Type Iib Full Sl2z S Duality

Use the ordered charge labels $Q=(p,q{)}^{\mathsf{T}}\in {\mathbb{Z}}^2$ and asymptotic modulus $\tau \in \mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$. For $\mathrm{\Lambda }=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in G:=\mathrm{S}\mathrm{L}(2,\mathbb{Z})$, the proposed bosonic action sends $\tau$ to $\mathrm{\Lambda }\cdot \tau =(a\tau +b)\mathrm{/}(c\tau +d)$ and $Q$ to $\mathrm{\Lambda }Q$. Construct an exact nonperturbative type-IIB quantum theory over this background family and establish equivalences on all its states and observables, including the NS–NS and RR sectors, fermions, branes, amplitudes and partition functions. Here nonperturbative data include contributions not determined by the expansion in the coupling $e^{{\mathrm{\Phi }}_{\mathrm{\infty }}}$; brane sectors are sectors carrying charges coupled to the corresponding differential-form gauge potentials.

Specify the fermionic lift rather than assuming that $G$ acts strictly on every state. A concrete candidate is the metaplectic group $\tilde{G}=\mathrm{M}\mathrm{p}(2,\mathbb{Z})$: its elements are pairs $(\mathrm{\Lambda },f)$ with $f:\mathbb{H}\to \mathbb{C}$ holomorphic and $f(\tau {)}^2=c\tau +d$, with product $(\mathrm{\Lambda },f)(\mathrm{\Gamma },h)=(\mathrm{\Lambda }\mathrm{\Gamma },(f\circ \mathrm{\Gamma })h)$, where $\mathrm{\Gamma }$ acts by the same fractional linear map. The projection $\rho (\mathrm{\Lambda },f)=\mathrm{\Lambda }$ has central kernel $\{(I,1),(I,-1)\}$. Determining the physically correct lift and any remaining quantum obstruction is part of the problem, not an established consequence of the bosonic action.

More precisely, a Hilbert-space formulation must supply, for each $\tau$, a complex Hilbert space ${\mathcal{H}}_{\tau }$, a unital complex associative observable algebra ${\mathcal{A}}_{\tau }$ with conjugate-linear involution $A\mapsto A^{\ast }$ satisfying $(AB{)}^{\ast }=B^{\ast }{A}^{\ast }$, and a unital *-representation ${\pi }_{\tau }:{\mathcal{A}}_{\tau }\to \mathcal{B}({\mathcal{H}}_{\tau })$. Here $\mathcal{B}({\mathcal{H}}_{\tau })$ is the algebra of bounded linear operators; unbounded observables require specified dense domains. Equivalence for $\tilde{g}\in \tilde{G}$ consists of a unitary $U_{\tilde{g},\tau }:{\mathcal{H}}_{\tau }\to {\mathcal{H}}_{\rho (\tilde{g})\cdot \tau }$ and a *-isomorphism $V_{\tilde{g},\tau }:{\mathcal{A}}_{\tau }\to {\mathcal{A}}_{\rho (\tilde{g})\cdot \tau }$ intertwining the representations, so that $U_{\tilde{g},\tau }{\pi }_{\tau }(A)U_{\tilde{g},\tau }^{-1}={\pi }_{\rho (\tilde{g})\cdot \tau }(V_{\tilde{g},\tau }(A))$. Require the charge transformation above and preservation of the specified dynamics and all observable data, together with the derived transformation laws for amplitudes and partition functions. General elements relate different backgrounds; only elements fixing $\tau$ act within ${\mathcal{H}}_{\tau }$.

In the absence of a residual anomaly, require identity maps at the group identity and $U_{\tilde{g},\rho (\tilde{h})\cdot \tau }{U}_{\tilde{h},\tau }=U_{\tilde{g}\tilde{h},\tau }$, with the analogous law for $V$. The central element $(I,-1)$ may act by fermion parity, an involution $P_{\tau }$ with $P_{\tau }^2=I$ whose $+1$ and $-1$ eigenspaces define even and odd states; its action must be specified. A residual anomaly means an obstruction to these exact composition laws after the lift. If the maps instead compose up to phases or require a further extension, determine those phases or extension and their associative composition law explicitly. The target is a complete, anomaly-consistent quantum equivalence; classical supergravity covariance and protected-sector matching alone do not establish it.

Iib Axiodilaton And Pq String Charge

Schwarz, Introduction and §§2–4

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

Uranga, §14.3.1 and Appendix B.2

Pantev and Sharpe, §§2 and 5

Status reviewed 2026-11-05 at the bounded source scope stated here: Schwarz's introduction and classical string construction and Hull–Townsend's abstract/introduction present full quantum duality as a proposal supported by restricted calculations. Pantev–Sharpe §5 identifies sign ambiguities for fermions and proposes a metaplectic lift; this is additional formulation data, not a construction of all states or a full equivalence proof. No solution is supplied here. This disposition does not claim an exhaustive survey of subsequent publications.

Iib Pq String Bps Tension Covariance proves conditional algebraic covariance of the adopted classical tension formula. It is a separate partial result and theoretical motivation, not a prerequisite for formulating or a proof of this open problem.