nLab duality between M-theory and F-theory

Idea

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 $S_1^A \times S_1^B$ to get type IIA on a circle and then T-dualizing that circle to get type IIB indeed only depends on the shape $\frac{R_A}{R_B}$ 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 $S^1_A$ – to obtain type IIA supergravity – and then applying T-duality along another circle $S^1_B$ to obtain type IIB supergravity.

To obtain type IIB sugra in noncompact 10 dimensions this way, also $S^1_B$ is to be compactified (since T-duality sends the radius $r_A$ of $S^1_A$ to the inverse radius $r_B = \ell_s^2 / R_A$ of $S^1_B$). Therefore type IIB sugra in $d = 10$ is obtained from 11d sugra compactified on the torus $S^1_A \times S^1_B$. 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 $d = 11$		F-theory in $d = 12$
$\downarrow$ KK-reduction along elliptic fibration		$\downarrow$ axio-dilaton is modulus of elliptic fibration
IIA string theory in $d = 9$	$\leftarrow$T-duality$\rightarrow$	IIB string theory in $d = 10$

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

$$Im(\tau) = \frac{R_A}{R_B} \,.$$

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 $S^1_A$ yields a type IIA string coupling

$$g_{IIA} = \frac{R_A}{\ell_s} \,.$$

Then the T-duality operation along $S^1_B$ divides this by $R_B$:

$$\begin{aligned} g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B} \\ & = \frac{R_A}{R_B} \\ & = Im(\tau) \end{aligned} \,.$$

References

See at F-theory for more