Localized currents and smeared source approximations

Use natural units $\mathrm{\hslash }=c=1$, length-valued coordinates, and ${\alpha }^{\mathrm{\prime }}>0$ of dimension length squared. Let $X$ be a compact oriented smooth six-manifold without boundary, fixed throughout as one smooth internal integration space; all source multiplicities and flux integrals refer to this same space. Model the internal charge of branes filling the four external dimensions by a finite signed degree-six current ${\rho }_{\mathrm{l}\mathrm{o}\mathrm{c}}={\sum }_a{q}_a{\delta }_{p_a}$, where $⟨{\delta }_{p_a},\psi ⟩=\psi (p_a)$ for $\psi \in C^{\mathrm{\infty }}(X,\mathbb{R})$. The coefficients $q_a$ are dimensionless signed D3-charge numbers in units ${\mu }_3=(2\pi {)}^{-3}{\alpha }^{\mathrm{\prime }-2}$: a positively oriented D3 has $q_a=1$, and any O3-plane coefficients and cover/quotient multiplicities are separately specified source-model data. Thus $Q=⟨{\rho }_{\mathrm{l}\mathrm{o}\mathrm{c}},1⟩={\sum }_a{q}_a$ is dimensionless, and ${\mu }_{3}Q$ is the charge in this normalization. This physical interpretation is an adopted source convention, not a consequence of the mathematical current definition.

Specifically adopt the current-valued Bianchi law $d{J}_5={\mathcal{B}}_6+c{\rho }_{\mathrm{l}\mathrm{o}\mathrm{c}}$, where $J_5$ is a degree-five current representing the internal gauge-invariant five-form, ${\mathcal{B}}_6$ is a specified degree-six current representing $H_3\wedge F_3$, and $c=2{\kappa }_0^2{\mu }_3=(2\pi {)}^4{\alpha }^{\mathrm{\prime }2}$ with $2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4}$. Here $F_3,H_3$ are real three-form field strengths with coordinate components of dimension length${}^{-1}$; for smooth fields $⟨{\mathcal{B}}_6,\psi ⟩={\int }_X\psi H_3\wedge F_3$, and singular fields require a separately specified continuous current extension. The pairings $⟨{\mathcal{B}}_6,1⟩$ and $cQ$ both have dimension length${}^4$; a smooth representative of the dimensionless source current has coordinate coefficients of dimension length${}^{-6}$. The Bianchi law, its sign and coefficient are conditional physical inputs. No field equation is asserted to hold smoothly through a point source. A smeared model replaces it by a smooth degree-six form ${\rho }_{\mathrm{s}\mathrm{m}}$ with ${\int }_X{\rho }_{\mathrm{s}\mathrm{m}}=Q$. This replacement preserves the source's contribution when the Bianchi identity is paired with the constant test function $1$, so it preserves that integrated charge constraint when the other terms are held fixed. It does not generally preserve the source as a distribution or a local sourced Bianchi equation. For any specified smooth dimensionless weight $w:X\to \mathbb{R}$, define $(w\rho )(\psi )=\rho (w\psi )$; equal total charge need not give equal $⟨\rho ,w⟩$. Thus it need not preserve an adopted warp equation's source term if that term involves this weighted pairing. For singular weights a multiplication or extension prescription is additionally required; equality of total charge supplies none. Source stress and charge are separate data, so this argument assumes no universal equality between tension and charge. In particular, equal total charge alone does not imply that a smeared solution is an exact localized ten-dimensional solution.

For example, orient the unit flat torus $X={\mathbb{R}}^6\mathrm{/}{\mathbb{Z}}^6$ by $d{y}^1\wedge \cdots \wedge d{y}^6$, let $p$ be a point, $q>0$, and let $\nu$ be its volume form normalized by ${\int }_X\nu =1$. Set ${\rho }_{\mathrm{l}\mathrm{o}\mathrm{c}}=q{\delta }_p$ and ${\rho }_{\mathrm{s}\mathrm{m}}=q\nu$. These mathematical currents have equal total coefficient but are unequal distributions; this example assigns no physical units to $q$.

Smeared compactification calculations can still provide controlled approximations when a stated model and scaling limit bound localized backreaction in a specified region away from source cores. Such control is model-dependent; a first-order profile or a smeared zero-mode solution does not by itself resolve the core or establish exact localization. Results about failure to localize are restricted to the source type and ansatz studied in the cited work.

De Rham currents of compact oriented source cycles

Giddings, Kachru, and Polchinski, equations (2.23)–(2.24)

Graña, §§4.1–4.2

DeWolfe et al., §§2–5

Cribiori et al., §§3–4

Blåbäck et al., §5.3

Van Riet and Zoccarato, §§4.2 and 5.3