Strong-Weak Coupling String Duality

We have described the massless spectrum of the five consistent superstring theories in ten dimensions. Additional theories can be constructed in lower dimensions by compactification of some of the ten dimensions. Thus the ten-dimensional spacetime $X$ looks like the product $X= {\cal K}^d \times {\bf R}^{1,9-d}$, with ${\cal K}$ a suitable compact manifold or orbifold. Depending on which compact space is taken, it will be the quantity of preserved supersymmetry. All five theories and their compactifications are parametrized by: the string coupling constant $g_S$, the geometry of the compact manifold ${\cal K}$, the topology of ${\cal K}$ and the spectrum of bosonic fields in the NS-NS and the R-R sectors. Thus one can define the string moduli space ${\cal M}$ of each one of the theories as the space of all associated parameters. Moreover, it can be defined a map between two of these moduli spaces. The dual map is defined as the map ${\cal S}: {\cal M} \to {\cal M}'$ between the moduli spaces ${\cal M}$ and ${\cal M}'$ such that the strong/weak region of ${\cal M}$ is interchanged with the weak/strong region of ${\cal M}'$. One can define another map ${\cal T}: {\cal M} \to {\cal M}'$ which interchanges the volume $V$ of ${\cal K}$ for ${1 \over V}$. One example of the map ${\cal T}$ is the equivalence, by T-duality, between the theories Type IIA compactified on ${\bf S}^1$ at radius $R$ and the Type IIB theory on ${\bf S}^1$ at raduis ${1 \over R}$. The theories HE and HO constitutes another example of the ${\cal T}$ map. In this section we will follows the Sen's review [20]. Another useful references are [21,22,23,24,25]. Type IIB theory is self-dual with respect the ${\cal S}$ map. truecm Type IIB-IIB Duality The Type IIB theory is self-dual. In order to see that write the bosonic part of the action of Type IIB supersting theory

$$S_{\bf IIB} = {1 \over 2 \kappa^2} \int_X d^{10}x \sqrt{-G_... ...\Phi \partial^{I} \Phi -{1 \over 2} H_{IJK} H^{IJK} \bigg)$$

$$-{1\over 4 \kappa^2} \int_X d^{10}x \sqrt{-G_{IIB}} \bigg... ...\over 4\kappa^2} \int_X A_{(4)} \land H_{(3)} \land F_{(3)},$$ (49)

where $\widetilde{F}_{(3)} = dA_{(2)} - a \land H_{(3)}$ and $\widetilde{F}_{(5)} = dA_{(4)} - {1 \over 2} A_{(2)} \land H_{(3)} + {1 \over 2} B_{(2)} \land F_{(3)}$. This action is clearly invariant under

$$\Phi' = - \Phi , \ \ \ \ \ \ G'_{IJ}= e^{-\Phi} G_{IJ},$$

$$B_{(2)} = A_{(2)} \ \ \ \ \ \ A'_{(2)}= - B_{(2)}, \ \ \ \ \ \ A'_{(4)} = A_{(4)}.$$ (50)

This symmetry leads to an identification of a fundamental string F1 with a D1-brane ( $B_{(2)} = A_{(2)}$) and the interchanging of a pair of D3-branes. truecm Type I-SO(32)-Heterotic Duality In order to analyze the duality between Type I and SO(32) heterotic string theories we recall the spectrum of both theories. These fields are the dynamical fields of a supergravity Lagrangian in ten dimensions. Type I string theory has in the NS-NS sector the following fields: the metric $G_{IJ}^{\bf I}$, the dilaton $\Phi^{\bf I}$ and in the R-R sector: the antisymmetric tensor $A_{IJ}^{\bf I}$. Also there are 496 gauge bosons $A^{a{\bf I}}_{I}$ in the adjoint representation of the gauge group SO(32). For the SO(32) heterotic string theory the spectrum consist of: the spacetime metric $G_{IJ}^{\bf H}$, the dilaton field $\Phi^{\bf H}$, the antisymmetric tensor $B_{IJ}^{\bf H}$ and 496 gauge fields $A^{a {\bf H}}_{I}$ in the adjoint representation of SO(32). Both theories have ${\cal N}=1$ spacetime supersymmetry. The effective action for the massless fields of the Type I supergravity effective action $S_{\bf I}$ is defined as

$$S_{\bf I} = {1 \over 2 \kappa^2} \int_X d^{10}x \sqrt{-G^{\... ...I})^2 -{1 \over 12} \vert\widetilde{F}_{(3)}\vert^2 \bigg)$$

$$- {1 \over g^2} \int_X d^{10}x \sqrt{-G^{\bf I}} e^{- \Phi^{\bf I}} Tr(F^{\bf I}_{IJ} F^{{\bf I}IJ})$$ (51)

where $\widetilde{F}_{(3)} = F_{(3)} - {\alpha ' \over 4} [\omega_{3Y}(A) - \omega_{3L} (\omega)].$ While the heterotic action $S_{\bf H}$ is defined as

$$S_{\bf H} = {1 \over 2 \kappa^2} \int_X d^{10}x \sqrt{-G^{\... ... {\alpha ' \over 8} Tr ( F^{\bf H}_{IJ} F^{{\bf H}IJ}) \bigg]$$ (52)

where $\widetilde{H}_{(3)} = dB_{(2)} - {\alpha ' \over 4} [\omega_{3Y}(A) - \omega_{3L} (\omega)].$ The comparison of these two actions leads to the following identification of the fields

$$G_{IJ}^{\bf I} = e^{-\Phi^{\bf H}} G_{IJ}^{\bf H}, \ \ \ \ \ \ \ \ \Phi^{\bf I} = - \Phi^{\bf H},$$

$$A_{I}^{a{\bf I}} = A_{I}^{a{\bf H}}, \ \ \ \ \ \ \ \ \ \widetilde{F}^{\bf I}_{(3)} = \widetilde{H}^{\bf H}_{(3)}.$$ (53)

This give us many information, the first relation tell us that the metrics of both theories are the same. The second relation interchanges the $B_{(2)}$ field in the NS-NS sector and the $A_{(2)}$ field in the R-R sector. That implies the interchanging of heterotic strings and Type I D1-branes. The third relation identifies the gauge fields coming from the Chan-Paton factors from the Type I side with the gauge fields coming from the 16 compactified internal dimensions of the heterotic string. Finally, the opposite sign for the dilaton relation means that the string coupling constant $g^{\bf I}_S$ is inverted $g^{\bf H}_S = 1/ g^{\bf I}_S$ within this identification, and interchanges the strong and weak couplings of both theories leading to the explicit realization of the ${\cal S}$ map. truecm