M-Theory

We have described how to construct dual pairs of string theories. By the uses of the ${\cal S}$ and the ${\cal T}$ maps a network of theories can be constructed in various dimensions all of them related by dualities. However new theories can emerge from this picture, this is the case of M-theory. M-theory (the name come from `mystery', `magic', `matrix', `membrane', etc.) was originally defined as the strong coupling limit for Type IIA string theory [21]. It is known that Type IIA theory can be obtained from the dimensional reduction of the eleven dimensional supergravity theory (a theory known from the 70's years) and given by the Cremmer-Julia-Scherk Lagrangian

$$I_{11} = {1 \over 2 \kappa^2_{11}} \int_Y d^{11}x \sqrt{-G_... ...) - {1\over 6} \int_Y A_{(3)} \land F_{(4)} \land F_{(4)},$$ (54)

where $Y$ is the eleven dimensional manifold. If we assume that the eleven-dimensional spacetime factorizes as $Y= X \times {\bf S}^1_R$, where the compact dimension has radius $R$. Usual Kaluza-Klein dimensional reduction leads to get the ten-dimensional metric, an scalar field and a vector field. $A_{(3)}$ from the eleven dimensional theory leads to $A_{(3)}$ and $A_{(2)}$ in the ten-dimensional theory. The scalar field turn out too be proportional to the dilaton field of the NS-NS sector of the Type IIA theory. The vector field from the KK compactification can be identified with the $A^{\bf IIA}$ field of the R-R sector. From the three-form in eleven dimensions are obtained the RR field $A_{(3)}$ of the Type IIA theory. Fin ally, the $A_{(2)}$ field is identified with the NS-NS B-field of field strength $H_{(3)} = d B_{(2)}$. Thus the eleven-dimensional Lagrangian leads to the Type IIA supergravity in the weak coupling limit ($\Phi \to 0$ or $R \to 0$). The ten-dimensional IIA supergravity describing the bosonic part of the low energy limit of the Type IIA superstring theory is

$$S_{\bf IIA} = {1 \over 2 \kappa^2} \int_X d^{10}x \sqrt{-G^... ...Phi^{\bf IIA})^2 -{1 \over 12} \vert H_{(3)}\vert^2 \bigg)$$

$$- {1 \over 4 \kappa^2} \int_X d^{10}x \sqrt{-G^{\bf IIA}} ... ...over 4 \kappa^2} \int_X B_{(2)} \land dA_{(3)} \land dA_{(3)}$$ (55)

where ${H}_{(3)} = dB_{(2)}$, $F_{(2)} = dA_{(1)}$ and $\widetilde{F}_{(4)} = dA_{(3)} - A_{(1)} \land H_{(3)}.$ It is conjectured that there exist an eleven dimensional fundamental theory whose low energy limit is the 11 dimensional supergravity theory and that it is the strong coupling limit of the Type IIA superstring theory. At the present time the degrees of freedom (dof's) are still unknown, through at the macroscopic level they should be membranes and fivebranes (also called M2-branes and M5-branes). truecm