A Kaluza–Klein / Quantum-Gravity Reformulation of the Yang–Mills Mass Gap

Why look at gravity at all?

The Clay Millennium “Yang–Mills existence and mass gap” problem asks for two things for any compact simple gauge group (G):

1. A mathematically precise construction of four-dimensional quantum Yang–Mills theory on (\mathbb R^4) satisfying standard field-theoretic axioms (in practice: something strong enough to justify reflection positivity / reconstruction and locality).
2. A proof that the physical spectrum has a strictly positive gap above the vacuum (equivalently: gauge-invariant correlators decay exponentially with a uniform rate (\Delta>0) at large separations).

This is hard partly because it is hard to define the theory nonperturbatively in the continuum, and partly because the gap is a deeply nonperturbative infrared property.

The gravitational instinct is: perhaps Yang–Mills should be viewed as an emergent sector of a higher-dimensional geometric theory, and perhaps “gap-ness” becomes a geometric or spectral statement.

That instinct has a close cousin in holography (gauge/gravity duality), where confinement and mass gaps often become statements about the infrared behaviour of a dual geometry. But holography typically comes with caveats (large (N), strong coupling regimes, extra fields, conjectural equivalence, etc.).

Here I want to sketch a different refocus: a Kaluza–Klein (KK) reformulation that tries to tie the gauge group directly to the isometries of a compact internal space, and tries to define the target QFT as a decoupling limit of a UV-complete quantum gravity.

The punchline is not “this solves the Clay problem”, but rather: it suggests a new way to package the problem in which the existence of the QFT might be inherited from quantum gravity, and where semisimple isometries are a natural geometric filter that removes obvious obstructions to a gap.

The KK mechanism: isometries become gauge symmetry

Start with a spacetime of the form

[ \mathcal M_{4+d} ;\approx; X_4 \times K_d, ]

where (K_d) is compact (the “internal space”) and (X_4) is the “external spacetime” that will become Minkowski space in the decoupling limit.

In KK reductions, components of the higher-dimensional metric with one leg along (X_4) and one along (K_d) behave like gauge fields in 4D. More precisely, if ({\xi_a}) are Killing vector fields on ((K_d,g_K)), then fluctuations along those directions give vector fields (A^a_\mu) on (X_4). Their non-abelian structure constants are the Lie brackets of the Killing fields:

[ [\xi_a,\xi_b] = f_{ab}{}^c,\xi_c \quad\Rightarrow\quad (A_\mu)^a\text{ gauge fields with Lie algebra }\mathfrak g. ]

So a compact semisimple connected isometry group (\mathrm{Isom}_0(K,g_K)) naturally produces a compact semisimple gauge group in the 4D zero-mode sector.

This is exactly why “non-abelian semisimple isometry group” is a relevant geometric property: it is a clean way to manufacture a non-abelian gauge symmetry without putting gauge fields in by hand.

Two different “gaps” appear immediately

Before discussing the Yang–Mills gap, it is useful to separate two notions that are often conflated:

1. KK tower gap (spectral gap on the internal space).
   Because (K) is compact, Laplace-type operators on (K) have discrete spectra. Nonzero KK modes come with masses (m_n^2\sim \lambda_n(K)). The first nonzero eigenvalue (\lambda_1) gives a perturbative gap above the zero modes.

2. Yang–Mills mass gap (nonperturbative confinement scale).
   Even after truncating to the zero modes, pure 4D Yang–Mills is expected to generate a dynamical scale (\Lambda) and produce massive glueball excitations with mass (\sim \Lambda). Proving (m_{\text{glueball}}>0) is the Clay problem.

The KK gap is relatively geometric; it can often be bounded in terms of curvature (e.g. lower Ricci bounds imply lower bounds on (\lambda_1) for scalars). But the Yang–Mills gap is an interacting QFT statement.

A promising reformulation would use KK geometry to:

• ensure that unwanted light fields are lifted (no extra massless scalars from moduli),
• isolate a pure Yang–Mills sector at low energies,
• and then argue that the low-energy theory inherits a mass gap.

The nontrivial part is the last bullet.

Why insist on semisimple isometries?

If the isometry group has a (U(1)) factor, there is an obvious obstruction to a gap: in the simplest cases the 4D theory contains a massless photon, so the gauge-invariant sector cannot be gapped in the strong sense demanded by the Clay statement.

Semisimplicity is a precise way to exclude abelian factors at the level of the continuous symmetry group.

There is also a deeper geometric reason: Euclidean factors in the universal cover of (K) lead to translation symmetries, which are abelian normal subgroups of the isometry group. Semisimplicity rules this out. In that sense, semisimplicity is a “no flat direction” filter, but phrased globally.

The decoupling limit and the “Minkowski question”

A KK model always has multiple scales:

• (M_{\mathrm{Pl},4}): the 4D Planck mass,
• (M_{\mathrm{KK}}\sim 1/\mathrm{diam}(K)): the KK scale,
• (g_4): the 4D gauge coupling of the isometry-generated gauge bosons,
• and the dynamical scale (\Lambda) generated by the 4D gauge dynamics.

A clean “pure QFT” limit demands something like:

[ M_{\mathrm{Pl},4}\to\infty,\qquad M_{\mathrm{KK}}\to\infty,\qquad g_4\to g_*\in(0,\infty), ]

and that all non-gauge degrees of freedom become heavy compared to (\Lambda).

Do we need (X_4) to be exactly Minkowski? If the external space has compact directions (say (\mathbb R^{1,3-k}\times T^k)), one introduces an infrared regulator. Finite-volume spectra are discrete and can exhibit an “artificial gap” (\sim 1/L) even for gapless theories. So a statement about a gap on (\mathbb R^{1,3-k}\times T^k) only becomes equivalent to the Clay gap if the gap is uniform as (L\to\infty).

For a Clay-equivalent reformulation, the cleanest target is that the decoupling limit produces the theory on (\mathbb R^{3,1}) (or Euclidean (\mathbb R^4)) directly.

A proposed quantum-gravity reformulation

Here is a candidate statement that is at least as strong as the Clay problem (and in practice stronger, because it assumes a UV-complete quantum gravity and a controlled decoupling limit).

QG–KK Mass Gap Conjecture (strong form)

For every compact simple Lie group (G), there exists a UV-complete, unitary quantum gravity theory (\mathcal Q) in dimension (4+d) and a one-parameter family of stable vacua ({\mathcal V_\epsilon}_{\epsilon>0}) such that:

1. Geometric regime. The low-energy classical background of (\mathcal V_\epsilon) is asymptotic to [ X_4(\epsilon)\times (K_\epsilon,g_\epsilon) ] with (K_\epsilon) compact.

2. Semisimple isometry gauge group. The connected isometry group (\mathrm{Isom}0(K\epsilon,g_\epsilon)) contains (G) (up to finite cover), and the corresponding KK zero modes produce a 4D gauge sector with gauge group (G).

3. Decoupling to pure YM. There exists a limit (\epsilon\to 0) in which

   • (X_4(\epsilon)\to \mathbb R^{3,1}) (flat Minkowski),
   • (M_{\mathrm{Pl},4}(\epsilon)\to\infty) (gravity decouples),
   • (M_{\mathrm{KK}}(\epsilon)\to\infty) (all KK excitations decouple),
   • all non-gauge massless fields are lifted (no moduli remain),
   • and the only light interacting sector that survives is pure 4D Yang–Mills with gauge group (G).

4. Mass gap. The resulting 4D theory has a spectral mass gap (\Delta>0) in the gauge-invariant sector (exponential decay of Euclidean correlators, or positive lower bound in the Hamiltonian spectrum above the vacuum).

This implies the Clay mass gap statement because it constructs the relevant QFT as a scaling limit of a well-defined UV theory and asserts the same kind of gap. It is stronger because it additionally demands a geometric UV completion and a precise KK realisation of the gauge symmetry.

A weaker “equivalent” packaging

If one wants a statement that is logically closer to Clay, one can separate the conjecture into two pieces:

• (A) Realisability: pure 4D Yang–Mills can be obtained as a decoupling limit of a UV-complete quantum gravity compactification with semisimple internal isometries.
• (B) Inheritance: whenever such a decoupling limit exists and produces pure YM, the resulting YM theory has a mass gap.

Then (A)+(B) implies Clay. Conversely, Clay implies (B) for any limit that truly yields pure YM, but says nothing about (A).

Where could curvature enter?

The “semisimple isometry” condition is global and algebraic, not a simple curvature inequality. But curvature can help in the decoupling step, especially to eliminate unwanted light fields.

Two places geometry is immediately useful:

1. Bounding the KK spectrum: lower curvature bounds can yield lower bounds on the first nonzero eigenvalues of Laplacians on (K), supporting a parametrically large KK scale.

2. Eliminating moduli: the dangerous massless scalars in KK reductions are often metric deformations of (K) (and, in more general theories, deformations of fluxes or additional fields). Infinitesimal metric moduli are controlled by the kernel of the Lichnerowicz Laplacian on symmetric 2-tensors, subject to gauge conditions. Certain classes of Einstein and locally symmetric spaces are known to be rigid (no such zero modes), which is exactly what one needs if the aim is “pure YM and nothing else light”.

These are the points where one might hope to turn “gap” into “geometry”: first make the spectrum above the YM sector large by curvature control, then attempt to argue that the YM sector itself confines.

Why this is not just holography in disguise

Holographic confining models already translate “gap” into a geometric statement on the gravity side, but usually:

• they describe families of gauge theories that are not exactly pure YM, or
• they require large (N) and strong coupling for clean control,
• and the duality is not (currently) a theorem in a form that would satisfy the axiomatic existence requirement.

The KK–QG reformulation is different in emphasis:

• it tries to define the 4D QFT via a decoupling limit of a UV-complete theory (so existence might be inherited),
• it ties the gauge group directly to compact internal isometries (a crisp geometric handle),
• and it aims at fixed gauge group (G) (not just large (N) limits).

It is, however, still speculative: the hard part becomes showing that the decoupled sector is exactly pure YM and then proving its gap.

The central difficulty: the gap step is still Yang–Mills

Even if one succeeds in building a KK compactification whose only light sector is pure 4D Yang–Mills, the statement “there is a mass gap” is then essentially the Clay assertion for that theory.

So why might this reformulation be worthwhile?

Because it changes what one tries to prove first.

Instead of constructing YM directly in the continuum, one tries to:

• construct a UV-complete QG theory (\mathcal Q),
• construct a family of vacua with strong geometric control,
• prove the existence of a scaling limit isolating YM,
• and only then address the infrared mass gap.

If the QG side provides a canonical nonperturbative definition (for example, via a path integral or Hamiltonian formulation that is better behaved), this might shift the “existence” part of Clay to a different domain.

But it is only an advantage if that definition remains meaningful and controlled in the decoupling limit, and if the properties required by QFT axioms can be shown to persist.

A concrete research programme (what would need to be shown)

To make the KK–QG reformulation genuinely useful, one would want results of the following flavour.

1. Geometric filter: classify compact ((K,g)) for which (\mathrm{Isom}_0(K,g)) is compact semisimple, and understand how stable this property is under deformations.

2. Vacuum engineering: find higher-dimensional actions and stabilisation mechanisms that admit Minkowski(_4) vacua with such (K), while lifting moduli.

3. Uniform decoupling theorem: prove that in an appropriate scaling limit, correlators of gauge-invariant operators converge to correlators of a 4D QFT satisfying the desired axioms.

4. Gap inheritance mechanism: identify a structural reason—unique to the QG origin—why the limiting YM sector must be gapped. This is the part that would make the approach more than a repackaging.

Without step (4), the programme reduces to “use QG to define YM, then solve Clay inside that definition”. That is still interesting, but it is not obviously easier.

Closing thought

Semisimple isometries are a natural geometric way to select KK compactifications whose emergent gauge symmetry has no abelian factors. That aligns well with the expectation that a confining pure gauge theory should have a mass gap.

The genuinely new ambition would be to turn the Clay problem into a statement about the existence and properties of decoupling limits of quantum gravity vacua. If quantum gravity supplies a constructive framework that survives the limit, this could change how one attacks the “existence” part of the problem.

Whether it can also make the “gap” part more tractable remains open—and identifying an actual inheritance mechanism is the key challenge.

Further reading (non-exhaustive)

• Yang–Mills existence and mass gap (Clay Millennium problem statement; Jaffe–Witten).
• Classic Kaluza–Klein reductions and isometry-gauge symmetry mechanism.
• Confining gauge theories from gravity: Witten’s model, AdS soliton ideas, holographic QCD.
• Rigidity of Einstein / locally symmetric spaces; Lichnerowicz Laplacian and moduli.
• Spectral geometry bounds for Laplacian eigenvalues (KK tower gaps).

The NCG Clay Mass Gap Problem: A Spectral-Action Reformulation

Aim

The Clay Millennium “Yang–Mills existence and mass gap” problem asks, in four dimensions, for a mathematically precise construction of pure Yang–Mills theory satisfying standard axioms and for a proof that the physical spectrum has a strictly positive gap above the vacuum.

This post states an NCG analogue in which gauge fields are not fundamental variables. Instead one varies the Dirac operator by inner fluctuations D ? D_A, uses the spectral action Tr ?(D_A/?) as the bare bosonic functional, and replaces “construct Yang–Mills” by “construct the renormalised spectral-action measure and show it defines a 4D QFT”.

Two extra constraints are built in. First, the condition “semisimple with no abelian factors” is imposed without naming any Lie group; it is encoded intrinsically in the finite algebraic data. Second, the almost-commutative input is not arbitrary: it is required to arise from a commutative spectral triple by a specified finite-resolution truncation (or quantisation) procedure.

Commutative and almost-commutative spectral triples

A commutative spectral triple (A, H, D; J, ?) satisfying the usual “manifold” axioms reconstructs a compact spin (or spin^c) manifold K with A ? C?(K), H ? L²(K, S), and D a Dirac-type operator. This is the standard route by which Riemannian geometry is packaged into spectral data.

An almost-commutative triple has the form A = C?(M) ? A_F with A_F finite-dimensional, H = L²(M, S) ? H_F, and a product Dirac operator D? = ??_M ? 1 + ?? ? D_F. One may think of this as a “discrete Kaluza–Klein” geometry M × F: inner fluctuations generate gauge bosons (and, typically, scalar fields), while the spectral action supplies the bosonic dynamics.

The formulation below is Euclidean. Take M = R?, or use the torus T?_L as an infrared regulator and pass to L ? ? at the end.

Gauge fields as inner fluctuations

Given an (even, real) spectral triple (A, H, D?; J, ?), Connes’ one-forms are generated by commutators with D?. Concretely, ?¹_{D?}(A) consists of finite sums of the form ? a_i [D?, b_i] with a_i, b_i in A. A “gauge potential” is a self-adjoint element A ? ?¹_{D?}(A).

The fluctuated Dirac operator is

D_A := D? + A + ?? JAJ^{-1},

with ?? determined by KO-dimension. This single object replaces the classical gauge field A_?.

Gauge transformations come from unitaries u ? U(A). In the real setting they are implemented by uJuJ^{-1} on H, and act by conjugation D_A ? u D_A u*.

Bare dynamics: the spectral action

Fix a positive cutoff function ? and an ultraviolet scale ?. The bosonic bare action is

S_?(D_A) := Tr ?(D_A/?).

In almost-commutative geometries the large-? expansion contains the Yang–Mills term for the gauge bosons produced by inner fluctuations, accompanied by higher-derivative gauge-invariant corrections. In the present problem statement, the spectral action is the defining bare functional for the Euclidean measure.

Encoding “semisimple with no abelian factors” without naming a group

The Clay problem fixes a compact simple gauge group. Here the aim is to avoid naming any group and impose instead a purely algebraic condition on the internal finite data.

Let (A_F, H_F, D_F; J_F, ?_F) be the finite real spectral triple, and let ?_F: A_F ? B(H_F) be the representation. Consider the real Lie algebra of skew-adjoint elements

u(A_F) := { x ? A_F : x* = ?x }.

Impose unimodularity using the trace on H_F:

u?(A_F, H_F) := { x ? nu(A_F) : Tr_{H_F}(?_F(x)) = 0 }.

Remove central directions that survive unimodularity by setting

z? := nu?(A_F, H_F) ? Z(A_F),

g_F := nu?(A_F, H_F) / z?.

The intrinsic translation of “semisimple with no abelian factors” is the Lie-algebra condition

g_F = [g_F, g_F],

equivalently that the solvable radical of g_F vanishes. This encodes the “no U(1)” content without mentioning any specific group.

Deriving almost-commutative data from commutative geometry

Almost-commutative models are often introduced as starting points. Here, instead, one requires the finite algebra to come from an underlying commutative internal geometry via a fixed finite-resolution procedure.

Begin with a commutative spectral triple (A_K, H_K, D_K) satisfying commutative reconstruction hypotheses, so A_K ? C?(K) for some compact spin manifold K. Fix a truncation scheme Trunc_N. A canonical choice uses the Dirac spectrum of K: let P_N be the spectral projection onto eigenspinors of D_K with |?| ? N, compress the commutative algebra to P_N ?_K(C?(K)) P_N ? B(P_N H_K), and define A_{F,N} to be the finite-dimensional C*-algebra generated by these compressed operators. Set H_{F,N} := P_N H_K, and take D_{F,N} to be a suitable finite operator derived from P_N D_K P_N (allowing standard finite-rank adjustments if needed to meet the finite-triple axioms).

For each N this gives finite internal data (A_{F,N}, H_{F,N}, D_{F,N}) extracted from the commutative geometry K. One then defines the almost-commutative triple at resolution N by

A_N := C?(M) ? A_{F,N},

H_N := L²(M, S) ? H_{F,N},

D?,N := ??_M ? 1 + ?? ? D_{F,N}.

The “no abelian factors” condition is lifted to a commutative-level admissibility requirement: the commutative internal triple is admissible if for all sufficiently large N, the intrinsic gauge Lie algebra g_{F,N} extracted from (A_{F,N}, H_{F,N}) satisfies g_{F,N} = [g_{F,N}, g_{F,N}].

The renormalised spectral-action measure

At fixed truncation level N and UV cutoff ?, the formal bare measure is

d?_{?,N}(D_{A,N}) ? exp(?S_?(D_{A,N})) ?D_{A,N},

where D_{A,N} ranges over inner fluctuations of D?,N modulo gauge transformations. A rigorous version replaces ?D by an explicit constructive definition of a gauge-invariant probability measure (for example via finite-dimensional cylindrical approximations) and then proves convergence of observables.

Renormalisation is expressed by allowing gauge-invariant local counterterms built from spectral invariants (the same local data that appears in the asymptotic expansion of the spectral action). One seeks counterterms C_?(D_{A,N}) such that

S^{ren}?(D{A,N}) := S_?(D_{A,N}) + C_?(D_{A,N})

defines renormalised measures

d?^{ren}{?,N}(D{A,N}) ? exp(?S^{ren}?(D{A,N})) ?D_{A,N}

whose gauge-invariant correlation functions have a continuum limit as ? ? ?.

Axioms: OS reconstruction and locality

To mirror the Clay problem’s insistence on an axiomatic construction, the Euclidean theory is required to satisfy the Osterwalder–Schrader axioms for gauge-invariant local observables built from D_{A,N}. Consider a class of gauge-invariant local observables O(D_{A,N}; x), for instance smeared local spectral invariants of D_{A,N} (after subtracting vacuum expectations) or gauge-invariant curvature polynomials encoded by D_{A,N}. Schwinger functions are defined by integrating products of such observables against ?^{ren}_{?,N}.

The requirement is that after taking ? ? ? and then removing the truncation N ? ? (and taking L ? ? if an IR torus regulator is used), the limiting Schwinger functions satisfy Euclidean invariance, symmetry, reflection positivity, suitable regularity, and clustering, so that OS reconstruction yields a Wightman QFT.

Explicit definition of the mass gap

Let OS reconstruction produce a Hilbert space ?, a vacuum vector ?, and a Hamiltonian H. A mass gap means that there exists ? > 0 such that

Spec(H) ? (0, ?) = ?.

Equivalently, there are no excitations with invariant mass below ?. In Euclidean terms, one can capture the same property by exponential decay of connected two-point functions: for every gauge-invariant local observable O with ?O? = 0, there exist constants C_O and ?_O > 0 with

|?O(x) O(0)?_c| ? C_O e^{??_O |x|}.

The mass gap is ? := inf_O ?_O, and “there is a mass gap” means ? > 0.

The NCG Clay Mass Gap Problem

Fix a truncation scheme Trunc_N from commutative triples to finite internal data, and fix a cutoff function ? defining the spectral action.

For every commutative spectral triple (A_K, H_K, D_K) satisfying commutative reconstruction hypotheses, assume it is admissible in the sense that its induced finite internal algebras A_{F,N} (via Trunc_N) yield intrinsic gauge Lie algebras g_{F,N} with no abelian factors for all sufficiently large N, namely g_{F,N} = [g_{F,N}, g_{F,N}].

For such input data, form the almost-commutative triples (A_N, H_N, D?,N) on M = R? (or on T?_L with L ? ? at the end), consider inner fluctuations D_{A,N}, and define the bare action S_?(D_{A,N}) = Tr ?(D_{A,N}/?).

The problem is to construct renormalised, gauge-invariant Euclidean measures ?^{ren}_{?,N} on the space of inner fluctuations modulo gauge, by adding gauge-invariant local counterterms built from spectral invariants, such that gauge-invariant Schwinger functions have a UV limit ? ? ? and a truncation-removal limit N ? ? which together produce Schwinger functions on R? satisfying the OS axioms and defining a nontrivial Wightman QFT. The final requirement is to prove that the reconstructed theory has a strictly positive mass gap ? > 0 as defined above.

Why the commutative origin constraint might matter

If one begins with an arbitrary finite algebra A_F, the space of internal gauge Lie algebras and representations is vast, and it is unclear which constraints are genuinely natural. Requiring instead that A_F arises from a commutative internal geometry K via a fixed truncation scheme ties A_F and D_F to an actual Dirac spectrum and its spectral constraints. It also introduces a built-in resolution parameter N that can be scaled relative to ?, potentially enabling sharper control of the continuum limit and of the decoupling of unwanted degrees of freedom.

Whether this ultimately makes the mass-gap step more tractable depends on finding an inheritance mechanism specific to the spectral-action formulation. Even without such a mechanism, the reformulation is meaningful as a precise axiomatic target for what it would mean to solve the mass gap problem inside Connes’ framework.

Kaluza–Klein Quantum Gravity as a Reformulation of the Yang–Mills Mass Gap Problem

1. Why revisit Kaluza–Klein at all?

The Clay Millennium “Yang–Mills existence and mass gap” problem asks for something both elementary to state and notoriously difficult to deliver: construct four?dimensional quantum Yang–Mills theory for any compact simple gauge group (G) (in a mathematically precise framework such as Wightman or Osterwalder–Schrader axioms) and prove that the theory has a strictly positive mass gap.

In physics culture there is a pervasive belief that pure non?abelian Yang–Mills in 4D confines and is gapped, while pure abelian gauge theory is gapless. Yet the Clay problem is not “argue plausibly”; it is “construct and prove.”

Holography and string dualities have generated an immense body of evidence for gapped dynamics from gravitational or geometric constructions, but those arguments are typically conditional on conjectural dualities and often live in special limits (large (N), strong ’t Hooft coupling, supersymmetric deformations, extra matter, compactified directions, etc.). That makes them invaluable intuition, but not a direct bridge to the Clay statement.

Kaluza–Klein (KK) offers a different angle: gauge fields can arise from higher?dimensional gravity in a way that is conceptually clean and entirely geometric. This suggests a potentially “orthogonal” reformulation: instead of proving the mass gap directly in a 4D gauge theory, prove it as a theorem about a class of quantum gravity compactifications whose Minkowski/flat limit forces the emergence of pure Yang–Mills.

This post lays out that reformulation precisely, explains what geometric conditions are relevant (especially conditions on the internal compact space that kill abelian gauge factors), and identifies the actual bottlenecks where the Clay difficulty reappears.

2. The KK mechanism: how gauge fields appear from a metric

Start with a ((4+n))-dimensional spacetime that is (at least semiclassically) a product

[ X = M_4 \times K, ]

where (M_4) is the “external” spacetime and (K) is a compact (n)-manifold with a Riemannian metric (h). In the simplest KK story, one expands the higher?dimensional metric (\hat g) in a basis adapted to the product. The mixed components (one leg on (M_4), one leg on (K)) behave like gauge potentials in 4D.

A standard ansatz looks like

[ \mathrm{d}\hat s^2 = g_{\mu\nu}(x),\mathrm{d}x^\mu\mathrm{d}x^\nu

• h_{ij}(y),\big(\mathrm{d}y^i + A^i_{\mu}(x,y),\mathrm{d}x^\mu\big)\big(\mathrm{d}y^j + A^j_{\nu}(x,y),\mathrm{d}x^\nu\big)
• \cdots, ]

where the dots include scalar fields (radions, shape moduli, etc.) depending on the precise truncation.

At the linearised level the massless gauge fields in 4D come from projecting the mixed metric fluctuations onto Killing vector fields on ((K,h)). Concretely, if ({\xi_a}) is a basis of Killing fields generating (\mathfrak{isom}(K,h)), one expands

[ A^i_{\mu}(x,y) \sim \sum_a A^a_{\mu}(x),\xi_a^i(y) + (\text{massive KK modes}). ]

Then (A^a_\mu(x)) are the would?be 4D gauge bosons, and their gauge symmetry is inherited from higher?dimensional diffeomorphisms along (K).

The headline is:

The connected isometry group (\mathrm{Isom}_0(K,h)) is the connected gauge group of the massless KK vectors.

This is the geometric core of the proposal.

3. Why “non?abelian semisimple isometry group” is the right geometric constraint

If (K) has a torus factor in its isometries (or, more generally, a positive?dimensional abelian normal subgroup in (\mathrm{Isom}_0(K,h))), then KK produces abelian gauge fields. Abelians are precisely where gaplessness is expected (Maxwell) and where confinement is not generic.

So one wants internal geometries whose continuous isometries are “as non?abelian as possible,” ideally with no (U(1)) factors at all.

The natural Lie?theoretic notion here is:

• (H := \mathrm{Isom}_0(K,h)) is compact semisimple (hence no torus factor in (H), no connected centre).
• Often one even wants simple rather than semisimple, but semisimple already kills the abelian pieces.

From the KK perspective this is appealing: the massless gauge group is then a compact semisimple group, i.e. precisely the class for which pure YM is expected to confine and have a gap.

Geometrically, the condition “(\mathrm{Isom}_0(K,h)) is compact semisimple” is not easy to express purely by pointwise curvature inequalities, because isometry groups are global symmetry objects. However, there are important geometric regimes where it is tightly controlled:

• If ((K,h)) is a compact Riemannian symmetric space of compact type and has no flat de Rham factor, then its connected isometry group is compact semisimple. In this case a curvature?level signature is (\nabla R = 0) (local symmetry) together with “no Euclidean factor” in the de Rham decomposition.

• In many homogeneous settings, one can compute (\mathrm{Isom}_0) from the reductive structure (K \simeq G/H) and check semisimplicity explicitly.

This suggests a workable “geometric class” of internal spaces: compact symmetric spaces (or selected homogeneous cosets) without flat factors.

4. The refocus: from a field theory theorem to a gravity decoupling theorem

A KK?gravity reformulation can be stated as a theorem/conjecture about a limit of a quantum gravity theory. The idea is:

1. Build a UV?complete quantum theory of gravity on (X = \mathbb{R}^4 \times K) (Euclidean) or (\mathbb{R}^{3,1} \times K) (Lorentzian) whose semiclassical vacuum is the product metric.

2. Prove that, in a scaling limit where gravity decouples and the KK tower becomes infinitely heavy, the remaining effective theory on (\mathbb{R}^4) is pure Yang–Mills with gauge group (H = \mathrm{Isom}_0(K,h)).

3. Prove that this limiting Yang–Mills theory exists in the required axiomatic sense and has a strictly positive mass gap.

The crucial twist is that this is stronger than Clay if it works: it constructs YM as a controlled limit of a quantum gravity model.

4.1. A precise statement (a “KK–QG Mass Gap” conjecture)

Fix a compact Riemannian ((K,h)) with (H := \mathrm{Isom}_0(K,h)) compact non?abelian semisimple.

Assume there exists an Osterwalder–Schrader (OS) quantum gravity on (X = \mathbb{R}^4 \times K) (a Euclidean formulation with reflection positivity and the other OS axioms for diffeomorphism?invariant observables), such that:

• The theory has a semiclassical phase around the background (\delta_{\mathbb{R}^4} \oplus h).
• The low?energy excitations admit a KK decomposition into a massless sector and a KK tower with mass scale (M_{\mathrm{KK}}).

Now impose existence of a double decoupling limit in which:

• (M_{\mathrm{Pl}} \to \infty) (so gravitational interactions on (\mathbb{R}^4) vanish).
• (M_{\mathrm{KK}} \to \infty) (so all nonzero KK modes decouple).
• The induced 4D gauge coupling (g_{\mathrm{YM}}) for the massless vectors corresponding to (\mathfrak{isom}(K,h)) is held fixed.

Assume that in this limit, the Schwinger functions of gauge?invariant composites built from the emergent field strength (F) converge (as distributions) to Schwinger functions of a 4D Euclidean Yang–Mills theory with gauge group (H), satisfying OS axioms and reconstructing to a Wightman QFT on Minkowski space.

Claim (KK–QG Mass Gap): The limiting Yang–Mills theory exists and has a strictly positive mass gap.

4.2. Why this implies the Clay problem

To recover the Clay statement for an arbitrary compact simple (G), one must arrange an internal geometry ((K_G,h_G)) whose connected isometry group (or a factor thereof) is (G). One clean choice is often to take (K_G = G) itself with a bi?invariant metric; left and right multiplications then generate a large semisimple isometry group containing (G) factors. If the KK–QG statement yields mass gap for the resulting gauge group, one extracts the mass gap for each simple factor.

This makes the KK–QG conjecture at least as strong as Clay.

5. Why this is not “just the same as assuming no Killing fields”

“No nontrivial Killing fields” means (\mathfrak{isom}(M,g)=0), hence (\mathrm{Isom}_0(M,g)) is trivial and continuous symmetries disappear. That is a generic condition and can be forced by curvature (e.g. negative Ricci on compact manifolds kills Killing fields).

Semisimple isometry is the opposite extreme: it asserts a large, non?abelian, rigid symmetry group, but excludes torus pieces. It is a fine?tuned property of very symmetric geometries.

From a KK perspective, “no Killing fields” produces no massless gauge fields. Semisimple isometry is the condition that produces only non?abelian gauge fields (no (U(1)) factors) at the massless level.

6. The true bottlenecks: where the Clay difficulty reappears

The reason you rarely see KK discussed as “progress toward the millennium problem” is that three hard problems stack on top of each other.

6.1. Defining quantum gravity in the needed axiomatic sense

The Clay problem already requires a rigorous construction of a 4D QFT. The KK–QG reformulation asks for something even more demanding: a rigorously defined quantum theory of gravity on (\mathbb{R}^4\times K) with OS?type control for diffeomorphism?invariant observables.

Even if one hopes for a constructive definition through an ultraviolet completion (string theory, asymptotic safety, a lattice/discrete approach, etc.), turning that into OS/Wightman axioms is its own research programme.

6.2. Consistent truncation: getting pure YM in the decoupling limit

KK reductions generically produce not only gauge fields but also scalar moduli (radii, shapes) and sometimes additional tensors. In a quantum theory, massless moduli are fatal to “pure YM” and complicate any mass gap statement.

Moreover, truncating to the massless sector is typically inconsistent beyond linear order: interactions among massless modes source the heavy KK tower. There are famous special cases (often sphere reductions in supergravity) where a fully non?linear consistent truncation exists, but “pure Einstein–Hilbert on (\mathbb{R}^4\times K)” is not generally one of them.

So the KK–QG conjecture must either:

• restrict to internal spaces and higher?dimensional theories where consistent truncation retaining the full semisimple gauge sector is known; or
• prove a new theorem: that in the double decoupling limit ((M_{\mathrm{Pl}},M_{\mathrm{KK}}\to\infty) with (g_{\mathrm{YM}}) fixed), the inconsistencies vanish and the heavy tower decouples in the strong OS sense.

6.3. Proving the mass gap of the limiting theory

Even if you succeed in defining the quantum gravity and proving the decoupling to pure 4D YM, the mass gap statement you must prove is essentially the Clay statement again—unless the gravity origin provides an additional structural inequality or monotonicity principle that forces a gap.

This is the part where one hopes gravity might add leverage: perhaps a geometric spectral gap on (K), a rigidity theorem, or a reflection?positivity argument inherited from the bulk could constrain the boundary correlators. But no established mechanism of that strength is currently known.

7. What is genuinely new in the KK reformulation?

Even if it does not immediately solve the gap, the KK reformulation changes the shape of the problem in ways that might be exploitable.

First, it supplies a geometric classification problem upstream: identify compact ((K,h)) such that (\mathrm{Isom}_0(K,h)) is compact semisimple and such that KK truncation/decoupling can plausibly yield pure YM. This is a concrete differential?geometric search.

Second, it invites a “geometric source” of the Yang–Mills coupling and scale. In KK, (g_{\mathrm{YM}}) is not a free parameter: it is built from gravitational parameters and the internal geometry (volume, metric normalisation along Killing directions, etc.). That may introduce constraints not visible in abstract YM.

Third, it creates a conceptual bridge to the idea that the YM gap should be an emergent spectral phenomenon of a more rigid object (a gravitational path integral) rather than a standalone gauge path integral.

8. A concrete research agenda (what would make this more than philosophy?)

The KK–QG reformulation becomes credible when it is pinned to explicit, checkable intermediate statements. Here is a reasonable sequence of milestones.

8.1. Pick a sharply controlled class of internal spaces

Work with compact symmetric spaces of compact type and no flat factor, or a carefully chosen family of homogeneous cosets (G/H) where (\mathrm{Isom}_0) is explicitly semisimple and the Killing algebra is known.

8.2. Identify higher?dimensional theories with consistent gauge truncations

Rather than pure Einstein–Hilbert, consider theories known to admit consistent Pauli/sphere reductions retaining the full non?abelian gauge sector. (Historically these often involve additional form fields; one can ask whether a “minimal” quantum gravity completion exists that still supports the needed truncation.)

8.3. Formulate a decoupling theorem as an OS convergence statement

Define precisely what it means for bulk correlators (of appropriate diffeo?invariant observables projecting onto the gauge sector) to converge to Yang–Mills correlators on (\mathbb{R}^4). This should be stated in distributional topology and include reflection positivity.

8.4. Search for a bulk principle forcing a boundary gap

This is the speculative heart. One wants a mechanism of the form:

• bulk locality + positivity + geometry (\Rightarrow) exponential decay of certain boundary correlators

or equivalently a uniform spectral gap in the reconstructed Minkowski Hamiltonian. In holography one often reads off gaps from discrete normal modes; in KK–QG one would need a rigorous analogue that survives the decoupling limit.

9. Closing thought: why this “amusing refocus” might matter

If the Yang–Mills mass gap is a problem about controlling a strongly coupled quantum field theory, then any route that replaces “strong coupling analysis” with “rigidity of geometry + positivity of a higher?dimensional path integral + controlled limits” is worth taking seriously.

The KK reformulation is a way of packaging the Clay problem inside a larger statement. That makes it strictly harder in one sense. Yet it may also expose hidden structures—geometric classification, moduli constraints, decoupling theorems, positivity/monotonicity principles—that are not visible in the gauge theory alone.

Whether that becomes a real alternate path or just a reframing depends on finding at least one genuinely new inequality or rigidity statement that links “semisimple internal isometries + quantum gravity” to “gap in the emergent 4D gauge sector.” If such a link exists, it would change the landscape of the problem.

NCG Clay Mass Gap Theorem (using ?_P instead of ?)

1. Aim

The Clay Yang–Mills problem asks: for any compact simple non?abelian gauge symmetry, construct 4D quantum Yang–Mills on Euclidean ?? in an axiomatic sense strong enough to reconstruct a Minkowski QFT, and prove a strictly positive mass gap.

A “Connes/NCG version” keeps the same logical shape but replaces the independent gauge potential A? by inner fluctuations D_A of a Dirac operator, and replaces “construct YM” by “construct the renormalised spectral?action measure and its Schwinger functions.”

Here all formulas are written without Planck’s constant; the quantum parameter enters only through the Planck length ?_P.

2. Dimensions and physical normalisations

It helps to distinguish a geometric Dirac operator from a physical one.

The geometric Dirac operator D_geom has dimensions 1/length.

Introduce the energy–length conversion constant

? := ?_P² c? / G.

Then the physical Dirac operator is

D_phys := ? · D_geom,

so D_phys has dimensions of energy. A spectral cutoff ? will be taken as an energy scale, so D_phys/? is dimensionless.

Planck mass and Planck energy expressed using ?_P are

M_P = ?_P c² / G,

E_P := M_P c² = ?_P c? / G.

If a compact direction has characteristic length R, the KK energy scale is of order

E_KK ? c · (? / R) = ?_P² c? / (G R).

(Equivalently, once one reintroduces the conventional constants, this is the usual E_KK ? (quantum of action)·c/R; here that “quantum of action” is encoded in ?_P² c³/G.)

3. Spectral triples, inner fluctuations, gauge symmetry

A (real, even) spectral triple is (?, ?, D; J, ?) with ? represented on ?, a Dirac-type operator D, a real structure J, and a grading ?.

The bimodule of one?forms generated by commutators with D is

?¹_D := { ?_j a_j [D, b_j] : a_j, b_j in ? }.

An inner fluctuation is a selfadjoint A in ?¹_D. The fluctuated Dirac operator is

D_A := D + A + ?? J A J?¹.

In physical units one uses

(D_A)_phys := ? · D_A.

Gauge transformations are implemented by unitaries u in U(?) acting by conjugation, D_A ? u D_A u*. The configuration space is therefore “gauge classes of fluctuated Dirac operators.”

4. Avoiding named groups: an intrinsic ‘simple gauge sector’ condition

Instead of quantifying over a named compact simple group G, impose a condition on the spectral triple itself.

Intrinsic Simple Gauge Sector (ISGS): the gauge Lie algebra determined by (?, ?; J) is nonzero, has trivial centre, and has no nontrivial ideals (i.e. it is simple; equivalently it is perfect and simple).

In almost?commutative examples this can be enforced by requiring the internal finite algebra (or algebra bundle) to be a single central simple block (no nontrivial direct sum), together with an intrinsic unimodularity restriction defined via traces in the representation on ?.

5. Characterising the commutative startpoint and forcing almost?commutative geometry from it

A stronger version does not start with an almost?commutative triple. It starts with a commutative triple and requires the almost?commutative one to be produced from it.

Assume a commutative real even triple (?_M, ?_M, D_M; J_M, ?_M) satisfying reconstruction hypotheses, so ?_M ? C?(M) for a compact spin (or spin?) manifold M and D_M is the corresponding Dirac operator.

Choose a commutative subalgebra ?_B ? ?_M corresponding to a smooth map ?: M ? B. Impose a “spectral KK?admissibility” condition: D_M decomposes into vertical and horizontal Dirac pieces over ?_B up to a bounded curvature correction. This is the Dirac-level proxy for “KK reduction exists.”

Let D_V be the vertical Dirac family. Pick an internal energy cutoff ?_int and define E_{?_int} ? B as the finite?rank bundle spanned fibrewise by vertical eigenspinors with |? · ?| ? ?_int. Define the internal algebra as

?_{?_int} := ?(End(E_{?_int})).

This produces an almost?commutative algebra over the commutative base B without choosing it arbitrarily.

6. Quantisation without heat?kernel expansion: use a spectral cutoff regulator

The step “heat?kernel expand to a local classical action, then quantise” is optional.

Instead, define the UV?regulated quantum theory directly at energy scale ?_UV by restricting to a finite spectral subspace. With a sharp cutoff projector ?,

?_{?_UV} := Range ?((D_A)_phys / ?_UV).

Then (D_A)phys restricted to ?{?_UV} is finite-dimensional, Tr f((D_A)_phys² / ?_UV²) is a finite trace, and the path integral becomes (after the gauge quotient) a finite-dimensional integral at fixed ?_UV.

Heat?kernel expansions can still be used later to classify counterterms or compare to local effective actions, but they are not needed to define the regulated measure.

7. The axioms-style NCG–Clay problem (with ?_P explicit)

Fix a KK?admissible commutative startpoint producing an almost?commutative triple (?, ?, D; J, ?) that satisfies ISGS.

Axiom A (configuration space). Let ? := { A in ?¹_D : A = A* }. Define D_A and (D_A)_phys = ? · D_A. The configuration space is {D_A}/gauge.

Axiom B (bare spectral action). Choose a positive cutoff function f and an energy cutoff ?_UV. Define a bosonic bare action

S_bos,?_UV(D_A) := S? · Tr f((D_A)_phys² / ?_UV²),

where S? is a fixed constant with units of action. The dimensionless weight parameter is obtained by dividing by the effective quantum of action

Q := ?_P² c³ / G.

So the bare weight is exp(?S_eff/Q).

Axiom C (renormalised spectral?action measure). Introduce an IR regulator scale L (e.g. base torus of size L). Define the UV?regulated partition functional formally by

Z_{?_UV,L} := ? exp( ? [ S_bos,?_UV + S_fer,?_UV + S_ct,?_UV ] / Q ) ?D_A ??,

where S_ct,?_UV is a local counterterm functional (local on the base and local in the internal algebra bundle) chosen to make limits exist.

For any finite collection of gauge?invariant local observables O_i, define Schwinger functions by inserting ? O_i and dividing by Z.

The axiom is: after choosing counterterms, the limits ?_UV ? ? (at fixed L) and then L ? ? exist and yield Schwinger functions on ??.

Axiom D (OS axioms). The limiting Schwinger functions satisfy the Osterwalder–Schrader axioms for the chosen algebra of gauge?invariant local observables.

Axiom E (nontriviality). The resulting OS theory is interacting (not Gaussian/free).

Definition (mass gap with ?_P). OS reconstruction yields a Minkowski Hilbert space, vacuum ?, and translation generators P_? with Hamiltonian H = P_0 (units: energy).

Energy gap: there exists ?_E > 0 such that Spec(H) ? (0, ?_E) is empty, and ?_E = inf(Spec(H) \ {0}).

Mass gap: ?_m := ?_E / c² > 0.

Equivalent Euclidean criterion: for any gauge?invariant local observable O with ??, O ?? = 0, the connected 2?point function decays for large Euclidean separation r = |x| as

?O(x) O(0)?_c ? C_O · exp( ? (G ?_m / (?_P² c²)) · r ).

So the correlation length is

? = ?_P² c² / (G ?_m).

8. Limits made explicit (now in terms of ?_P)

Classical limit. The stationary-phase regime is S_eff/Q ? ?. Formally, this is the limit ?_P ? 0 with c and G fixed (so Q ? 0). In this limit the measure concentrates on critical points of S_eff.

Gravity decoupling. Gravitational interactions are controlled by ratios to the Planck energy E_P = ?_P c?/G. A practical “decouple gravity” regime at characteristic energy E is

E / E_P ? 0, equivalently (G E) / (?_P c?) ? 0.

(If one insists on a strict parameter limit at fixed c, this can be phrased as G ? 0 with ?_P²/G held fixed, i.e. with Q fixed; the point is that gravity decoupling is not the same as the classical limit.)

KK decoupling. If a compact length R is present, the KK tower decouples when the KK energy scale diverges:

E_KK ? ?_P² c? / (G R) ? ?, equivalently R ? 0 with ?_P, G, c fixed.

Relativistic structure. OS reconstruction provides the Euclidean-to-Minkowski bridge at finite c. A nonrelativistic limit c ? ? is separate and not required for the mass-gap statement.

9. NCG–Clay Mass Gap Theorem (problem statement)

Given a commutative KK?admissible spectral triple that functorially produces an almost?commutative triple satisfying ISGS, construct the renormalised spectral?action Schwinger functions on ?? satisfying OS axioms and nontriviality, and prove a strictly positive energy gap ?_E (equivalently a positive mass gap ?_m).

A stronger variant adds a decoupling clause: in a regime where gravity effects are negligible (E/E_P ? 0) and the KK tower decouples (E_KK ? ?), the NCG theory reduces (in the sense of convergence of Schwinger functions of gauge?invariant observables) to pure 4D Yang–Mills. Then the NCG statement implies the Clay statement and is strictly stronger because it produces YM as a controlled limit of a spectral?action QFT.