The Monster at the Heart of Quantum Gravity might Explain the Hubble Tension and Vacuum Catastrophe

Physicist Edward Witten and others have argued that a two-dimensional bosonic quantum field theory with Monster symmetry (explored by my own undergraduate mathematics professor Dr. Richard Borcherds) might be the ultraviolet anchor of a consistent quantum gravity theory (“Three-Dimensional Gravity Revisited,” arXiv:0706.3359). What this means is that this strange mathematical object, the Monster CFT (also known as the Moonshine Module, which connects to the Monster Group by the j-function), may be the missing link needed to complete a theory of quantum gravity, describing the quantum Riemannian Dirac–Kähler-like entropic dilation operator governing spacetime itself as emergent from entanglement entropy of other particles that collapses in the gravitational action (by the spectral action principle into the Einstein-Hilbert action).

As I've argued in previous blog posts, modular invariance, the mathematical rule that tucks a function into an unbreakable symmetry under twisting and inversion, could force the zeta zeros to fall exactly where the Riemann Hypothesis predicts if the Riemann zeta function is the spectrum of this missing quantum operator, thus solving the Riemann Hypothesis by realizing a self-adjoint operator predicted by the Hilbert-Polya conjecture in a physical system that has Li's criterion. The symmetries imposed by the Monster CFT could be responsible. As a system approaches this ultraviolet fixed point predicted by Asymptotically Safe Gravity, effective dimensionality reduces to 2, and can be described with only bosonic degrees of freedom by means of fermionic condensation. In Einstein–Cartan theory, the spins of fermions induce a repulsive four-fermion interaction at extreme densities, preventing a singular crunch and triggering a finite bounce that renders the space asymptotically safe. At that Planck-scale pivot point, fermions could condense into a bosonic phase whose only consistent fixed point is the Monster CFT itself, lifting the construction from an ad-hoc assumption to a natural consequence of gravity’s spin structure. Several studies have shown numerical evidence for the existence of this UV completion of quantum gravity.

In his 2007 paper “Three-Dimensional Gravity Revisited,” Edward Witten showed that pure gravity in a three-dimensional anti-de Sitter background, that is, a theory with negative cosmological constant and no local degrees of freedom beyond black holes, can only be consistent at discrete values of the coupling. At the extreme end of that coupling range, he argued, the unique holomorphic two-dimensional conformal field theory dual would have central charge 24 and exactly the Monster module as its partition function. The Monster CFT (also known as the Moonshine Module) appears as the ultraviolet completion of pure AdS₃ gravity. While this is intriguing, our universe is not an anti-de Sitter (AdS) space - it has a small positive cosmological constant - indicating its expansion, and also has time, which could be considered a fourth dimension.

If we were to think about fixing this with a hypothesis, we might start by embarking on an audacious reinterpretation of the Riemann Hypothesis, by drawing on Alain Connes’s noncommutative-geometry approach, where we construct an abstract “spectral triple” - essentially an operator on a Hilbert space - whose only allowable vibrations coincide with those mysterious zeros. In this setup, any frequency straying off the critical line where the zeros must lie would break the Monster symmetry and become physically inadmissible. In effect, the Riemann Hypothesis follows if that Monster-symmetrized operator can exist without inconsistency. In order to match the sort of universe we live in (3 dimensions of space and 1 dimension of time in a de-Sitter universe) a four-dimensional spacetime is split between an Anti-de Sitter interior and a de Sitter exterior, joined by a thin spherical wall. By studying how a quantum field waves across that wall, enforcing the usual continuity of the field but a precise jump in its derivative, we might find a quantization condition. Remarkably, that condition mirrors the same equation whose roots are the nontrivial zeros of the Riemann zeta function. In other words, the allowed “notes” of this exotic universe line up exactly with the primes’ hidden rhythms.

The Monster CFT’s own invariances lock in the spectral symmetry that enforces the critical-line condition. This elegant holographic picture unites the abstract operator we have discussed and the wave model here into a single unified framework: the Monster lives on the wall, and its symmetry dictates the bulk physics all the way from high-energy ultraviolet behavior to the deep infrared spectrum. In a series of numerical experiments, we can solve the wave equations in the AdS–dS toy universe and recover hundreds of discrete frequencies, and compare them to the known Riemann zeros. The match is stunning. We can extract the Li coefficients - alternate expressions of the Riemann Hypothesis - from the computed spectrum and find them all positive, as required. We can even nudge the wall’s modular parameter slightly away from perfect self-duality and see immediately how the spectrum loses its prime-number alignment, underscoring that the Monster’s exact symmetry is essential. In one influential study attributed to D. B. Kaplan and collaborators (2022, unpublished or in preprint), the authors investigated modular-invariant partition functions of two-dimensional conformal field theories and found that the asymptotic behavior of these partition functions—particularly near the high-temperature (τ → i0⁺) limit—can be expressed in terms involving the Riemann zeros.

Vacuum energy in quantum field theory is normally catastrophically large, but the symmetric pairing of positive and negative modes enforced by the Monster causes nearly complete cancellations. What remains can be as small as the observed cosmological constant (which describes the expansion of the universe, caused by a concept called dark energy) with no fine-tuning required. The discrepancy between predicted large vacuum energy and the small measured cosmological constant is known as the vacuum catastrophe, and the Hubble tension is the discrepancy of values measured for the cosmological constant across the universe which seems to vary. The presence of an AdS core inside our de Sitter universe naturally yields different expansion rates inside and outside a cosmic “bubble,” offering a fresh explanation for why early-universe measurements of H₀ (from the CMB) differ from local observations today.

A 2024 preprint by S. Khaki explored the idea that the “Hilbert space dimension” in quantum gravity might take only special discrete values (inspired by Witten’s comment that quantum de Sitter space might be associated with sporadic group structures). Khaki assumed a de Sitter universe whose entropy (or Hilbert space dimension) is fixed by the Monster CFT, and then computed the consequences for vacuum energy. Interestingly, it was reported that this construction could address the hierarchy and cosmological constant problems: the scale of the resulting vacuum energy (from a certain twisted sector of the theory) comes out “close to the cosmological constant” in magnitude

In this framework we use “centaur” and “minotaur” geometries as two complementary ways of thinking about the same hybrid spacetime that stitches together an Anti-de Sitter region and a de Sitter region via a thin, self-dual domain wall. The notion of a “centaur” geometry - an asymptotically AdS spacetime that in its deep infrared smoothly opens up into a dS patch - was first introduced by Dionysios Anninos and Diego M. Hofman in their 2017 paper Infrared Realization of dS₂ in AdS₂. In that work, they exhibited a two-dimensional dilaton–gravity solution that interpolates between an AdS₂ boundary and a static-patch dS₂ core, coining the term “centaur” to describe this hybrid geometry. Subsequent authors (e.g. Iizuka & Sake’s “A note on Centaur geometry,” 2025) have extended the idea to JT gravity and explored its holographic implications, but the original centaur construction traces back to Anninos & Hofman’s 2017 paper.

In the centaur geometry, the universe looks like an AdS exterior whose usual conformal boundary at infinity is occupied by the Monster CFT. Deeper in, behind the wall, the cosmological constant flips sign and you find yourself in a dS interior whose finite horizon is then interpreted as an infrared cutoff on that same CFT. From this vantage the Monster module plays the role of the ultraviolet anchor, you read its partition function at spatial infinity, while the dS core regulates the long-distance behavior and guarantees the Riemann–zero spectrum emerges in the infrared.

The minotaur geometry is a geometry we may construct to be the inverse, and simply flips that assignment: here the AdS region lies inside the wall and dS lives outside. There is no traditional asymptotic AdS boundary, so the Monster CFT must instead reside directly on the wall itself, now thought of as the cosmological horizon of the dS exterior. In this view the Monster CFT encodes the degrees of freedom responsible for de Sitter entropy, and the same τ=i modular self-duality that enforces the critical-line spectrum also ensures there is no conical singularity or stress discontinuity at the join.

An interesting 2025 work by Tamburini et al. effectively found a physical scenario (Majorana fermion in Rindler space) that produces the Riemann zeros as a spectrum in Rindler spacetime, which can be thought of as a patch of flat space akin to an observer horizon – conceptually related to a dS-like horizon, although technically different (Rindler is flat but not global AdS or dS). Their formulation can be seen as matching conditions in different regions (left and right Rindler wedges), which is somewhat analogous to matching across a domain wall, and may provide further insights into Majorana physics which can be explored also with out minotaur/centaur/monster geometric setup.

Viewed together, the centaur and minotaur geometries illustrate a single principle: no matter whether you treat the Monster CFT as living at infinity or on the horizon, interpolated together its full modular invariance at the τ=i interface glues the two phases on a thin domain wall into one self-consistent whole. Either way, the Monster symmetry enforces both the functional-equation duality needed to pin every mode to ℜ(s)=½ and the extreme cancellations that yield a tiny cosmological constant, and thus provides the unifying glue for prime numbers, quantum gravity, and cosmic expansion.