The Born connection

The Born connection is the connection of a Born geometry, phase space with the three compatible structures ω, η and H of Born geometry: the rule for comparing vectors at neighbouring points that respects all three structures. A connection defines how fields are differentiated, and its curvature is what the field equations of a geometry are built from. Freidel, Rudolph and Svoboda showed that a Born geometry has exactly one connection compatible with all three under the conditions of their theorem, so its curvature R_B[H] is well defined. The program’s first action for the phase-space metric was built on that curvature. In the current reference the connection appears in a citation and in the argument that frames, not metrics, are the natural variables for the mass term of the momentum graviton. Under the corrected postulate only ω and H are fundamental, and whether a unique compatible connection still exists then has not been shown.

Status: Established — literature [T] (#1 §2), cited from model memory as #11 ref. [7]† and not yet checked against the source (the † literature battery) — not machine-checked. Uniqueness for the (ω, H)-only structure of P2′ is open [O] (#5 §F item 5); it is recorded below and is not among #11’s numbered open problems.

What a connection is [T]#

Vectors at different points of a curved space live in different tangent spaces, and comparing them needs a rule. A connection ∇ supplies it: it defines the covariant derivative of a field and parallel transport along a curve. A connection is compatible with a structure S, such as a metric or a symplectic form, when ∇S = 0, which means that parallel transport preserves S. Its torsion measures the failure of infinitesimal parallelograms to close, and its curvature measures how a vector rotates when transported around a small loop.

Compatibility alone does not fix a connection. The standard uniqueness statement is the Levi-Civita theorem: a metric has exactly one compatible connection with zero torsion. General relativity’s Einstein–Hilbert action, the integral of √|g| R over spacetime, is built from that connection’s curvature scalar R.

A Born geometry carries three structures, and a connection on it has to preserve ω, η and H at once. Whether any connection does so, and what further condition makes one unique, as zero torsion does for a single metric, is the question Freidel, Rudolph and Svoboda answered.

The theorem [T]#

The lab notebook (#1 §2) states the result and what follows from it:

A theorem of Freidel, Rudolph & Svoboda (2018) establishes the existence and uniqueness of a connection compatible with the full triple (ω,η,H) — the Born connection. [T] Its curvature R_B[H] is therefore well-defined, and a diffeomorphism-style invariant action for H can be written.

Checkpoint #11 §5 cites it in one clause, “a unique compatible connection exists [7†]”, and the v1 program paper as “[T: Freidel–Rudolph–Svoboda 2018]” (#9 §3). None of the documents states the theorem’s hypotheses beyond compatibility with the full triple; in particular they do not say which condition on the torsion makes the connection unique.

The reference. #11 [7]† L. Freidel, F. J. Rudolph, D. Svoboda, “A unique connection for Born geometry,” Commun. Math. Phys. (2019), arXiv:1806.05992. The lab notebook and #9 date the result 2018, #11 dates it 2019. No document reconciles the years; the difference is probably an arXiv year against a journal year, and it is left to the † literature battery (conflicts between sources). The toy-model note’s ref. [2], arXiv:1706.07089, is a different paper by the same authors, “Generalised kinematics for double field theory”.

How the program used it#

The lab notebook’s geometric action (v0.1, superseded)#

The notebook wrote an Einstein-like action for the phase-space metric from the Born curvature (#1 §3):

S_H = ½ M_B² ∫_Σ √|H| R_B[H]        over the section Σ, with ω, η frozen (P2)

“On the x-tilted vacuum, H decomposes into a spacetime metric g_μν(x) and a momentum-space metric g̃^μν; the g-block reproduces the Einstein–Hilbert action with M_P fixed by M_B and the section volume. [D, schematic]” (#1 §3). The section Σ comes from a section condition borrowed from double field theory: “Field dependence is restricted to a Lagrangian leaf by a section condition, as in double field theory; consistency of the section condition on general backgrounds is open there and open here. [O]” (#1 §2).

The notebook’s first open problem records how far this went: “The field equation for H beyond minisuperspace — Born’s 88-year debt. The action exists (Born connection [T]), but solutions, the section condition on curved backgrounds, and the full ghost analysis do not. [O]” (#1 §11 item 1).

The compatibility theorem then emptied the action of new content. With ω and η both rigid, H = diag(g, g⁻¹), and “S_H is a functional of g alone: a pure metric theory” (calc note #5 §B). That finding forced the change from the original P2 to P2′. The superseded action is recorded on the geometric action S_H.

After P2′#

With the modular field Z = X + iY in place, calc note #5 restated the old debt as “the field equation for momentum-space geometry is the Re Z sector of S_H” (§C) and posed the decisive computation: “expand the generalized curvature scalar of (ω, H_Z) to quadratic order in X about X = 0 on the section, with ω-generalized diffeomorphisms as the gauge symmetry” (§D). The Debt-1 notes then settled the character of X by symmetry arguments and the tuning test instead (why X must be massive), and the step back from the frame-tier potential to S_H has not been made. #11 lists it as open problem 1: “First-principles reduction of the frame-tier wedge sector from S_H (class selected, not derived) [O]” (Open: deriving the wedge sector from S_H).

The first-order tier#

The connection’s remaining role in #11 is in an argument about variables. A vielbein, or frame, E is a square root of the metric, H = EᵀE. Calc note #7 §3 sorts the program’s objects by how they enter: “J and E live on the same geometric tier (first-order structures on the tangent, like the Born connection), while H = EᵀE is second-order.” The tuning test fails for potentials built from metrics and passes for potentials built from frames, and the shared tier is the reason given for preferring the frame formulation (#11 §7.2): “J, E, and the Born connection share the first-order tier, so this is the geometrically minimal formulation.” The notebook’s v0.5 addendum says the same. See the frame tier.

Open: connection uniqueness under P2′#

Under P2′ only ω and H are fundamental. η is not a background structure: “the vacuum spontaneously selects a real structure J (an involution anticommuting with I), and η ≡ HJ” (#5 §B). Freidel, Rudolph and Svoboda’s theorem covers the full triple (ω, η, H). Whether ω and H alone determine a unique compatible connection is item 5 of calc note #5’s list of what is “Now precisely open”:

(5) connection uniqueness for the (ω, H)-only structure ([Freidel–Rudolph–Svoboda] proved it for the full triple; the para-Hermitian literature — Svoboda, Marotta–Szabo — is the nearest neighbor).

The question bears on the computation #5 §D posed, which is phrased in terms of “the generalized curvature scalar of (ω, H_Z)”, a curvature for exactly the (ω, H) structure. The item is not carried into #11 §14 and no later document reports work on it. This wiki keeps it open [O] (open problems). The para-Hermitian literature it points to is described on the structures ω, η, H.

Used by#

Sources#

  • papers/broken-born-checkpoint-v2.md §5 and ref. [7] — “a unique compatible connection exists [7†]”; the reference and its 2019 date.
  • papers/broken-born-checkpoint-v2.md §7.2, §14 item 1 — the first-order tier shared with the vielbein; the wedge sector not derived from S_H.
  • papers/reciprocity-toy-models.md ref. [2] — the other Freidel–Rudolph–Svoboda paper.
  • notes/debt1a-geometric-core.md §B, §C, §D, §F — S_H reduced to g under strict P2; η ≡ HJ under P2′; the curvature of (ω, H_Z); item 5, connection uniqueness for (ω, H) alone.
  • notes/debt1a-iii-tuning-test.md §3 — J, E and the Born connection on the first-order tier.
  • notes/broken-born-theory.md §2, §3, §11 item 1 — the theorem as #1 states it, with R_B[H] and the 2018 date; S_H; the field equation for H as open; ‘Addendum (v0.5)’ — the first-order tier.
  • notes/broken-born-program-v1.md §3 — the same attribution, dated 2018.
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