# An Investigation in Strings Theory to outline Supersymmetric Wormholes Nature

## Low-Dimensions vs High-Dimensions Theories

In recent years, **significant progress** has been made in **understanding** the **Euclidean Path Integral** (further information in ** here**, Ref. 3) [

__1__] for

**low-dimensional quantum gravity theories**like

__(__

**Jackiw–Teitelboim**__)__

**JT**__. These insights often involve summing over__

**Gravity**__s with varying__

**Saddle Point**__(ies). In contrast, for__

**Topology****higher-dimensional theories**like those involving

__coupled with matter, the situation is__

**Einstein - Hilbert Gravity****less straightforward**.

## What About Wormholes?

__ Wormhole__s (

__Figure 1__), which are

__s connecting separate regions of Space - Time, are key to__

**Geometrical Structure****understanding**certain

**quantum gravity aspects**. However, their

**presence**in the Euclidean path integral can lead to

**puzzles**regarding

__and__

**Unitarity****Non-Factorization**[

__2__] of

__s. This raises__

**Correlation Function****questions**about the

**consistency**of the

**Holographic Duality**(

__Figure 2__) [

__3__] in such cases.

Additionally, wormholes suggest the **possibility** of **Baby Universe**s [__4__], impacting the __ Swampland Program__'s guidelines. To address these issues, a

**deeper exploration**of higher-dimensional Euclidean wormholes within

**String Theory**(further information in

**, Section 1) is needed.**

__here__

## Constructing Euclidean Wormholes in Supergravity

Building Euclidean wormhole geometries generally requires a **source** of **negative Euclidean Energy** [__5__]. In string theory, this is achieved using **axion** (further information in ** here**, Section 2) fields.

**Analytical continuation**of a

**Lorentzian Theory**[

__6__] with axions to the

**Euclidean Regime**can yield a

**negative energy-momentum tensor**(further information in

**, Section 1), providing the**

__here__**necessary conditions**for

**wormhole formation**.

**Figure 1**. A Pictorial Representation of a Wormhole Interior

## Step by Step from a Five-Dimensional Framework

One can start with a **five-dimensional metric**, which is written in the form:

**Equation 1**. The form of a Five-Dimensional Metric

where [math]\small{ds^2_{\mathbf{R}^4}}[/math] is the __ Flat Metric__ on

__, [math]\small{\mathbf{R}^4}[/math], and the__

**Euclidean Space**__, [math]\small{A}[/math] ([math]\small{r}[/math]) depends solely on the__

**Metric Function****radial coordinate**, [math]\small{r}[/math]. The Euclidean gravity path integral compactifies [math]\small{\mathbf{R}^4}[/math] to the

__, [math]\small{T^4}[/math].__

**Normal Space**__ Supersymmetry__ is

**broken**from [math]\small{N = 4}[/math] to [math]\small{N = 1}[/math], with distinct

**(**

__(ies)__**Quantum Field Theory**__s) on__

**QFT****each side**of the wormhole,

**differentiated**by their

**Yang-Mills****coupling constants**and

**other parameters**. A

__exists where the__

**Special Point****wormhole's "neck"**shrinks to

**zero**, leading to a

**Singular Metric**resembling the well-known

**GPPZ Solution**[

__7__].

## ... Here is Supersymmetry!

To **ensure supersymmetry**, the **solutions** must **satisfy** the **first-order** __ Bogomol'nyi - Prasad - Sommerfield__
(

__)__

**BPS**__(__

**Equations**__Equation 2__and

__Equation 3__). These equations

**guide**the

**construction**of the supersymmetric Euclidean wormhole

**solutions**. A

**key point**in this analysis is whether the wormhole solutions

**dominate**over corresponding

**disconnected geometries**with the same

__s.__

**Boundary Condition****Equation 2**. First BPS Equation: the form of Total Energy in the Euclidean Space

in which [math]\small{A'}[/math] is the **derivative** of the **metric function** and [math]\normalsize{W}[/math] is a **real** __ Supepotential__.

**Equation 3**. Second BPS Equation: the Energy for each point on the Euclidean Space

[math]\Large{i}[/math] is a generic **Point** on Euclidean space; ( [math]\Large{z^{i}}[/math])[math]\Large{'}[/math] is the **first derivative** of the __ Scalar Field__, [math]\Large{z^{i}}[/math] of each generic point, with respect to the radial coordinate; and [math]\Large{K^{i\bar{J}}}[/math] is the

__.__

**Kähler Metric**Using a **Consistent** __ Truncation__

^{7}of

**maximal five-dimensional**

__with an__

**Supergravity****SO(6)**

__, a__

**Gauge Group****specific set**of BPS equations is derived. A

**simplified set**of field variables and a constant of motion

**help**in solving these equations. A

**critical aspect**is the

**interpretation**of these solutions within the context of the holographic duality and the

**implications**for non-factorization.

**Figure 2**. Illustration of the Hologram for a Sphere

## Implications of Wormholes Discovery

The **existence** of **Euclidean wormholes** and their **role** in **quantum gravity path integrals** raise **significant questions** about **unitarity**, **holographic duality**, and the **Swampland program**. Future work involves **uplifting** these **wormhole solutions** to **ten dimensions** and **exploring** the **implications** of **imaginary scalar**s in the **Euclidean regime**. The **factorization puzzle** in __ Holography__ remains an

**open question**, with

**possible solutions**involving

**supersymmetry considerations**and potential

**Fermion Zero-Mode [**. This area of research holds

__8__] effects**great potential**for advancing our

**understanding**of quantum gravity and its

**holographic connections**.

- Springer. "Euclidean Path Integrals"
__https://link.springer.com/chapter/10.1007/978-1-4612-0009-3_14__ - Inspire HEP. "Factorization and Non-Factorization of In-Medium Four-Quark Condensates"
__https://inspirehep.net/literature/676262__ - arXiv.org. "Editorial: New frontiers in holographic duality"
__https://arxiv.org/abs/2210.03315__ - Big Think. "Are we living in a baby universe that looks like a black hole to outsiders?"
__https://bigthink.com/hard-science/baby-universes-black-holes-dark-matter/__ - Wiley Online Library. "Energy-Efficient Memristive Euclidean Distance Engine for
Brain-Inspired Competitive Learning"
__https://onlinelibrary.wiley.com/doi/full/10.1002/aisy.202100114__ - Nature. "Strongly enhanced effects of Lorentz symmetry violation in entangled Yb+ ions"
__https://www.nature.com/articles/nphys3610__ - ResearchForLife7 (revisited from arXiv). "Discussion on Consistent Truncations: Uplifting the GPPZ Solutions"
__https://httpsresearchforlife7.com/wp-content/uploads/2024/05/Discussion_on_Consistent_Truncations__Uplifting_the_GPPZ_Solutions.pdf__ - Inspire HEP. "Fermion Zero Modes and Topological-charge on a
Domain Wall of the D-brane-like Dot"
__https://inspirehep.net/literature/1353725__