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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 

<img src='content_image/311951.jpg'>

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: 

<img src='content_image/311952.jpg'>

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: 

<img src='content_image/311953.jpg'>

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- 

