Nonperturbative Definition Proposals: Objects, Evidence, and Falsifiers

A nonperturbative-definition proposal must specify an exact object, an observable algebra, a regulator and limiting prescription, and the spacetime sector it claims to recover. Agreement with one protected quantity or one low-energy saddle is evidence, not completeness. Fixed criteria let us compare matrix models, string field theories, and holographic definitions without erasing their different targets.

Required background. Nonperturbative definition and completion criteria supplies the definition standard used here.

Helpful background. Top-down and bottom-up claim contracts separates embeddability from completion. Claim status and research handoffs supplies the evidence and update discipline.

Objects and target sectors

BFSS proposes supersymmetric $U(N)$ matrix quantum mechanics, with Gauss constraint, as discrete light-cone M-theory with $N$ units of longitudinal momentum and an uncompactified target only after a large-$N$ limiting argument. BMN adds a plane-wave mass deformation and targets M-theory in that maximally supersymmetric background. IKKT proposes a zero-dimensional type-IIB matrix integral Ishibashi et al. 1997, whose contour and Lorentzian definition are part of the object rather than implementation details.

Matrix string theory uses a two-dimensional supersymmetric matrix gauge theory; perturbative second-quantized strings emerge in an infrared strong-coupling limit. Open string field theory defines off-shell open strings about a chosen boundary conformal field theory. Closed string field theory aims to cover closed-string moduli space about a chosen background through an infinite set of vertices. AdS/CFT can define a bulk sector if a complete finite-$N$ boundary theory, global dictionary, state space, and boundary conditions are already defined.

These targets are not interchangeable. A Hamiltonian matrix model may be nonperturbative in its coupling while remaining restricted to a light-front sector. A string field theory can be exact in $\alpha'$ yet organized perturbatively in $g_s$. A boundary CFT can be nonperturbative at finite $N$ while fixing asymptotically AdS boundary conditions.

First application: apply one recovery grid

For each proposal, test four layers.

1. Internal definition: Does the regulated partition function or Hamiltonian exist, with gauge quotient, contour, and supersymmetry specified?
2. Controlled recovery: Does a limit reproduce the perturbative string spectrum, interactions, and duality parameter map with quantified errors?
3. Gravitational recovery: Are long-distance scattering, black-object thermodynamics, or geometric observables reproduced outside protected sectors?
4. Completeness: Are global charges, topology-changing sectors, finite-$N$ states, and the claimed asymptotics included without importing missing data?

BFSS has strong first-layer control at finite $N$ and notable scattering and thermodynamic tests, while its uncompactified large-$N$ completeness remains conjectural Banks et al. 1997. Closed string field theory has a precise perturbative moduli-space construction but does not currently provide a universally convergent, background-independent nonperturbative sum Zwiebach 1993. A fully defined CFT can meet the internal standard, but bulk uniqueness still depends on complete global dictionary data.

Adversarial control: ask outside the recovered sector

Take a calculation that reproduces the D0-brane potential at large separation and ask it to establish a finite-$N$ cosmology with no light-front compactification. Or take an open-string tachyon solution and ask it to define all closed-string vacua. In each case the evidence is real, but the requested observable lies outside the demonstrated target. This is a failure of inference rather than a falsification of the narrower result.

A true falsifier is proposal specific: violation of the target supersymmetry Ward identities after controlled regulator removal; failure to reproduce a protected spectrum; incompatible long-distance scattering in a shared regime; regulator-dependent continuum observables; or two inequivalent bulk completions with the same allegedly complete defining data. The evidence ceiling remains the highest recovery layer actually passed; subsequent pages apply these tests proposal by proposal.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Banks, T., Fischler, W., Shenker, S. H., and Susskind, L. (1997), “M Theory as a Matrix Model: A Conjecture,” Physical Review D 55, 5112–5128. arXiv:hep-th/9610043.
• Ishibashi, N., Kawai, H., Kitazawa, Y., and Tsuchiya, A. (1997), “A Large-$N$ Reduced Model as Superstring,” Nuclear Physics B 498, 467–491. arXiv:hep-th/9612115.
• Zwiebach, B. (1993), “Closed String Field Theory: Quantum Action and the Batalin–Vilkovisky Master Equation,” Nuclear Physics B 390, 33–152. arXiv:hep-th/9206084.