Flux-vacuum ensemble and tadpole-bound data

A flux-vacuum ensemble datum is a record

Here $\mathcal{C}$ is the supplied compactification-model data, not a set over which this record automatically sums. It includes a compact smooth internal manifold $Y$ of declared dimension and stated boundary status, orientation and smooth geometric structures; a declared field-index set $J$ and smooth vector bundles $E_j\to Y$ for $j\in J$; and each field $s_j$ as a $C^{r_j}$ section of $E_j$ on a stated open domain $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}\subseteq Y$, with $r_j\in {\mathbb{N}}_0\cup \{\mathrm{\infty }\}$. If a field is singular at a source, its distributional extension and support are supplied; no smoothness across that support is assumed. The admissible field configurations form a declared set $\mathcal{F}$ of such sections. The field and critical-point equations are supplied as maps ${\mathcal{E}}_{\mathrm{\ell }}:\mathcal{F}\to W_{\mathrm{\ell }}$ into declared target spaces, with their regularity and boundary conditions; solutions are elements of ${\bigcap }_{\mathrm{\ell }}{\mathcal{E}}_{\mathrm{\ell }}^{-1}(0)$. An equivalence relation $\sim$ on this solution set records the adopted dualities. A localized-source record specifies a closed support $S_s\subseteq Y$, a source current $J_s$ (a distributional section of a declared bundle supported on $S_s$), a charge in a declared charge space, and an orientation convention. A brane label is a declared source type on such a record. Orientifold data specify a smooth involution $\sigma :Y\to Y$ with ${\sigma }^2=\mathrm{i}\mathrm{d}$, its declared bundle maps on the fields, and the associated source records on the fixed set. All these are model inputs; this definition asserts neither that the equations have a solution nor that the supplied data form a consistent string background. Pooling distinct model records requires an additional counting or weighting rule and is not part of this datum. The moduli space $\mathcal{M}$ and equation maps below are the specified moduli and critical-equation data for this model. The manifold $\mathcal{M}$ is second-countable and smooth complex of dimension $m\ge 1$, with local complex coordinates $z^1,\dots ,z^m$; $U\subseteq \mathcal{M}$ is the root-counting region, measurable in every local real coordinate chart.

The real flux space $V_{\mathbb{R}}$ has dimension $K$. Its flux lattice $\mathrm{\Lambda }\subset V_{\mathbb{R}}$ is a discrete subgroup with an ordered integral basis $e_1,\dots ,e_K$; a lattice element has coordinate vector $N=(N^1,\dots ,N^K)\in {\mathbb{Z}}^K$. The chosen basis fixes the Lebesgue measure $d^{K}N$ with unit lattice covolume. The nondegenerate symmetric bilinear pairing $\eta :V_{\mathbb{R}}\times V_{\mathbb{R}}\to \mathbb{R}$ has matrix $({\eta }_{\alpha \beta })$ in this basis. Define the dimensionless flux tadpole $L(N)=\frac{1}{2}{N}^{\alpha }{\eta }_{\alpha \beta }{N}^{\beta }$. The pairing need not be positive definite on all of $V_{\mathbb{R}}$; positivity on a supersymmetric solution locus is separate model data. The allowed source set $\mathcal{S}$ fixes localized charges, orientation, cover-versus-quotient multiplicity, and the resulting tadpole bound $L_{\ast }>0$.

The equation data include a jointly $C^1$ map $\mathrm{\Phi }:V_{\mathbb{R}}\times \mathcal{M}\to {\mathbb{C}}^m$ whose slices ${\mathrm{\Phi }}_N(z):=\mathrm{\Phi }(N,z)$ are smooth in $z$; the map is understood in local real coordinates on $V_{\mathbb{R}}$ and $\mathcal{M}$. Write ${\varphi }_N:\mathcal{M}\to {\mathbb{R}}^{2m}$ for its real-coordinate form. A finite-lattice mode uses only the restrictions with $N\in \mathrm{\Lambda }$. A continuum-approximation mode uses the supplied map for every real $N\in V_{\mathbb{R}}$; these values are additional model data and are not determined by the restriction to $\mathrm{\Lambda }$. Thus lattice maps alone do not define the continuum integral below. The selected region $U$ is measurable with respect to local Lebesgue measure in these charts; roots are counted only at $z\in U$. Such a root is regular when the real derivative $D{\varphi }_N(z):T_z\mathcal{M}\to {\mathbb{R}}^{2m}$ is an isomorphism, equivalently when its matrix in local real coordinates is invertible. For a regular root set, the local zero-counting density is the coordinate expression ${\delta }_0^{(2m)}({\varphi }_N(z))\mathrm{\mid }{\det }_{\mathbb{R}}D{\varphi }_N(z)\mathrm{\mid }{d}^{2m}z$, where ${\delta }_0^{(2m)}$ is the Dirac distribution defined by ${\int }_{{\mathbb{R}}^{2m}}{\delta }_0^{(2m)}(y)\psi (y)d^{2m}y=\psi (0)$ for smooth compactly supported $\psi$.

The record $\kappa$ identifies which quantity is being counted: total roots, supersymmetric roots, signed index, finite-lattice count, or continuum approximation. The signed index uses the signed real Jacobian rather than its absolute value. The delta-density prescription is admissible only for regular counted roots: in a lattice count using that prescription, require every counted root at each allowed lattice vector to be regular. For a continuum count, supply a Lebesgue-null measurable exceptional set $E\subseteq V_{\mathbb{R}}$ and require every counted root to be regular for every $N\mathrm{\notin }E$ with $L(N)\le L_{\ast }$. For such $N$, the density means the atomic measure ${\sum }_{z\in U:{\varphi }_N(z)=0}{\delta }_z$, each root carrying unit mass; it is not a delta pullback at a singular zero. For the index, replace each unit mass by $\mathrm{sgn}{\det }_{\mathbb{R}}D{\varphi }_N(z)$, using the complex orientation of $\mathcal{M}$ and the standard orientation of ${\mathbb{R}}^{2m}$. A mode that includes singular roots must supply a separate counting rule for them; the displayed density assigns them no count. A mode called finite requires finitely many counted lattice-root pairs; the tadpole inequality alone does not ensure this for an indefinite $\eta$. In the continuum-approximation mode, replace the lattice sum by the Lebesgue integral of the root count over $L(N)\le L_{\ast }$ in the fixed unit-covolume basis, with integrand set to zero on $E$. Require that count to be Lebesgue measurable. Nonnegative total counts may take the value $+\mathrm{\infty }$; a finite signed index requires absolute integrability of its positive and negative root contributions. These are admissibility conditions, not asserted properties of arbitrary smooth equation data. This is a counting approximation, not a probability distribution on vacua; its error and regime of validity must be stated separately and are not supplied by this definition.

The flux coordinates and tadpole are dimensionless in the chosen charge normalization. Different flux lattices, duality quotients, source conventions, or zero-counting modes define different ensemble data. A finite continuum integral does not by itself define how cosmological histories populate vacua or how observers select them.