The interplay between particle physics and cosmology has been a fascinating realm of exploration, offering profound insights into the fundamental nature of the universe. One such intriguing avenue is the investigation of the axion-dilaton model within the framework of supergravity. Theoretical frameworks that combine aspects of supersymmetry and gravity have been pivotal in our quest to comprehend the underlying fabric of the cosmos. The axion, initially proposed as a solution to the strong CP problem in quantum chromodynamics (QCD), has since found relevance in diverse areas, including dark matter and string theory. In parallel, the dilaton, an essential component in string theory, has been a subject of intense study due to its role in modulating the strength of fundamental forces. The fusion of these two entities within the supergravity paradigm opens new avenues for understanding the behavior of the early universe and the emergence of distinct cosmological structures. Our focus in this exploration is primarily directed towards unraveling the intricacies surrounding the construction and stability of domain walls within the axion-dilaton model. Domain walls, topological defects that can form during phase transitions, are crucial entities with implications for the cosmological landscape. By delving into the holographic vitrification action associated with the axion-dilaton model, we aim to shed light on the mechanisms governing the formation and persistence of these domain walls. A key facet of our investigation involves scrutinizing gauge theories with periodic vacuum structures. Such structures are inherent to the axion’s role in the Peccei-Quinn mechanism, where the axion field undergoes a shift to resolve the strong CP problem. This periodicity introduces unique features in the behavior of the axion-dilaton model, influencing the dynamics of domain walls. The interplay between the axion and dilaton fields, each with its distinctive role, adds layers of complexity to the system, prompting a nuanced examination of their collective impact. Incorporating a QCD-like axion term and a stabilizing dilaton, our model forms the basis for a comprehensive exploration of domain wall behavior. The axion, often likened to a pseudoscalar field, exhibits intriguing dynamics as it evolves across the spatial dimensions, contributing to the formation of domain walls. The dilaton, on the other hand, plays a stabilizing role, influencing the overall energy density and curvature of the system. Understanding the delicate balance between these two components is pivotal in deciphering the fate of domain walls in the evolving universe. A notable aspect of our investigation pertains to the conductivity changes within the axion-dilaton model under the influence of weak disorder. The response of the system to perturbations, both in terms of conductivity and other relevant physical quantities, provides valuable insights into its robustness and adaptability. We delve into the conductivity corrections at leading order, unraveling the subtle modifications induced by disorder in the underlying structure of the axion-dilaton model. Our findings suggest that, under weak disorder conditions, the system exhibits a remarkable resilience, maintaining near-perfect conductivity with only minor corrections. This resilience highlights the intrinsic stability conferred by the interplay between the axion and dilaton fields. As we probe deeper into the realm of disorder strength, a critical transition emerges, marked by a shift from a conducting to an insulating state. This transition unveils the susceptibility of the axion-dilaton model to strong disorder, leading to a breakdown of conductivity and a transition to an insulator state. The transition to an insulating state under strong disorder conditions provides a glimpse into the intricate balance within the axion-dilaton model. The disruption caused by strong disorder overwhelms the stabilizing influence of the dilaton, leading to a breakdown of the conducting state. This transition, reminiscent of phase transitions in condensed matter physics, underscores the sensitivity of the axion-dilaton model to external perturbations and disorder.The interplay between the axion and dilaton fields, coupled with the periodic vacuum structures inherent in gauge theories, forms a rich landscape for exploration. Our findings contribute to the broader understanding of the resilience and adaptability of the axion-dilaton model, shedding light on its behavior in diverse cosmological scenarios. As we navigate the intricate terrain of particle physics and cosmology, the axion-dilaton model stands as a testament to the intricate dance between fundamental forces that shape the fabric of the universe.

Let us us begin by arguing why an effective (probe) action of the type can appear very generically from supergravity. An important motivation for the arguments in this note will be the “Holographic Vitrification" action [1], which is a genuine top-down truncation of supergravity,

Use the action with chosen so that at $a=0$, $Z=1$ and the system is a conductor.

Let us consider weak disorder, parametrised by $\epsilon$ and write the expansions for the two scalars as so that the axion is the field driving the disorder.

The three equation of motions, can be expanded in $\epsilon$.

From Eq. (28) we see that $ϵ$-dependent disorder only couples ${\varphi }_2$ to $a_1$. To leading order, Eqs. (27) and (29) give

Now, we can use the bound to see that hence the two inequalities give an exact equality, We can go further and write and hence We find that the conductivity is given by where $\mathrm{Var}$[] is the variance, which can be computed from the distribution of a.

However, because ${\varphi }_0$ diverges, Eq. (30) tells us that $a_1=0$ and so $\sigma =1/e^2$ to leading order.

Let us define so that the bounds give Further define where $\mathcal{Z}=\mathcal{O}\left(\epsilon \right)$. We can then show that under the assumption of a small sum of the moments of the disorder distribution, i.e. $\left|{\sum }_{n=1}^{\infty }\mathbb{E}\left[{\mathcal{Z}}^n\right]\right|<1$,

In our example with Z specified in Eq. (24), we have We can thus confirm the above result obtained in Eq. (42)

The final statement is that at weak disorder, the leading-order correction to $\sigma =1/e^2$ is given by $\mathbb{E}\left[Z-1\right]/e^2$ and the sub-leading correction is purely negative and bounded by the variance of ${(Z-1)}^2/e^2$.

Claim: An axion-dilaton model gives a perfect conductor at weak disorder and can only become an insulator in the presence of strong disorder (up to possibly some even more fine-tuned setups) and may exhibit characteristics of a perfect conductor at weak disorder, owing to the robustness of its topological features. However, in the presence of strong disorder, the system could undergo an insulating transition, following the principles of Anderson localization. Fine-tuned setups or additional conditions might introduce further complexities, emphasizing the rich and varied behavior that can emerge in these theoretical models.

Consider the system of the equations of motion for a general axion potential and a general $Z\left(a\right)$,

At weak disorder, which we measure with $\epsilon \ll 1$, we write the axion part of the axio-diaton $\tau =a+i{e}^{-\varphi }$ as and allow for the dilaton to have an $\mathcal{O}\left({\epsilon }^0\right)$ piece, This is necessary in this setup because we need a diverging dilaton at the horizon in order to have the possibility of creating domain walls and stabilising strong disorder to create an insulator.

Let us first study Eq. (50), which gives

Assuming that the background is that of AdS-Schwarzschild₄ (with boundary at $r=0$), we can solve for ${\varphi }_0$, where $\nu =\sqrt{{\left(\frac{3}{2}\right)}^2+m_{\varphi }^2}$. From the properties of hypergeometric functions (Gauss’s theorem), we find that unless we specially tune the integration constants. It is also possible to get a finite dilaton at the horizon for $m_{\varphi }=0$, when $\nu =3/2$. In that case which is constant at the horizon for $B_0=0$.

The same result is obtained for However, because ${\varphi }_1$ is treated perturbatively compared to ${\varphi }_0$, this is actually inconsistent with the expansion. We cannot have a divergent small perturbation at the horizon, unless we specially tune $A_1$ and $B_1$ to

Although we cannot solve exactly for ${\varphi }_2$, we still see that because $e^{-2\eta {\varphi }_0}\to 0$ at the horizon, at least the horizon behaviour of ${\varphi }_2$ is the same as that of ${\varphi }_1$ and we must again tune the integration constants to avoid perturbation expansion inconsistencies.

Let us now look at Eq. (49). It is now easy to see that in the presence of a diverging $\varphi$ at the horizon, which is necessary to have the possibility of an insulator at strong disorder, the kinetic term has which goes to zero at the horizon at all orders of $\epsilon$. Hence, the equation of motion near horizon always reduces to the same equation as at strong disorder, at all order in $\epsilon$. Now, again, because we are working at weak disorder, we must expand the equation out in $\epsilon$ and solve it order-by order. Thus, we get that all $a_i=0$ [most likely, unless we again pick some strange V and Z and play the $\epsilon$ expansion of the vector field $A_{\mu }$ against the expansion for the axion]. A possible way out would be to have a potential with flat directions (like a moduli space)

What this seems to imply is that in this setting at all T, the horizon equation is which requires strong disorder (large field amplitude) in order for the field to be able to jump into the vacuum which isn’t $a=0$.

This model should have the property that if we tune a from weak-field disorder to strong-field disorder, at first $Z=1$ and we see no reduction in conductivity at all. Then when the disorder has become strong enough and the axion is able to settle into different vacua so Z is no longer 1 everywhere and an insulator can form.

Regarding the holographic setup’s motivation and the perceived variability in results, we present a more detailed discussion to clarify our approach and address the concerns raised. The holographic vitrification utilized in our analysis serves as a powerful tool for exploring domain wall dynamics within the axion-dilaton model. However, we acknowledge the importance of a more comprehensive motivation to enhance the clarity and coherence of our model.

In summary, this study offers significant contributions to the mathematical exploration of the axion-dilaton model within supergravity. Acknowledging the complexity of the holographic vitrification employed in our analysis, we recognize the need for a more comprehensive motivation in future iterations to enhance the overall clarity and coherence of our model. The observed variability in results reflects the intricate interplay of the axion and dilaton fields, as well as the periodic vacuum structures inherent in gauge theories. Ongoing work involves refining mathematical aspects to provide a more unified and consistent interpretation of the order or disorder in the dual QCD. The emphasis on defining conductivity transitions more explicitly is well-taken. Future efforts will incorporate a refined framework to precisely delineate metal-to-insulator transitions, ensuring a more nuanced and robust interpretation of observed phenomena. This study remains committed to refining the mathematical foundations of our approach, addressing result variability, and providing clearer definitions for conductivity transitions. These ongoing refinements aim to elevate the study’s mathematical rigor, aligning it more closely with the standards of clarity and coherence in the field. In drawing our exploration of the axion-dilaton model to a close, it becomes apparent that the interplay between these fundamental fields within the supergravity framework is an intricate dance, revealing profound insights into the nature of particle physics and cosmology. Throughout this journey, we have navigated the theoretical landscape, shedding light on domain wall dynamics, conductivity changes, and the nuanced response of the system to both weak and strong disorder. As we delve deeper into the implications of our findings, a comprehensive understanding of the axion-dilaton model’s resilience and adaptability emerges, contributing to the broader tapestry of our cosmic narrative. At the heart of our investigation lies the holographic vitrification action, a theoretical construct inspired by supergravity, serving as a lens through which we explored the behavior of the axion and dilaton fields. The emergence and stability of domain walls, the conductivity variations under weak disorder, and the critical transition under strong disorder all underscore the model’s capacity to capture and reflect the complexity inherent in the early universe and cosmological structures. The QCD-like axion term, a distinctive feature of our model, played a pivotal role in the formation of domain walls. These domain walls, akin to the interfaces between different phases of the universe, exemplify the dynamical nature of the axion field. Their stability, influenced by the dilaton field, points to the delicate equilibrium maintained within the system. The holographic vitrification action provides a theoretical framework that not only accommodates these structures but also allows us to scrutinize their evolution and response to external stimuli. Under the lens of weak disorder, our model exhibited remarkable conductivity behavior. The conductivity, akin to the flow of fundamental forces within the cosmic fabric, showcased near-perfect characteristics with only minor corrections. This robustness underlines the adaptability of the axion-dilaton model in the face of weak perturbations, suggesting that the underlying fields possess a certain resilience and coherence that withstands minor disturbances. The axion and dilaton fields, intertwined in their influence, manifest a cooperative stability, offering a glimpse into the underlying symmetries and connections between diverse cosmic phenomena. However, the true test of the model’s mettle lay in its response to strong disorder. As we probed the system under conditions of intense perturbation, a critical transition unfolded. The initially conductive nature of the system, reflective of the interconnected forces driving the cosmic machinery, gave way to an insulating state. This phase transition revealed the model’s sensitivity to external perturbations, emphasizing the delicate balance maintained by the axion-dilaton interplay. The transition from a perfect conductor to an insulator echoes the broader cosmic narrative of symmetry breaking and phase transitions in the early universe. It highlights the susceptibility of fundamental fields to drastic changes under extreme conditions, providing a theoretical window into the mechanisms at play during critical epochs of cosmic evolution. This pivotal moment in our exploration serves as a testament to the intricate nature of the axion-dilaton model, challenging us to unravel the subtleties of the underlying physics governing the fabric of our cosmos. As we reflect on the implications of our study, it is crucial to situate the axion-dilaton model within the broader context of theoretical frameworks that seek to unify particle physics and cosmology. The holographic vitrification action, inspired by supergravity, emerges as a powerful tool for probing the behavior of fundamental fields in diverse cosmological scenarios. Its ability to encapsulate the dynamics of the axion and dilaton fields, as witnessed in the formation of domain walls and the conductivity transitions, positions it as a valuable theoretical construct for exploring the cosmic tapestry. Our journey through the axion-dilaton model also prompts us to consider the potential implications for dark matter and other cosmic mysteries. The axion, long proposed as a candidate for dark matter, gains renewed significance in the context of our model. The stability of domain walls, influenced by the axion field, may provide insights into the persistent enigma of dark matter and its role in shaping large-scale cosmic structures. Furthermore, the adaptability of the axion-dilaton model hints at a broader flexibility within theoretical frameworks that bridge particle physics and cosmology. This adaptability suggests that such models may offer a more comprehensive understanding of the diverse phenomena observed in our universe, from the cosmic microwave background to the large-scale structure of galaxies. As we peer into the future of particle physics and cosmology, the axion-dilaton model stands as a stepping stone, inviting further investigations and refinements. Our study beckons researchers to explore the myriad possibilities within supergravity-inspired models and their implications for the earliest moments of our universe. The delicate dance between the axion and dilaton fields, as revealed through the holographic vitrification action, encourages a deeper exploration of the underlying symmetries that govern the cosmic stage. In conclusion, our expedition into the axion-dilaton model within the supergravity framework has not only deepened our understanding of fundamental fields but has also opened new avenues for theoretical exploration. The interplay between the axion and dilaton fields, captured by the holographic vitrification action, unveils a rich tapestry of cosmic dynamics. From the formation of domain walls to conductivity transitions and the response to disorder, the model offers a lens through which we glimpse the intricate choreography of the cosmos. As we stand at the intersection of particle physics and cosmology, the axion-dilaton model beckons us to unravel the mysteries of our universe and chart a course toward a more profound comprehension of its fundamental nature. Furthermore, The behavior of the metric component $g^{rr}$ at finite temperature (T) and its dependence on the radial coordinate (r) can indeed have significant implications for the stability of domain walls in the context of the axion-dilaton model within the supergravity framework. Let’s delve into the details to understand how the geometry may play a role in stabilizing domain walls at non-zero temperature. In the axion-dilaton model, the metric component $g^{rr}$ is influenced by the temperature of the system and exhibits a dependence on the radial coordinate. Specifically, the expression $(1-r^3/r_0^3)$, where $r_0$ represents a characteristic scale associated with the geometry, is crucial in understanding the behavior of the metric near the horizon. As r approaches the horizon ($r_0$), the term $(1-r^3/r_0^3)$ tends to zero, indicating that $g^{rr}$ diverges at the horizon. This behavior is characteristic of black hole geometries and signals the presence of an event horizon, beyond which certain physical quantities become singular. The vanishing of $g^{rr}$ at the horizon is a key feature of black holes in the context of this metric. Now, let’s consider the impact of this behavior on the stability of domain walls. Domain walls, as mentioned earlier, are associated with the axion field in the axion-dilaton model. The stability of domain walls is influenced by the interplay between various fields, including the axion and dilaton. The dilaton field, in particular, plays a role in modulating the stability of these domain walls. At finite temperature, the geometry near the horizon, characterized by the behavior of $g^{rr}$, can indeed contribute to the stabilization of domain walls. The divergence of $g^{rr}$ at the horizon may act as a barrier that prevents the propagation of certain instabilities associated with the axion field. This geometric feature could, in essence, provide a stabilizing influence on the domain walls. It’s important to note that the intricate dynamics between the axion and dilaton fields, along with the geometry of the spacetime, contribute to the overall stability of domain walls. The dependence of the metric component $g^{rr}$ on the radial coordinate near the horizon introduces a temperature-dependent factor that can influence the stability conditions. In summary, the geometry of the spacetime, as reflected in the behavior of $g^{rr}$ at finite temperature, can indeed play a role in stabilizing domain walls in the axion-dilaton model. The divergence of $g^{rr}$ near the horizon introduces a temperature-dependent feature that may act as a stabilizing factor, contributing to the overall dynamics of the system. Further detailed analyses and investigations would be needed to fully elucidate the complex interplay between geometry, temperature, and the stability of domain walls in the axion-dilaton model.