Nonperturbative String- and M-Theory Definition Proposals

This chapter compares proposals that aim to define string or M-theory beyond a background-specific perturbative expansion. The central question is not whether a formulation reproduces some string amplitudes, but which mathematical object it defines, which spacetime sector and observables it claims to recover, which limits are essential, and what result would falsify the recovery claim.

Helpful background. Duality Checks, Evidence Independence, Status, and Failure Modes supplies the general comparison method. Nonperturbative Definition and Completion Criteria fixes the target, while String Theory as a Holographic Construction Interface and Decoupling Limits and the Original AdS/CFT Proposal supply the perturbative and decoupling baselines.

Enter this chapter

The chapter assumes the distinction between a top-down construction and a complete microscopic definition. Matrix models, matrix strings, string field theories, and boundary CFTs solve different parts of the problem: they should not be combined into a single generic “nonperturbative string theory.”

Follow the pages in order:

1. Nonperturbative Definition Proposals: Objects, Evidence, and Falsifiers establishes common comparison criteria.
2. String and M-Theory Duality Webs and Parameter Maps distinguishes exact parameter maps from simultaneous calculational control.
3. BFSS Matrix Quantum Mechanics and M-Theory Conjectures treats the light-front matrix proposal and its large-$N$ limit.
4. BMN Plane-Wave Matrix Model and Controlled Sectors studies the mass-deformed plane-wave sector and fuzzy-sphere vacua.
5. IKKT Type-IIB Matrix Model and Emergent-Spacetime Claims separates a zero-dimensional integral from claims of Lorentzian spacetime emergence.
6. Matrix-String Constructions and Second-Quantized Strings derives long-string sectors in a two-dimensional gauge theory’s infrared limit.
7. Open String Field Theory and Tachyon Dynamics develops a background-dependent off-shell theory and its D-brane decay tests.
8. Covariant and Closed String Field Theory explains moduli-space vertices, gauge structure, and loop coverage.
9. AdS/CFT as a Conditional Nonperturbative Definition states what complete boundary data would define for fixed asymptotics.
10. Background Dependence, Background Independence, and Emergence gives operational meanings to three often ambiguous terms.
11. Numerical Matrix Evidence, Classical-Spacetime Recovery, and Limits specifies a regulator-to-continuum protocol for quantitative tests.

Preparation and comparison method

For each proposal, record five items before interpreting a calculation: the finite-regulator variables and measure; the gauge constraint or quotient; the observable algebra; the limiting sequence in $N$, temperature, volume, coupling, and cutoff; and the target sector, including asymptotics and conserved charges. A formal action without these items is not yet a nonperturbative definition.

The proposals have complementary strengths. BFSS has an ordinary Hamiltonian at finite $N$ but a delicate infinite-momentum limit Banks et al. 1997. BMN improves spectral control by adding a plane-wave scale but narrows the target. IKKT makes spacetime emergent from matrices but introduces contour and Lorentzian-measure questions. Matrix strings recover second-quantized strings only in a specified infrared limit Dijkgraaf, Verlinde, and Verlinde 1997. String field theory is explicitly off shell on a chosen background and organizes closed-string moduli spaces through a BV action Zwiebach 1993, while AdS/CFT can be exact if a complete boundary theory and global dictionary are independently specified.

Synthesis: recovery before interpretation

Evidence should be ordered from internal consistency to target recovery:

• convergence or exact definition at finite regulator;
• regulator removal with controlled Ward identities and gauge constraints;
• recovery of perturbative spectra, amplitudes, brane charges, and dualities in overlapping limits;
• reproduction of genuinely gravitational observables, including black-hole thermodynamics or long-distance scattering; and
• uniqueness or completeness of the claimed target, including global sectors and nonperturbative states.

Passing an earlier test does not imply the later ones. In particular, a large-$N$ eigenvalue distribution may be a useful collective variable without yet being a metric, and agreement with a low-energy supergravity coefficient does not prove full M-theory.

Review the chapter

A satisfactory review answer should be able to:

1. state the exact object and limiting sequence proposed by BFSS, BMN, IKKT, matrix string theory, open and closed string field theory, and AdS/CFT;
2. derive the D0 longitudinal-momentum map and explain why finite $N$ is a sector rather than uncompactified M-theory;
3. identify the BMN mass scale and show how fuzzy-sphere vacua solve the matrix equations;
4. distinguish IKKT matrix eigenvalue observables from gauge-invariant evidence for Lorentzian geometry;
5. explain why matrix-string permutation cycles become long strings only in an infrared limit;
6. distinguish open-string tachyon-vacuum evidence from a closed-string nonperturbative definition;
7. explain how closed-string vertices cover moduli space without overcounting; and
8. design a numerical extrapolation that separates cutoff, finite-temperature, finite-$N$, and sign or contour systematics.

Answers must state the target asymptotics and evidence ceiling. “Nonperturbative” without an observable set and limiting prescription is insufficient.

Exits

Continue to Holographic Renormalization and Radial Dynamics to turn a specified AdS construction into finite boundary observables. Return to String, Brane, and Top-Down Constructions when a proposal is being tested only in a perturbative or supergravity corner, or to Quantum-Gravity Claims, Observables, and Evidence to classify the resulting claim.

Chapter-scale structure and validity checks

The chapter-scale structure map locates this page’s result inside the full reasoning chain. Follow the solid arrows through the declared inputs and checks; the dashed final arrow marks the point where an additional inference would be required.

BFSS, BMN, IKKT, matrix strings, string field theory, and AdS/CFT define different objects and satisfy different completeness tests. The diagram is an original schematic, is not to scale, and uses the dashed final arrow to mark the claim boundary.

Accessible figure data (JSON)

The companion validity map turns three common overclaims into explicit failure tests. Read each row from its declared object to the diagnostic, then compare the licensed conclusion with the dashed “not” endpoint.

BFSS, BMN, IKKT, matrix strings, string field theory, and AdS/CFT define different objects and satisfy different completeness tests. Each row pairs a diagnostic with the strongest supported conclusion and an explicitly unsupported promotion. The diagram is an original schematic and is not to scale.

Accessible figure data (JSON)

Claim-domain comparison

The table below gives a screen-reader-friendly comparison of three representative claims. It keeps the required declaration, approximation status, evidence timing, counterevidence, falsifier, failure condition, and licensed conclusion in one reading order.

Representative claim domains and validity boundaries for Nonperturbative String- and M-Theory Definition Proposals

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
matrix quantum mechanics	Declare large-N limit and compactification sector; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: defining variables and action → Hamiltonian or path integral → observable and sector map → spacetime recovery tests → typed definition claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “spectrum and scattering benchmarks” check is counterevidence to the promoted claim.	spectrum and scattering benchmarks	manifest covariant completeness	a candidate M-theory sector
matrix or string field model	Declare gauge fixing, background, and observables; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: defining variables and action → Hamiltonian or path integral → observable and sector map → spacetime recovery tests → typed definition claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “unitarity and spacetime-recovery tests” check is counterevidence to the promoted claim.	unitarity and spacetime-recovery tests	all nonperturbative backgrounds	the model-defined amplitudes or states
AdS/CFT as definition	Declare exact boundary theory and global data; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: defining variables and action → Hamiltonian or path integral → observable and sector map → spacetime recovery tests → typed definition claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “bulk dictionary completeness” check is counterevidence to the promoted claim.	bulk dictionary completeness	a definition of arbitrary quantum gravity	conditional definition of its AdS sector

Download the structured table data (JSON).

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.
• Dijkgraaf, R., Verlinde, E., and Verlinde, H. (1997), “Matrix String Theory,” Nuclear Physics B 500, 43–61. arXiv:hep-th/9703030.
• 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.