Silverstein Westphal Nil Background Data For Wrapped D4 Reduction

Take dimensionless coordinates $(u_1,u_2,x^{\mathrm{\prime }})\in {\mathbb{R}}^3$, integer $M\ge 1$, and quotient by the lattice transformations ${\tau }_1(u_1,u_2,x^{\mathrm{\prime }})=(u_1+1,u_2,x^{\mathrm{\prime }}-M{u}_2),{\tau }_2(u_1,u_2,x^{\mathrm{\prime }})=(u_1,u_2+1,x^{\mathrm{\prime }}),{\tau }_x(u_1,u_2,x^{\mathrm{\prime }})=(u_1,u_2,x^{\mathrm{\prime }}+1)\mathrm{.}$ The adopted Nil metric block is $d{s}_{N^3}^2={\alpha }^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}(L_{u_1}^{2}d{u}_1^2+L_{u_2}^{2}d{u}_2^2+L_x^2(d{x}^{\mathrm{\prime }}+M{u}_{1}d{u}_2{)}^2),$ with $L_{u_1},L_{u_2},L_x>0$ dimensionless and $[{\alpha }^{\mathrm{\prime }}]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^2$. Define $L_u^2=L_{u_1}{L}_{u_2}$, $\beta =L_{u_2}\mathrm{/}{L}_{u_1}>0$, so $L_{u_1}=L_u\mathrm{/}\sqrt{\beta }$, $L_{u_2}=L_u\sqrt{\beta }$, and $L^3=L_u^2{L}_x$. Define the second factor ${\tilde{N}}^3$ as an isomorphic copy of $N^3$ with dimensionless coordinates $({\tilde{u}}_1,{\tilde{u}}_2,{\tilde{x}}^{\mathrm{\prime }})\in {\mathbb{R}}^3$, quotient transformations ${\tilde{\tau }}_1({\tilde{u}}_1,{\tilde{u}}_2,{\tilde{x}}^{\mathrm{\prime }})=({\tilde{u}}_1+1,{\tilde{u}}_2,{\tilde{x}}^{\mathrm{\prime }}-M{\tilde{u}}_2)$, ${\tilde{\tau }}_2({\tilde{u}}_1,{\tilde{u}}_2,{\tilde{x}}^{\mathrm{\prime }})=({\tilde{u}}_1,{\tilde{u}}_2+1,{\tilde{x}}^{\mathrm{\prime }})$, and ${\tilde{\tau }}_x({\tilde{u}}_1,{\tilde{u}}_2,{\tilde{x}}^{\mathrm{\prime }})=({\tilde{u}}_1,{\tilde{u}}_2,{\tilde{x}}^{\mathrm{\prime }}+1)$, and metric $d{s}_{{\tilde{N}}^3}^2={\alpha }^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}(L_{u_1}^{2}d{\tilde{u}}_1^2+L_{u_2}^{2}d{\tilde{u}}_2^2+L_x^2(d{\tilde{x}}^{\mathrm{\prime }}+M{\tilde{u}}_{1}d{\tilde{u}}_2{)}^2)\mathrm{.}$ Let $X_4$ be a smooth oriented boundaryless four-manifold with a smooth Lorentzian string-frame metric $g_{4,s}\in \mathrm{\Gamma }({\mathrm{Sym}}^2{T}^{\ast }{X}_4)$ of signature $(-,+,+,+)$; write $d{s}_{4,s}^2$ for its local line element, with units of length squared. On $X^{10}:=X_4\times N^3\times {\tilde{N}}^3$, take the smooth product metric $G^{(s)}:={\pi }_4^{\ast }{g}_{4,s}+{\pi }_N^{\ast }{g}_N+{\pi }_{\tilde{N}}^{\ast }{g}_{\tilde{N}}$ of signature $(-,+,\dots ,+)$, where the $\pi$ are the three projections and $g_N,g_{\tilde{N}}$ are the metric tensors just specified. Thus $d{s}_{10,s}^2=d{s}_{4,s}^2+d{s}_{N^3}^2+d{s}_{{\tilde{N}}^3}^2$. The dilaton is the constant field $\mathrm{\Phi }={\mathrm{\Phi }}_{\mathrm{\infty }}$, and the dimensionless string coupling is $g_{\mathrm{s}\mathrm{t}\mathrm{r}}:=e^{{\mathrm{\Phi }}_{\mathrm{\infty }}}>0$. For the DBI contribution isolated in the cited calculation, set the pulled-back $B_2$ and $F_A$ to zero and keep Chern–Simons, stabilization, loop, and higher-${\alpha }^{\mathrm{\prime }}$ contributions separate. These are adopted background data for a probe calculation, not proof that the compactification equations have a global solution.

Silverstein and Westphal, §2.2–§2.3

Silverstein–Westphal §2.3, equation (2.6), describes the second Nil factor as isometric to the first; the tilded metric and quotient above make that adopted copy explicit. The lattice preserves $d{x}^{\mathrm{\prime }}+M{u}_{1}d{u}_2$: under ${\tau }_1$, $d{x}^{\mathrm{\prime }}\mapsto d{x}^{\mathrm{\prime }}-Md{u}_2$ and $u_1\mapsto u_1+1$, so the added terms cancel. The same calculation holds in the tilded copy. This local metric postulate does not specify the global orientifold image or the net D4 charge.