Toroidal U Duality

Choose a real basis $v_1,\dots ,v_6$ of ${\mathbb{R}}^6$ and set $\mathrm{\Gamma }:={\sum }_{i=1}^6\mathbb{Z}{v}_i$ and $T^6:={\mathbb{R}}^6\mathrm{/}\mathrm{\Gamma }$. Use $\mathrm{\hslash }=c=1$ and work in four-dimensional Einstein frame: use a metric $g_{\mu \nu }^{(E)}$ for which the gravitational action has a fixed Einstein--Hilbert coefficient $(2{\kappa }_4^2{)}^{-1}\int R[g^{(E)}]{\mathrm{vol}}_{g^{(E)}}$, with ${\kappa }_4>0$ of length dimension. Use the Type IIB presentation for the toroidal scalar coordinates. Use $E_{7(7)}(\mathbb{R})$ for the connected split real group of type $E_7$ in its faithful real 56-dimensional charge representation. Adopt the source's classical scalar model with maximal compact subgroup $K\cong SU(8)\mathrm{/}\{\pm I_8\}$, where $SU(8):=\{A\in {\mathrm{Mat}}_8(\mathbb{C}):A^{†}A=I_8,\det A=1\}$. The compact action on the charge space is the action on ${\bigwedge }^2{\mathbb{C}}^8$ viewed as a real vector space; its kernel is $\{\pm I_8\}$, so $SU(8)$ itself is a double cover of $K$, not the subgroup used for the global coset. Define the classical 70-scalar space by ${\mathcal{M}}_{\mathrm{c}\mathrm{l}}:=E_{7(7)}(\mathbb{R})\mathrm{/}K$, the set of right cosets $hK$; a point $m=hK$ represents one constant scalar background, and the classical duality action is $g\cdot (hK):=(gh)K$. In toroidal variables, these 70 real scalars comprise a positive-definite inner product on ${\mathbb{R}}^6$ descending to the flat torus metric (21 independent components), 15 internal NS--NS two-form components, the constant ten-dimensional RR axion and dilaton, 15 internal RR two-form components, 15 internal RR four-form potential components (whose five-form field strength obeys the type-IIB self-duality condition), and two scalars dual to the four-dimensional NS--NS and RR two-form fields.

Let $V_{56}$ be the real 56-dimensional electric/magnetic charge representation of $E_{7(7)}(\mathbb{R})$, with representation map ${\rho }_{56}$, and fix a rank-56 integral lattice ${\mathrm{\Lambda }}_{56}\subset V_{56}$ (a free abelian subgroup spanning $V_{56}$ over $\mathbb{R}$). Define the candidate discrete group $G:=\{g\in E_{7(7)}(\mathbb{R}):{\rho }_{56}(g){\mathrm{\Lambda }}_{56}={\mathrm{\Lambda }}_{56}\},$ denoted $E_{7(7)}(\mathbb{Z})$ for this lattice convention. A Hilbert-space formulation at $m$ must supply a complex Hilbert space ${\mathcal{H}}_m$, a unital complex observable *-algebra ${\mathcal{A}}_m$ (an associative algebra with conjugate-linear involution $A\mapsto A^{\ast }$ satisfying $(AB{)}^{\ast }=B^{\ast }{A}^{\ast }$), and a unital *-representation ${\pi }_m:{\mathcal{A}}_m\to \mathcal{B}({\mathcal{H}}_m)$, where $\mathcal{B}({\mathcal{H}}_m)$ denotes the bounded linear operators with Hilbert adjoint. Any additional unbounded observables require specified dense domains and intertwining on those domains. An equivalence from $m$ to $g\cdot m$ consists of isomorphisms $U_{g,m}:{\mathcal{H}}_m\to {\mathcal{H}}_{g\cdot m}$ and a *-isomorphism $V_{g,m}:{\mathcal{A}}_m\to {\mathcal{A}}_{g\cdot m}$ satisfying $U_{g,m}{\pi }_m(A)U_{g,m}^{-1}={\pi }_{g\cdot m}(V_{g,m}(A))(A\in {\mathcal{A}}_m);$ require $U_{g,m}$ to be unitary in this Hilbert-space formulation. The maps must send each charge label $Q$ to ${\rho }_{56}(g)Q$ in ${\mathrm{\Lambda }}_{56}$, brane sectors (state sectors labelled by charges under the corresponding p-form gauge potentials), and amplitudes/partition functions. Here nonperturbative observables are observables not determined by the perturbative expansion in the string coupling. An anomaly means a quantum obstruction, if any, to the duality maps forming a strict group action; the exact composition law and its anomaly data are part of the problem.

Construct such complete nonperturbative quantum formulations and establish the equivalences for every $m\in {\mathcal{M}}_{\mathrm{c}\mathrm{l}}$ and $g\in G$. Include the proposed eleven-dimensional M-theory-on-$T^7$ description of the same four-dimensional scalar family, where $T^7:={\mathbb{R}}^7\mathrm{/}{\mathrm{\Gamma }}_7$ and ${\mathrm{\Gamma }}_7:={\sum }_{i=1}^7\mathbb{Z}{w}_i$ for a real basis $w_1,\dots ,w_7$ of ${\mathbb{R}}^7$. This description is part of the conjectured duality web. Use of the separately adopted IIA circle dictionary remains conditional and does not itself establish an exact M/IIA equivalence or construct M-theory. At a fixed background $m$, only the stabilizer $G_m:=\{g\in G:g\cdot m=m\}$ acts within that fixed-background theory; a general $g$ relates different scalar backgrounds.

Classical supergravity symmetry and protected charge/BPS matching do not prove these full equivalences. “U-duality” here means the conjectured exact action combining the relevant S- and T-duality subgroups on the compactified quantum-theory family, with identifying those subgroup embeddings for the chosen charge lattice included in the problem, not merely an automorphism of the low-energy equations. The relation to the M-theory description uses the separately adopted circle dictionary below and remains part of the proposed nonperturbative duality web, not a construction of M-theory or a proved M/IIA equivalence.

Free Closed String Graded Spectrum Under Radius Inversion

Hull and Townsend use $E_7(\mathbb{Z})$ for the subgroup preserving the integral charge lattice; their four-dimensional type-II-on-$T^6$ discussion identifies the 56 charge representation.

Hull and Townsend, Abstract, §§1–3, 5, 7–8

Giveon, Porrati, and Rabinovici, §2.2

Uranga, §14.3.2

Status reviewed 2026-10-30: the cited treatments distinguish classical continuous supergravity groups, charge-lattice-preserving discrete candidates, and conjectured exact quantum equivalences. The current Free Closed String Graded Spectrum Under Radius Inversion establishes only the flat free-circle oscillator-weight and level-matching calculation in its stated scope; it does not establish full quantum U-duality. M Theory Type IIA Circle Limit Dictionary is an adopted physical-assumption supplier: its string-frame metric ansatz, positive length/coupling domains, and D0/first-KK mass match are conditional on the stated sector identification. It claims neither tower completeness nor exact M/IIA equivalence.

The earlier missing-readiness note is superseded: the retained run's canonical historical-authoring projection authenticates that supplier's original ready Step 1 record, and the present supplier source matches the projection's bound target artifact. There is no newly authored successor readiness row; absence of that new row is not absence of the authenticated historical record. This reconciles authoring provenance only, not independent proof acceptance or a new Step 2 judgment. The full quantum U-duality question remains open, and neither the adopted circle dictionary nor protected classical calculations settle it.