Lab 19 — String Theory

T-Duality — A Circle Too Small Is a Large Circle in Disguise

Roll an extra dimension into a circle of radius R. A point particle can do exactly one thing with it: carry quantised momentum n around the loop. A closed string can also wind around it, w times. Momentum modes cost n/R, so they get heavier as the circle shrinks; winding modes cost wR/α′, so they get lighter. Swap the two roles and the spectrum does not change at all. A circle of radius R and a circle of radius α′/R are the same physics.

T-duality is not an approximation but an exact identity. Compute the whole spectrum (every level-matched state with |n|,|w| ≤ 3 and N+Ñ ≤ 3) at R and at α′/R, sort both, and they agree below the rounding you get from adding R² to 1/R² (≲1e-13). Two consequences follow. First, a minimum length: taking R toward zero just rewrites a large circle in other words, so a string cannot resolve any distance shorter than √α′. Second, at the self-dual radius the four states (n,w) = (±1,±1) become exactly massless (M² = 1/R² + R² − 2 → 0). That is two extra gauge bosons for each chirality, so the symmetry is enhanced from U(1)_L × U(1)_R to SU(2)_L × SU(2)_R — visible below as that curve touching M² = 0. This is a bosonic and heterotic phenomenon; type II strings get no enhanced gauge symmetry at the self-dual radius. Nor are those four the only massless states there: the (±2,0) and (0,±2) tachyonic levels pass through zero at R = √α′ as well. The ground state does sit at M² = −4/α′ — a tachyon, a feature of the bosonic string that supersymmetry removes — and it is the flat line on the plot. Note the momentum and winding curves are the towers built on the massless level (N = Ñ = 1), so their tachyonic N = Ñ = 0 partners are not drawn. T-duality also acts on the dilaton, g_s → g_s√α′/R.