Strong Constraint DFT Generalized Metric Data

For $D=10$, let $\mathcal{U}\subset {\mathbb{R}}^{20}$ be an open doubled-coordinate domain with coordinates $X^M=(x^i,{\tilde{x}}_i)$, $i=0,\dots ,9$, each of length dimension. Let ${\eta }_{MN}=\left(\begin{array}{cc}0& I_{10}\\ I_{10}& 0\end{array}\right)$ be the constant dimensionless bilinear form. A doubled scalar is a function in $C^{\mathrm{\infty }}(\mathcal{U},\mathbb{R})$, and a doubled tensor is a smooth array of components on $\mathcal{U}$ with the indicated upper and lower index slots. A generalized vector parameter is $\xi \in C^{\mathrm{\infty }}(\mathcal{U},{\mathbb{R}}^{20})$ with components of length dimension; set ${\xi }_M:={\eta }_{MN}{\xi }^N$ and ${\mathrm{\partial }}^M:={\eta }^{MN}{\mathrm{\partial }}_N$, so ${\mathrm{\partial }}_M$ has inverse-length dimension. A scalar $\chi \in C^{\mathrm{\infty }}(\mathcal{U},\mathbb{R})$ of length-squared dimension defines a gradient-form parameter by ${\xi }^M={\mathrm{\partial }}^M\chi$. A generalized metric field is a smooth symmetric covariant tensor ${\mathcal{H}}_{MN}$ on $\mathcal{U}$ satisfying ${\mathcal{H}}_{MP}{\eta }^{PQ}{\mathcal{H}}_{QN}={\eta }_{MN}$; its components and those of its inverse ${\mathcal{H}}^{MN}:={\eta }^{MP}{\eta }^{NQ}{\mathcal{H}}_{PQ}$ are dimensionless. The strong constraint requires ${\eta }^{MN}{\mathrm{\partial }}_M{\mathrm{\partial }}_{N}A=0$ and ${\eta }^{MN}({\mathrm{\partial }}_{M}A)({\mathrm{\partial }}_{N}B)=0$ for every component $A,B$ of every field, gauge parameter, and product under consideration.

After restricting to a product coordinate neighborhood $U_x\times \tilde{U}\subset \mathcal{U}$, the standard local physical section is the slice $s(x)=(x,{\tilde{x}}_0)$ for fixed ${\tilde{x}}_0\in \tilde{U}$, and fields on the section are independent of all ${\tilde{x}}_i$ (equivalently, ${\tilde{\mathrm{\partial }}}^i=0$). Let $g$ be an invertible symmetric Lorentzian metric and $B_2$ an antisymmetric two-form on $U_x$, both with dimensionless components in these length-valued coordinates. The restriction of the generalized metric to this section is parameterized by

This block matrix is symmetric and obeys ${\mathcal{H}}_{MP}{\eta }^{PQ}{\mathcal{H}}_{QN}={\eta }_{MN}$. Let $\mathrm{\Phi }\in C^{\mathrm{\infty }}(U_x,\mathbb{R})$ be the real dimensionless physical dilaton on this physical-coordinate domain. On $U_x\times \tilde{U}$ its section-compatible extension is $\mathrm{\Phi }\circ {\mathrm{pr}}_x$, independent of every ${\tilde{x}}_i$. Let $d\in C^{\mathrm{\infty }}(\mathcal{U},\mathbb{R})$ be the generalized-dilaton field; on the physical section define its restriction by $e^{-2s^{\ast }d}=\sqrt{-\det (g_{ij})}{e}^{-2\mathrm{\Phi }}$, where the determinant is taken in the stated length-valued coordinates. Thus $d$ is dimensionless there. The notation $\sqrt{-g}$ in this density relation means $\sqrt{-\det (g_{ij})}$, not the metric tensor itself. The metric $g$ here is the local metric parameter in this DFT definition; identifying it with a particular supergravity frame is a separate convention. All doubled indices are raised and lowered with $\eta$.

For a generalized vector parameter ${\xi }^M$, define its action on a covariant generalized tensor $A_{MN}$ by

and use the following rank-zero and rank-one rules:

For a mixed tensor $A_{M_1\dots M_r}{}^{N_1\dots N_s}$ with $r,s\ge 0$, define

For $r=s=0$ this is the scalar rule; for one lower or upper slot it is the displayed covector or vector rule. This componentwise definition obeys the Leibniz rule and is compatible with raising and lowering by $\eta$.

Define the generalized-dilaton transformation separately by ${\delta }_{\xi }d={\xi }^P{\mathrm{\partial }}_{P}d-\frac{1}{2}{\mathrm{\partial }}_P{\xi }^P$; equivalently its density $\rho :=e^{-2d}$ transforms by ${\delta }_{\xi }\rho ={\mathrm{\partial }}_P({\xi }^P\rho )$. The two-derivative generalized-metric action density is

For later conditional calculations, define the generalized-scalar candidate

The expression $\mathcal{R}$ is only defined here; its scalar transformation law and relation to the action density are separate results. These are local field and transformation conventions only. They assert neither gauge closure nor gauge invariance of the action.

Hohm, Hull, and Zwiebach, Introduction and §3.1

Aldazabal, Marqués, and Núñez, §§3.3–3.4

The strong constraint includes pairwise derivative contractions, not only the weak equation ${\eta }^{MN}{\mathrm{\partial }}_M{\mathrm{\partial }}_{N}A=0$ on each field separately. Choosing ${\tilde{\mathrm{\partial }}}^i=0$ is one local solution; this definition does not assert that a physical background globally admits that section.