M Theory Brane Web Tbrane Frontier 2026

Collinucci’s local setup uses the source’s flat ten-dimensional Minkowski coordinates $x_0,\dots ,x_9$ with $x_0$ time and length-valued coordinates in natural units. The coordinates $x_7,x_8$ are noncompact, and a chosen circle $x_9\sim x_9+2\pi R$ of radius $R>0$ lies along one D7-brane and transverse to the D5-branes. A D5 spans $x_0,\dots ,x_4$ and a half-line $x_5\in (-\mathrm{\infty },0]$ ending at a codimension-three point of the D7, which spans $x_0,\dots ,x_4,x_7,x_8,x_9$; in the source normalization its endpoint is a unit magnetic source, ${\int }_{S^2}F=2\pi$, through a positively oriented linking sphere. On the D7 with the endpoints removed, $F$ is the smooth real curvature two-form of a $U(1)$ line-bundle connection: $F=d{A}_a$ only on trivializing patches, with $A_a$ real local one-forms. No single smooth potential on the linking sphere is assumed. Put $z=x_7+i{x}_8$ and take ${\alpha }^{\mathrm{\prime }}>0$ to have units of length squared, so the T-dual radius $\tilde{R}={\alpha }^{\mathrm{\prime }}\mathrm{/}R$ is a length. For the curve description retain only the circle zero mode, namely the average $(2\pi R{)}^{-1}{\int }_0^{2\pi R}{\varphi }_5(z,x_9)d{x}_9$ of the D7 position scalar ${\varphi }_5$ along $x_5$. The source drops the nonzero Fourier modes before T-dualizing; localized endpoints do not give an exact circle isometry. In this truncation, after T-duality on $x_9$, define the dimensionless coordinates $w_{\pm }=\exp (\pm (x_5+i{\tilde{x}}_9)\mathrm{/}\tilde{R})$, so $w_{+}{w}_{-}=1$. In the no-NS5 setup of the source’s §§3–5, let $q_L,q_R\in {\mathbb{Z}}_{\ge 0}$ count the D5 endpoints on the left and right of the D7, and let $z_i^{(-)},z_j^{(+)}\in \mathbb{C}$ be their positions. Within this zero-mode description these endpoints map to one D6-brane on the holomorphic curve $w_{+}{\prod }_{j=1}^{q_R}(z-z_j^{(+)})=C{\prod }_{i=1}^{q_L}(z-z_i^{(-)}),$ where $C$ is a nonzero complex constant of dimension length${}^{q_R-q_L}$ (its numerical value in a fixed length unit lies in ${\mathbb{C}}^{\ast }$). More explicitly, $C=c_0{\mathrm{\Lambda }}^{q_R-q_L}$ for a reference length $\mathrm{\Lambda }>0$ and a dimensionless complex constant $c_0\mathrm{\ne }0$ incorporating the position and circle phase integration constants. Both sides then have dimension length${}^{q_R}$. A single left endpoint gives $w_{+}=C(z-z_0)$ with $[C]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$. For the source’s coherent-sheaf description, take a complex patch $U\subset {\mathbb{C}}_z\times {\mathbb{C}}_{w_{+}}^{\ast }$ and let ${\mathcal{O}}_U$ be its sheaf of holomorphic functions. For $k\in {\mathbb{Z}}_{>0}$, a finite-rank presentation $0\to {\mathcal{O}}_U^{\oplus k}\stackrel{\mathcal{T}(z,w_{+})}{\to }{\mathcal{O}}_U^{\oplus k}\to \mathcal{F}\to 0$ encodes the D6 support. Here the local T-brane data mean the non-diagonal matrix entries of the holomorphic tachyon map $\mathcal{T}$; in the source’s examples, nilpotent entries encode the partition. In the sheaf presentation, the local uplift is the hypersurface $uv=\det \mathcal{T}(z,w_{+})$ in ${\mathbb{C}}_{u,v}^2\times U$; its M-theory circle collapses where $\det \mathcal{T}=0$. These are source-reported local constructions, not a universal web/geometry equivalence.

Collinucci, Abstract, Introduction, §§2–3.2, §§4–5.4

Witten, §§2.2–2.3

M Theory Type IIA Circle Limit Dictionary

The source reports the local profile and holomorphic D6 curve for D5 endpoints on a D7 and extends the construction to multiple D7-branes and partition data. Section 3.1, Eqs. (6)–(8), separates the circle zero mode from nonzero modes and explicitly retains the former for the subsequent curve; the latter are exponentially suppressed far from each endpoint, at transverse distance much larger than $R$, but are not identically zero. Its T-brane/sheaf discussion represents the local support by $\det (\mathcal{T})=0$ and gives the uplift $uv=\det (\mathcal{T})$ in that presentation. Sections 3–5 assume there is no NS5-brane; the source treats NS5-containing configurations separately using Taub–NUT data. It explicitly warns that mutually nonlocal $(p,q)$ branes cannot be handled by one global duality frame and require patchwise constructions and gluing. The retained M/IIA circle-dictionary prerequisite supplies the adopted circle-scaling and metric convention as formulation context for interpreting the local lift; it does not supply the local curve or T-brane map. The removed flat open-string D$p$ T-duality theorem concerns Neumann/Dirichlet boundary data and right-moving chirality; it does not formulate or derive this seven-brane/T-brane chain. The source-status review dated 2026-10-30 is limited to the cited version and sections; it makes no universal or global claim.

The diagram records the source’s local chain after the stated circle zero-mode truncation. The arrows summarize source-reported duality/uplift steps and supply no proof; the source text and limits above carry the claim.