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Undecidability in Spacetime Geometry via the AdS/CFT Correspondence

Sameer Ahmad Mir, Francesco Marino, Arshid Shabir, Lawrence M. Krauss, Mir Faizal

TL;DR

The paper shows that undecidability in spectral-gap problems for quantum many-body systems can be holographically transmitted to bulk geometry selection in AdS/CFT. By embedding a translationally invariant, undecidable spin Hamiltonian into a large-$N$ gauge theory, the dual AdS geometry competition between Poincaré AdS$_4$ and the AdS$_4$ soliton becomes undecidable, controlled by a halting predicate via an effective mass term $m_{ m eff}^2(u)=m^2+\Sigma_{\rm loop} + g_{\rm halt}$ with $g_{\rm halt}=\vartheta(u)\mu_h^2$. In the planar, strongly coupled regime, the dominant saddle is determined by the sign of $m_{ m eff}^2(u)$, yielding either gapless (Poincaré AdS$_4$) or gapped (AdS$_4$ soliton) phases, while the classical on-shell actions satisfy $I[g_P]=0$ and $I[\text{Sol}]<0$ in a low-temperature window; the halting bit fixes which saddle dominates, and no algorithm can determine this from the boundary data alone. This constitutes a Gödel–Turing obstruction to spacetime emergence under standard holographic assumptions (large $N_c$, large $\lambda$, classical gravity). The work suggests undecidability could be a pervasive feature of quantum gravity beyond this explicit construction.

Abstract

Undecidability, a hallmark of Gödel incompleteness theorems, has recently emerged in quantum many-body physics through the spectral gap problem. We demonstrate how this logical limitation can be holographically transmitted to a class of gravitational theories via the AdS/CFT correspondence. By embedding a translationally invariant spin Hamiltonian with undecidable gap status into a large-N gauge theory, we generate an AdS dual in which the selection of dominant bulk saddle (Poincaré AdS or AdS soliton) is itself undecidable. Consequently, under standard semiclassical holographic assumptions, even determining which smooth spacetime geometry emerges from quantum gravity can be beyond the limits of computability.

Undecidability in Spacetime Geometry via the AdS/CFT Correspondence

TL;DR

The paper shows that undecidability in spectral-gap problems for quantum many-body systems can be holographically transmitted to bulk geometry selection in AdS/CFT. By embedding a translationally invariant, undecidable spin Hamiltonian into a large- gauge theory, the dual AdS geometry competition between Poincaré AdS and the AdS soliton becomes undecidable, controlled by a halting predicate via an effective mass term with . In the planar, strongly coupled regime, the dominant saddle is determined by the sign of , yielding either gapless (Poincaré AdS) or gapped (AdS soliton) phases, while the classical on-shell actions satisfy and in a low-temperature window; the halting bit fixes which saddle dominates, and no algorithm can determine this from the boundary data alone. This constitutes a Gödel–Turing obstruction to spacetime emergence under standard holographic assumptions (large , large , classical gravity). The work suggests undecidability could be a pervasive feature of quantum gravity beyond this explicit construction.

Abstract

Undecidability, a hallmark of Gödel incompleteness theorems, has recently emerged in quantum many-body physics through the spectral gap problem. We demonstrate how this logical limitation can be holographically transmitted to a class of gravitational theories via the AdS/CFT correspondence. By embedding a translationally invariant spin Hamiltonian with undecidable gap status into a large-N gauge theory, we generate an AdS dual in which the selection of dominant bulk saddle (Poincaré AdS or AdS soliton) is itself undecidable. Consequently, under standard semiclassical holographic assumptions, even determining which smooth spacetime geometry emerges from quantum gravity can be beyond the limits of computability.
Paper Structure (7 sections, 4 theorems, 134 equations, 9 figures)

This paper contains 7 sections, 4 theorems, 134 equations, 9 figures.

Table of Contents

  1. Supplemental Material
  2. Spectral-Gap Undecidability (Full Details)
  3. Euclidean Path-Integral Representation
  4. Large-Nc Matrix Completion
  5. A Concrete Ultraviolet Completion
  6. Holographic Duality Setup
  7. Undecidability Theorem (Holographic Form)

Key Result

Theorem 1

There exists a Turing machine that, given any input $u$, outputs a translation-invariant, finite-dimensional, nearest-neighbour interaction $H(u)$ on $\mathbb Z^2$ such that, for the torus-restricted Hamiltonians $H_L(u)$ and their gaps $\gamma_L(u)$ in eqA:finite_size_gap, exactly one holds: No algorithm can decide from the finite description of $H(u)$ which branch holds.

Figures (9)

  • Figure 1: Anisotropic $3$D lattice $\Lambda_L\times \mathbb Z_{N_\tau}$ from second-order Suzuki--Trotter; spatial links carry $J_s\sim a_\tau$, temporal links $J_\tau\sim a_\tau^{-1}$.
  • Figure 2: Hubbard-Stratonovich linearisation of a generic Boltzmann factor.
  • Figure 3: Tadpole diagram: one-loop correction to the two-point function (self-energy) used for $\delta m^2$.
  • Figure 4: Fish (bubble) diagram: one-loop correction to the four-point vertex used for $\delta\lambda$.
  • Figure 5: Schematic one-loop RG flows in $(m^{2},\lambda)$ with the Wilson-Fisher fixed point. For $\vartheta(u){=}1$, the relevant coupling $g_{\rm halt}$ drives the flow to a massive phase. The critical-tuning trajectory ($\vartheta{=}0$) approaches but does not land on WF.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Theorem 1: Cubitt-Pérez-García-Wolf Cubitt:2015xsa
  • Lemma 1: Halting inputs yield a uniform gap
  • Lemma 2: Non-halting inputs are gapless
  • Theorem 1: Holographic undecidability, conditional form
  • proof