String and M-Theory Duality Webs and Parameter Maps

String dualities identify observables in theories that use different radii, couplings, charges, or even spacetime descriptions. They turn isolated perturbative corners into a connected web, but a chain of exact parameter maps need not contain one frame in which every step is simultaneously weakly coupled. The web is evidence for a common underlying theory; it is not by itself a constructive global definition.

Required background. The proposal comparison supplies the distinction between a map and a definition.

Helpful background. Duality claims, dictionaries, and evidence supplies standards for exact maps. Charge lattices and duality local systems supplies the global charge data that local coupling maps can miss.

T-, S-, and dimensional maps

T-duality on a circle exchanges momentum and winding and maps Polchinski 1998, Vol. 2, Ch. 8

$$R' = \frac{\alpha'}R, \qquad g_s'=g_s\frac{\sqrt{\alpha'}}R.$$

It interchanges type IIA and type IIB and changes D-brane dimension according to whether the dualized direction is longitudinal or transverse. Type-IIB S-duality acts on $\tau=C_0+ie^{-\Phi}$ by $SL(2,\mathbb Z)$, exchanging fundamental strings with D1-branes and NS5-branes with D5-branes. Compactification combines these transformations into U-duality groups acting on an integral charge lattice Hull and Townsend 1995.

At strong type-IIA coupling an eleventh circle opens:

$$R_{11}=g_s\ell_s, \qquad \ell_p^3=g_s\ell_s^3.$$

The D0-brane mass $1/(g_s\ell_s)$ equals one unit of momentum $1/R_{11}$. This is a parameter-and-charge identification, not merely dimensional analogy Witten 1995.

First application: D0-branes through the web

Begin with $N$ D0-branes in type IIA. Their low-energy open strings give $U(N)$ supersymmetric matrix quantum mechanics. Lifting to eleven dimensions identifies the conserved D0 charge with $P_{11}=N/R_{11}$. T-dualizing a compact transverse circle turns D0-branes into D1-branes and converts matrix quantum mechanics into a two-dimensional gauge theory; its infrared permutation sectors underlie matrix strings.

Every arrow transports more than a coupling: boundary conditions, brane charges, spin structures, compact radii, and the observable being compared must be mapped. A protected BPS mass can often be followed through strong coupling because the supersymmetry algebra fixes it. An unprotected scattering amplitude can only be calculated where at least one frame has a controlled expansion.

Adversarial control: compose through no weak frame

Choose moduli so that the IIA description is strongly coupled, the eleven-dimensional circle is comparable to the Planck length, and after T-duality the dual circle is also string scale. The algebraic duality maps remain meaningful, but neither ten-dimensional perturbation theory nor eleven-dimensional supergravity gives a small expansion parameter. Composing the arrows does not manufacture calculational control. Monodromies can also act nontrivially on the charge lattice, so local parameter formulas alone do not specify global sectors.

The evidence ceiling is an exact or conjecturally exact equivalence for mapped observables with all global data included. The web strongly constrains any nonperturbative completion, yet does not choose a regulator, Hilbert space, or sum over sectors. BFSS matrix quantum mechanics uses the D0/Momentum map as the basis of a concrete 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

• Hull, C. M., and Townsend, P. K. (1995), “Unity of Superstring Dualities,” Nuclear Physics B 438, 109–137. arXiv:hep-th/9410167.
• Polchinski, J. (1998), String Theory, Vol. 2: Superstring Theory and Beyond, Cambridge University Press. doi:10.1017/CBO9780511618123.
• Witten, E. (1995), “String Theory Dynamics in Various Dimensions,” Nuclear Physics B 443, 85–126. arXiv:hep-th/9503124.