Dualities Clause Samples

Dualities. ‌ As discussed earlier, after dimensional reduction on an n torus the ▇▇▇▇▇▇▇▇-▇▇▇▇▇▇▇ action possesses a global GL(n, R) symmetry. For type IIA/B supergravity on an n torus or, equivalently, eleven dimensional supergravity on an n + 1 torus, this symmetry is enhanced to an En+1(R) global symmetry [8–11]. However, in the full quantum string theory and M-theory the global En+1(R) symmetry is broken, due to the quantisation conditions on the brane charges, to an En+1(Z) subgroup. This En+1(Z) subgroup is the U-duality group of type IIA/B string theory and M- theory compactified on a torus to d = 10 — n dimensions [14]. From a type IIB string theory point of view, U-duality may be thought of as being generated by T-duality [55], which relates type IIA string theory and type IIB string theory wrapped on a circle, and S-duality, which relates type IIB string theory in different coupling regimes. We will begin by reviewing T-duality and S-duality before discussing the U-duality group and in particular how the effective actions of dimensionally reduced type IIA/B string theory and M-theory are formulated as non-linear realisations of the U-duality group. 3.4.1 T-duality‌ Type IIA and type IIB string theory compactified on a circle S1 of radius r are related by T-duality. Under compactification on a circle of S1 the momentum p of the closed type IIA/B string along r the compact dimension, is quantised in units of K where K ∈ Z is the ▇▇▇▇▇▇-▇▇▇▇▇ excitation number. The winding number W ∈ Z is the number of times the string wraps around the circle. The mass M of the string is given by M 2 = K2 r2 + r2W 2 α′2 + 4 (NL + NR — 2) ,