M5 brane to D4 brane via cyclification of rational relative 3-cohomotopy

In this article, we start by re-deriving the equations of motion and Bianchi identities for the abelian D4 brane worldvolume. Noting that the 4-flux in M-Theory is rationally flux quantized in a non-abelian cohomology theory called 4-cohomotopy, and the three-flux on the M5 brane worldvolume in rational (twisted) 3-cohomotopy, we compute the minimal model for the cyclification of the quaternionic Hopf fibration which encodes the Bianchi identities for the fluxes on the D4 brane worldvolume after double dimensional reduction. The two pictures can be mapped to each other, and thus at the rational level, we conjecture a non-abelian relative cohomology theory for the D4 brane, fibered over the 10d Type IIA spacetime fluxes.

💡 Research Summary

The paper investigates the relationship between the M‑theory M5‑brane and the Type IIA D4‑brane by employing the machinery of rational homotopy theory, specifically the cyclification of relative 3‑cohomotopy. Starting from the hypothesis that the M‑theory four‑form flux G₄ is quantized in the non‑abelian cohomology theory known as 4‑cohomotopy (π₀ Map(X, S⁴)), the authors recall that the eleven‑dimensional Bianchi identities d G₄ = 0 and d G₇ = −½ G₄∧G₄ coincide with the minimal Sullivan model of the 4‑sphere, written as the Chevalley‑Eilenberg (CE) algebra CE(l S⁴) = (Λ(g₄,g₇), dg₄ = 0, dg₇ = −½ g₄²).

Next, they consider the M5‑brane world‑volume theory. The three‑form field strength H₃ on the M5 couples to the bulk G₄ via the sourced Bianchi identity d H₃ = ϕ*G₄. This coupling is captured by the relative 3‑cohomotopy of the quaternionic Hopf fibration h : S⁷→S⁴, whose rational minimal model is CE(l S⁴ S⁷) = (Λ(h₃,g₄,g₇), dh₃ = g₄, dg₄ = 0, dg₇ = −½ g₄²).

To connect to the D4‑brane, the authors perform two operations simultaneously: (i) free loop space construction (L) on the target spheres, and (ii) quotient by the circle action (//S¹) which implements double dimensional reduction. The CE algebra of the cyclified 4‑sphere, CE(l LS⁴//S¹), contains a degree‑2 generator ω₂ that will later be identified with the D4 world‑volume electric field F₂, together with higher‑degree generators g₄, g₇, g₃, g₆ encoding the Type IIA RR fluxes (F₂, F₄, F₆) and the NS‑NS three‑form H₃.

The central technical result is the computation of the cyclified relative model for the quaternionic Hopf fibration, denoted CE(l L S⁷/S⁴//S¹). This algebra includes generators (ω₂, h₃, g₄, g₇, h₂, g₃, g₆) with differential relations:

• d g₄ = ω₂ g₃,
• d g₇ = −½ g₄² + ω₂ g₆,
• d h₃ = g₄ + ω₂ h₂,
• d h₂ = −g₃,
• d g₃ = 0,
• d g₆ = g₃ g₄,
• d ω₂ = 0.

Mapping the abstract generators to physical fields via the pull‑back ϕ* (ω₂ → F₂, g₄ → F₄, g₇ → H₇, g₃ → −H₃, h₃ → f₃ = −₅G, h₂ → F) reproduces exactly the Bianchi identities derived from the D4 world‑volume action. The authors first re‑derive the D4 equations of motion from the abelian DBI action plus the Chern‑Simons term, obtaining d F = ϕH₃, d ₅G = −(ϕF₄ + ϕF₂∧F).
These are precisely the relations obtained from the cyclified relative CE algebra after the above identification, confirming that the cyclization of rational relative 3‑cohomotopy encodes the full set of D4 flux Bianchi identities, including the non‑linear term ϕF₂∧F that cannot be captured by the DBI action alone.

When background fluxes are turned off, the cyclized relative model collapses to the CE algebra of LS³, which is simply (h₃, h₂) with trivial differential. This corresponds to K(ℚ,2)×K(ℚ,3) and reproduces the familiar five‑dimensional SYM Bianchi identities d f₂ = 0, d f₃ = 0 (with f₃ ∼ *₅f₂ after duality).

Based on these observations, the authors propose a “non‑abelian relative cohomology” theory for a single D4‑brane fibered over the Type IIA spacetime fluxes. Formally, they define a relative de Rham complex Σ_D4 Ω¹_dR(−; l(L S⁷/S⁴//S¹)) → Ω¹_dR(−; l(LS⁴//S¹)), with the pull‑back of the spacetime fluxes (F₂ᵢ, H₃, H₇) and the world‑volume fields (F, f₃) as coefficients. While an integral lift of this rational picture is not addressed, the paper demonstrates that at the rational level the cyclized relative cohomotopy provides a mathematically rigorous and physically accurate description of D4‑brane fluxes derived from M‑theory.

In summary, the work shows that the minimal model of the cyclified quaternionic Hopf fibration reproduces the D4‑brane Bianchi identities obtained from conventional world‑volume actions, thereby establishing a concrete bridge between M‑theory’s non‑abelian cohomotopy quantization and Type IIA D‑brane physics. This contributes a novel perspective on how higher‑cohomotopical structures survive dimensional reduction and suggests a broader framework for non‑abelian relative cohomology in string/M‑theory.