simple dualities, meaning that there was only one theory after all, called M-theory. One of those dualities is T-duality. T-duality is a symmetry between small and large distances. The ‘T’ stands for topology , since T-duality transformations work on spaces in which at least one dimension has the topology of a circle. Let us consider a string that is wrapped around the circle, ie. it is closed. It’s energy spectrum is given by The first term is the Kaluza-Klein energy, and the second term is the wind- ing number. (1.49) is invariant under the following transformation: Therefore a theory with only closed modes will be invariant under (1.50). But what if a theory contained open strings too? These aren’t invariant under (1.50). The open strings will have a Kaluza-Klein contribution to their energy spectrum, but no winding term. Hence T-duality will change the spectrum of the open strings. Under T-duality, the boundary conditions of the open string change: We go from Neumann boundary conditions (LHS) to Dirichlet boundary conditions (RHS). For a long time this was result was ignored by the string community because the Dirichlet boundary condition is not Lorentz invari- ant - a special surface in the 10D spacetime has been picked out. It was Polchinski who first embraced this result, and extolled the modern explana- 31 m 2 ∼ n 2 R 2 + ω $^{2}$R2 α ′ 2 (1.49) R → $_{α}$′$_{/R,}$ n → ω (1.50) ∂$_{σ}$$_{2}$X $^{µ}$( σ$_{1}$, (0 , l )) = 0 ←→ X $^{µ}$( σ$_{1}$, (0 , l )) = c µ (1.51)