are a doublet of $(D-1)$ forms, the $\lambda_a$ are two scalars, and $$M = {\mathrm{e}}^{\phi} \begin{pmatrix} 1 & - \chi \\[2pt] - \chi & {\mathrm{e}}^{\phi}
  + \chi^2 \end{pmatrix}\ .$$ The two scalars $\phi$ and $\chi$ parametrize the coset SL$(2,\fieldR)/
\mathrm{O}(2)$; in terms of the complex combination $$\label{tau}
  \tau \equiv \chi + 2{\mathrm{i}}\, {\mathrm{e}}^{\phi/2}$$ the scalar part of ${\mathcal{L}}_\mathrm{DT}$ can be written as $2|d\tau/\operatorname{Im}\tau|^2$. The tensor terms, however, break the global SL$(2,\fieldR)$ symmetry, leaving only shift symmetries of $\phi$ and $\chi$. The shift in $\chi$ acts as $$\label{shift-chi}
  \tau \rightarrow \tau + b\ ,\quad \begin{pmatrix} H^1 \\[2pt] H^2
  \end{pmatrix} \rightarrow \begin{pmatrix} 1 & b \\[2pt] 0 & 1
  \end{pmatrix} \begin{pmatrix} H^1 \\[2pt] H^2 \end{pmatrix}\ ,
  \quad \begin{pmatrix} \lambda^1 \\[2pt] \lambda^2 \end{pmatrix}
  \rightarrow \begin{pmatrix} 1 & 0 \\[2pt] -b & 1 \end{pmatrix}
  \begin{pmatrix} \lambda^1 \\[2pt] \lambda^2 \end{pmatrix}\ ,$$ whereas the shift in $\phi$ acts as $$\label{shift-phi}
  \tau \rightarrow {\mathrm{e}}^\kappa \tau\ ,\quad \begin{pmatrix} H^1 \\[2pt]
  H^2 \end{pmatrix} \rightarrow \begin{pmatrix} {\mathrm{e}}^{-\kappa} H^1
  \\[2pt] {\mathrm{e}}^{-2\kappa} H^2 \end{pmatrix}\ ,\quad \begin{pmatrix}
  \lambda^1 \\[2pt] \lambda^2 \end{pmatrix} \rightarrow \begin{pmatrix}
  {\mathrm{e}}^\kappa \lambda^1 \\[2pt] {\mathrm{e}}^{2\kappa} \lambda^2 \end{pmatrix}\ .$$ Note that the latter acts like an SL$(2,\fieldR)$ transformation on $\tau$, but not on the $H^a$. The full type IIB theory compactified to 