Type-IIB axiodilaton and elliptic-fiber dictionary

Adopt the following geometric dictionary for a type-IIB compactification model. Let $B_3$ be a smooth connected complex algebraic threefold representing the six real internal dimensions, and let $Y_4$ be the corresponding projective elliptically fibered Calabi–Yau fourfold input. Here the broad Calabi–Yau convention is the one in Compact Calabi–Yau manifold conventions: connected, compact, Kähler, and with holomorphically trivial canonical bundle. Let $\pi :Y_4\to B_3$ be the proper flat genus-one fibration with a specified regular section as in Elliptic fibration with a section. The adopted regime is geometric and long-wavelength: ${\alpha }^{\mathrm{\prime }}$ has dimensions of length squared, its square-root string scale is small relative to the curvature and variation scales used in the model, and string-loop corrections are controlled or explicitly retained. No numerical bound is asserted. This physical input does not assert that a singular Weierstrass presentation is smooth or that a global resolution exists.

On a local patch where the holomorphic line bundle $L\to B_3$ in Weierstrass model over a line bundle is trivialized, use the affine chart $z=1$ and write the model as $y^2=x^3+fx+g$, where $f\in H^0(B_3,L^4)$ and $g\in H^0(B_3,L^6)$ are represented by holomorphic coefficient functions in the chosen trivialization, and the discriminant section is $\mathrm{\Delta }=4f^3+27g^2\in H^0(B_3,L^{12})$. Under a line-bundle change of frame both $f$ and $g$ acquire the same corresponding fourth/sixth-power weights, so the numerator and denominator of $4f^3\mathrm{/}\mathrm{\Delta }$ have equal weight twelve and their ratio is independent of that local frame. On the smooth-fiber locus $B_3^{\circ }=B_3\setminus \{\mathrm{\Delta }=0\}$, choose a simply connected analytic cover $\{U_i\}$ and, on each $U_i$, locally constant cycles $A_i,B_i$ forming a basis of $H_1(E_b,\mathbb{Z})$ with oriented intersection $A_i\cdot B_i=1$, where $E_b={\pi }^{-1}(b)$. Shrink each patch to trivialize $L$, and let ${\mathrm{\Omega }}_i$ be the relative holomorphic one-form whose restriction to the affine smooth fiber is $dx\mathrm{/}y$ and which extends across its point at infinity. Define the period ratio by ${\tau }_i(b)=\frac{{\int }_{B_i(b)}{\mathrm{\Omega }}_i{\mathrm{\mid }}_{E_b}}{{\int }_{A_i(b)}{\mathrm{\Omega }}_i{\mathrm{\mid }}_{E_b}}\mathrm{.}$ The denominator is nonzero and this ratio is holomorphic with values in $\mathbb{H}=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$, using the complex orientation. Rescaling ${\mathrm{\Omega }}_i$ by a nowhere-zero holomorphic function cancels between the two periods. Use the ordered cycle column $(B_i,A_i{)}^T$: on overlaps $(B_i,A_i{)}^T=M_{ij}(B_j,A_j{)}^T$. Thus the same matrix gives the fractional-linear transformation below, even if the two local forms differ by a nonzero factor. In this local duality frame define the dimensionless real fields $C_{0,i}=\mathrm{Re}{\tau }_i,{\varphi }_i=-\log (\mathrm{Im}{\tau }_i),g_{s,i}=e^{{\varphi }_i}>0.$ Thus $C_{0,i}$ (the Ramond–Ramond axion), ${\varphi }_i$ (the dilaton), and $g_{s,i}$ are smooth real functions on $U_i$; ${\tau }_i=C_{0,i}+i{e}^{-{\varphi }_i}$ is dimensionless and holomorphic. These fields are not asserted to extend across $\mathrm{\Delta }=0$. On each connected overlap a locally constant basis-change matrix $M_{ij}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathrm{S}\mathrm{L}(2,\mathbb{Z})$ acts by the convention ${\tau }_i=\frac{a{\tau }_j+b}{c{\tau }_j+d},C_{0,i}=\mathrm{Re}\frac{a{\tau }_j+b}{c{\tau }_j+d},e^{-{\varphi }_i}=\frac{e^{-{\varphi }_j}}{\mathrm{\mid }c{\tau }_j+d{\mathrm{\mid }}^2}\mathrm{.}$ The matrices satisfy the local-system cocycle convention $M_{ij}{M}_{jk}=M_{ik}$. In particular ${\varphi }_i={\varphi }_j+2\log \mathrm{\mid }c{\tau }_j+d\mathrm{\mid }$; the axion and dilaton need not patch separately as scalar functions on $B_3^{\circ }$. The central matrix $-I$ acts trivially on $\tau$ but is retained in the homology local system. In the adopted dictionary, $\tau$ is the complex structure modulus of the elliptic fiber and

$j_{\mathrm{s}\mathrm{t}\mathrm{d}}(\tau )=1728\frac{4f^3}{4f^3+27g^2}\mathrm{.}$

The standard invariant is normalized by $q=e^{2\pi i\tau }$ and $j_{\mathrm{s}\mathrm{t}\mathrm{d}}(\tau )=q^{-1}+744+O(q)$ near the cusp; in particular $j_{\mathrm{s}\mathrm{t}\mathrm{d}}(i)=1728$. A loop in $B_3^{\circ }$ acts on the fiber homology lattice $H_1(E_b,\mathbb{Z})$ by a matrix in $\mathrm{S}\mathrm{L}(2,\mathbb{Z})$ preserving its oriented intersection pairing. With the conventional modular action, $\tau$ changes by $\tau \mapsto (a\tau +b)\mathrm{/}(c\tau +d)$ for the corresponding matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$. Thus $\tau$ is generally a local lift with duality monodromy, not a globally single-valued scalar.

The fiber is auxiliary geometric data encoding the axiodilaton and its duality. Its extra complex dimension is not an asserted pair of ordinary spacetime dimensions. This is an adopted physical dictionary in the stated geometric regime, not a mathematical consequence of the fibration definition and not evidence for a constructed nonperturbative or 12-dimensional spacetime theory.

Elliptic fibration with a section

Weierstrass model over a line bundle

Compact Calabi–Yau manifold conventions