DGKT classical smeared four-dimensional effective solution

Use natural units $\mathrm{\hslash }=c=1$ and set ${\mathrm{\ell }}_s:=\sqrt{{\alpha }^{\mathrm{\prime }}}=1$ in the four-dimensional DGKT reduction formulas below. Every reduced flux coefficient, Kähler coordinate, axion, and volume appearing in those formulas denotes its dimensionless numerical value in this source string-unit convention; $\kappa$ and $\beta$ are dimensionless intersection coefficients, and $e^D$ and $e^{\varphi }$ are dimensionless couplings. In the associated ten-dimensional convention, coordinate components of the R-R fields and $H_3$ have units of inverse length; a raw R-R $p$-form period has units of length${}^{p-1}$, while an $H_3$ period has units of length squared. The flux integers are dimensionless, and equations (3.8)–(3.9) fix the period normalizations in DGKT's R-R basis, including its $1\mathrm{/}\sqrt{2}$ rescaling relative to the unrescaled R-R basis. The scalar coefficients in the reduction formulas are those basis-normalized source numbers after setting ${\mathrm{\ell }}_s=1$. In particular, the source hierarchy $\mathrm{\mid }{\widehat{e}}_i\mathrm{\mid }\gg \mathrm{\mid }{\widehat{f}}_A\mathrm{\mid }\gg \mathrm{\mid }{m}_0\mathrm{\mid }$ compares dimensionless string-unit numbers, not dimensionful quantities.

Adopt the source-reported classical four-dimensional $N=1$ constant-mode solution of DeWolfe–Giryavets–Kachru–Taylor (DGKT) for the $T^6\mathrm{/}{\mathbb{Z}}_3$ orientifold with a further freely acting ${\mathbb{Z}}_3$ quotient. This is a product of two ${\mathbb{Z}}_3$ actions; it is not a quotient by three independent ${\mathbb{Z}}_3$ factors. In the resolved Calabi–Yau model of the source, $h^{1,1}=12$ gives three bulk Kähler moduli and nine blow-up Kähler moduli, while $h^{2,1}=0$. The source adopts a smeared/averaged O6 term in the four-dimensional reduction.

Let $m_{0,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ and $p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ denote the dimensionful Romans and NS-flux parameters before setting ${\mathrm{\ell }}_s=1$; they have units of inverse length and length squared, respectively. The source normalization is $m_{0,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=\frac{f_0}{2\sqrt{2}\pi {\mathrm{\ell }}_s},p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(2\pi {\mathrm{\ell }}_s{)}^2{h}_3,H_3^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=-p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{\beta }^0\mathrm{.}$ Thus the positively oriented dual cycle used by the source has ${\int }_{{\mathrm{\Sigma }}_3}{H}_3^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=-p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, so its normalized free period is $-h_3$. In the reduced string-unit equations write $m_0:={\mathrm{\ell }}_s{m}_{0,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=f_0\mathrm{/}(2\sqrt{2}\pi )$ and $p:=p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{\mathrm{\ell }}_s^2=(2\pi {)}^2{h}_3$. On the source cover its D6/O6 signed source charge is $-2$, and the DGKT source-model tadpole input is $f_0{h}_3=-2$. The standard Graña Romans parameter is $F_0^{\mathrm{G}\mathrm{r}\mathrm{a}\tilde{\mathrm{n}}\mathrm{a},\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=\sqrt{2}{m}_{0,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=f_0\mathrm{/}(2\pi {\mathrm{\ell }}_s)$.

The bulk four-form coefficients also obey the explicit DGKT lattice, in the source cycles and R-R basis of equations (3.8)–(3.9): $e_{i,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=\frac{{\kappa }^{1\mathrm{/}3}}{\sqrt{2}}(2\pi {\mathrm{\ell }}_s{)}^3{f}_4^i,f_4^i\in \mathbb{Z},e_i:=\frac{e_{i,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}{{\mathrm{\ell }}_s^3}=c_4{f}_4^i,c_4:=\frac{{\kappa }^{1\mathrm{/}3}}{\sqrt{2}}(2\pi {)}^3>0.$ Thus setting ${\mathrm{\ell }}_s=1$ does not make $e_i$ an arbitrary integer: it leaves the fixed spacing $c_4$. The factor $1\mathrm{/}\sqrt{2}$ is the source R-R normalization and ${\kappa }^{1\mathrm{/}3}$ is the source bulk-cycle/basis normalization. The source notes possible shifted lattices generally, and states that this shift does not affect the cycles of its specified Calabi–Yau example; this adopted integer lattice is model specific. In particular, the allowed equal-bulk sequence is $f_4^i=-N$ for $N\in \mathbb{N}$, so $e_i=-c_{4}N=-\bar{e}$ with $\bar{e}:=c_{4}N\to \mathrm{\infty }$ and $c_4$ fixed.

For cyclic $(i,j,k)$ in $\{1,2,3\}$, write the bulk Kähler coordinates as $t_i=b_i+i{v}_i$, with $\kappa >0$, and define ${\widehat{e}}_i=e_i+\frac{\kappa m_j{m}_k}{m_0},b_i=\frac{m_i}{m_0},v_i=\frac{1}{\mathrm{\mid }{\widehat{e}}_i\mathrm{\mid }}\sqrt{-\frac{5{\widehat{e}}_1{\widehat{e}}_2{\widehat{e}}_3}{3\kappa m_0}}\mathrm{.}$ For each of the nine exceptional divisors, let $f_A\in \mathbb{Z}$ be the source-normalized exceptional $F_4$ flux, let $t_A=b_A-i{v}_A$, and define ${\widehat{f}}_A=f_A+\frac{\beta n_A^2}{2m_0},b_A=\frac{n_A}{m_0},v_A=\sqrt{-\frac{10{\widehat{f}}_A}{3\beta m_0}},\beta >0.$ Here $v_i>0$ is the bulk imaginary coordinate, whereas $v_A>0$ is the magnitude of the negative exceptional imaginary coordinate: $\mathrm{Im}{t}_A=-v_A$ (DGKT Eq. (5.8), footnote 9). The real flux coefficients $m_i,n_A,e_0$ and the displayed lattice coefficients specify the branch. Require $m_{0}p<0$, common sign of the ${\widehat{e}}_i$, $m_0{\widehat{e}}_1{\widehat{e}}_2{\widehat{e}}_3<0$, and $m_0{\widehat{f}}_A<0$ for every $A$, together with the source's Kähler-cone hierarchy below and positive total volume.

To fix the volume convention, let $w_i,w_A$ be the source's real degree-two cohomology basis represented by smooth closed forms on the resolved threefold $X$, with symmetric triple intersections ${\int }_X{w}_1\wedge w_2\wedge w_3=\kappa$, ${\int }_X{w}_A^3=\beta$ and all other independent triple intersections zero. The Kähler form $J$ is a smooth closed real $(1,1)$-form with $J(u,Iu)>0$ for every nonzero tangent vector $u$, where $I$ is the complex structure; its class is $[J]={\sum }_i{v}_i[w_i]-{\sum }_A{v}_A[w_A]$. Consequently the dimensionless volume and four-dimensional dilaton are $\mathcal{V}:=\frac{1}{6}{\int }_X{J}^3=\kappa v_1{v}_2{v}_3-\frac{\beta }{6}{\sum }_A{v}_A^3>0,e^D=\frac{5\mathrm{\mid }p\mathrm{\mid }}{4\sqrt{2}\mathrm{\mid }{m}_0\mathrm{\mid }\mathcal{V}},e^{\varphi }=e^D\sqrt{\mathcal{V}}\mathrm{.}$ The factor six in the bulk contraction counts the permutations of three distinct indices; there is one exceptional term per $A$, with negative sign because $\mathrm{Im}{t}_A=-v_A$. The dilaton expression follows from DGKT Eq. (5.10) and $e^D=e^{\varphi }\mathrm{/}\sqrt{\mathcal{V}}$ in Eq. (4.12). These defining equations resolve the sign inconsistency of the printed positive exceptional term in Eq. (5.11); that term cannot be used with the negative exceptional coordinate of Eq. (5.8). The bulk-only expression $\mathrm{\mid }p\mathrm{\mid }\sqrt{27\kappa \mathrm{\mid }{m}_0\mathrm{\mid }\mathrm{/}(160\mathrm{\mid }{\widehat{e}}_1{\widehat{e}}_2{\widehat{e}}_3\mathrm{\mid })}$ is the leading approximation when the exceptional-volume contribution is negligible, not the exact dilaton with nine nonzero exceptional moduli. On the branch $m_i=n_A=0$, the RR axion is fixed by $\xi =e_0\mathrm{/}p$. These equations are adopted source-model data, not a derivation from a ten-dimensional localized solution.

In the controlled scaling regime of the source, scale the dimensionless bulk four-form coefficients together, $\mathrm{\mid }{e}_i\mathrm{\mid }\sim \bar{e}\to \mathrm{\infty }$, while the Romans and NS quanta and blow-up fluxes remain fixed with $\mathrm{\mid }{\widehat{e}}_i\mathrm{\mid }\gg \mathrm{\mid }{\widehat{f}}_A\mathrm{\mid }\gg \mathrm{\mid }{m}_0\mathrm{\mid }\mathrm{.}$ The source reports $\mathcal{V}\sim {\bar{e}}^{3\mathrm{/}2}$ and $e^{\varphi }\sim {\bar{e}}^{-3\mathrm{/}4}$. With dimensions restored, its four-dimensional Einstein-frame scalings mean $\frac{\mathrm{\mid }{\mathrm{\Lambda }}_4\mathrm{\mid }}{M_P^4}\sim {\bar{e}}^{-9\mathrm{/}2},M_P{\mathrm{\ell }}_s\sim {\bar{e}}^0,{\left(\frac{R_{KK}}{{\mathrm{\ell }}_s}\right)}^2\sim {\bar{e}}^{7\mathrm{/}2},$ where ${\mathrm{\Lambda }}_4$ is the vacuum-energy density (units of length${}^{-4}$), $M_P$ is the reduced four-dimensional Planck mass (units of length${}^{-1}$), and $R_{KK}$ is the characteristic internal Kaluza–Klein length (units of length). The volume $\mathcal{V}$ is the dimensionless volume in string units. Defining $H_{\mathrm{A}\mathrm{d}\mathrm{S}}^2:=\mathrm{\mid }{\mathrm{\Lambda }}_4\mathrm{\mid }\mathrm{/}(3M_P^2)$ gives $(H_{\mathrm{A}\mathrm{d}\mathrm{S}}{R}_{KK}{)}^2\sim {\bar{e}}^{-1}$. These are source-reported scalings within the classical constant-mode truncation; the hierarchy and all reduced formulas use the source string-unit normalization.

Massive IIA Romans-mass and NS three-form flux quantization