invariant with all orders in *α*�

Here we have chosen for the currents *J* and *J*, *∂φ* and *∂φ* correspondingly, as their

The background corresponding to the perturbation (7) is determined by background fields,

*e*−2Φ*Hμνρ*

These are derived from the variations of the following effective 4-dimensional gauge theory

where *C* is the l.h.s. of the first equation in (8). Here *gstr* = 1, the gauge coupling

field. One can observe that the term in the square bracket in *Hμνρ* is the Chern-Simons term

Now, the background complying with these equations and which respects the deformation

<sup>4</sup> , *<sup>G</sup>αγ* <sup>=</sup> *<sup>k</sup>*

Similarly, when gravitational marginal deformations as in the vertex *Vgr* <sup>=</sup> <sup>R</sup>(*J*<sup>3</sup> <sup>+</sup> *<sup>ψ</sup>*1*ψ*2)*<sup>J</sup>*

are included, where R is the curvature parameter of the deformation, one can derive

*<sup>R</sup>* <sup>+</sup> <sup>4</sup>(∇Φ)<sup>2</sup> <sup>−</sup> <sup>1</sup>

*μA<sup>c</sup>*

corresponding exact background of string theory via *σ*-model calculations, Hassan & Sen (1993); Kiritsis & Kounnas (1995b). Again, the fields in this background which solve the effective field theory equations (8), are [Kiritsis & Kounnas (1995b)]:

$$G\_{00} = 1 \; \sigma\_{\beta\beta} = \frac{k}{4}$$

$$G\_{\text{flax}} = \frac{k}{4} \frac{(\lambda^2 + 1)^2 - (8H^2 \lambda^2 + (\lambda^2 - 1)^2) \cos^2 \beta}{(\lambda^2 + 1 + (\lambda^2 - 1)\cos \beta)^2}$$

$$G\_{\gamma\gamma} = \frac{k}{4} \frac{(\lambda^2 + 1)^2 - (8H^2 \lambda^2 - (\lambda^2 - 1)^2) \cos^2 \beta}{(\lambda^2 + 1 + (\lambda^2 - 1)\cos \beta)^2}$$

$$G\_{\alpha\gamma} = \frac{k}{4} \frac{\lambda^2 (1 - 2H^2) \cos \beta + (\lambda^4 - 1) \sin^2 \beta}{(\lambda^2 + 1 + (\lambda^2 - 1)\cos \beta)^2} \tag{11}$$

$$B\_{\alpha\gamma} = \frac{k}{4} \frac{\lambda^2 - 1 + (\lambda^2 + 1)\cos \beta}{(\lambda^2 + 1 + (\lambda^2 - 1)\cos \beta)^2}$$

$$A\_{\alpha} = 2g\sqrt{k} \frac{H\lambda\cos\beta}{(\lambda^2 + 1 + (\lambda^2 - 1)\cos \beta)^2}$$

$$\Phi = \frac{t}{\sqrt{k} + 2} - \frac{1}{2} \log \left[\lambda + \frac{1}{\lambda} + (\lambda - \frac{1}{\lambda})\cos \beta\right]$$

The dependence on *λ* shows the existence of gravitational backreaction which was absent in the purely magnetic deformed background (10).

### **3.2 Field theory vs. string theory spectra of charged particles in standard 4-space**

In the case of field theory in 4 dimensions we introduce the magnetic field on *S*<sup>3</sup> which agrees with the magnetic part of the string background (10) as:

$$A\_{\mathfrak{A}} = H \cos \mathfrak{F},\ A\_{\mathfrak{F}} = 0,\ A\_{\mathfrak{I}} = H. \tag{12}$$

The Hamiltonian for a particle with electric charge *e* moving on *S*3, is

$$\mathbf{H} = \frac{1}{\sqrt{\det \mathbf{G}}} (\partial\_{\mu} - ieA\_{\mu}) \sqrt{\det \mathbf{G}} G^{\mu \nu} (\partial\_{\nu} - ieA\_{\nu}) \,. \tag{13}$$

where we assume at the beginning that *Gμν* is standard metric on *S*3.

The energy spectrum for **H** is then given by:

$$
\Delta E\_{j,m} = \frac{1}{R^2} \left[ j(j+1) - m^2 + (eH - m)^2 \right] \tag{14}
$$

where *<sup>R</sup>* is the radius of *<sup>S</sup>*3, *<sup>j</sup>* <sup>∈</sup> **<sup>Z</sup>** and <sup>−</sup>*<sup>j</sup>* <sup>≤</sup> *<sup>m</sup>* <sup>≤</sup> *<sup>j</sup>*, as is the case for *SO*(3). In the flat limit we retrieve the Landau spectrum in 3-dimensional space of spinless particles:

$$
\Delta E\_{n,p\_3} = \varepsilon \tilde{H}(2n+1) + p\_3^2 + \mathcal{O}(R^{-1}) \tag{15}
$$

where magnetic field is pointing into 3-rd direction and the re-scaling of *eH* is performed as *eH* <sup>=</sup> *eH*˜ <sup>+</sup> *<sup>κ</sup><sup>R</sup>* <sup>+</sup> <sup>O</sup>(1), *<sup>m</sup>* <sup>=</sup> *eHR*˜ <sup>2</sup> + (*p*<sup>3</sup> <sup>+</sup> *<sup>κ</sup>*) + <sup>O</sup>(1). This follows from rewriting the spectrum (14) as Δ*En*,*<sup>m</sup>* = <sup>1</sup> *<sup>R</sup>*<sup>2</sup> [*n*(*n* + 1) + |*m*|(2*n* + 1)] + *eH*−*m*<sup>2</sup> *R* 2 by introducing new parameter *n*: *j* = |*m*| + *n* for |*m*|,*n* ∈ **N**.

**3.3 The exotic 4D interpretation of string calculations**

[*j*(*<sup>j</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *<sup>m</sup>*2] + (<sup>2</sup>

gravitational backreaction on exotic smooth **R**<sup>4</sup>

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

*b*

**<sup>R</sup>**<sup>4</sup> <sup>×</sup> *<sup>W</sup>*<sup>6</sup>

background resulting in the deformed spectrum Δ*E<sup>k</sup>*

*k*

*<sup>S</sup>*[Φ, *<sup>U</sup>*] =

backgrounds; *d* assigns **R**<sup>4</sup>

when *H*˜ , *G*˜*μ*,*<sup>ν</sup>* are on exotic **R**<sup>4</sup>

bosonic field *T* in a universal fashion:

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

level *k* is written explicitly:

as the result of exotic **R**<sup>4</sup>

*k* + 2

*<sup>j</sup>*,*m*,*<sup>m</sup>* <sup>=</sup> <sup>1</sup>

Δ*E<sup>k</sup>*

Given the dictionary (21), we can rewrite (20) in a way where the dependance on the even

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 65

<sup>√</sup>*<sup>k</sup>* <sup>+</sup> <sup>2</sup>*eH* <sup>−</sup> (*<sup>λ</sup>* <sup>+</sup> <sup>1</sup>

Thus this is the 4D spectrum of a scalar particle with charge *e* which is modified by the magnetic field *H* and its gravitational backreaction *λ* (20). The spectrum depends on *k* = 2*p* which indicates the relevance of the stringy regime. One can interpret this dependance on *k*

4-geometry, in the QG limit of string theory, generates the quantum gravity effects in 4D. In deriving the spectrum (22) we commence with the flat standard smooth **R**<sup>4</sup> which is a part of the exact string background. Then we switched to another exact string background where the 4D part is now *SU*(2)*<sup>k</sup>* × **R***φ*. This new 4D part ceases to be flat. Its curvature has defined gravitational meaning in superstring theory such that the QG calculations are possible. The effects are derived in the regime of QG, i.e. heterotic string theory. The same deformed spectrum could be obtained, in principle, via including magnetic field *H*˜ and its

fields, however, are not explicitly specified but still the effects in QG regime are derived from string theory as above. Such an approach serves as a way of quantization of gravity while on exotic **R**4. The relations between various ingredients appearing here are presented in Fig. 2.

> *k d*

*SU*(2)*<sup>k</sup>* × **R**

*<sup>c</sup> SU*(2)*<sup>k</sup>* <sup>×</sup> **<sup>R</sup>***<sup>φ</sup>* <sup>×</sup> *<sup>W</sup>*<sup>6</sup>

embedding of flat smooth **R**<sup>4</sup> into the string background; *c* is the change of the string

*<sup>k</sup> SU*(2)*<sup>k</sup>* <sup>×</sup> **<sup>R</sup>** the end of exotic **<sup>R</sup>**<sup>4</sup>

embedding of *SU*(2)*<sup>k</sup>* × **R** into the string background; *H*, *Gμ*,*<sup>ν</sup>* is the deformation of the CFT

Let us turn to the appearance of the mass gap in the spectrum when the theory is formulated

*e*

*<sup>k</sup>* rather than standard smooth **<sup>R</sup>**4. In field theory a dilaton <sup>Φ</sup> couples to a massless

<sup>−</sup>2Φ*∂MT∂MT*.

*<sup>∂</sup>MU∂MU* + [*∂*2<sup>Φ</sup> <sup>−</sup> *<sup>∂</sup>M*Φ*∂M*Φ]*U*.

Fig. 2. *a* is the change of smoothness on **R**<sup>4</sup> from standard one to exotic **R**<sup>4</sup>

*<sup>S</sup>*[Φ, *<sup>T</sup>*] =

One can introduce a new field *U* = *e*−Φ*T* hence the above action becomes:

*e* 

*<sup>H</sup>*˜ ,*G*˜*μ*,*<sup>ν</sup>*

*H*,*Gμ*,*<sup>ν</sup>*

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

*<sup>λ</sup>* )*<sup>m</sup>* <sup>−</sup> (*<sup>λ</sup>* <sup>−</sup> <sup>1</sup>

*<sup>k</sup>* geometry of a 4-region where the particle travells. However, this

*λ* )

*<sup>k</sup>* where modified metric *<sup>G</sup>*˜*μν* emerges. These

*<sup>k</sup>* , via GV invariant; *e* is the

*<sup>j</sup>*,*m*,*m*; the same spectrum is obtained

Δ*E<sup>k</sup> j*,*m*,*m*

*<sup>k</sup>* ; *b* is the

<sup>4</sup>(*<sup>k</sup>* <sup>+</sup> <sup>2</sup>)(<sup>1</sup> <sup>−</sup> <sup>2</sup>*H*2) . (22)

(1 + 2/*k*)*m*)<sup>2</sup>

Let us, again following Kiritsis & Kounnas (1995b), calculate the spectrum in the case of full exact string background (10) as our starting point. One takes the metric components from the background (10) and derive the eigenvalues of the Hamiltonian (14). The result is [Kiritsis & Kounnas (1995b)]:

$$
\Delta E\_{j,m} = \frac{1}{R^2} \left[ j(j+1) - m^2 + \frac{(eHR - m)^2}{(1 - 2H^2)} \right] \,. \tag{16}
$$

Again, introducing *n* ∈ **N** by *j* = |*m*| + *n*, |*m*| = 0, 1/2, 1, ... we can rewrite the spectrum (16) as:

$$
\Delta E\_{n,m} = \frac{1}{R^2} [n(n+1) + |m|(2n+1)] + \left(\frac{eHR - m}{R\sqrt{1 - 2H^2}}\right)^2 \tag{17}
$$

which is the energy spectrum containing the corrections due to *H* field appearing in the string exact background (10), but the Hamiltonian (13) is field theoretic 4-dimensional one.

One can also calculate the exact string spectrum of energy in this exact background (see Kiritsis & Kounnas (1995a;b)) and when compared with (17) gives rise to the following dictionary rules enabling passing between the spectra:

$$\begin{aligned} R^2 \to k + 2, \ m \to \mathcal{Q} + J^3, \ e \to \sqrt{\frac{2}{k\_3}} \overline{\mathcal{Q}}\\ H \to \frac{F}{\sqrt{2}(1 + \sqrt{1 + F^2})} = \frac{1}{2\sqrt{2}} \left[ F - \frac{F^3}{4} + \mathcal{O}(F^5) \right] \,. \end{aligned} \tag{18}$$

Here *F*<sup>2</sup> = *Fa μνFμν a* is the integrated (square of) field strength where *H<sup>a</sup> <sup>i</sup>* <sup>=</sup> *�ijkFa jk* as before.

For a particle with spin *S* setting *S* = *Q* the following modification of the spectrum appear due to the above rules [Kiritsis & Kounnas (1995b)]:

$$
\Delta E\_{\rm j,m,S} = \frac{1}{k+2} \left[ j(j+1) - (m+S)^2 + \frac{(eHR - m - S)^2}{(1 - 2H^2)} \right].\tag{19}
$$

Next step is the inclusion of gravitational backreactions. One begins with the string background (11) and compute again the eigenvalues of (13). The result for scalar particles is [Kiritsis & Kounnas (1995b)]:

$$
\Delta E\_{j,m,\overline{m}} = \frac{1}{R^2} \left[ j(j+1) - m^2 + \frac{(2\text{Re}H - (\lambda + \frac{1}{\lambda})m - (\lambda - \frac{1}{\lambda})\overline{m})^2}{4(1 - 2H^2)} \right] \tag{20}
$$

where now −*j* ≤ *m*, *m* ≤ *j*. Again, comparing with exact string spectra for even *k* we have the corresponding dictionary rules in the case where gravity backreactions are included:

$$\begin{aligned} R^2 \to k+2, \ m \to \mathcal{Q} + f^3, \ e \to \sqrt{\frac{2}{k\_{\mathcal{S}}}} \overline{\mathcal{P}}, \ \overline{m} \to \overline{f}^3\\ H^2 \to \frac{1}{2} \frac{F^2}{F^2 + 2(1 + \sqrt{1 + F^2 + \mathcal{R}^2})}, \ \lambda^2 = \frac{1 + \sqrt{1 + F^2 + \mathcal{R}^2} + \mathcal{R}}{1 + \sqrt{1 + F^2 + \mathcal{R}^2} - \mathcal{R}} \end{aligned} \tag{21}$$

where <sup>R</sup><sup>2</sup> <sup>=</sup> *RμνρσRμνρσ* is the integrated squared scalar curvature and *Rμνρσ* is the Riemann tensor of the "squashed" *SU*(2) = *S*<sup>3</sup> in the deformed background.

### **3.3 The exotic 4D interpretation of string calculations**

12 Will-be-set-by-IN-TECH

Let us, again following Kiritsis & Kounnas (1995b), calculate the spectrum in the case of full exact string background (10) as our starting point. One takes the metric components from the background (10) and derive the eigenvalues of the Hamiltonian (14). The result is [Kiritsis &

Again, introducing *n* ∈ **N** by *j* = |*m*| + *n*, |*m*| = 0, 1/2, 1, ... we can rewrite the spectrum (16)

which is the energy spectrum containing the corrections due to *H* field appearing in the string

One can also calculate the exact string spectrum of energy in this exact background (see Kiritsis & Kounnas (1995a;b)) and when compared with (17) gives rise to the following

is the integrated (square of) field strength where *H<sup>a</sup>*

For a particle with spin *S* setting *S* = *Q* the following modification of the spectrum appear

Next step is the inclusion of gravitational backreactions. One begins with the string background (11) and compute again the eigenvalues of (13). The result for scalar particles

*<sup>j</sup>*(*<sup>j</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> (2*ReH* <sup>−</sup> (*<sup>λ</sup>* <sup>+</sup> <sup>1</sup>

corresponding dictionary rules in the case where gravity backreactions are included:

*<sup>R</sup>*<sup>2</sup> <sup>→</sup> *<sup>k</sup>* <sup>+</sup> 2 , *<sup>m</sup>* → Q <sup>+</sup> *<sup>J</sup>*<sup>3</sup> , *<sup>e</sup>* <sup>→</sup>

*F*2

Riemann tensor of the "squashed" *SU*(2) = *S*<sup>3</sup> in the deformed background.

*F*<sup>2</sup>+2(1+

where now −*j* ≤ *m*, *m* ≤ *j*. Again, comparing with exact string spectra for even *k* we have the

<sup>√</sup>1+*F*<sup>2</sup>+R<sup>2</sup>) , *<sup>λ</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup><sup>+</sup>

*<sup>j</sup>*(*<sup>j</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> (*<sup>m</sup>* <sup>+</sup> *<sup>S</sup>*)<sup>2</sup> <sup>+</sup> (*eHR* <sup>−</sup> *<sup>m</sup>* <sup>−</sup> *<sup>S</sup>*)<sup>2</sup>

*<sup>R</sup>*<sup>2</sup> [*n*(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>) + <sup>|</sup>*m*|(2*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)] +

exact background (10), but the Hamiltonian (13) is field theoretic 4-dimensional one.

*<sup>R</sup>*<sup>2</sup> <sup>→</sup> *<sup>k</sup>* <sup>+</sup> 2 , *<sup>m</sup>* → Q <sup>+</sup> *<sup>J</sup>*<sup>3</sup> , *<sup>e</sup>* <sup>→</sup>

<sup>√</sup>1+*F*<sup>2</sup>) <sup>=</sup> <sup>1</sup> 2 √2 *<sup>F</sup>* <sup>−</sup> *<sup>F</sup>*<sup>3</sup>

*F* 2(1+

*<sup>j</sup>*(*<sup>j</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> (*eHR* <sup>−</sup> *<sup>m</sup>*)<sup>2</sup>

(<sup>1</sup> − <sup>2</sup>*H*2)

*R* √

 2 *kg* Q

<sup>4</sup> <sup>+</sup> <sup>O</sup>(*F*5)

<sup>1</sup> − <sup>2</sup>*H*<sup>2</sup>

 .

(<sup>1</sup> − <sup>2</sup>*H*2)

*<sup>λ</sup>* )*<sup>m</sup>* <sup>−</sup> (*<sup>λ</sup>* <sup>−</sup> <sup>1</sup>

P , *m* → *J*

<sup>√</sup>1+*F*<sup>2</sup>+R<sup>2</sup>+<sup>R</sup>

<sup>√</sup>1+*F*<sup>2</sup>+R<sup>2</sup>−R

3

<sup>4</sup>(<sup>1</sup> − <sup>2</sup>*H*2)

 2 *kg*

1+

*RμνρσRμνρσ* is the integrated squared scalar curvature and *Rμνρσ* is the

*<sup>i</sup>* <sup>=</sup> *�ijkFa*

*<sup>λ</sup>* )*m*)<sup>2</sup>

<sup>2</sup>

*eHR* <sup>−</sup> *<sup>m</sup>*

. (16)

(17)

(18)

(20)

(21)

*jk* as before.

. (19)

<sup>Δ</sup>*Ej*,*<sup>m</sup>* <sup>=</sup> <sup>1</sup>

<sup>Δ</sup>*En*,*<sup>m</sup>* <sup>=</sup> <sup>1</sup>

dictionary rules enabling passing between the spectra:

*H* → <sup>√</sup>

due to the above rules [Kiritsis & Kounnas (1995b)]:

*k* + 2

<sup>Δ</sup>*Ej*,*m*,*<sup>S</sup>* <sup>=</sup> <sup>1</sup>

*R*2

*<sup>H</sup>*<sup>2</sup> <sup>→</sup> <sup>1</sup> 2

*R*2 

Kounnas (1995b)]:

as:

Here *F*<sup>2</sup> =

 *Fa μνFμν a* 

is [Kiritsis & Kounnas (1995b)]:

where <sup>R</sup><sup>2</sup> <sup>=</sup>

<sup>Δ</sup>*Ej*,*m*,*<sup>m</sup>* <sup>=</sup> <sup>1</sup>

Given the dictionary (21), we can rewrite (20) in a way where the dependance on the even level *k* is written explicitly:

$$\Delta E\_{j,m,\overline{m}}^{k} = \frac{1}{k+2} [j(j+1) - m^2] + \frac{(2\sqrt{k+2}\epsilon H - (\lambda + \frac{1}{\lambda})m - (\lambda - \frac{1}{\lambda})\sqrt{(1+2/k)\overline{m}})^2}{4(k+2)(1-2H^2)} . \tag{22}$$

Thus this is the 4D spectrum of a scalar particle with charge *e* which is modified by the magnetic field *H* and its gravitational backreaction *λ* (20). The spectrum depends on *k* = 2*p* which indicates the relevance of the stringy regime. One can interpret this dependance on *k* as the result of exotic **R**<sup>4</sup> *<sup>k</sup>* geometry of a 4-region where the particle travells. However, this 4-geometry, in the QG limit of string theory, generates the quantum gravity effects in 4D.

In deriving the spectrum (22) we commence with the flat standard smooth **R**<sup>4</sup> which is a part of the exact string background. Then we switched to another exact string background where the 4D part is now *SU*(2)*<sup>k</sup>* × **R***φ*. This new 4D part ceases to be flat. Its curvature has defined gravitational meaning in superstring theory such that the QG calculations are possible. The effects are derived in the regime of QG, i.e. heterotic string theory. The same deformed spectrum could be obtained, in principle, via including magnetic field *H*˜ and its gravitational backreaction on exotic smooth **R**<sup>4</sup> *<sup>k</sup>* where modified metric *<sup>G</sup>*˜*μν* emerges. These fields, however, are not explicitly specified but still the effects in QG regime are derived from string theory as above. Such an approach serves as a way of quantization of gravity while on exotic **R**4. The relations between various ingredients appearing here are presented in Fig. 2.

Fig. 2. *a* is the change of smoothness on **R**<sup>4</sup> from standard one to exotic **R**<sup>4</sup> *<sup>k</sup>* ; *b* is the embedding of flat smooth **R**<sup>4</sup> into the string background; *c* is the change of the string backgrounds; *d* assigns **R**<sup>4</sup> *<sup>k</sup> SU*(2)*<sup>k</sup>* <sup>×</sup> **<sup>R</sup>** the end of exotic **<sup>R</sup>**<sup>4</sup> *<sup>k</sup>* , via GV invariant; *e* is the embedding of *SU*(2)*<sup>k</sup>* × **R** into the string background; *H*, *Gμ*,*<sup>ν</sup>* is the deformation of the CFT background resulting in the deformed spectrum Δ*E<sup>k</sup> <sup>j</sup>*,*m*,*m*; the same spectrum is obtained when *H*˜ , *G*˜*μ*,*<sup>ν</sup>* are on exotic **R**<sup>4</sup> *k*

Let us turn to the appearance of the mass gap in the spectrum when the theory is formulated on exotic **R**<sup>4</sup> *<sup>k</sup>* rather than standard smooth **<sup>R</sup>**4. In field theory a dilaton <sup>Φ</sup> couples to a massless bosonic field *T* in a universal fashion:

$$\mathcal{S}[\Phi, T] = \int e^{-2\Phi} \partial\_M T \partial^M T.$$

One can introduce a new field *U* = *e*−Φ*T* hence the above action becomes:

$$S[\Phi, \mathcal{U}] = \int \partial\_M \mathcal{U} \partial^M \mathcal{U} + [\partial^2 \Phi - \partial\_M \Phi \partial^M \Phi] \mathcal{U}.$$

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

development, do not refer to AdS/CFT techniques (cf. Król (2005)).

*H* = ∑ → *k* ,*α*

conduction electron of spin *<sup>α</sup>* and momentum <sup>→</sup>

*<sup>T</sup>* <sup>+</sup> ...<sup>2</sup>

screening the spin impurity (see eg. Potok et al. (2007)).

that between spin *<sup>s</sup>* impurity <sup>→</sup>

*λ* + *νλ*<sup>2</sup> ln *<sup>D</sup>*

*ρ*(*T*) ∼

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):

> *<sup>ψ</sup>*+*<sup>α</sup>* <sup>→</sup> *k ψ*<sup>→</sup> *k α*

the only interaction which restrains quantization.

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

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 67

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

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

*�*(*k*) + *λ*

explaining the growth of the resistivity *ρ*(*T*). Here *ψ* is the annihilation operator for the

of Pauli matrices. From this Hamiltonian one can derive, in the Born approximation, that

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

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

→ *<sup>S</sup>* · ∑ → *k* , → *k*�

*S* with spins of conducting electrons, at <sup>→</sup>

*ψ*+ → *k*

→ *σ* <sup>2</sup> *<sup>ψ</sup>*<sup>→</sup>

where *D* is the 'width of the band' parameter and the second

*<sup>k</sup>*� . (24)

*σ* is the vector

*k* , the antiferromagnetic interaction term is

*x* = 0; <sup>→</sup>

Thus, for a linear dilaton Φ = *qMX<sup>M</sup>* the field *U* gets a mass square *M*<sup>2</sup> = *qMq<sup>M</sup>* for *qM* spacelike. This way the massless boson *T* is mapped to the boson *U* with the mass *M*. However, this mechanism does not work in the case of massless free fermions. In four dimensional spacetime the chiral fermion *ψ* can be coupled to an antisymmetric tensor *Hμνρ* as follows:

$$S[\psi, H] = \int \overline{\psi}\gamma^{\mu} \left[ \overleftrightarrow{\partial\_{\mu}} + H\_{\mu} \right] \psi$$

where *H<sup>μ</sup>* = *εμνρσHνρσ* is the dual of the antisymmetric tensor *Hνρσ*. If one can embed this system into a string background with the fields: Φ and *HMNP*, then using one-loop string equations:

$$\begin{aligned} R\_{MN} &= -2\nabla\_M \nabla\_N \Phi + \frac{1}{4} H\_{MPR} H\_N{}^{P\mathbf{R}} , \\ \nabla\_L \left( e^{-2\Phi} H\_{MN}^L \right) &= 0 , \\ \nabla^2 \Phi - 2 \left( \nabla \Phi \right)^2 &= -\frac{1}{12} H^2 . \end{aligned} \tag{23}$$

one gets for the linear dilaton Φ = *qMX<sup>M</sup>* the following relation:

$$q\_M q^M = \frac{1}{6} H^2$$

and the scalar curvature *R* is:

$$\mathcal{R} = \frac{3}{2} q\_M q^M \dots$$

If non-vanishing components of *qM* and *HMNP* are in four dimensional space, one obtains that:

$$
\eta^{\mu} \sim \varepsilon^{\mu\upsilon\rho\sigma} H\_{\upsilon\rho\sigma} \cdot
$$

Thus, the Dirac operator acquires a mass gap proportional to *qμqμ*.

The problem of embedding a four dimensional fermion system in the exact string background was considered in Kiritsis & Kounnas (1995c). In the case when four dimensional space is represented by the **R***<sup>φ</sup>* × *SU*(2)*<sup>k</sup>* part of the string background, than the linear dilaton is Φ = *QX*<sup>0</sup> and *Q* is given by the level *k* of the WZW model on *SU* (2) as *Q* = (*k* + 2) <sup>−</sup>1/2 so that the CFT has the same central charge as flat space. Hence, the massless bosons acquire the mass gap Δ*M*<sup>2</sup> = *μ*<sup>2</sup> = (*k* + 2) −1 .

That way we arrive at the important feature of the theory when on exotic **R**<sup>4</sup> *k* :

*The theory predicting the energy spectra of charged particles as in (22) in the flat smooth 4D limit, <sup>k</sup>* <sup>→</sup> <sup>∞</sup> *does not show the existence of the mass gap in the energy spectra. However, in the exotic* **<sup>R</sup>**<sup>4</sup> *k limit the theory acquires the mass gap <sup>μ</sup>*<sup>2</sup> <sup>∼</sup> <sup>1</sup> (*k*+2).

The mass gap which appears here in 4D theory is the result of QG computations i.e. those on linear dilaton background in superstring theory. Such overlapping QG with field theory in 4D is a special new feature of the approach via exotic open 4-smooth spaces which bridges 10D superstring and 4D matter fields. We will see in the next section that there are also bottom–up arguments where exotic **R**4's emerge from the regime of low energy effective states of condensed matter. The latter means that gravity is present in the description of the effective entangled matter, since **R**<sup>4</sup> *<sup>k</sup>* is not flat and Einstein equations can be written on these 4-manifolds.

14 Will-be-set-by-IN-TECH

Thus, for a linear dilaton Φ = *qMX<sup>M</sup>* the field *U* gets a mass square *M*<sup>2</sup> = *qMq<sup>M</sup>* for *qM* spacelike. This way the massless boson *T* is mapped to the boson *U* with the mass *M*. However, this mechanism does not work in the case of massless free fermions. In four dimensional spacetime the chiral fermion *ψ* can be coupled to an antisymmetric tensor *Hμνρ*

where *H<sup>μ</sup>* = *εμνρσHνρσ* is the dual of the antisymmetric tensor *Hνρσ*. If one can embed this system into a string background with the fields: Φ and *HMNP*, then using one-loop string

*ψγ<sup>μ</sup>* ←→*∂μ* <sup>+</sup> *<sup>H</sup><sup>μ</sup>*

<sup>2</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup>

6 *H*2

*μνρσHνρσ*.

 *ψ*

<sup>4</sup> *HMPRH PR*

<sup>12</sup> *<sup>H</sup>*2,

*<sup>N</sup>* ,

(23)

<sup>−</sup>1/2 so that the

*k*

*k* :

*<sup>k</sup>* is not flat and Einstein equations can be written on these

*S*[*ψ*, *H*] =

*RMN* <sup>=</sup> <sup>−</sup>2∇*M*∇*N*<sup>Φ</sup> <sup>+</sup> <sup>1</sup>

<sup>∇</sup>2<sup>Φ</sup> <sup>−</sup> <sup>2</sup> (∇Φ)

*qMq<sup>M</sup>* <sup>=</sup> <sup>1</sup>

If non-vanishing components of *qM* and *HMNP* are in four dimensional space, one obtains

The problem of embedding a four dimensional fermion system in the exact string background was considered in Kiritsis & Kounnas (1995c). In the case when four dimensional space is represented by the **R***<sup>φ</sup>* × *SU*(2)*<sup>k</sup>* part of the string background, than the linear dilaton is Φ =

CFT has the same central charge as flat space. Hence, the massless bosons acquire the mass

*The theory predicting the energy spectra of charged particles as in (22) in the flat smooth 4D limit, <sup>k</sup>* <sup>→</sup> <sup>∞</sup> *does not show the existence of the mass gap in the energy spectra. However, in the exotic* **<sup>R</sup>**<sup>4</sup>

The mass gap which appears here in 4D theory is the result of QG computations i.e. those on linear dilaton background in superstring theory. Such overlapping QG with field theory in 4D is a special new feature of the approach via exotic open 4-smooth spaces which bridges 10D superstring and 4D matter fields. We will see in the next section that there are also bottom–up arguments where exotic **R**4's emerge from the regime of low energy effective states of condensed matter. The latter means that gravity is present in the description of the

(*k*+2).

*<sup>R</sup>* <sup>=</sup> <sup>3</sup> 2 *qMq<sup>M</sup>* .

*<sup>q</sup><sup>μ</sup>* <sup>∼</sup> *<sup>ε</sup>*

*QX*<sup>0</sup> and *Q* is given by the level *k* of the WZW model on *SU* (2) as *Q* = (*k* + 2)

That way we arrive at the important feature of the theory when on exotic **R**<sup>4</sup>

∇*L e*−2Φ*H<sup>L</sup> MN* = 0,

one gets for the linear dilaton Φ = *qMX<sup>M</sup>* the following relation:

Thus, the Dirac operator acquires a mass gap proportional to *qμqμ*.

−1 .

*limit the theory acquires the mass gap <sup>μ</sup>*<sup>2</sup> <sup>∼</sup> <sup>1</sup>

effective entangled matter, since **R**<sup>4</sup>

4-manifolds.

as follows:

equations:

that:

and the scalar curvature *R* is:

gap Δ*M*<sup>2</sup> = *μ*<sup>2</sup> = (*k* + 2)
