A string duality identifies apparently different string theory descriptions as descriptions of the same physical system, with a map between their states, parameters and observables. It can exchange quantities interpreted very differently in the two descriptions, such as momentum and winding around a compact dimension.
T-duality relates a circle of radius to one of radius by interchanging momentum and winding quantum numbers. A small geometrical circle in one description can therefore correspond to a large one in another; comparing only the classical geometry would miss the matching spectrum. S-duality instead relates coupling regimes and can turn a difficult strong-coupling question into a weak-coupling one. A proposed duality needs a concrete dictionary and consistency checks; a coincidental equality of one calculation is insufficient to establish equivalence of full theories.
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