Investigate whether the imaginary part of the Chern-Simons action for flat connections can be understood as an integration of a pullback of the Cartan 3-form | Step-by-Step Solution

Hello! This is a fascinating problem that sits at the beautiful intersection of differential geometry, topology, and representation theory.

1. What We're Solving:

You're exploring whether there's a fundamental geometric relationship between two important mathematical objects: the imaginary part of the Chern-Simons functional for flat SL_n(ℂ) connections and the Cartan 3-form. This is essentially asking if we can interpret a topological invariant (coming from gauge theory) in terms of classical Lie group geometry.

2. The Approach:

This is a research-level investigation that requires you to:

• Understand the geometric meaning of both objects
• Explore how they might be related through pullback operations
• Investigate whether this relationship holds for flat connections specifically
• Consider what this tells us about the topology of the underlying manifold and connection space

3. Step-by-Step Research Framework:

Step 1: Understand Your Objects

• Chern-Simons Functional: For a connection A on a principal SL_n(ℂ) bundle over a 3-manifold M, CS(A) = ∫_M Tr(A ∧ dA + (2/3)A ∧ A ∧ A)
• Cartan 3-form: The canonical 3-form on the Lie group SL_n(ℂ), typically ω = Tr(g⁻¹dg ∧ g⁻¹dg ∧ g⁻¹dg)
• Flat connections: Connections with zero curvature (dA + A ∧ A = 0)

Step 2: Identify the Geometric Setup

• Consider what maps might create the pullback relationship
• Think about holonomy maps from the fundamental group π₁(M) → SL_n(ℂ)
• Explore how flat connections correspond to representations of π₁(M)

Step 3: Analyze the Imaginary Part

• Decompose the Chern-Simons action into real and imaginary components
• Understand why we focus on the imaginary part specifically
• Consider how this relates to the hermitian structure

Step 4: Investigate the Pullback Relationship

• Examine how the holonomy representation might pull back the Cartan form
• Consider whether ∫_M pullback(Cartan 3-form) equals Im(CS(A)) for flat A
• Look into existing literature on this relationship (Reznikov, Dupont-Zocca work)

Step 5: Explore Consequences and Examples

• Test your findings on simple examples (like lens spaces, knot complements)
• Consider what this relationship tells us about both objects
• Investigate implications for topology and representation theory

4. Research Framework Structure:

Introduction Approach:

• Start with the fundamental question: "What geometric meaning does the imaginary part of Chern-Simons carry?"
• Motivate why connecting to Cartan forms is natural and important

Main Investigation Structure:

• 1. Setup and Definitions: Carefully define all objects involved
• 2. The Conjectured Relationship: State precisely what relationship you're investigating
• 3. Theoretical Analysis: Work through the mathematics step by step
• 4. Examples and Verification: Test on concrete cases
• 5. Implications and Conclusions: What does this tell us about the broader theory?

Key Questions to Address:

• When does this relationship hold exactly?
• What role does flatness play in making this work?
• How does this connect to classical results in topology?

5. Memory Tip:

Think of this as asking: "Can we read off topological information about 3-manifolds by looking at how their fundamental groups 'twist' the geometry of the Lie group SL_n(ℂ)?" The Chern-Simons functional captures topological data, while the Cartan form captures the intrinsic geometry of the group - you're investigating how these perspectives connect!

This is cutting-edge mathematics, so approach it with curiosity and patience. The journey of understanding these connections is just as valuable as any final result you might reach!