nLab duality between M-theory and F-theory



String theory

Duality in string theory



A duality in string theory between M-theory and F-theory:

The following line of argument shows why first compactifying M-theory on a torus S 1 A×S 1 BS_1^A \times S_1^B to get type IIA on a circle and then T-dualizing that circle to get type IIB indeed only depends on the shape R AR B\frac{R_A}{R_B} of the torus, but not on its other geometry.

By the dualities in string theory, 10-dimensional type II string theory is supposed to be obtained from the UV-completion of 11-dimensional supergravity by first dimensionally reducing over a circle S A 1S^1_A – to obtain type IIA supergravity – and then applying T-duality along another circle S B 1S^1_B to obtain type IIB supergravity.

To obtain type IIB sugra in noncompact 10 dimensions this way, also S B 1S^1_B is to be compactified (since T-duality sends the radius r Ar_A of S A 1S^1_A to the inverse radius r B= s 2/R Ar_B = \ell_s^2 / R_A of S B 1S^1_B). Therefore type IIB sugra in d=10d = 10 is obtained from 11d sugra compactified on the torus S A 1×S B 1S^1_A \times S^1_B. More generally, this torus may be taken to be an elliptic curve and this may vary over the 9d base space as an elliptic fibration.

Applying T-duality to one of the compact direction yields a 10-dimensional theory which may now be thought of as encoded by a 12-dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity vacuum is the input data that F-theory is concerned with.

A schematic depiction of this story is the following:

M-theory in d=11d = 11F-theory in d=12d = 12
\downarrow KK-reduction along elliptic fibration\downarrow axio-dilaton is modulus of elliptic fibration
IIA string theory in d=9d = 9\leftarrowT-duality\rightarrowIIB string theory in d=10d = 10

In the simple case where the elliptic fiber is indeed just S A 1×S B 1S^1_A \times S^1_B, the imaginary part of its complex modulus is

Im(τ)=R AR B. Im(\tau) = \frac{R_A}{R_B} \,.

By following through the above diagram, one finds how this determines the coupling constant in the type II string theory:

First, the KK-reduction of M-theory on S A 1S^1_A yields a type IIA string coupling

g IIA=R A s. g_{IIA} = \frac{R_A}{\ell_s} \,.

Then the T-duality operation along S B 1S^1_B divides this by R BR_B:

g IIB =g IIA sR B =R AR B =Im(τ). \begin{aligned} g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B} \\ & = \frac{R_A}{R_B} \\ & = Im(\tau) \end{aligned} \,.


See at F-theory for more

Created on May 15, 2019 at 12:15:21. See the history of this page for a list of all contributions to it.