M Theory Extensions of T Duality

TL;DR

The paper investigates non-perturbative extensions of T-duality within M theory by proposing two dualities: M theory on a two-torus $T^2$ is equivalent to Type IIB theory on a circle $S^1$, and M theory on a cylinder $C$ is equivalent to the SO(32) string on $S^1$. The circle size scales as $L ∝ A^{-3/4}$ (at fixed shape), and the duality is tested by detailed matching of wrapped p-branes, yielding tension relations such as $T_5^{(M)} = \frac{1}{2π}(T_2^{(M)})^2$ and $(T_1^{(B)} L_B^2)^{-1} = \frac{1}{(2π)^2} T_2^{(M)} A_M^{3/2}$. A central identification $\rho_0 = τ$ links the IIB S-duality to the torus modular group, and nine-dimensional brane spectra are shown to match across the dual theories. For the cylinder duality, the shape parameter $σ$ encodes strong/weak coupling by mapping $σ = λ_H^{(O)} = (λ_I^{(O)})^{-1}$ and obeys $(T_1^{(O)} L_O^2)^{-1} = \frac{1}{(2π)^2} T_2^{(M)} A_C^{3/2} σ^{-1/2}$, with brane tensions related by $T_5^{(O)} = \frac{1}{(2π)^2} (L_2/L_1)^2 (T_1^{(O)})^3$, consistent with heterotic and Type I limits. Overall, the work frames M-theory as a unifying, non-perturbative description that connects all known superstring theories via brane dynamics and dual geometries, suggesting further generalizations.

Abstract

T duality expresses the equivalence of a superstring theory compactified on a manifold K to another (possibly the same) superstring theory compactified on a dual manifold K'. The volumes of K and K' are inversely proportional. In this talk we consider two M theory generalizations of T duality: (i) M theory compactified on a torus is equivalent to type IIB superstring theory compactified on a circle and (ii) M theory compactified on a cylinder is equivalent to SO(32) superstring theory compactified on a circle. In both cases the size of the circle is proportional to the -3/4 power of the area of the dual manifold.