# What is U-duality? and Why U-duality is important?

In string theory, we know three dualities.

S-duality: Extension of Electric-magnetic duality, Duality between strongly coupled qft and weakly coupled qft. (One of the typical example is Seiberg duality in sqft)

T-duality: In short $R\leftrightarrow \frac{1}{R}$

U-duality: In M theory perspective, U-duality usually means T+S duality.(???)

But I am wondering about the last part, U-duality.

What is U-duality and why it is important in string theory?

• are you aware of the mandelstam variables in simple QFT? en.wikipedia.org/wiki/Mandelstam_variables . Seems to me this will be an extension of the concept in the string theoretical formulation (other wise why confuse the issue using u as variable) – anna v Aug 10 '18 at 17:50
• @anna v Actually there's no connection. String theory has S-duality for weak coupling to strong coupling, and T-duality for small radius to large radius, and U-duality was Ashoke Sen's name for dualities combining S and T dualities. – Mitchell Porter Aug 11 '18 at 0:50
• @MitchellPorter thanks for clarifying. Very naughty and unimaginative of those who christened these dualities. – anna v Aug 11 '18 at 4:27
• @anna v it’s not entirely unrelated: string theory has worldsheet duality (which generalises the result that, e.g. s+u mandelstam channels contain t channel to arbitrary amplitudes, associated to ope associativity and modular invariance, or equivalently the ability to cut open the path integral across different cycles without changing the result), and STU dualities are analogous relations in target space, where the corresponding path integral one cuts open is the unknown target space path integral associated to non-perturbative string theory – Wakabaloola Aug 11 '18 at 8:58
• @Wakabaloola thanks for the clarification, ( there is a reason in the madness :) ) – anna v Aug 11 '18 at 12:08