DBI Analysis of Open String Bound States on Non-compact D-branes

CHAPTER 5. BRANES 64meaning that the level-matching condition got somewhat altered, and[ (K ) 2 ( ) ]WR2α ′ M 2 = α ′ +R α ′ + 2N L + 2N R − 4. (5.17)This last equation hides a symmetry, namely it is invariant under the simultaneoustransformations {R −→ ˜R = α ′ R −1 ,(5.18)W ←→ K.This symmetry is exactly what is called T-duality, and it teaches us that for closedstrings compactification on a circle <strong>of</strong> radius R is physically equivalent to, and evenindistinguishable from, compactification on a circle <strong>of</strong> radius α ′ R −1 . You might wonderwhy it is called a duality and not a symmetry. The reason is simple: it is only asymmetry in the closed string case, but not in the open string case as we will soon findout. However, we will still be able to map solutions <strong>of</strong> one theory to solutions <strong>of</strong> another,hence we are left with a duality. According to [11] (p.38), the “T” stands for “targetspace,” being as we map different space-time configurations into one another.At any rate, the effect <strong>of</strong> performing a T-duality transformation on the closed stringalso entails that one changesX 25R −→ −X 25R ; X 25L −→ X 25L . (5.19)This mapping is obvious for the zero modes, in which case it simply follows from Eqs.5.8 and 5.9, but for the other modes we need to impose this. We can do this, as thischanges nothing to the physics <strong>of</strong> our theory. This map leaves all physical quantities<strong>of</strong> the theory unchainged, and hence truely constitutes a symmetry. This allows one toexpress the theory using an equivalent coordinate ˜X defined aswith the generic expansion˜X 25 (σ, τ) = X 25L (τ + σ) − X 25R (τ − σ), (5.20)˜X 25 (σ, τ) = ˜x 25 + 2α ′K R σ + 2RWτ25 + oscillators. (5.21)We note that R can take on a specific value at which the theory is self-dual, namelyR = √ α ′ . At this radius, the massless closed string spectrum gets modified so as toacquire two SU(2) gauge groups, one left-moving and one right-moving. For a shortdiscussion, see [12] §3.3.It is worth pointing out a major difference between Kaluza-Klein (KK) compactificationand T-duality. With KK compactification, what one does is compactify adimension, so that one can make it “disappear” by choosing it infinitally small (R −→ 0limit), in which case its physics decouple from that in the other dimensions. This ispossible, because in KK theory, one only has a mass-squared-like factor going as ∼ R −2 .With T-duality however, two mass-squared-like terms appear, one going as ∼ R −2 andanother going as ∼ R 2 . You simply cannot make them both disappear simultaneously!