## 7 Compactification and Dimension Dynamics  
In this section we use the idea of interpolating (fractal) space dimension in another class of models: with compactified dimensions. We implement the idea in the simplest case of one compact dimension and then give some general arguments for cases with several compact dimensions. Let us take one of the dimensions \(y\) as circle with radius \(R\). This corresponds to the periodic structure with a point charge sources at each point \(y_{n}=y+2\pi Rn,n=0,\pm 1,\pm 2,\ldots\)  
\[\Delta\varphi=e\sum_{n}\delta^{D}(x)\delta(y_{n}),\varphi(D,r,y) =\sum_{n}\varphi(D,r,y_{n}),\] \[V(D,r,y)=-\alpha(D+1)\sum_{n=-\infty}^{\infty}(r^{2}+(2\pi Rn+y) ^{2})^{(1-D)/2}. \tag{35}\]  
When \(D=3\), the potential (35) can be writen in a closed form [1]  
\[V_{3}(r,y)=-\frac{\alpha(4)}{2 Rr}\frac{\sinh(r/R)}{\cosh(r/R)-\cos(y/R)}= \begin{cases}\alpha(4)/(2 Rr),&r\gg R\\ \alpha(4)/(r^{2}+y^{2}),&r,\ y\ll R\end{cases}  \tag{36}\]  
where \(\alpha(4)/(2R)=\alpha(3)\). Alternatively, we can rewrite (36) as  
\[V_{3}(r,y)=-\frac{\alpha(4)}{4 Rr}\left[\coth\left(\frac{r+\mathrm{i}y}{2R} \right)+\coth\left(\frac{r-\mathrm{i}y}{2R}\right)\right], \tag{37}\]  
or, using  
\[A^{-\alpha}=1/\Gamma(\alpha)\int_{0}^{\infty}dt\,t^{\alpha-1}e^{-tA}, \tag{38}\]  
by means of the Theta function as  
\[V_{3}(r,y)=-\alpha(4)\int_{0}^{\infty}dte^{-tr^{2}}\sum_{-\infty}^{\infty}e^{ -t(2\pi Rn+y)^{2}}=-\alpha(4)\int_{0}^{\infty}dte^{-tr^{2}}\frac{\theta\left( \frac{\mathrm{i}y}{2\pi R},\exp\left(\frac{\mathrm{i}}{4R^{2}t}\right)\right) }{2R\sqrt{\pi}\sqrt{t}}, \tag{39}\]  
where \(\theta(z,\tau)=\) is given in Appendix D.  
_Dimension Dynamics with One Compact Dimention_  
For \(y=0\), the potential takes the following simple form  
\[V_{3}(r,y=0)=-\frac{\alpha(3)}{r}\coth\frac{r}{2R}. \tag{40}\]  
For \(y=\pi R\), we have  
\[V_{3}(r,y=\pi R)=-\frac{\alpha(3)}{r}\tanh\frac{r}{2R}. \tag{41}\]From (36), we see that for big \(r\), the effective dimension of space is 3 and for small \(r\) is 4. For intermediate scales, the effective dimension might change smoothly from 3 to 4. Integrating by coordinate \(y\) (or angle \(\theta\), see Appendix E) we define mean potential depending only on the variable \(r\)[1],  
\[\bar{V}_{3}(r)=\frac{1}{2\pi}\int_{0}^{2\pi}d\vartheta V_{3}(r, \theta)=-\frac{\alpha_{3}}{r},\quad\alpha_{3}\equiv\alpha(3). \tag{42}\]  
As in the Cornell potential case (cf. Equality (15)), we define the dimension dynamics from equality between the Coulomb potentials (8) and (36)  
\[\frac{\alpha(4)}{2r}\frac{\sinh(r/R)}{\cosh(r/R)-\cos(y/R)}= \alpha(D)(x)^{2-D},\;\alpha(4)=2R\alpha(3)\] \[\mu=1/R,\;x=\mu r, \tag{43}\]  
We may take some interpolating dimension, e.g.,  
\[D(x)=4-\tanh x, \tag{44}\]  
and define the coupling function dynamics as  
\[\alpha(x,\varphi)=\alpha_{3}x^{D-3}\frac{\sinh x}{\cosh x-\cos \varphi},\;x=r/R,\;\varphi=y/R\in[0,2\pi),\] \[\alpha(x,0)=\alpha_{3}x^{D-3}\coth(x/2)=\left\{ \begin{array}{ ccc}2\alpha_{3}&x\ll 1\\ \alpha_{3}&x\gg 1\end{array} \right.,\] \[\alpha(x,\pi)=\alpha_{3}x^{D-3}\tanh(x/2)=\left\{ \begin{array}[] {cc}\alpha_{3}x^{2}/2&x\ll 1\\ \alpha_{3}&x\gg 1\end{array} \right.. \tag{45}\]  
The last case corresponds to the asymptotic freedom at small \(r/R\) scales, see Figure 4. From the point of view of the compact subspace, it corresponds to infrared asymptotic freedom. The global maximum value is \(\alpha_{s}(x,\varphi)/\alpha_{3}=2.87644\) at \(\varphi=0,x_{c}=0.360028,D(x_{c})=3.65476\), see Figure 5. So, we have ultraviolet and infrared fixed points and intermediate fixed point of renormdynamics. Beta functions for the normed coupling constant: \(a=\alpha_{s}(x,\varphi)/\alpha_{3}\) are  
\[a_{\varphi}=\frac{\partial a}{\partial\varphi}=-\frac{\sin \varphi}{\cosh x-\cos\varphi}a,\] \[a_{x}=x\frac{\partial a}{\partial x}=\left(xD_{x}\ln x+D-3+x \left(\coth x-\frac{\sinh x}{\cosh x-\cos\varphi}\right)\right)a,\]\[D=4-\tanh x,D_{x}=-1/\cosh^{2}x. \tag{46}\]  
From the first equation of the system we see that \(a_{\varphi}=0\) for \(a\
eq 0\) at \(\varphi=0,\pi\).