You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: '[*The coupling of the Higgs field through the Ricci tensor, put forward by Balakrishna and Wali, is derived using a conformal rescaling of the metric. Earlier results on “Bogomolny-type” equations in curved space, by Comtet, and others, are recovered. The procedure can be generalized to any static background metric.* ]{}'
author:
- 'P-M Zhang'
- 'P. A. Horváthy'
title: Bogomolny equations and conformal transformations in curved space
---

Introduction
============

In the study of solitons in flat space, the ‘Bogomolny’, or ‘self-duality’ equations [@Bogom], $ 
D_i\Phi=\2\epsilon_{ijk}F^{jk},
$ play two, complementary roles. On the one hand, they provide the absolute minima of the energy, yielding static solutions. But they also allow us to reduce the second-order field equations to first-order ones.

In curved space, an obstruction arises, though : a self-dual field may fail to solve the field equations.

To be specific, consider a purely magnetic, static Yang-Mills-Higgs system $(A_i,\phi)$ in $3$ space dimensions, where the YM potential $A_i$ takes its values in the Lie algebra of the gauge group $G$ (a compact Lie group), $F_{ij}=\partial_iA_j-\partial_jA_i+[A_i,A_j]$ is the YM field strength and $\phi$, the Higgs field, belongs to the adjoint representation of the gauge group. Let $g_{\mu\nu}=\big(g_{oo},-\hat{g}_{ij}\big)$ be a static background metric. For vanishing Higgs potential, the Lagrangian is L\_[YMH]{}=(-1/4 F\_[ij]{}F\^[ij]{}+D\_i D\^i), \[1.1\] where $g=g_{oo}\hat{g}$ is the determinant of metric with $\hat{g}=\det(g_{ij})$ the determinant of the space-metric alone. The associated field equations read {

[lll]{} D\_i(F\^[ij]{}) &=&\[,D\^j\],\
D\_i(D\^i) &=&0.

. \[1.2\]

The natural generalization to curved space of the Bogomolny equations would be D\_i=\_[ijk]{}F\^[jk]{} \[1.3\] However, inserting Eq. (\[1.3\]) into the r.h.s. of the second equation in (\[1.2\]) yields, using the Bianchi identities, $
\2\partial^i\big(\ln g_{oo}\big)D_i\phi,
$ which does not vanish, unless $g_{oo}$ is a constant.

A clever way of removing this obstruction is to add a suitable curvature term to the Lagrangian [@BW; @Edel]. One assumes that the metric is of the Papapetrou-Majumdar form V\^2dt\^2-, \[2.4-5\] and one seeks static fields $\phi, A_i$ which extremize S=V\^[-2]{}d\^4[x]{}, \[3.1\]where $R$ is the Ricci scalar of the background metric. This expression differs from (\[1.1\]) in the term $(R/4)\tr\psi^2$, which makes it indefinite. The associated field equations, {

[lll]{} D\_i(V\^2F\_[ij]{})&=&\[, D\_j\],\
V\^2D\_i\^2&=&R.

. \[3.2\] can be obtained by solving instead [^1], F\_[ij]{}=\_[ijk]{}D\^k(V). \[3.3\]

To explain why this works, we note first that (\[3.3\]) are in fact (\[1.3\]) for the conformally rescaled metric $
G_{\mu\nu}=V^{-2}\,g_{\mu\nu}
=\big(1,-V^{-4}\II_3\big),
$ for which no obstruction arises. But such a rescaling, implemented on the fields $(A_i,\phi)$ as $
A_i\to A_i,
\,
\phi\to\psi=V^{-1}\phi,
$ changes (\[1.1\]) written in the rescaled metric $G_{ij}$ into & +\_j\
&-\^2, \[3.7\] where $\widehat{\bigtriangleup}=\frac{1}{\sqrt{\hat{g}}}\,
\partial_i\big(\sqrt{\hat{g}}\,\partial^i\big)$ is the Laplacian associated with the space metric $\hat{g}_{ij}$. Our clue is now that, for the class (\[2.4-5\]) of metrics, =R, \[3.8\] one half of the Ricci scalar. In the last term in (\[3.7\]) we recognize, hence, precisely the curvature term in (\[3.1\]). The rescaled expression, (\[3.7\]), only differs from the density (\[3.1\]) by a surface term; they are, therefore, equivalent. At last, the self-duality equations are conformally invariant.

Our results here shed some new light on those, obtained earlier by Comtet, and others [@CFH]. In that approach, the original field equations, (\[1.2\]), are kept unchanged but the Bogomolny equations are modified. D\_i+\_i=\_[ijk]{}F\^[jk]{} \[Bogotype\] is readily shown to solve the second-order field equations (\[1.2\]), provided the metric satisfies the constraint =0. \[2.2\] Indeed, if the constraint (\[2.2\]) holds then the Ricci scalar vanishes by (\[3.8\]), so that the curvature-modified model (\[3.1\]) reduces to (\[1.1\]). Note that for (\[2.2\]) $V$ takes the Papapetrou-Majumdar form $V^{-1}=1+\sum_a^N\frac{m_a}{|\vec{r}-\vec{r}_a|}
$, whose particular case is the extreme Reissner-Nordström metric $V^{-1}=1-1/r$ considered in [@CFH].

On the other hand, the Bogomolny equation (\[1.3\]) for the rescaled metric $G_{\mu\nu}=g_{\mu\nu}/g_{oo}$ becomes, for $\psi=\phi/\sqrt{g_{oo}}$, precisely (\[Bogotype\]).

Which one is the ‘natural’ theory: that in (\[1.1\]), or the curvature-modified one in (\[3.1\]) ? An argument in favor of the second choice is the following: a static and purely magnetic theory can be converted, by putting $A^a_0=\sqrt{g_{oo}}\,\psi^a$, into a pure Yang-Mills configuration on euclidean four-space. Then essentially the same calculation as for (\[3.7\]) shows that the curvature-modified action (\[3.1\]) \[and *not* the simple expression (\[1.1\])\] becomes, up to a surface term, the pure Yang-Mills expression $-\smallover{1}/{4}
\tr F_{\mu\nu} F^{\mu\nu}\sqrt{g}$, whereas the $4D$ self-duality equations, $F_{\mu\nu}=\pm\2\sqrt{g}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$, become (\[Bogotype\]) [@CFH].

Our results have the following intuitive explanation: the self-duality equations are conformally invariant, but the field theory defined by (\[1.1\]) are not. However, adding the curvature term restores the conformal invariance, as it has been observed before in the Chern-Simons context [@DHP1].

It is worth mentioning that the procedure considered here can be further generalized. Let indeed $g_{\mu\nu}=(g_{oo},-\hat{g}_{ij})$ be an arbitrary static spacetime, and consider the rescaled metric $G_{\mu\nu}=(1,-\widehat{G}_{ij})$, $\widehat{G}_{ij}=\hat{g}_{ij}/g_{oo}$. Then for the YMH theory (\[1.1\]) with the rescaled metric $G_{\mu\nu}$, the Bogomolny (alias self-dual) equations (\[1.3\]) work. Then implementing the rescaling as $\psi=(g_{oo})^{-1/2}\phi$ provides us with a YMH theory on the original space with Lagrangian , \[genrescac\]\
=- \[Omega\] where a surface term has been dropped. The Bogomolny equations (\[1.3\]) become “Bogomolny-type” , Eqns. (\[Bogotype\]).

In the Papapetrou-Majumdar case, $\Omega=-R/2$, and we recover the results in [@BW].

Another interesting case can be that of AdS space [^2], $$ds^2=\left( 1-\frac \Lambda 3r^2\right) dt^2-\frac{dr^2}{1-\frac \Lambda 3r^2}-r^2d\omega,
\label{AdSmetric}$$ where $d\omega=d\theta^2+\sin^2\theta d\phi^2$. Then we find =. \[AdSomega\] For large $r$, (\[AdSomega\]) becomes approximately $\Omega\approx R/6$, where $R=4\Lambda$ is the scalar curvature.

Similarly in Schwarzschild space, $$ds^2=\left(1-\frac{r_c}r\right)dt^2-\frac{1}{\left( 1-\dfrac{r_c}r\right)}dr^2-r^2d\omega,
\label{Schwarzschild}$$ we have =. In the Reissner-Nordström case, at last, $$ds^2=(1-\frac{r_s}r+\frac{r_Q^2}{r^2})dt^2-\frac{dr^2}{(1-\dfrac{r_s}r+\dfrac{
r_Q^2}{r^2})}-r^2d\omega\,,$$ we find =, which, in the extreme case $r_s^2=4r_Q^2$, vanishes, leaving us with the result of [@CFH].

P.A.H is indebted to the *Institute of Modern Physics* (Lanzhou) of the Chinese Academy of Sciences for hospitality.

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[^1]: The Einstein equations can also be solved when $
    \tr\,\psi^2\equiv1
    $ everywhere in space [@BW; @FHH].

[^2]: Monopoles in AdS space were considered by Lugo et al [@LMS]. The do not have Bogomolny equations, however, since their model is different. In particular, they don’t have our $\Omega$-term here.