14 TCS Spectral Bounds

Twisted Connected Sum (TCS) manifolds provide a concrete construction for \(G_2\)-holonomy manifolds. This chapter formalizes the spectral theory connecting neck geometry to the mass gap \(\lambda _1 = 14/99\).

14.1 TCS Manifold Structure

Definition 14.1 TCS Manifold

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A TCS manifold \(K = M_1 \cup _N M_2\) consists of:

Two asymptotically cylindrical manifolds \(M_1, M_2\) (building blocks)
A cylindrical neck \(N \cong Y \times [0, L]\) with cross-section \(Y\)
Neck length parameter \(L {\gt} 0\)
Volume normalization: \(\mathrm{Vol}(K) = 1\)

Definition 14.2 TCS Hypotheses Bundle

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The six hypotheses (H1)–(H6) for spectral bounds:

Volume normalization: \(\mathrm{Vol}(K) = 1\)
Bounded neck volume: \(v_0 \leq \mathrm{Vol}(N) \leq v_1\)
Product metric: \(g|_N = dt^2 + g_Y\)
Block Cheeger bound: \(h(M_i \setminus N) \geq h_0\)
Balanced blocks: \(\mathrm{Vol}(M_i) \in [1/4, 3/4]\)
Neck minimality: \(\mathrm{Area}(\Gamma ) \geq \mathrm{Area}(Y)\)

14.2 Spectral Bound Constants

Definition 14.3 Lower Bound Constant

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\(c_1 = v_0^2\) (from Cheeger inequality)

Definition 14.4 Upper Bound Constant

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\(c_2 = \frac{16 v_1}{1 - v_1}\) (from Rayleigh quotient)

Definition 14.5 Threshold Length

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\(L_0 = \frac{2 v_0}{h_0}\) (neck dominates for \(L {\gt} L_0\))

14.3 Model Theorem

Theorem 14.6 TCS Spectral Bounds

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For TCS manifold \(K\) with neck length \(L {\gt} L_0\) satisfying (H1)–(H6):

\[ \frac{c_1}{L^2} \leq \lambda _1(K) \leq \frac{c_2}{L^2} \]

Theorem 14.7 Inverse Square Scaling

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The spectral gap scales as \(\lambda _1 = \Theta (1/L^2)\) as \(L \to \infty \).

Theorem 14.8 Typical Parameter Values

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For typical TCS with \(v_0 = v_1 = 1/2\), \(h_0 = 1\):

\(c_1 = 1/4\)
\(c_2 = 16\)
\(L_0 = 1\)
Bound ratio: \(c_2/c_1 = 64\)

14.4 Literature Axioms (Langlais/CGN)

Definition 14.9 Cross-Section

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Cross-section of TCS cylindrical end with dimension and Betti numbers.

Definition 14.10 K3 x S1 Cross-Section

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Standard Kovalev construction with \(\dim = 5\) and \(b_0 = 1, b_1 = 1, b_2 = 22, b_3 = 22, b_4 = 23, b_5 = 1\).

Theorem 14.11 Langlais Spectral Density

For TCS family \((M_T, g_T)\) with cross-section \(X\):

\[ \Lambda _q(s) = 2(b_{q-1}(X) + b_q(X))\sqrt{s} + O(1) \]

Reference: Langlais 2024, Comm. Math. Phys.

Theorem 14.12 K3 x S1 Density Coefficients

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For K3 \(\times \) S\(^1\):

2-forms: \(2(b_1 + b_2) = 2(1 + 22) = 46\)
3-forms: \(2(b_2 + b_3) = 2(22 + 22) = 88\)

Theorem 14.13 CGN No Small Eigenvalues

For TCS manifold with neck length \(L\): \(\exists c {\gt} 0\) such that no eigenvalues in \((0, c/L)\). Reference: Crowley-Goette-Nordstrom 2024, Inventiones Math.

Theorem 14.14 GIFT Prediction Structure

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The GIFT prediction \(\lambda _1 = 14/99 = \dim (G_2)/H^*\) is consistent with TCS bounds: \(1/100 {\lt} 14/99 {\lt} 1/4\).

14.5 Connection to Mass Gap

Theorem 14.15 TCS Bounds Certificate

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Complete verification of TCS spectral bounds and GIFT prediction compatibility.

Remark 14.16 Three-Level Derivation

The physical spectral gap \(\lambda _1 = 13/99\) emerges from three levels:

Model Theorem (PROVEN): \(\lambda _1 \sim 1/L^2\) from TCS geometry
Canonical Length (CONJECTURE): \(L^2 \sim H^*= 99\) from volume principle
Holonomy Coefficient (PROVEN): \(\mathrm{dim}(G_2) - h = 14 - 1 = 13\) (Berger)

The bare ratio \(14/99\) receives a correction of \(-1/99 = -h/H^*\) from the parallel spinor (see Chapter 15).