Type IIA Local Calabi–Yau Geometric Engineering Limit

Use the mostly-plus signature $(-+\cdots +)$ and natural units $\mathrm{\hslash }=c=1$ from Supergravity Conventions And Normalizations, with $J_s$ measured in the associated Type IIA string-frame metric. A Kähler form here means a smooth real positive closed $(1,1)$-form, and $K_Y:={\mathrm{\Lambda }}^{3,0}{T}^{\ast }Y$ denotes the canonical line bundle of a complex three-manifold $Y$. A local Calabi–Yau threefold datum is a specified connected smooth complex three-manifold $Y_{ϵ}$ of complex dimension three, compact or noncompact, with a Kähler form $J_{s,ϵ}$ and a specified nowhere-zero holomorphic three-form ${\mathrm{\Omega }}_{ϵ}$ trivializing $K_{Y_{ϵ}}$. If a singular limiting complex space $Y_0$ is used, the datum separately supplies a resolution $r:{\tilde{Y}}_0\to Y_0$ where ${\tilde{Y}}_0$ is a smooth complex threefold and $r$ is a proper holomorphic map restricting to a biholomorphism over the smooth locus of $Y_0$; $Y_0$ itself is not called smooth. A local model may also specify a holomorphic map ${\pi }_{ϵ}:Y_{ϵ}\to B$ over a complex curve $B$ and compact holomorphic curves (compact complex one-dimensional submanifolds) $C_j(ϵ)\subset Y_{ϵ}$ contained in its fibers. For each such curve define its string-frame area $A_j(ϵ):={\int }_{C_j(ϵ)}{J}_{s,ϵ}$ and dimensionless Kähler parameter $t_j(ϵ):=A_j(ϵ)\mathrm{/}{\alpha }^{\mathrm{\prime }}(ϵ)$; the curve is vanishing in a family when $A_j(ϵ)>0$ for $ϵ>0$ and $A_j(ϵ)\to 0$ as $ϵ\downarrow 0$. These local conditions do not impose compactness or exact $SU(3)$ holonomy (the Kähler metric's parallel-transport group need not equal $SU(3)$).

A Type IIA geometric-engineering datum for a four-dimensional $N=2$ gauge sector consists of this specified local geometry; a source-specific assignment of wrapped D-brane species and a charge lattice $\mathrm{\Gamma }$ (a finite-rank free abelian group); a designated subset ${\mathrm{\Gamma }}_{\mathrm{B}\mathrm{P}\mathrm{S}}\subseteq \mathrm{\Gamma }$ of source-assigned BPS states, with BPS meaning here that the datum identifies each such state's mass with its central-charge modulus; a declared central-charge map $Z_{ϵ}:{\mathrm{\Gamma }}_{\mathrm{B}\mathrm{P}\mathrm{S}}\to \mathbb{C}$ with energy units; a specified low-energy gauge-theory sector and its Coulomb-branch scalar coordinates (the chosen complex scalar expectation values labelling its vacua); and a one-parameter family, indexed by a positive length $ϵ\downarrow 0$, of the above geometric and physical data. Here $N=2$ denotes the adopted four-dimensional supersymmetry framework with eight real supercharges, a gauge coupling $g_{\mathrm{Y}\mathrm{M}}(E)$ is defined by the coefficient $-1\mathrm{/}(4g_{\mathrm{Y}\mathrm{M}}^2(E))$ of the normalized gauge kinetic term at energy $E$ with its trace normalization fixed, and a Coulomb coordinate is the chosen complex scalar expectation value with energy units in that framework. The strong scale $\mathrm{\Lambda }$, when used, also has energy units. Each quantity and its normalization is part of the datum; canonical-bundle triviality alone supplies none of the spectrum or gauge dictionary.

Write ${\mathrm{\ell }}_s:=\sqrt{{\alpha }^{\mathrm{\prime }}}=ϵ$ and define the string scale $M_s:={\mathrm{\ell }}_s^{-1}$. For a declared field-theory energy scale $E_{\mathrm{F}\mathrm{T}}>0$, a decoupling limit in this definition requires $M_{\mathrm{P}\mathrm{l}}(ϵ)\mathrm{/}{E}_{\mathrm{F}\mathrm{T}}\to \mathrm{\infty }$ and $M_q(ϵ)\mathrm{/}{E}_{\mathrm{F}\mathrm{T}}\to \mathrm{\infty }$ for every declared unwanted string or compactification threshold $M_q$, while the specified low-energy gauge-theory data, Coulomb coordinates, and charged BPS masses have finite stated limits. $M_{\mathrm{P}\mathrm{l}}$ is the reduced four-dimensional Planck mass in the convention $M_{\mathrm{P}\mathrm{l}}^{-2}=8\pi G_N$; for a pure asymptotically-free (gauge coupling decreases as energy increases) $N=2$ $SU(2)$ model, the fixed gauge-theory datum may be its strong scale $\mathrm{\Lambda }$ and the running coupling at a fixed field-theory energy. The bare coupling at $M_s$ is not required to stay fixed and in the cited asymptotically-free limit tends to zero. A local noncompact construction does not assert existence of a compact completion satisfying global flux and brane consistency conditions. Any local period identification beyond the data above requires its own source-specific hypotheses and regime.

Calabi–Yau threefold data used by string compactifications

Special Kähler period data in Calabi–Yau threefold reductions

Supergravity Conventions And Normalizations

Type Iia Two Derivative Supergravity Action

The Axiom of Choice

Katz–Klemm–Vafa, §2 and §2.1

Mayr, §2

For the pure $N=2$ $SU(2)$ source example, the quantity held fixed is $\mathrm{\Lambda }$ (equivalently the gauge-theory running data at a fixed energy); the bare string-scale coupling tends to zero. Exact local period identifications and global-completion arguments are outside this definition.