Nonperturbative Mtheory Definition

Here “M-theory” names the sought complete eleven-dimensional quantum theory; its existence and construction are not assumed by this item. “Controlled nonperturbative theory” means a construction that specifies its admissible background class, quantum observables, global gauge and anomaly data, and completion, rather than only a low-energy effective action. The Cremmer–Julia–Scherk (CJS) target is the adopted source-free, classical, two-derivative bosonic action on an oriented Lorentzian eleven-manifold; brane currents, higher-derivative curvature corrections, and quantum corrections are outside that action’s stated scope. The adopted type-IIA circle interface takes ${\alpha }^{\mathrm{\prime }}>0$, ${\mathrm{\ell }}_s=\sqrt{{\alpha }^{\mathrm{\prime }}}$, $g_s=e^{{\mathrm{\Phi }}_{\mathrm{\infty }}}>0$, $R_{11}=g_s{\mathrm{\ell }}_s$, and ${\mathrm{\ell }}_p^3=g_s{\mathrm{\ell }}_s^3$. In the string-frame metric ansatz $d{s}_{11}^2=g_s^{2\mathrm{/}3}{e}^{-2\mathrm{\Phi }\mathrm{/}3}d{s}_{10,s}^2+e^{4\mathrm{\Phi }\mathrm{/}3}({\mathrm{\ell }}_{p}d\theta +C_1{)}^2$, $\theta$ has period $2\pi$, $C_1$ is length-valued, and $R_{11}$ is the asymptotic proper circle radius; the unit D0/first-Kaluza–Klein mass match is conditional on the stated sector identification and is not a complete duality. The stated classical bosonic M2 interface takes a smooth embedding $X:{\mathrm{\Sigma }}_3\to Y^{11}$ of a closed oriented three-manifold with nondegenerate Lorentzian induced metric $g$, and $S_{\mathrm{M}2}=-T_{\mathrm{M}2}{\int }_{{\mathrm{\Sigma }}_3}{d}^3\xi \sqrt{-\det g}+T_{\mathrm{M}2}{\int }_{{\mathrm{\Sigma }}_3}{X}^{\ast }{C}_3$. Here $Y^{11}$ is oriented Lorentzian CJS spacetime with metric $G$, $T_{\mathrm{M}2}>0$ is independent with units length${}^{-3}$ in $\mathrm{\hslash }=c=1$, and $\int C_3$ has units length${}^3$. Its Green–Schwarz completion is on-shell $11\mathrm{\mid }32$ CJS superspace with the $+C_3$ Wess–Zumino sign fixed by the adopted superspace constraints. The separate conditional local-$\kappa$ theorem uses a smooth superembedding of closed oriented ${\mathrm{\Sigma }}_3$ with timelike body and nondegenerate metric, plus its exact torsion/four-form conventions; the cited local-kappa theorem fixes these conventions and uses coefficientwise smooth fields over a finite complex Grassmann algebra, with real embedded timelike body, one target potential superchart, and compactly supported odd local gauge parameters. It does not supply a global quantum membrane sector.

The single-brane M5 action interface is the single abelian Pasti–Sorokin–Tonin (PST) five-brane on a smooth oriented timelike embedding $\iota :{\mathrm{\Sigma }}_6\to Y^{11}$ whose image lies in one smooth potential patch for $C_3,C_6$. In this paragraph use dimensionless worldvolume coordinates and the source-normalized dimensionless fields, with $g={\mathrm{\ell }}_p^{-2}{\iota }^{\ast }G$ of signature $(-+++++)$ and $P={\iota }^{\ast }$. Let $A_2\in {\mathrm{\Omega }}^2({\mathrm{\Sigma }}_6)$ be real and smooth and $a\in C^{\mathrm{\infty }}({\mathrm{\Sigma }}_6,\mathbb{R})$. Set $F_3=2d{A}_2$, $H_3=F_3-P[C_3]$ and $H^{\ast mnl}={\epsilon }^{mnlpqr}{H}_{pqr}\mathrm{/}(3!\sqrt{-\det g})$, with ${\epsilon }^{012345}=+1$ and indices raised using $g$. Restrict to $(\mathrm{\partial }a{)}^2:=g^{mn}{\mathrm{\partial }}_{m}a{\mathrm{\partial }}_{n}a>0$ and define ${\tilde{H}}_{mn}=H_{mnl}^{\ast }{\mathrm{\partial }}^{l}a\mathrm{/}\sqrt{(\mathrm{\partial }a{)}^2}$. Pointwise, restrict $H_3$ to the connected component containing zero of the real three-forms for which $\det (g+i\tilde{H})<0$, with $g,a$ fixed, and use the positive real square root. The local kinetic density is $\sqrt{-\det (g+i\tilde{H})}+\sqrt{-\det g}{\mathrm{\partial }}_{m}a{H}^{\ast mnl}{H}_{nlp}{\mathrm{\partial }}^{p}a\mathrm{/}[4(\mathrm{\partial }a{)}^2]$, with Wess–Zumino term $\int (P[C_6]+(1\mathrm{/}2)F_3\wedge P[C_3])$ and suppressed overall tension. A finite total functional requires absolute integrability of its densities; otherwise only compactly supported action differences are intended. The supplier reports nonlinear self-duality only after appropriate local PST gauge fixing, not as an independently proved consequence of the CJS action. This is one local abelian five-brane interface, without global patching or multiple-M5 dynamics.

Construct a controlled nonperturbative theory with eleven-dimensional supergravity as its low-energy limit, a full type-IIA circle limit, globally defined membrane/five-brane sectors and consistent charge quantization. CJS supergravity, the Bogomolny–Prasad–Sommerfield (BPS) mass and charge matches, the M/IIA dictionary and matrix proposals do not establish such a theory.

M Theory Type IIA Circle Limit Dictionary

M2 Brane Worldvolume Action

Single M5 worldvolume tensor action

Pasti, Sorokin, and Tonin, §§2–3, Eqs. (1), (9)–(10), (16)–(18)

M2 Kappa Symmetry on a CJS Background

Witten, §§1 and 2.2–2.3

Taylor, The M(atrix) model of M-theory, Introduction and §3.2

Banks, Fischler, Shenker, and Susskind, §§1, 3–5, 8–9

Taylor, M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory

Witten, §1, equation (1.2)

The source-status check dated 2026-11-05, restricted to the listed versions available by 2026-10-30, keeps this as an unresolved foundational problem in that record. Witten argues that strong coupling of type IIA produces an eleven-dimensional low-energy supergravity sector and motivates a circle interpretation; the paper does not construct a complete quantum theory. Banks, Fischler, Shenker, and Susskind explicitly propose the matrix equivalence as a conjecture and identify assumptions and limits that remain. Taylor’s 2001 review says that identifying M-theory with the strong-coupling limit of type IIA is incomplete without a nonperturbative definition of type IIA; it also reports that the evidence was circumstantial and that no complete background-independent formulation had been given at that review date.

The bounded 2026 source record adds Chen et al.’s M2/M5 index conjecture (Abstract and §1.1), Herderschee–Maldacena’s compactified matrix three-graviton amplitude (Abstract, §§1 and 4; version 4 dated 2026-10-14), and Collinucci’s local noncompact web geometries (Abstract, §§1 and 3.1–3.2); none supplies the complete construction asked for here. This is a targeted comparison of these source versions, not an exhaustive proof that no construction exists.

Witten’s shifted-flux condition is a source-reported global anomaly/quantization input, not a consequence of the local CJS equations. On a spin eleven-manifold $Y$, supply the integral spin class $\lambda (Y)\in H^4(Y;\mathbb{Z})$ with $2\lambda (Y)=p_1(TY)$ and a smooth closed real four-form ${\mathcal{G}}_4$ in unit-membrane-coupling normalization. On each potential patch its relation to the dimensionful membrane potential above is ${\mathcal{G}}_4=T_{\mathrm{M}2}d{C}_3$, with constant $T_{\mathrm{M}2}$; no global potential is assumed. Its periods are dimensionless. Let ${\rho }_Y:H^4(Y;\mathbb{Z})\to H_{\mathrm{d}\mathrm{R}}^4(Y;\mathbb{R})$ be coefficient extension followed by the de Rham comparison. The curvature-level condition is $[{\mathcal{G}}_4\mathrm{/}(2\pi ){]}_{\mathrm{d}\mathrm{R}}-{\rho }_Y(\lambda (Y))\mathrm{/}2\in \mathrm{im}{\rho }_Y$. This condition alone does not specify a differential-cohomology refinement or flat torsion data required for a global field configuration.

The BFSS, matrix-amplitude, M2/M5-index, and local brane-web items on the companion page are partial proposals or source reports. They are not logical suppliers for this open problem.